Giancarlo Bernacchi
TENSORS made easy An informal introduction to Maths of General Relativity
To Sophie, Alexandre and Elena
2017
4dt edition: March 2017 Giancarlo Bernacchi All rights reserved
Giancarlo Bernacchi Rho (Milano ! "talia giancarlo#ernacchi$li#ero%it Text edited by means of Sun Microsystem OpenOffice Writer 3 (except some figures Edition !ecember !ecember "#$% &e'ision March March "#$
4dt edition: March 2017 Giancarlo Bernacchi All rights reserved
Giancarlo Bernacchi Rho (Milano ! "talia giancarlo#ernacchi$li#ero%it Text edited by means of Sun Microsystem OpenOffice Writer 3 (except some figures Edition !ecember !ecember "#$% &e'ision March March "#$
Contents "ntroduction
&
'otations and conventions
1
Vectors and Covectors
7
1%1 1%2 1%& 1%4 1% 1% 1%7
)ectors Basis!vectors and com*onents +ovectors -calar *roduct (heterogeneous .he /ronecer Meta*hors and models .he 3.!mosaic model
7 7 , 11 12 1& 14
2
Tensors
19
2%1 2%2 2%& 2%4 2% 2% 2%7 2%8 2%, 2%10 2%11 2%12 2%1& 2%14 2%1 2%1 2%17 2%18 2%18
5uter *roduct #et6een vectors and covectors Matri re*resentation of tensors -um of tensors and *roduct #y a num#er -ymmetry and se6!symmetry Re*resenting tensors in .!mosaic .ensors in .!mosaic model: definitions .ensor inner *roduct 5uter *roduct in .!mosaic +ontraction "nner *roduct as outer *roduct 9 contraction Multi*le connection of tensors 3-calar *roduct or 3identity tensor "nverse tensor )ector!covector 3dual s6itch tensor )ectors covectors homogeneous scalar *roduct G a**lied to #asis!vectors G a**lied to a tensor Rel Relati ations ons #et #et6e 6eeen I G, δ
21 2& 2 2 2 27 2, &7 &8 40 40 41 42 4 4 4, 4, 1
3
Change of basis
53
&%1 &%2 &%&
Basis change in .!mosaic "nvariance of the null tensor "nvariance of tensor e;uations
, 0
4
Tensors in manifolds
61
4%1 4%2 4%& 4%4 4%
+oordinate systems +oordinate lines and surfaces +oordinate #ases +oordinate #ases and non!coordinate #ases +hange of the coordinate system +ontravariant and covariant tensors
& & 4 7 , 71
,
4%7 Affine tensors 4%8 +artesian tensors 4%, Magnitude of vectors 4%10
ed distances 4%1& .ensors and not ? 4%14 +ovariant derivative ̃ at 6or 4%1 .he gradient ∇ 4%1 Gradient of some fundamental tensors 4%17 +ovariant derivative and inde raising lo6ering 4%18 +hristoffel sym#ols 4%1, +ovariant derivative and invariance of tensor e;uations 4%20 .!mosaic re*resentation of gradient@ divergence and covariant derivative
72 7& 7& 7& 74 74 7 7 80 8 8 87 8,
5
Curved nanifolds
92
%1 %2 %& %4 % % %7 %8 %, %10 %11 %12 %1& %14
-ym*toms of curvature
,2 ,4 ,8 ,, 100 102 104 10 10 111 11 117 117 118
!!endi"
121
Bi#liogra*hic references
12
,0
Introduction Tensor Analysis has a particular, though not exclusive, interest for the theory of General Relativity, in which curved spaces have a central role; so we cannot restrict ourselves to Cartesian tensors and to usual 3D space. n fact, all texts of General Relativity include so!e treat!ent of Tensor Analysis at various levels, "ut often it reduces to a sche!atic su!!ary that is al!ost inco!prehensi"le to the first approach, or else the topic is distri"uted along the text to "e introduced piece!eal when needed, inevita"ly disorgani#ed and frag!ented. $n the other hand, wor%s devoted to the tensors at ele!entary level are virtually !issing, neither is worth e!"ar%ing on a preli!inary study of speciali#ed texts& even if one !ight overco!e the !athe!atical difficulties, he would not "e on the shortest way leading to relativity '"ut perhaps to differential geo!etry or surroundings(. )hat we ulti!ately need is a si!ple introduction, not too for!al * !ay"e heuristic * suita"le to "e used as a first approach to the su"+ect, "efore dealing with the study of specific texts of General Relativity. )e will cite as references those articles or handouts that we found closer to this approach and to which we have referred in writing these notes. $ur goal is not to do so!ething "etter, "ut !ore easily afforda"le; we payed attention to the funda!entals and tried to !a%e clear the line of reasoning, avoiding, as it often happens in such circu!stances, too !any i!plicit assu!ptions 'which usually are not so o"vious to the first reading(. As a general approach, we fully agree 'and we do it with the faith of converts( on the appropriateness of the geo!etric approach- to Tensor Analysis& really, it does not reuire !ore effort, "ut gives a strength to the concept of tensor that the old traditional approach "y co!ponents-, still adopted "y !any texts, cannot give. Addressing the various topics we have adopted a top/down- strategy. )e are convinced that it is always "etter, even didactically, address the pro"le!s in their generality 'not Cartesian tensors as introduction to general tensors, not flat spaces "efore curved ones(; to shrin%, to go into details, etc. there will always "e ti!e, "ut it would "e "etter a later ti!e. After all, there is a strategy "etter than others to open Chinese "oxes& to "egin with the larger one. This should so!ehow contain the irresisti"le te!ptation to unduly transfer in a general context results that apply only in particular instances. )e also tried to apply a little/%nown corollary of $cca!0s ra#or-& do not introduce restrictive assu!ptions "efore necessary 'that is why, for exa!ple, we introduce a dual switch- "efore the !etric tensor(. The desire not to appear pedestrian is the factor that increases "y an order of !agnitude the illegi"ility of !athe!atics "oo%s& we felt that it is "etter to let it go and accept the ris% of "eing +udged so!ewhat dull "y those who presu!e to %now the story in depth. This "oo% does not clai! to originality, "ut ai!s to didactics. )ith an exception& the !odel 'or !etaphor( that represents the tensor as a piece or tessera- of a !osaic 'we0ll call it T/!osaic-(. )e are not aware that this !etaphor, useful to write without e!"arrass!ent tensor euations and even a "it o"vious, has ever "een presented in a text. That !ight "e even surprising. 1ut the !agicians never reveal their tric%s, and !athe!aticians so!eti!es rese!"le the!. f it will "e appreciated, we will feel honored to have "een +ust us to uncover the tric%. To read these notes should "e enough so!e differential calculus 'until partial 3
derivatives(; for what concerns !atrices, it suffices to %now that they are ta"les with rows and colu!ns 'and that swapping the! you can create a great confusion(, or slightly !ore. nstead, we will assiduously use the 2instein su! convention for repeated indexes as a great si!plification in writing tensor euations. As 2uclid said to the haraoh, there is no royal road to !athe!atics; this does not !ean that it is always co!pulsory to follow the !ost i!pervious pathway. The hope of the author of these notes is they could "e useful to so!eone as a conceptual introduction to the su"+ect. G.B.
4
Notations and conventions n these notes we0ll use the standard notation for co!ponents of tensors, na!ely upper 'apices( and lower 'su"script( indexes in Gree% letters, for exa!ple T , as
usual in Relativity. The coordinates will "e !ar%ed "y upper indexes, such as x , "asis/vectors will "e represented "y e , "asis/covectors "y e .
n a tensor for!ula such as P 5 g V the index 6 that appears once at left side !e!"er and once at right side !e!"er is a free- index, while 7 that occurs twice in the right !e!"er only is a du!!y- index. )e will constantly use the 2instein su! convention, where"y a repeated index means an implicit summation over that index. 8or exa!ple, P 5 g V is for
P 5 g 9 V
9
A B 5
:
n
g : V ... g n V . n
n
∑ ∑ A B =9 = 9
8urther exa!ples are& A B 5
9
:
n
∑ =9
A B ,
n
and A 5 A9 A:... An .
)e note that, due to the su! convention, the chain rule- for partial derivatives can f f x 5 "e si!ply written and the su! over co!es auto!atically. x x x A du!!y index, unli%e a free index, does not survive the su!!ation and thus it does not appear in the result. ts na!e can "e freely changed as long as it does not collide with other ho!ony!ous indexes in the sa!e ter!. n all euations the dummy indexes must always be balanced updown and they cannot occur !ore than twice in each ter!. The free indexes appear only once in each ter!. n an euation "oth left and right !e!"ers !ust have the sa!e free indexes. These conventions !a%e it !uch easier writing correct relationships. The little needed of !atrix alge"ra will "e said when necessary. )e only note that the !ultiplication of two !atrices reuires us to devise- a du!!y index. Thus, for instance, the product of !atrices [ A ] and [ B ] "eco!es [ A ]⋅[ B ] 'also for !atrices we locate indexes up or down depending on the tensors they represent, without a particular !eaning for the !atrix itself(. The !ar% will "e used indiscri!inately for scalar products "etween vectors and covectors, "oth heterogeneous and ho!ogeneous, as well as for inner tensor products. $ther notations will "e explained when introduced& there is so!e redundancy in notations and we will use the! with easiness according to the convenience. n fact, we had "etter getting fa!iliar with all various alternative notations that can "e found in the literature. The indented paragraphs !ar%ed < are inserts- in the thread of the speech and they can, if desired, "e s%ipped at first reading 'they are usually +ustifications or proofs(. =ne!o- "oxes suggest si!ple rules to re!ind co!plex for!ulas.
>
?
1 Vectors and covectors 1.1 Vectors
A set where we can define operations of addition "etween any two ele!ents and !ultiplication of an ele!ent "y a nu!"er 'such that the result is still an ele!ent of the set( is called a vector space and its ele!ents vectors.@@ The usual vectors of physics are oriented uantities that fall under this . definition; we will denote the! generically "y V n particular, we will consider the vector space for!ed "y the set of vectors defined at a certain point P. 1.2 Basis-vectors and components
The !axi!u! nu!"er n of vectors independent to one another that we can put together is the dimension of a vector space. These n vectors, chosen ar"itrarily, for! a basis of vectors 'it doesn0t !atter they are unit or eual length vectors(. e : , ... e n 'the generic "asis/vector )e denote the "asis/vectors "y e9 , "y e , = 9,:,3,... n and the "asis as a whole "y {e } (. Any further vector can "e expressed as a linear co!"ination of the n "asis/vectors. n other words, any vector can "e written as an expansion 'or deco!position( on the "asis&
= e9 V V
9
e: V : ... en V n
9.9
that, using 2instein0s su! convention, we write&
V
= e V
9.:
The n nu!"ers V '6 5 9, :, ... n( are the components of the vector on the "asis we have chosen ∀ V ∈ ℝ . V
is si!ply its recipe-& This way of representing the vector V e9 , e : ,... e n 'i.e. the e ( is the list of the ingredients, and 9 : n V ,V , ...V 'i.e. the V ( are the uantities of each one. The choice of the n "asis/vectors is ar"itrary. $"viously, the change with the chosen "asis co!ponents V of a given vector V 'unli%e the vector itself( = a Ab B , where A , B ,! are ele!ents of the @ n fact, it !ust "e defined ! sa!e set and a, b nu!"ers ∈ ℝ . )e gloss over other reuests !ore o"vious. B
The expansion e".#.# allows us to represent a vector as an n/tuple&
V 9 , V : , ...V n e : ,... e n fixed in advance. provided there is a "asis e9 , comp As an alternative to e".#.$ we will also write V V with the sa!e !eaning. et0s represent graphically the vector/space of plane vectors ' n 5 :( e!anating fro! P , drawing only so!e vectors a!ong the infinite ones&
!
≡ e: B
≡ e9 A
P
≡ e9 , B ≡ e: as "asis/vectors, a third vector ! $nce selected A is no !ore independent, "ut it can "e expressed as a linear co!"inaE tion of the first two; in this case& = !
− 9 e9 : e: :
on the esta"lished "asis are thus 'F, :(. The co!ponents of ! )e o"serve that the expansion 'or recipe-( of a "asis/vector on its own "asis 'i.e. the "asis !ade "y the!selves( cannot "e other than&
9,H,H ,...H e: H,9,H , ...H e9
IIIIIIIIIIII
en
H,H,H , ...9
regardless whether they are unit "asis/vector or not and their length. J
1.3 Covectors
'or dual vector or one/for!-( a linear scalar )e define covector P applied to a vector n other words& P function of the vector V . results in a nu!"er& V = nu!"er ∈ ℝ 9.3 P
can thus "e seen as an operator- that for any vector ta%en in P input gives a nu!"er as output. 1y what ruleK $ne of the possi"le rules has a particular interest "ecause it esta"lishes a sort of reciprocity or duality "etween the , and it is what we ai! to for!ali#e. and covectors P vectors V to a "asis/vector 'instead To "egin with, we apply the covector P of a generic vector(. 1y definition, let0s call the result the 6/th & co!ponent of P e = P P 9.4 )e will then say that the n nu!"ers P '6 5 9, :, ... n( are the co!ponents of the covector 'as well as the V were the (. co!ponents of the vector V n operational ter!s we can state a rule 'which will "e generali#ed later on(& Apply the covector ̃ to the "asis/vectors e to get the co!ponents of the covector ̃ itself. Lote that this definition of co!ponent sets up a precise relationship "etween vectors and covectors. Due to that definition, also a covector can "e represented as an n/tuple
P 9 , P : , ... P n associated with a "asis e 9 , e : ,... e n of covectors. 2ven a covector can then "e expressed as an expansion on its own "asis& = e P P '6 5 9, :, ... n( 9.> using the co!ponents P 6 as coefficients of the expansion. 1y itself, the choice of the "asis of covectors is ar"itrary, "ut at this and '"y !eans of point, having already fixed "oth the covector P e".#.%( its co!ponents, the "asis of covectors {e } follows, so that M
the last euation 'e".#.&( can "e true. n short, once used the vector "asis {e } to define the co!ponents of the covector, the choice of the covector "asis {e } is forced. 1efore giving a !athe!atical for! to this lin% "etween the two "ases, we o"serve that, "y using the definition e".#.% given a"ove, can "e the rule according to which acts on the generic vector V specified& V = P P e V = V P e = V P 9.?
!eans !ultiplying orderly "y couples the Applying ̃ to V co!ponents of "oth and su! up. is co!pletely defined. n this way the !eaning of P it is now possi"le to define operations Also on the set of covectors P of su! of covectors and !ultiplication of a covector "y a nu!"er. This confers also to the set of covectors the status of a vector space.@ @ )e have thus defined two separate vector spaces, one for the vectors, the other for the covectors '"oth related to a point P ( , eual in di!ension and in dual relationship "etween each other. )e can now clarify how the duality relationship "etween the two vector spaces lin%s the "ases {e } and {e }. rovided
V
= e P we can write& = e V and P V P
= e P V e = P V e e
"ut, on the other hand, 'e".#.' (&
V P
= P V
1oth expansions are identical only if& β
ẽ ( e⃗
= 9 for β=
H otherwise
1y defining the (ronec)er symbol as&
!β =
9 for β= H otherwise
9.B
is V it follows that& i( the application P @ 8ro! the definition given for P a A b B = P a A b B = a P A b P B = a P A b P B ; linear& P is a vector space, too& a P b * V = ii( the set of the covectors P V b * V = a P V b * V = a P b * V = + V = + V = a P 9H
we can write the duality condition&
e e
= "
9.J
A vector space of vectors and a vector space of covectors are dual if and only if this relation "etween their "ases holds good.
V = V P 'e".#' ( lends itself to an )e o"serve now that P alternative interpretation. 1ecause of its sy!!etry, the product V P can "e interpreted not V "ut also as V P , interchanging operator and only as P operand& P = V P = nu!"er ∈ ℝ V 9.M
n this reversed- interpretation a vector can "e regarded as a linear scalar function of a covector ̃ . @ @ As a final step, if we want the duality to "e co!plete, it would "e possi"le to express the co!ponents V of a vector as the result of the to the "asis/covectors e '"y sy!!etry with the application of V definition of co!ponent of covector e".#.%(& V
V e =
9.9H
< t 0s easy to o"serve that this is the case, "ecause&
P V
e P = P V e = V
P = P V "ut since V the assu!ption follows.
'e".#.(, euating the right !e!"ers
-".#.# is the dual of e".#.% and together they express a general rule& To get the co!ponents of a vector or covector apply the vector or covector to its dual "asis. 1.4 Scalar product (heterogeneous
The application of a covector to a vector or vice versa 'the result does V or V P has not change( so far indicated with notations li%e P the !eaning of a scalar product "etween the two ones. An alternative notation !a%es use of parentheses # ... $ e!phasi#ing the sy!!etry of the two operands. is linear& @ t0s easy to verify that the application V a P b * = V a P b * = a V P b V * = a b V * V V P 99
The linearity of the operation is already %nown. All writings&
V P
P = # P , V $ = # V , P $ = V P = V
9.99
are euivalent and represent the heterogeneous scalar product "etween a vector and a covector. 1y the new introduced notation the duality condition "etween "ases e".#./ is currently expressed as& e $ = " # e ,
9.9:
The ho!ogeneous scalar product "etween vectors or covectors of the sa!e %ind reuires a different definition that will "e introduced later. 1.! "he #ronec$er %
-".#.0 defines the Nronec%er O&
!β =
9 for β= H otherwise
A re!ar%a"le property of the Nronec%er O often used in the calculations is that " acts as an operator that identifies the two indexes 6, 7 turning one into the other. 8or exa!ple&
" V = V ;
" P = P
9.93
Lote that the su!!ation that is i!plicit in the first !e!"er collapses to the single value of the second !e!"er. This happens "ecause " re!oves fro! the su! all ter!s whose indexes 6, 7 are different, !a%ing the! eual to #ero. Roughly spea%ing& " changes- one of the two indexes of its operand into the other one; what survives is the free index 'note the "alance- of the indexes in "oth e".#.#1(. < )e prove the first of e".#.#1& e " = e 'e".#./( "y V gives& =ultiplying
= V " V = e e V = e V 'the last euality of the chain is the rule e".#.#(. Pi!ilarly for the other euation e".#.#1. n practice, this property of Nronec%er O turns useful while calculating. 2ach ti!e we can !a%e a product li%e e e to appear in an expression, we can replace it "y " , that soon produces a change of the index in one of the factors, as shown in e".#.#1. 9:
Po far we have considered the Nronec%er sy!"ol " as a nu!"er; we will see later that " is to "e considered as a co!ponent of a tensor.
1.& 'etaphors and models
The fact that vectors and covectors can "oth "e represented as a dou"le array of "asis ele!ents and co!ponents e9 e: e3 ... en 9 : 3 n V V V ... V
9 : 3 n e e e ... e P 9 P : P 3 ... P n
and
suggests an useful !etaphor to display graphically the for!ulas. Qectors and covectors are represented as interloc%ing cu"es or "uilding "loc%s- "earing pins and holes that allow to hoo%up each other.
P
P 1
≡ e
P 3
9
e9
V
P 2
e
:
e
3
e3
e:
V3
≡
V2 V1
93
Qectors are cu"es with pins upward; covectors are cu"es with holes downward. )e0ll refer to pins and holes as connectors-. ins and holes are in nu!"er of n, the di!ension of space. 2ach pin represents a e '6 5 9, :, ... n(; each hole is for a e . n correspondence with the various pins or holes we have to i!agine the respective co!ponents written sideways on the "ody of the cu"e, as shown in the picture. The exa!ple a"ove refers to a 3D space 'for n 5 4, for instance, the representation would provide cu"es with 4 pins or holes(. The n pins and the n holes !ust connect together si!ultaneously. Their connection e!ulates the heterogeneous scalar product "etween vectors and covectors and creates an o"+ect with no exposed connectors 'a scalar(. The heterogeneous scalar product
e P
e V
scalar product
P V
!ay "e indeed put in a pattern that !ay "e interpreted as the !etaphor of interloc%ing cu"es& e
9
%
e
:
%
3 n e e % .... %
P 9 P : P 3
P n
e 9 e : e 3 e n % % % .... % 9 : 3 n
V
V
V
P 9 P : P 3
P n % % % .... % 9 : 3 n V V V V
scalar product
V
as shown in the previous picture, once we i!agine to have !ade the fit "etween the two cu"es.
1. "he )"-mosaic* model
n the following we will adopt a !ore si!ple drawing without perspective, representing the cu"e as seen fro! the short side, li%e a piece of a !osaic 'a tessera-( in two di!ensions with no depth, so that all the pins and holes will "e aligned along the line of sight and !erge into one which represents the! collectively. The generic na!e of the array e for pins or e for holes is thought as written upon 94
it, while the generic na!e of the co!ponent array ' V for vectors or P for covectors( is written in the piece. This representation is the "asis of the !odel that we call "-mosaic- for its a"ility to "e generali#ed to the case of tensors. Po, fro! now on we will use the two/di!ensional representation& e
Qector V
Covector P
≡
≡
P
V
e
1loc%s li%e those in the figure a"ove give a synthetic representation of the expansion "y co!ponents on the given "asis 'the recipe- e".#.$, e".#.&(. The connection of the "loc%s, i.e. the application of a "loc% to another, represents the heterogeneous scalar product2 P
V P
e
P = # P , V $ ≡ = V
P
e
+
P V 5
V
V
The connection is !ade "etween ho!ony!ous connectors ' 5 with the sa!e index& in this exa!ple e with e (. )hen the "loc%s fit together, the connectors disappear and "odies !erge multiplying to each other . e 1asis/vector will "e drawn as& and "asis/covector as&
e 9>
&
8or si!plicity, nothing will "e written in the "ody of the tessera which represents a "asis/vector or "asis/covector, "ut we have to re!e!"er that, according to the perspective representation, a series of H together with a single 9 are inscri"ed in the side of the "loc%. 2xa!ple in >D& ,
,
e:
,
≡
1
,
This !eans that, in the scalar product, all the products that enter in the su! go to #ero, except one. The application of a covector to a "asis/vector to get the covector co!ponents 'e".#.%( is represented "y the connection& P
e P
= P &
e
P
+
P
e
Pi!ilarly, the application of a vector to a "asis/covector to get the co!ponents of the vector 'e".#.#( corresponds to the connection&
e V
= V &
e
e
V
V
9?
+
V
A s!ooth- "loc%, i.e. a "loc% without free connectors, is a scalar ' 5 a nu!"er(. Lotice that& n T/!osaic representation the connection always occurs "etween connectors with the sa!e index 'sa!e na!e(, in contrast to what happens in alge"raic for!ulas 'where it is necessary to diversify the! in order to avoid the su!!ations interfere with each other(. Sowever, it is still possi"le to perfor! "loc%wise the connection "etween different indexes, in which case it is necessary to insert a "loc% of Nronec%er O as a plug adapter -&
A P
'
= P e A e = P A e e = P A " = P A
P
P e
e
A
e
e
'
P
e
e
"
P e
A
e
A
'
"
"
e
e
'
A
P A
The chain of eualities a"ove is the usual seuence of steps when using the conventional alge"raic notation, which !a%es use of different indexes and the sy!"ol Nronec%er O. Lote the correspondence "etween successive steps in alge"raic for!ulas and "loc%s. 9B
t is worth noting that in the usual T/!osaic representation the connection occurs directly "etween P e and A e "y !eans of the ho!ony!ic 6/connectors, s%ipping the first 3 steps. The T/!osaic representation of the duality relation 'e".#./ or e".#.#$( is a significant exa!ple of the use of Nronec%er O as a plug adapter-&
e
e
e
e $≡ # e ,
" e
e
"
+
"
e
n practice, it does not !atter plugging directly connectors of the sa!e na!e or connectors with different index via an interposed plug adapter; the first way is easier and faster, the second allows a "etter correspondence with alge"raic for!ulas. )e will see later that using the Nronec%er O as a plug adapter- "loc% is +ustified "y its tensor character.
9J
2 "ensors The concept of tensor T is an extension of those of vector and covector. A tensor is a linear scalar function of h covectors and ) vectors 'h, ) 5 H, 9, :, ...(. )e !ay see T as an operator that ta%es h covectors and ) vectors in input to give a nu!"er as a result&
, B , ... P , * , ... T A
= nu!"er ∈ ℝ
:.9
P h number
T
V
ℝ
)
h 1y ) or even r = h ) we denote the ran% or order of the tensor.
H 9 H A tensor H is a scalar; a tensor H is a vector; a tensor 9 is a covector.
3arning & in the notation T , , ... the parenthesis following the sy!"ol of tensor contains the list of argu!ents that the tensor ta%es as input 'input list ( . t does not si!ply ualify- the tensor, "ut it represents an operation already perfor!ed. n fact, T , , ... is the result of applying T to the list within parentheses. The na%ed tensoris si!ply written T. The components of a tensor T are defined in a way si!ilar to vectors and covectors "y applying the tensor to "asis/vectors and covectors. n si!ple words& we input into the tensor T the reuired a!ount of "asis/vectors and "asis/covectors ' h "asis/covectors and ) "asis/vectors(; the nu!"er that co!es out is the co!ponent of the H tensor on the given "ases. 8or exa!ple, the co!ponents of a : e & tensor result "y giving to the tensor the various couples e ,
e T e ,
= T
:.:
in order to o"tain n: nu!"ers !ar%ed "y the dou"le index 6 7 'for e : we get T 9: , and so on(. instance, fro! e9 , 9M
n general&
e , ... e T e ,
, e , ...
= T .. ....
:.3
This euation allows us to specify the calculation rule left undeter!ined "y e".$.# 'which nu!"er does co!e out "y applying the 9 tensor to its input listK(. 8or exa!ple, given a tensor S of ran% 9 for
which e".$.1 "eco!es S e , e
SV
= 4 , we could get&
= S , P e V , e P = V P S e , e
= V P 4
:.4
n general e".$.# wor%s as follows&
, B , ... P , * , ... T A
= A B⋅⋅⋅ P *⋅⋅⋅T
..
.
..
.
:.>
and its result is a nu!"er 'one for each set of values , , 6, 7(.
An expression li%e A B P * T that contains only "alanced du!!y indexes is a scalar "ecause no index survives in the result of the i!plicit !ultiple su!!ation. -".$.%, e".$.& appear as an extension to tensors of e".#.' , e".#. valid for vectors and covectors. Applying the tensor T to its input list P or P V of a vector or covector ' , , ...( is li%e the application V to its respective argu!ent, !eaning heterogeneous scalar product. t !ay "e seen as a sort of !ultiple scalar product, i.e. a seuence of scalar products "etween the tensor and the vectors U covectors of the list 'indeed, it is a tensor inner product- si!ilar to the scalar product of vectors and covectors as we0ll explain later on(. Ppea%ing of tensor co!ponents we have so far referenced to "ases of vectors and covectors. Sowever, it is possi"le to express the tensor as an expansion "y co!ponents on its own basis. That is to say, in the H case of a : tensor&
T
= e T
:.?
'it0s the recipe- of the tensor, si!ilar to that of vectors(. This ti!e the "asis is a tensorial one and consists of "asis/tensors e with dou"le index, in nu!"er of n:. This expression has only a for!al !eaning until we specify what sort of "asis is that and how it is related to the "asis of vectors U covectors defined "efore. To do this we need to define first the outer product "etween vectors and Uor covectors. :H
2.1 uter product et/een vectors and covectors
, B and the covectors P , * we define vector Given the vectors A and B & outer product "etween the vectors A B P , * = A P * B such that& A B A
:.B
B is an operator acting on a couple of covectors 'i.e. La!ely& A vectors of the other %ind( in ter!s of a scalar products, as stated in the right !e!"er . The result is here again a nu!"er ∈ ℝ . t can "e i!!ediately seen that the outer product is non B ) B A 'indeed, note that B A against the commutative& A P * (. A sa!e operand would give as a different result B
B is a ran% :H tensor "ecause it !atches the Also note that A given definition of tensor& it ta%es : covectors as input and gives a nu!"er as result. Pi!ilarly we can define the outer products "etween covectors or H 9 "etween vectors and covectors, ran%ed : and 9 rispectively&
, B * such that& P * A P
* B and also A = P B , * = P * etc. such that& P A B A A P
Ptarting fro! vectors and covectors and !a%ing outer products "etween the! we can "uild tensors of gradually increasing ran%. n general, the inverse is not true& not every tensor can "e expressed as the outer product of tensors of lower ran%. )e can now characteri#e the tensor/"asis in ter!s of outer product of "asis/vectors and U or covectors. To fix on the case ( H: , it is& e
= e e
< n fact, @@ fro! the definition of co!ponent T 'e".$.$(, using the expansion T T
:.J
=
e T e ,
= T e 'e".$.' ( we get&
= T e e , e , which is true only if&
@ The !ost direct de!onstration, "ased on a co!parison "etween the two for!s * = P e * e = P * e e = T e e and T = T e T = P holds only in the case of tensors deco!posa"le as tensor outer product. :9
e
e , e = " " "ecause in that case&
T
= T " " .
Pince " = e e and "eco!es& e
" = e e the second/last euation
e , e = e e e e
which, "y definition of 'e".$.0 (, is euivalent to& e
= e e , .e.d.
The "asis of tensors has "een so reduced to the "asis of 'co(vectors. H The : tensor under consideration can then "e expanded on the "asis of covectors& T
= T e e
:.M
t0s again the recipe- of the tensor, as well as e".$.' , "ut this ti!e it uses "asis/vectors and "asis/covectors as ingredients-. n general, a tensor can "e expressed as a linear co!"ination of 'or as an expansion over the "asis of( ele!entary outer products e e ... e e ... whose coefficients are the co!ponents. 3 8or instance, a 9 tensor can "e expanded as&
T
= T e e e e
:.9H
which is usually si!ply written& T
= T e e e e
:.99
Lote the "alance- of upper U lower indexes. The sy!"ol can "e nor!ally o!itted without a!"iguity.
⃗ B ⃗ or V ⃗ P ̃ can "e una!"iguously interpreted < n fact, products A
⃗ B ⃗ or V ⃗ P ̃ "ecause for other products other explicit as A ⃗ (̃ P , #⃗ V ,̃ P $ or V ⃗ ⃗ 3 for scalar sy!"ols are used, such as V products, T ( ... or even for inner tensor products. The order of the indexes is i!portant and is stated "y e".$.# or e".$.## which represent the tensor in ter!s of an outer product& it is understood that changing the order of the indexes !eans to change the order of factors in the outer product, in general non/co!!utative. ::
Thus, in general T ) T ) T .. . . To preserve the order of the indexes is often enough a notation with a dou"le seuence for upper and lower indexes li%e 5 . Sowever, this notation is a!"iguous and turns out to "e i!proper when indexes are raised U lowered. Actually, it would "e convenient using a scanning with reserved colu!ns li%e 5 * ** * ; it aligns in a single seuence upper and lower indexes and assigns to each index a specific place wherein it !ay !ove up and down without colliding with other indexes. To avoid any a!"iguity we ought to use a notation such as is used to %eep "usy the colu!n, or 5 , where the dot si!ply 5
replacing the dot with a space.
2.2 'atri0 representation o tensors + A tensor of ran% H 'a scalar( is a nu!"er 9 H + a tensor of ran% 9, that is 9 or H , is an n/tuple of nu!"ers : 9 H + a tensor of ran% :, that is H , 9 , : , is a suare !atrix n 6 n + a tensor of ran% 3 is a cu"ic lattice- of n 6 n 6 n nu!"ers, etc...
'n each case the nu!"ers are the co!ponents of the tensor(. A tensor of ran% r can "e thought of as an r /di!ensional grid of nu!"ers, or co!ponents 'a single nu!"er or n/array or n6n !atrix, etc.(; in all cases we !ust thin% the co!ponent grid as associated with an underlying "ases grid- of the sa!e di!ension. $n the contrary, not every n/tuple of nu!"ers or n 6 n !atrix, and so on, is a vectors, tensor, etc. $nly tensors of ran% 9 and :, since represented as vectors or !atrices, can "e treated with the usual rules of !atrix calculus 'the interest to do that !ay "e due to the fact that the inner product "etween tensors of these ran%s is reduced to the product of !atrices(. : 9 H n particular, tensors H , 9 or : can all "e represented "y
!atrices, although "uilt on different "ases grids. 8or instance, for : tensors of ran% H the "asis grid is&
:3
e9 e9 e: e9
e9 e: e: e:
, e9 en e: en en e: , en en
en e9
$n this "asis the !atrix of the tensor is&
T
≡
[
99
T
:9
T
9:
T
, T 9 n
::
:n
T
T
n9
T
n:
T
nn
, T
]
= [ T ]
:.9:
Pi!ilarly& 9 for 9 tensor&
T
≡ [ T ] on grid e e
H for : tensor&
T
≡ [ T ] on grid e e
The outer product "etween vectors and tensors has si!ilarities with the Cartesian product '5 set of pairs(. 3 generates a "asis grid e e that includes all 8or exa!ple V the couples that can "e for!ed "y e and e and a corresponding !atrix of co!ponents with all possi"le couples such as V 3 = T ordered "y row and colu!n. : produces a f X is a ran% H tensor, the outer product X V 3/di!ensional cu"ic grid e e e of all the triplets "uilt "y e , e , e and a si!ilar cu"ic structure of all possi"le couples 7 T .
t should "e stressed that the di!ension of the space on which the tensor spreads itself is the ran% r of the tensor and has nothing to do with n, the di!ension of geo!etric space. 8or exa!ple, a ran% : tensor is a !atrix ':/di!ensional( in a space of any di!ension; what varies is the nu!"er n of rows and colu!ns. Also the T/!osaic "loc%wise representation is invariant with the di!ension of space& the nu!"er of connectors does not change since they are not individually represented, "ut as arrays. :4
2.3 Sum o tensors and product a numer
As in the case of vectors and covectors, the set of all tensors of a h certain ran% ) defined at a point P has the structure of a vector
space, once defined operations of su! "etween tensors and !ultiplication of tensors "y nu!"ers. The su! of tensors 'of the sa!e ran%( gives a tensor whose co!ponents are the su!s of the co!ponents&
. ! ...... = A ...... B ......
A B = C
:.93
The product of a nu!"er a "y a tensor has the effect of !ultiplying "y a all the co!ponents& comp ... :.94 a A a A ... 2.4 Smmetr and s$e/-smmetr
A tensor is told symmetric with respect to a pair of indexes if their exchange leaves it unchanged; skew-symmetric if the exchange of the two indexes involves a change of sign 'the two indexes !ust "e both upper or both lower indexes(. 8or exa!ple& T is .
.
.
= T + s%ew/sy!!etric with respect to indexes 7, V if T . . . =−T . . .
+ sy!!etric with respect to indexes 6, V if T
.
.
.
.
.
.
Lote that the sy!!etry has to do with the order of the argu!ents in H the input list of the tensor. 8or exa!ple, ta%en a : tensor&
, B = A T A
, A = B T B
B T e , e = A B T
A T e , e = A B T
/
the order of the argu!ents in the list is not relevant if and only if the tensor is sy!!etric, since& T = T H
.
, B =T B , A T A
:.9>
:
8or tensors of ran% : or H , represented "y a !atrix, sy!!etry U s%ew/sy!!etry reflect into their !atrices&
/ sy!!etric !atrix& [ T ] = [ T ] + s%ew/sy!!etric tensor / s%ew/sy!!etric !atrix [ T ] =−[ T ] + sy!!etric tensor
:>
H : Any tensor T ran%ed : or H can always "e deco!posed into a
sy!!etric part S and s%ew/sy!!etric part A & T = S A
T = 4 A
that is
where&
4 = 9 T T and A = 9 T −T : :
:.9? :.9B
Tensors of ran% greater than : can have !ore co!plex sy!!etries with respect to exchanges of 3 or !ore indexes, or even groups of indexes. The sy!!etries are intrinsic properties of the tensor and do not depend on the choice of "ases& if a tensor has a certain sy!!etry in a "asis, it %eeps the sa!e in other "ases 'as it will "eco!e clear later(. 2.! epresenting tensors in "-mosaic
Tensor T “naked” & it consists of a "ody that "ears the indication of the co!ponents 'in the shaded "and(; at edges the connectors& e 'in the top edge(
+ pins
+ holes e
'in the "otto! edge( e
T
≡
e
T
T e
2
e
= T 0 β 1 ẽ 0 e⃗⃗ eβ⃗ e 1
co!ponent W connectors
< )hen necessary to !a%e explicit the order of indexes "y !eans of reserved colu!ns a representation "y co!part!ents will "e used, as& e
e e T
e
for
:?
T
= T e e e e
The exposed connectors correspond to free indexes; they deter!ine h nu!"er of pins e the ran%, given "y ) = nu!"er of holes e or even "y r = h ) .
1loc%s representing tensors can connect to each other with the usual rule& pins or holes of a tensor can connect with holes or pins 'with eual generic index( of vectors U covectors, or other tensors. The !eaning of each single connection is si!ilar to that of the heterogeneous scalar product of vectors and covectors& connectors disappear and "odies !erge multiplying to each other . )hen 'at least( one of the factors is a tensor 'r X 9( we properly refer to it as an inner tensor product . The shapes of the "loc%s can "e defor!ed for graphic reasons without changing the order of the connectors. t is convenient to %eep fixed the orientation of the connectors 'pins on-, holes down-(. 2.& "ensors in "-mosaic model deinitions
reviously stated definitions for tensors and tensor co!ponents have si!ple graphical representation in T/!osaic& Definition of tensor plug all r connectors of the tensor with vectors or covectors&
, P , * T A
=
T A
P
e
e , P e , * e
e
P *
e
e
T
T e , e
T
+
7
A
e
A
, e
= A P * T = 7
*
e
= A P *
setting&
03
T A P 0 * 3 :B
= 7
=
The result is a single nu!"er 7 'all the indexes are du!!y(. !omponents of tensor T & o"tained "y saturating with "asis/vectors and "asis/covectors all connectors of the na%ed- tensor. 8or exa!ple&
e e
e e
T e
T
+
T
e
T e
, e , e
= T
The result T are n3 nu!"ers, the co!ponents '3 are the indexes(.
nter!ediate cases "etween the previous two can occur, too& "nput list with basis- and other vectors # covectors&
T e , P
, e
=
T e , P e
, e
= P T e , e , e = P T = 5
P
e
e
e
P
e
T e
T
5
+
setting&
e
T P
:J
= 5
3
The result 5 are n: nu!"ers 'survived indexes are :(& they are the 3 co!ponents of a dou"le tensor Y = 5 e⃗3 ẽ ; "ut it is i!proper to say that the result is a tensor +emar)s2 The order of connection, stated "y the input list, is i!portant. Lote , e = 5 = 8 ) T e , P that a different result T e , e , P with e instead of e . would "e o"tained "y connecting P n general a coated- or saturated tensor- , that is a tensor with all connectors plugged '"y vector, covectors or other( so as to "e a s!ooth o"+ect- is a nu!"er or a !ultiplicity of nu!"ers. t is worth e!phasi#ing the different !eaning of notations that are si!ilar only in appearance&
= T 0 β e⃗⃗ eβ ẽ 0 is the tensor na%ed β β T (̃ e , ẽ ,⃗ e0 = T 0 is the result of applying the tensor to the T
list of "asis/vectors 'i.e. a co!ponent(. nput list& T applied to ...-
The input list !ay also "e incomplete 'i.e. it !ay contain a nu!"er of argu!ents Y r , ran% of the tensor(, so that so!e connectors re!ains unplugged. The application of a tensor to an inco!plete list cannot result in a nu!"er or a set of nu!"er, "ut is a tensor lowered in ran%.
2. "ensor inner product
The tensor inner product consists in the connection of two tensors T and Y reali#ed "y !eans of a pair of connectors 'indexes( of different %ind "elonging one to T , the other to Y. The connection is a single one and involves only one ⃗e and only one ẽ , "ut if the tensors are of high ran% can "e reali#ed in !ore than one way. To denote the tensor inner product we will use the sy!"ol and we0ll write T 9 Y to denote the product of the tensor T on Y , even if this writing can "e a!"iguous and reuires further clarifications. The two tensors that connect can "e vectors, covectors or tensors of ran% r X 9.
:M
A particular case of tensor inner product is the fa!iliar heterogeneous dot product "etween a vector and a covector. et0s define the total ran% + of the inner product as the su! of ran%s of the two tensors involved& + $ r 9 Z r : ' + is the total nu!"er of connectors of the tensors involved in the product(. 8or the heterogeneous scalar 'or dot( product is + =9 Z 9=: . n a less trivial or strict sense the tensor inner product occurs "etween tensors of which at least one of ran% r X 9, that is, + X :. n any case, the tensor inner product lowers "y : the ran% + of the result. )e will exa!ine exa!ples of tensor inner products of total ran% + gradually increasing, detecting their properties and peculiarities. Tensor inner product tensor % vector # covector
⃗ 8or the !o!ent we li!it ourselves to the case T V or T ̃ P for which + + : Z9 + 3. et0s exe!plify the case T ̃ P . )e o"serve that !a%ing the tensor 03 : inner product of a ran% ( H tensor T = T e⃗0⃗ e 3 with a covector P
̃ , !eans applying the for!er, instead of a co!plete list li%e T (̃ P , * ̃ to an inco!plete list for!ed "y the single ele!ent P .
)e see i!!ediately that this product T ̃ P can "e !ade in two ways, depending on which one of the two connectors ⃗ e of T is involved, and the results are in general different; we will distinguish the! "y writing T (̃ P , or T ( ,̃ P where the space is for the !issing argu!ent. The respective sche!es are& P
T (̃ P ,
≡
e
P
e
e
e
e
03
T
e
03
+
T P 0
V
03
T
setting& 3 T P 0 = V 03
and& 3H
P P
e e T(
̃ , P
≡
e
e
e
03
e
03
T
03
T
+
T P 3
3
setting&
03
T P 3
= 3 0
$nly one connector of the tensor is plugged and then the ran% r of the tensor decrease "y 9 'in the exa!ples the result is a vector ( 9H (. Alge"raic expressions corresponding to the two cases are&
= T T 9̃ P
03
e⃗0⃗ e 3 ( P ẽ
= T 0 3 P e⃗0⃗ e 3 (̃e =
03
= T 0 3 P ⃗ e 3 !0 = T 0 3 P 0⃗ e 3 = V 3 e⃗3
= T 0 3 P ⃗ e0 !3 =T 0 3 P 3⃗ e0 = 3 0 e⃗0
T P ⃗ e 0⃗ e 3 (̃e
=
" 03
T P ⃗ e 0⃗ e 3 (̃e
"
̃ !ay designate "oth cases, "ut turns to The notation T (̃ P or T 9 P "e a!"iguous "ecause it does not specify the indexes involved in the inner product. i%ewise a!"iguous is the writing e e e in euations written a"ove. n fact, in alge"ra as well as in T/!osaic "loc% connection, we need to %now which indexes connect 'via connectors0 plugging or indirectly "y Nronec%er O(, i.e. on which indexes the inner product is perfor!ed. $nly if T is sy!!etrical with respect to , the result is the sa!e and there is no a!"iguity.
: n general, a writing li%e e e e e e 'which stands for e e e e e with i!plied( has the !eaning of an inner product of the covector e "y one ele!ent a!ong those of the different %ind aligned in the chain of outer products, for exa!ple&
e e e e e
= e e e "
or
e e e e e "
" 39
= e e e "
The results are different depending on the ele!ent 'the index( hoo%ed-, that !ust "e %nown fro! the context 'no !atter the position of e or e in the chain(. i%ewise, when the internal product is !ade with a vector, as in e e e e e we have two chances&
e e e e e = e e e "
or
e e e e e = e e e "
"
"
β
depending on whether the dot product goes to hoo% ẽ or ẽ . 4
8or!ally, the inner product of a vector or covector e⃗4 or ẽ addressed to a certain ele!ent 'different in %ind( inside a tensorial chain e e e e⋅⋅⋅ re!oves the hoo%ed- ele!ent fro! the chain and insert a O with the vanished indexes. The chain welds with a ring less, without changing the order of the re!aining ones.
̃ T on fixed indexes cannot give a result different Lote that product P ̃ T = T P ̃ and the tensor inner product is . Sence P fro! T ̃ P commutattive in this case.the Pa!e as a"ove is the case of tensor inner product tensor [ "asis/ covector. Sere as well, the product can "e reali#ed in two ways& 0 T( e ̃ or T ( ẽ . )e carry out the latter case only, using the sa!e T of previous exa!ples& T(
, ẽ
= T ẽ = T 0 3 e⃗0⃗ e 3 ( ẽ = T 0 3 e⃗0 !3 = T 0 e⃗0 "
which we represent as&
e e
e
e
e
0
T
+
0
T
0
T
0 T , the co!ponent of T , would "e the result if the input list had "een co!plete. Saving o!itted a "asis/covector in the input list has e , so that the result is the effect of leaving uncovered a connector ⃗ here a vector.
3:
Pi!ilar considerations apply to inner products tensor [ vector where T is ran%ed ( H: . 9 ⃗ and T ̃ P have an f T is ran%ed ( 9 the tensor inner products T V
uniue 'not a!"iguous( !eaning "ecause in "oth cases there is only an o"liged way to perfor! the! 'it is clear in T/!osaic(. ⃗ =⃗ V T . 2ven here the products co!!ute& for exa!ple T V n conclusion& inner tensor products with ran% + + 3 !ay or !ay not "e twofold, "ut they are anyways co!!utative. "nner tensor product tensor % tensor ( + 5 % )hen the total ran% + of the inner tensor product is + 5 4 the !ultiplicity of possi"le products increases and further co!plications arise, due to the non/co!!utativity of the products and for what concerns the order of the indexes that survive 'when they co!e fro! "oth tensors, how order the!K(. et0s specify "etter. Given two tensors A , B we !ain "y& 9 multiplicity the nu!"er of inner product possi"le upon various inE dexes 'connectors( after fixed the order of the factors 'A 5 B or B 5 A(. 9 commutativity the eventuality that is A 5 B + B 5 A after fixed the pair of indexes 'connectors( on which the products are performed. H : 2xa!ple of a case + + : Z : + 4 is the following, with A ( : and B ( H . u ;ultiplicity
4 different tensor inner products A 5 B are possi"le, depending on which connectors plug 'that is, which indexes are involved(& 0 3
A B = ( A0 3 e ̃
0 , 3 ,
=
0 , β 3 , β
ẽ
( Bβ e⃗⃗ eβ = = A = A = A = A
!0 = A B β ẽ 3 e⃗ = ! β3 ẽ 3 e⃗ β 0 3 β 0 β 0 B ẽ e⃗ ! = A B ẽ e⃗ = < ẽ e⃗ β 3 0 β 3 3 B ẽ e⃗ ! = A B ẽ e⃗ = - 3 ẽ e⃗ β 0 3 β 0 0 B e e A B e e = ẽ e⃗ ⃗ ! = ⃗ = ̃ ̃ 3 B
β
3
ẽ e⃗
33
which correspond rispectively to the following T/!osaic patterns& @ @ A0 3
e e
e
B
e
!0β
!β3
ẽ β
ẽ
e
β
e
e
B
e
e
β
B
e
e e
e
e e⃗3
"
e
A0 3
A0 3
" e
e
A0 3
e⃗3
e⃗β
e
β
B
β
e β
The 4 results of A 5 B are in general different. 1ut it !ay happen that& A sy!!etrical
(/ ẽ 0 , ẽ 3 interchangea"le / 9\ product 5 :\
B sy!!etrical
(/ ⃗e , ⃗e β interchangea"le / 9\ product 5 3\
3\ product 5 4\ :\ product 5 4\
The 4 products coincide only if "oth A and B are sy!!etrical. v >oncommutativity
and indexes? arrangement
B 5 A represent
4 !ore inner products, distinct fro! one another and different fro! the previous @@; @@e.g. the first one is 'product on 6 ,(& B A =( B
β
0 3
e⃗⃗ e β ( A0 3 ẽ ẽ
= ! β3 e⃗β̃ e 3 .
Co!paring with the first a!ong the products a"ove we see that the inner tensorial product on a given pair of indexes "etween tensors of ran% r X9 is in general non/co!!utative& A5 B ] B 5 A. @ )e use here the artifice of O as plug adapter- to %eep the correspondence of inE dexes "etween euations and "loc%s. @@ $f course, the 4 possi"le pairs of indexes 'connectors( are the sa!e as the previous case, "ut their order is inverted. Lote that the !ultiplicities of A5B are counted separately fro! those of B 5 A . 34
Sow to account for the different results if T/pu##le graphical representation for A 5 B and B 5 A is uniueK The distinction lies in a different reading of the result that ta%es into account the order of the ite!s and give the! the correct priority. n fact, the result !ust "e constructed "y the following rule& @ @ list first connectors 'indices( of the first operand that survive free, then free connectors 'indices( survived of the second one. Reading of the connectors 'indexes( is always fro! left to right. $"viously indexes that !a%e the connection don0t appear in the result. f "oth A and B are sy!!etrical, then A 5 B + B 5 A , and in addition the 4Z4 inner products achieva"le with A and B converge in one, as is easily seen in T/!osaic. < As can "e seen "y T/!osaic, the co!plete variety of products for + + 4 includes, in addition to the 4 cases +ust seen, the following&
⃗ T with T ( H , (9 , ( : [ B cases of inner product V 3 : 9
̃ T with T (3 , ( : , ( 9 [ B cases of inner product P H 9 : with !ultiplicity 3, : or 9, all co!!utative; [ ? cases of tensor product A B with A (99 and B (:H , ( 99 , (H: all with !ultiplicity :, non/co!!utative, giving a total of :4 distinct inner products 'Z :4 co!!utated( for + + 4 'including 4 cases considered "efore(. The nu!"er of products grow very rapidly with +; only the presence of sy!!etries in tensors reduces the nu!"er of different products. "ncomplete input lists and tensor inner product There is a partial overlap of the notions of inner product and the application of a tensor to an input list- @@ @@The co!plete input list is used to saturate all the connectors 'indexes( of the tensor in order to give a nu!"er as a result; however, the list !ay "e inco!plete, and @ This rule is fully consistent with the usual interpretation of the intner tensor product as ^external tensor product Z contraction^ which will "e given later on. @@At least for what concerns the product tensor [ vector or covector- 'input lists contain only vectors or covectors, not tensors. 3>
then one or !ore connectors 'indexes( residues of the tensor are found in the result, which turns out to "e a tensor of lowered ran%. The extre!e case of inco!plete input list- is the tensor inner product tensor [ vector or covector, where all the argu!ents of the list are !issing, except one. Conversely the application of a tensor to a list 'co!plete or not( can "e seen as a succession of !ore than one single tensor inner products with vectors and covectors. The higher the ran% r of the tensor, the larger the nu!"er of possi"le inco!plete input lists, i.e. of situations inter!ediate "etween the co!plete list and the single tensor inner product with vector or covector. 8or exa!ple, for r + 3 there are ? possi"le cases of : inco!plete list.@@ n the case of a ( 9 tensor one a!ong the! is&
P
T
, e , P
P
e e
≡
3
e e
5
T
T
e
e
e
5
setting T P = 5
The result is n covectors, one for each value of @. This sche!e can "e interpreted as a tensor inner product "etween the , followed "y a second tensor inner tensor T and the covector P 3 product "etween the result and the "asis/covector ẽ , treated as such in alge"raic ter!s& i( T (̃ P = T ẽ e⃗0⃗ e 3 ( P 4 ẽ = T P 4 ẽ e⃗3 !0 = T P 0̃ e e⃗3 = 5 ẽ e⃗3 " 03
4
03
ii( 5 e e e = 5 e " = 5 e
4
03
! 13 @ 3 lists lac%ing of one argu!ent Z 3 lists lac%ing of : argu!ents. 3?
3
2.6 uter product in "-mosaic
According to T/!osaic !etaphor the outer product !eans union "y side gluing-. t !ay involve vectors, covectors or tensors. $f the !ore general case is part the outer product "etween vectors andUor covectors; for exa!ple& e
e
B
A
e 7
A
e
e
B
+
e
A B
T
B A
e = = A B e e = A B e e = T e
T
or else& e
e P e
A
7
P A
e
e
+
P A
e
Y
A P
= P A e e = P A e e = 5 e e = Y
The outer product !a%es the "odies to "lend together and co!ponents !ultiply 'without the su! convention co!es in operation(. t is clear fro! the T/!osaic representation that the sy!"ol has a !eaning si!ilar to that of the con+unction and- in a list of ite!s and can usually "e o!itted without a!"iguity. The sa!e logic applies to tensors of ran% r X9, and one spea%s a"out outer tensor product. The tensor outer product operates on tensors of any ran%, and !erge the! into one for side gluing- The result is a co!posite tensor of ran% eual to the su! of the ran%s. 8or exa!ple, in a case A ( 99 B( 9: = C ( :3 & 3B
A e
B
e
e
e
e
e
e
e
e
A
e
_
B
e
e
!
+
e
e
e
setting&
A B = !
A e e B e e e = A B e e e e e = = ! e e e e e
usually written ! e e e e e
.
Lote the non/co!!utativity& A B ) B A .
2.8 Contraction
dentifying two indexes of different %ind ' one upper and one lower (, a tensor undergoes a contraction 'the repeated index "eco!es du!!y and appears no longer in the result(. Contraction is an unaryoperation. h The tensor ran% lowers fro! ) to
9 h) − −9 .
n T/!osaic !etaphor "oth a pin and a hole "elonging to the sa!e tensor disappear. 2xa!ple& contraction for 6 5 ` & e
e
e
! !
e
e
0
contraction
6
!
! β3
=6
e
e
e
+
e
3J
e
e
e
e
t is worth noting that the contraction is not a si!ple canceling or si!plification- of eual upper and lower indexes, "ut a su! which is triggered "y repeated indexes. 8or exa!ple&
9
:
n
! = ! 9 ! :... ! n = ! n a sense, the contraction consists of an inner product that operates inside the tensor itself, plugging two connectors of different %ind. The contraction !ay "e seen as the result of an inner product "y a tensor carrying a connector 'an index( which is already present, with opposite %ind, in the operand. Typically this tensor is the Nronec%er O. 8or exa!ple&
!
" = ! = ! where contraction is on index 6 .
The T/!osaic representation of the latter is the following& e
e e
!
e
e
e
e
5
!
e
!
+
e
e
" e
e
e e
e
8or su"seuent repeated contractions a tensor ( h can "e reduced )
h until h 5 H or ) 5 H. A tensor of ran% ( h can "e reduced to a scalar. 9
The contraction of a 9 tensor gives a scalar, the su! of the !ain diagonal ele!ents of its !atrix, and is called the trace& A = A 99 A::... Ann The trace of I is
:.9J
" = 99... = n , di!ension of the !anifold. @@
@ I is the tensor identity- whose co!ponents are Nronec%er/O & see later. 3M
2.1, Inner product as outer product 9 contraction
The inner tensor product A B can "e interpreted as tensor outer product A B followed by a contraction of indexes& A B = contraction
of ( A B .
The !ultiplicity of possi"le contractions of indexes gives an account of the !ultiplicity of inner tensor product represented "y A B . 8or exa!ple, in a case A ( H: B( 3H= C( :9 , with total ran% + 5 >& A B
5 Aβ 3⋅ B β 1 = ! β. .3 β1 = ! 3. 1 e
Aβ 3 ẽ
β
e
e B
e
e
β1
'product on 7(& e⃗β e
. . β1
+
! β 3
ẽ
β
e
e
e
. 1
+
! 3 ẽ
3
The 9st step 'outer product( is univocal; the : nd step 'contraction( i!plies the choice of indexes 'connectors( on which contraction !ust ta%e place. Gradually choosing a connector after the other exhausts the choice of possi"ilities '!ultiplicity( of the inner product A B . The sa!e considerations apply to the switched product B A . This :/steps !odality is euivalent to the rules stated for the writing of the result of the tensor inner product. ts utility stands out especially in co!plex cases such as& A
03 7 β β B 1
= ! 0 3β 7 β 1 = ! 0 3 7 1
:.9M
2.11 'ultiple connection o tensors
n T/!osaic the tensor inner product is always perfor!ed "y !eans of a si!ple connection "etween a pin e of a tensor and a hole e of the other. =ultiple connections "etween two tensors, easy to fulfill in T/!osaic, can "e alge"raically descri"ed as an inner product followed "y contractions in nu!"er of m 9 , where m is the nu!"er of plugged connectors( . 4H
2.12 )Scalar product* or )identit* tensor
The operation of 'heterogeneous( scalar product ta%es in input a vector 9 and a covector to give a nu!"er / it is a tensor of ran% 9 . )e denote it "y I &
, V I P I
and can expand it as&
≡ 8 P , V $ = P V
= e e , that is
I
:.:H
= e e .
Sow is it IK et us calculate its co!ponents giving to it the dual "asis/ vectors as input list&
e = 8 e , e $ = " = I e ,
I
hence&
:.:9
= " e e
:.::
/ the Nronec%er O sy!"ol is a tensor&
δ
≡
I
na!ely&
[
]
9 9 e 9 $ # e , e : $ , # e , en $ # e 9 , : : : e , e 9 $ # e , e:$ e , en $ # # ≡ [ " ] = = n n e 9 $ # e , e : $ , # e , en $ # e n ,
δ
≡
I
[
9 H ... H 9 ... ... ... 9
comp
"
]
:.:3 :.:4
The heterogeneous scalar product is the tensor I, identity tensorits co!ponents are the Nronec%er0s " its related !atrix is the unit !atrix 5 diag 'Z9( Sow does I actK ts co!plete input list contains two argu!ents& a vector and a covector. et0s apply it to a partial list that contains only one of the two; there are : possi"ilities& u
⃗ I ( V
= !β ẽ β e⃗ ( V 1 ⃗e 1 = !β V 1 ẽ β e⃗ ( ⃗e 1 = !β ! 1β V 1 e⃗ = β
! 1 v
I (̃ P
⃗ 1 e⃗ = V e⃗ =⃗ V = ! 1 V
= !β ẽ β e⃗ ( P 1̃ e 1 = !β P 1 ẽ β e⃗ ( ẽ 1 = !β ! 1 P 1 ẽ β = ! 1
49
̃ = ! 1β P 1̃ eβ = P β̃ eβ = P
'note&
!β ! 1β = ! 1 (
⃗ is& The T/!osaic "loc%wise representation of I ( V e e I
" e
V
e
e
"
V
V
V
V
Actually, I transfor!s a vector into itself and a covector into itself&
I V V
= V V
= P I P
;
:.:>
hence its na!e identity tensor-. The sa!e tensor is also called funda!ental !ixed tensor-.@ @ )e also o"serve that, for ∀ T & T I = I T
=T
:.:?
2.1, Inverse tensor
Given a tensor T , if there exists a uni"ue uni"ue te tensor Y such such that -1 T Y = I , we say that Y is the inverse of T , i.e. Y = T and T
T-1 =
I
:.:B
$nly tensors T of ran% :, and a!ong the! only 9 type or symmetric or symmetric 9
( H or ( : type tensors, can satisfy these conditions @@@@ and have an :
H
inverse T-1 . @ Sere Sereaf afte terr the the not notatio ation n δ will "e a"andoned in favor of I. @@ That e".$.$0 is is satisfied "y tensors T and T
-1
"oth ran%ed r 5 5 : can easily "e H
:
shown in ter!s of T/!osa T/!osaic ic "loc%s. f T e T -1 are tensors ran%ed ( : or ( H , e".$.$0 stands stands for 4 different different inner inner products and, for a given T , can "e satisfied -1 for !ore than one T -1 ; only if we as% that T is sy!!etric ' / T si!!etric, too( the tensor T -1 that satisfies it is uniue and the uniueness of the inverse is guaranteed. 8or this reason one restricts the field to sy!!etric tensors only. 4:
8or tensors of ran% r ] : the inverse is not defined. The The invers inversee T-1 of a sy!!e sy!!etric tric tensor tensor T of ran% H or : has the : H following properties& + the indexes position interchanges upper b lower @ @ + it is represented "y the inverse !atrix + it is sy!!etric : 8or exa!ple& given the H sy!!etric tensor T
is the
H :
comp
T , its inverse
comp
tensor T-1
T 0 3 such that& T
T =
"
:.:J
8urther!ore, denoting T and T and T the !atrices associated to T and -1 , we have& T T ⋅ T = :.:M where where is the unit !atrix; na!ely na!ely,, the !atrices !atrices T and T and T are inverse to each other. : < ndeed, for H tensors tensors the definition definition e".$.$0 turns turns into e".$.$/. e".$.$/.
1ut 1ut the the latt latter er,, tran transc scri ri"e "ed d in !atr !atrix ix ter! ter!s, s, ta%e ta%ess the the for! for! [T ]⋅ [ T ] = , which is euivalent to e".$.$. e".$.$. Pince the inverse of a sy!!etric !atrix is sy!!etric as well, the sy!!etry of T-1 follows& if a tensor is sy!!etric, then its inverse is sy!!etric, too. The The corre corresp spon onde denc ncee "etw "etwee een n the the inve invers rsee !atri !atrix x and and the inve invers rsee tensor forwards to the tensor other properties of the inverse !atrix& + the co!!uta co!!utativi tivity ty T ⋅T
hich appl applie iess to inve invers rsee = T ⋅T = = , whic !atrices, is also true for inverse tensors& T T-1 = T-1 T = I
+ in order that T
-1
exists, the inverse !atrix !ust also exist and for @@ @@ that it is reuired to "e det T ) H . @@
Curr Curren entl tly y we say say that that tens tensor orss T and T are are inve invers rsee to each each @ This This fact is often often expre expresse ssed d "y saying saying that that the inver inverse se of a dou"le/ dou"le/con contra travar varian iantt tensor is a dou"le/covariant tensor, and vice versa. @@ A !atrix !atrix can have an inverse inverse only if its its deter!inant deter!inant is not #ero. #ero. 43
other& usually we call the co!ponents of "oth tensors with the sa!e sy!" sy!"ol ol T and and distinguish the! only "y the position of the indexes. Sowe Soweve verr, it is wort worth h real reali# i#in ing g that that we are are deali dealing ng with with diff differe erent nt tensors, and they cannot run "oth under the sa!e sy!"ol T 'if we call T one of the two, we !ust use another na!e for the other& in this -1 instance T ( . t shou should ld also also "e re!i re!ind nded ed that that 'onl 'only y( if T is diag diagon onal al 'i.e 'i.e.. T ) H only for = ( its inverse will "e diagonal as well, with co!ponents T = 9 9 T , and viceversa. The property of a tensor to have an inverse is intrinsic to the tensor itself and does not depend on the choice of "ases& if a tensor has inverse in a "asis, it has inverse in any other "asis, too 'as will "eco!e clear later on(.
9
An o"vious property "elongs to the !ixed tensor T of ran% 9 related- with "oth T and T , defined "y their inner product&
T = T T
Co!paring this relation with e".$.$/ 'written e".$.$/ 'written as T 7 in place of ( we see that& T β = !β ndeed ndeed,, the !ixed !ixed funda! funda!ent ental al tensor tensor 9 9
T =
" with :.3H
" , or tensor
I
, is the
β
co!!on relative- of all couples of inverse tensors T β , T . (.@@ comp
The !ixed !ixed funda! funda!ent ental al tensor tensor I the property to "e the inverse of itself.
! β has 'with few others @@( @@
< ndeed, an already noticed @@@ @@@ property of Nronec%er0s O&
! β ! 1β = ! 1
is the condition of inverse 'si!ilar to e".$.$/( e".$.$/( for ! β . 'T/!osaic 'T/!osaic icastically shows the !eaning of this relation(.
@ That That does does not !ean, !ean, of course course,, that it is the the only exis existi ting ng !ixed !ixed dou" dou"le le tenso tensorr 'thin% to ! = A B when A e B are not related to each other(. 9 @@ Also auto/inverse auto/inverse are the tensors tensors ( 9 , whose !atrices are !irror i!ages of I. @@@Already noticed a"out the calculation of I (̃ P P .
44
2.14 Vector-covector “dual switch” tensor 0 A tensor of rank 2 needs two vectors as input to give a scalar. If the
input list is incomplete and consists in one vector only, the result is a covector:
⃗ )= G β̃ e G ( V
β
ẽ
having set G V
̃ ( V γ e⃗ γ )= G β V γ ẽ ẽ β ⃗ e γ = G β V ẽ β = P β ẽ β = P δ γα
= P .
The representation in T-mosaic is:
G
G
e
e
!
V
to "e read:
P
=
e
e
≡ V
G
e
setting:
V
G V
G V
= P
= G V e = P e = P
0
into a #y means of a 2 tensor we can then transform a vector V ̃ "elonging to the dual space. covector P 0
$et us pick out a 2 tensor G as a %dual switch& to "e used from : into its %dual& covector V now on to transform any vector V
G V
G
2.2' = V V "etween the two dual esta"lishes a correspondence G : V
vector spaces (we use here the same name V with different marks a"ove in order to emphasi)e the relationship and the term %dual& as %related through * in the dual space&+. The choice of G is ar"itrary, nevertheless we must choose a tensor which has inverse G-1 in order to perform the %switching& in the to V . In addition, if we want the opposite sense as well, from V inverse G-1 to "e uniue, we must pick out a G which is symmetric, as we know. ote that, in this manner, using one or the other of the two ẽ connectors of G "ecomes indifferent. /
Applying G-1 to the switch definition eq.2.31:
G V = G-1 V = I V V , then:
-1
G
"ut
G
-1
G =
I
and
V -1
G
-1
=
G
V
2.2
thus esta"lishes an inverse correspondence G-1 : V
. V
In T-mosaic terms:
V
V
e
-1
G
V
e
e
!
G
e
e
G
=
V
setting: G V = V
to "e read
-1
G
V =
G
V e
=
V e
= V
1 The vector covector switching can "e e3pressed componentwise. = V is written as: In terms of components G V
G V which is roughly interpreted: 4onversely, G-1 V
= V
2.
G %lowers the index ”.
can "e written: = V G
V
=
V
2.
and interpreted as: G %raises the index &. 2.15 Vectors / covectors homogeneous scalar product
It presupposes the notion of switch tensor G. 5e define the homogeneous scalar product between two vectors as the scalar product "etween one vector and the dual covector of the other: 6
⃗ ⃗ A
! ̃ 〉 ≝ ⃗ A
or, likewise,
⃗〉 ≝ ̃ A !
2./
7rom the first euality, e3panding on the "ases and using the switch G we get:
A
! 〉 = A e ! e 〉 = A e ! G e 〉 = = A
e ! e 〉 = G A = G A = = G A
A
In short:
=
! G A
! G A
2.6
8ence, the dual switch tensor G is also the % scalar product between two vectors& tensor . The same result follows from the second euality eq.2.3". The symmetry of G guarantees the commutative property of the scalar product: = A A
! = G ! A G A
or
2.9
(note this is ust the condition of symmetry for the tensor G eq.2.1"+. 1 The homogeneous scalar product between two covectors is then -1 defined "y means of G :
= G A ! A -1
G
-1
2.;
, the inverse dual switch, is thus the % scalar product between two
covector & tensor .
1 In the T-mosaic metaphor the scalar (or inner+ product "etween two vectors takes the form:
G
G !
e e A
e
=
e
A
G A
A
9
= G A
while the scalar (or inner+ product "etween two covectors is:
A
A
e e
e
=
G
A
A
e
!
=
G
A
G
G
The result is a num"er in "oth cases. 1 8ow is it G< $et=s compute its components using as arguments the "asis ! : vectors instead of A
G > G e ! e > e e
⇒
G
⇒
≡ [ G β ]
[
⃗ e ? ⃗e? e⃗2 ⃗ e? = ⋮ e⃗n ⃗e?
e⃗? e⃗2 e⃗2 ⃗ e 2
⋯ e⃗? e⃗n e⃗2 ⃗ e n ⋮ e⃗n e⃗2 ⋯ e⃗n e⃗n
]
2.'
It =s clear that the symmetry of G is related to the commutativity of the scalar product: e e > e e
G > G
G symmetric
and that its associated matri3 is also symmetric. In a similar way for the inverse switch:
G
-1
> G
⇒
e ! e > e e
-1
G
≡ [G
β
]
⇒
[
ẽ ? 2 ? ẽ ẽ = ⋮ n ? ẽ ẽ ẽ
?
;
?
2
ẽ
2
ẽ 2
ẽ
n
2
ẽ ẽ
ẽ
⋯ ẽ ? ẽ n 2 n ẽ ẽ ⋮ ⋯ ẽ n ẽ n
]
2.0
Notation
Various equivalent writings for the homogeneous scalar product are: ▪ between vectors: > A > 〉 > G A ! ! A 〉 > A ! A A ! > G
1 between covectors: > A > A ! 〉 > G-1 > G-1 ! A A 〉 > A ! A ! The notation 〈 , 〉 is reserved to heterogeneous scalar product vector-covector
2.16 G applied to asis-vectors G transforms
a "asis-vector into a covector, "ut in general not into a "asis-covector: G (⃗ e )= G ! e ̃
!
ẽ (⃗ e )= G ! ẽ
!
δ = G ẽ
2.?
and this is not =ẽ "ecause G ẽ is a sum over all ẽ and can=t collapse to a single value ẽ (e3cept particular cases+. $ikewise:
G
-1
( ẽ ) = G ! ⃗e ⃗e ! (̃e ) = G ! ⃗e δ! = G ⃗e "e⃗
2.2
@ otice that in eq.2.#1 we have made use of the duality condition (eq.1.$+ that link the "ases of vectors and covectors it is a relation different from that stated "y the %dual switching&. 2.1! G applied to a tensor
4an we assume that the converter G acts on the single inde3 of the tensor (i.e. on the single connector e⃗ + as if it were isolated, in the same way it would act on a vector, that is, changing it into a e # without changing other indices and their seuence< ot uite. (The same uestion concerns its inverse G-1 +. βγ
5e see that the application of G to the tensor ( 0 ) X = % G ! %
βγ
= % !
e⃗⃗ eβ⃗ e γ :
βγ
has, in this e3ample, really the effect of lowering the inde3 B involved in the product without altering the order of the remaining others. 8owever, this is not the case for any inde3. 5hat we need is to e3amine a series of cases more e3tensive than a single e3ample, having in mind that the application of G to a tensor according to usual rules of the tensor inner product G ( T ) = G T gives rise to a '
multiplicity of different products (the same applies to G-1 +. To do so, let=s think of the inner product as of the various possi"le contractions of the e3ternal product: in this interpretation it is up to the different contractions to produce multiplicity. @ Ceferring to the case, let=s e3amine the other inner products you can do on the various indices, passing through the outer product βγ = % ! βγ and then performing the contractions. G ! % β γ βγ The D-B contraction leads to the known result % ! = % ! A in addition: βγ = % ! γ E contraction D-F gives % β! β γ
β
= % ! E contraction D-G gives % γ ! (since G is symmetric, contractions of H with B, F or G give the same results as the contractions of D+. ther results rise "y the application of G in post-multiplication: βγ βγ βγ γ β G ! = % ! we get % ! , % ! , % ! and other from % similar contractions for H . 5e o"serve that, among the results, there are lowerings of indices βγ β (e.g. % ! , lowering B( H , and % ! , lowering G( H +, together with other results that are not lowerings of indices (for e3ample γ γ % ! : F is lowered F( H "ut also shifted+. % ! does not appear among the results: it is therefore not possible to get "y means of G the transformation: e
e e
e
e
!
%
αβγ
!
%
α γ !
#
e
as if e⃗β were isolated.
It=s easy to see that, given the usual rules of the inner tensor product, only indices placed at the beginning or at the end o& the string can be raised ' lowered by G (a similar conclusion applies to G-1 +. It is clear, however, that this limitation does not manifest itself until the inde3es= string includes only two, that is, for tensors of rank r = 2 . 1 7or tensors of rank r =2 and, restricted to e3treme inde3es for tensors
/0
of higher rank, we can enunciate the rules: G applied
to a tensor lowers an index (the one "y which it connects+. % nly the writing "y components, e.g. G % # ( = ( # , clarifies which inde3es are involved. -1
G
applied to a tensor raises an index (the one "y which it connects+. β ! β ! 7or e3ample ( γ G = ( γ . Mnemo
Examples: G ! ( β
(
'
β
G
= ( ! β
!
:
hook J , raise it, rename it H
= ( β ' !
: hook D, raise it, rename it H
1 A special case is when the tensor G applies to its inverse G-1: G G
=
G
which can "e thought to as a raising as well as a lowering of an inde3. #ut since the two tensors G and G-1 are inverse to each other: G G
=
and from the last two it follows (not surprisingly, given the eq.2.3)K+ that: G = 2. 2.1" #elations etween I$ G$ δ
! 〉 = 〈 A ! 〉 = = A and to one another All euivalent to 〈 A A are the following e3 pressions:
! I A
! = = I A
! G A
=
G
-1
! A
2.
(easy to see using T-mosaic+. 1 The componentwise writing for G G-1 = I gives:
G = * , that, together with eq.2.#3, leads to:
=
2./
= G = * =
2.6
G
=
*
$ikewise: G
=
*
/?
@ That does not mean that G and I are the same tensor. 5e o"serve that the name G is reserved to the tensor whose components are G the other two tensors, whose components are , are different tensors and cannot "e la"eled "y G and G the same name. In fact, the tensor with components G is G-1, while that one whose components are G coincides with I. It is thus matter of three distinct tensors: LL comp
I
comp
G G
-1
comp
G
= * =
G
= * =
= * =
G
even if the components= names are somewhat misleading.
ote that only is the Mronecker delta, represented "y the matri3 diag (N?+. β
β
#esides, neither * β nor * (and not even δ β and δ + deal with the identity tensor I ("ut rather with G their matri3 may "e diag (N?+ or not+. The conclusion is that we can la"el "y the same name tensors with inde3es moved up O down when using a componentwise notation, "ut we must "e careful, when switching to the tensor notation, to avoid identifying as one different tensors. β
Pust to avoid any confusion, in practice the notations * β , * , δ β , δ β are almost never used.
?
0
2
L The rank is also different: ? , 2 , 0 respectively. Their matri3 representation can "e formally the same if G diag (N?+, "ut on a di&&erent +basis grid”, /2
% &hange o' asis
has e3pansion V e on the "asis of vectors e . The vector V Its components will vary as the "asis changes. In which way< It=s worth noting that the components change, "ut it does not the K vector V $et us denote "y V ' the components on the new "asis e it is = V ' e ' , hence: now V '
V
= V ' e ' = V e
.?
In other words, the same vector can "e e3panded on the new "asis as well as it was e3panded on the old one (from now on the upper ' will denote the new "asis+. 1 $ike all vectors, each "asis-vector of the new "asis e3panded on the old "asis-vectors e :
e ' = * ' e
e ' can "e
' = ?,2,... n
.2
* ' are the coefficients of the e3pansion that descri"es the %recipe& e ' on the old "asis e (i.e. on the "asis of the old of the new %ingredients&+. Taken as a whole, these coefficients e3press the new "asis in terms of the old one they can "e arranged in a matri3 n - n
[ * ] '
1 onversely we can e3press the old "asis-vectors on the "asis of new ones as: . = ?,2,... n e = * e '
'
The two matrices that appear in eq.3.2 and eq.3.3 are inverse to each other: e3changing the inde3es the matri3 is inverted "ecause the sense of the transformation reverses applying "oth them one after the other we return to the starting point. 1 5e now aim to find the relation "etween the new components of a and the old ones. Qsing eq.3.3: given vector V
V
and since:
= V e = = V ' e ' V
V
* e '
/
'
⇒ V ' = * ' V
.
1 5hat a"out covectors< 7irst let us deduce the transformation law for e eq.1.#, which also holds good in the components. 7rom P = P new "asis, "y means of the transformation of "asis-vectors eq.3.2 that we already know, we get:
e = P * e = * P e = * P = P ./ and using the eq.3." a"ove, 7rom the definition of component of P P
'
'
'
'
'
we deduce the inverse transformation for "asis-covectors (from new to old ones+:
= P e P
= P e P
'
'
= * P e
'
⇒ e = * e
'
'
'
and hence the direct from old to new: e
'
= * ' e
.6
1 To summari)e, all direct transformations (from old to new "asis+ are ruled "y two matrices:
[ * ]
e3presses the transformation of the components of vectors (eq.3.#+ and of "asis-covectors (eq.3./ +
[ +β ]
e3presses the transformation of the components of covectors (eq.3."+ and of "asis-vectors (eq.3.2+
'
'
The two matrices are one the transposed inverse LL of the other. L The transpose 0 .( of a matri3 0 is o"tained "y interchanging rows and columns ,? ( ( ,? the transposed inverse 0 coincides with the inverse transpose 0 . The notation we use cannot distinguish a matri3 from its transpose (and the transR posed inverse from the inverse, too+ "ecause the upper O lower inde3es do not ualify rows and columns in a fi3ed way. 8ence, the two matrices * in eq.3.3 and eq.3./ are not the same: they are the inverse in the first case and the transposed inverse in the second. It can "e seen "y e3panding the two euations: '
eq.3.3:
e = *
'
( (
e? =* ? e ? - * ? e 2
e ⇒ '
-... ? 2 e2 =* 2 e ? - * 2 e 2 -... ?
2
'
'
'
'
'
'
'
............
eq.3./ :
e
'
=* ' e ⇒
'
) )
= * ?? e ?- * ?2 e 2- ... 2 2 ? 2 2 e = * ? e - * 2 e - ... ?
e
'
'
'
'
'
'
............ /
[ [
* ?? * 2? ⇒ * = * ?2 * 22 ⋯ '
'
'
'
'
] ]
⋯ ⋯
* ??' * ?2 ' ⋯ ⇒ * ' = * 2?' * 22 ' ⋯ ⋯
n the contrary, the inverse transformation (from old to new "asis+ is ruled "y two matrices that are the inverse of the previous two. So, ust one matri3 (with its inverse and transposed inverse+ is enough to descri"e all the transformations of (components of+ vectors and covectors, "asis-vectors and "asis-covectors under change of "asis. 1 $et us agree to call * (without further specification+ the matri that trans&orms vector components and basiscovectors from the old "ases system with inde3 to the new system inde3ed :
* ≝ [ * ' ]
.9
*iven this position, the transformations of "ases and components are summari)ed in the following ta"le: (
*,?
e
V
*
e'
e
*
e
'
.;
(
* ,?
'
P
V
P '
Coughly speaking: if the "ases vary in a manner, the components vary in the opposite one in order to leave the vector unchanged (as well as the num"er that e3presses the measure of a uantity increases "y making smaller the unit we use, and vice versa+. The transformations in the opposite sense (+ reuire to invert all the matrices of the ta"le (remind that the inverse of
* ,?
(
is
* ( +.
1 7or a transformation to "e reversi"le, it is reuired that its matri3 * is inverti"le, that means det * " 0 . 1 The duality relation "etween "ases of vectors and covectors holds good in the new "asis, too:
e # ' ! e ' 〉 = * #% ' e % ! * ' e 〉 = * #% ' * ' e % ! e 〉 = = * #% ' * ' % = * # ' * ' = # ' ' (the two matrices are inverse to each other, #
'
#
,?
[ * ] = [* ] # %
'
'
and hence
'
related "y * * = +. '
'
1 The matri3 * that rules the transformations can "e e3pressed in terms of "oth old and new "asis: //
% = e ! e % 〉 = e ! * %# e # 〉 = * %# e ! e # 〉 , '
'
'
'
e ! e # 〉 = * # since then % = * %# * # . 8ence, the element of the matri3 * is: '
only possi"le if
'
'
'
+ ! ' = e⃗ ! ẽ ! ' 〉
.'
and the transformation matri3 is thus "uilt up "y crossing old and new "ases. 1 In practice, it is not convenient trying to remind the transformation matrices to use in each different case, nor reasoning in terms of matri3 calculus: the "alance of the inde3es inherent in the sum convention is an automatic mechanism that leads to write the right formulas in every case. Mnemo
The transformation of a given obect is correctl! written b! simpl! taking care to balance the indexes" #or example, the transformation of components of covector from new to old basis, provisionall! # written P % = * P # , can onl! be completed as P % = * % P # ⇒ '
'
'
the matrix element to use is
*
# ' %
%.1 (asis change in )-mosaic
In the T-mosaic metaphor, the change of "asis reuires each connector to be converted . The conversion is performed "y applying a %"asis converter& e3tra-"lock to the connector for convenience we represent it as a standard "lock with a pin and a hole ("ut "eware: it is not a tensorK+. The "asis-converter "lock is one and only, in two variants distinguished only "y the different position of the indices with ape3 ' : e
e
'
* '
* '
e
'
'
e
'
/6
The connection is done %wearing& the "asis-converters "locks as %shoes& on the connectors of the tensor, docking them on the pin-side or the hole-side as needed, Uape3 on ape3U or Unon-ape3 on non-ape3U. The "ody of the converter "locks will "e marked with the element of ' the transformation matri3 * or * ' with the apices ' up or down oriented in the same way with those of connectors (this implicitly leads to the correct choice of + or + +. 7or e3ample, to "asis-transform the components of the vector V e we must apply on the e connector a converter "lock that hooks it and replaces with e : '
'
'
e
'
e * ' * e
'
'
!
e
e =
'
'
V
V
since:
* ' V = V '
V
Similarly, to "asis-transform the components of the covector P e the appropriate converter "lock applies to the old connector e :
P
e
e
* '
P
!
=
* ' e
P ' e '
'
since:
e '
/9
* P = P '
'
,
1 This rule doesn=t apply to "asis-vectors (the converter "lock would contain in that case the inverse matri3, if one+ however the "lock representation of these instances has no practical interest. 1 Su"ect to a "asis transformation, a tensor needs to convert all its connectors "y means of the matrices * or * an appropriate conversion "lock must "e applied to each connector. '
'
7or e3ample: e
'
* '
e
e
'
e
e
* '
( % # e
%
e
* %% *
e# '
e % '
=
#
e%
* %%
( % #
!
* ## e #
'
# #
e % ' e #
'
'
( %
' '
#
'
e % ' e #
'
'
'
* ' * %% ' * ## ' ( % # = ( % ' ' # '
since: '
The position ust made:
( %
'
#
'
= * * %% * ##
'
'
'
'
( % #
.?0
e3emplifies the transformation rule for tensors: to old ! new "asis transform the components of a tensor we must ' apply as many * as upper inde3es and as many * ' as lower inde3es.
/;
1 A very special case is that of tensor I = e e whose components never change: = diag -? is true in any basis : e
*
'
e '
'
e
* '
e
e
! e
=
'
' '
*
e
* '
e '
'
e ' e '
* * = * * = '
since:
'
'
'
' '
%.2 *nvariance o' the null tensor
A tensor is the null tensor when all its components are )ero: C=0
. . % = 0 ..
.
L
.. .
L
.??
If a tensor is null in a certain "asis, it is also null in any other "asis. @ In fact: .?2 ⇒ . % = 0 ⇒ . % = * * %% . .. . % = 0 % for all * , * % ... , that is for any transformation of "ases. C=0
' .
. ..
..
' .
.
..
..
'
'
..
.
..
.
'
'
7rom that the invariance of tensor euations follows. L It makes no sense eualing a tensor to a num"er writing C = 0 is only a conventional notation for . . % = 0. therwise written 0. . . .
. ..
/'
%.% *nvariance o' tensor e+uations
Vectors, covectors and tensors are the invariants of a given %landscape&: changing the "asis, only the way "y which they are represented "y their components varies. Then, also the relationships "etween them, or tensor euations, are invariant. 7or tensor euations we mean euations in which only tensors (vectors, covectors and scalars included+ are involved, no matter whether written in tensor ! ecc. or componentwise. notation T ! V Ceduced to its essential terms, a tensor euation is an euality "etween two tensor like: A =B
or
...
.. .
A% . .. = % . ..
.?
ow, it=s enough putting A - B = C to reduce to the euivalent form .. . eq.3.12 C = 0 or % .. . = 0 that, as is valid in a "asis, is valid in all "ases. It follows that eq.3.13, too, as is valid in a "asis is valid in all "ases. So the euations in tensor form are not affected "y the particular "asis chosen, "ut they apply in all "ases without changes. In practice an euation derived in a system of "ases, once e3pressed in tensor form, applies as well in any system of "ases. Also some properties of tensors, if e3pressed "y tensor relations, are valid regardless of the "asis. It is the case, inter alia, of the properties of symmetry O skew-symmetry and inverti"ility. 7or e3ample, eq.2.1" that e3presses the symmetry properties of a dou"le tensor is a tensor euation, and that ensures that the symmetries of a tensor are the same in any "asis. The condition of inverti"ility eq.2.24 , eq.2.2$, is a tensor euation too, and therefore a tensor which has inverse in a "asis has inverse in any other "asis. As will "e seen in the following paragraphs, the change of "ases can "e induced "y a transformation of the coordinates of space in which tensors are set. Tensor euations, inasmuch invariant under change of "asis regardless of the reason that it is due, will "e invariant under coordinate trans&ormation (that is, valid in any reference accessi"le via coordinate transformation+. In these features reside the strength and the most interest of the tensor formulation. 60
4 Tensors in manifolds We have so far considered vectors and tensors defined at a single point P . This does not mean that P should be an isolated point: vectors and tensors are usually given as vector fields or tensor fields defined in some domain or continuum of points. Henceforth we will not restrict to ℝ n space, but we shall consider a wider class of spaces retaining some basic analytical properties of ℝ n , such as the differentiability (of functions therein defined). Belongs to this larger class any ndimensional space M whose points may be put in a onetoone (! bi"ective) and continuous correspond# ence with the points of ℝ n (or its subset). $ontinuity of correspond# ence means that points close in space M have as image points close in ℝ n , that is a re%uisite for the differentiability in M . &nder these conditions we refer to M as a differentiable manifold . ('ometime well also use, instead of the term manifold*, the less technical space* with the same meaning). +oughly speaing, a differentiable manifold of dimension n is a space that can be continuously mapped* in ℝ n (with the possible e-cep# tion of some points). /n more precise terms: /t is re%uired that every infinitesimal neighborhood of the generic point P ∈ M has as image an infinitesimal neighbor# hood of the corresponding point P' in ℝ n or, turning to finite terms, that for every open set U M there is a continuous correspondence φ: U → ℝ n . 0 couple of elements ( U, φ) is called a chart*. 1 2ne complication comes from the fact that, in general, there is not a correspondence that transports the whole space M into ℝ n , i.e. there is not a single chart ( M , φ). To map M com# pletely we need a collection of charts (U i , φi), called atlas*. 1 The charts of the atlas must have overlapping areas at the edges so as to ensure the transition between one chart to the other. 'ome points (or rather, some neighborhoods) will appear on more than one charts, each of which being the result of a different correspondence rule. 3or e-ample, a point Q will be mapped as point φ(Q) on a chart and as ψ (Q) on another one: it 45
is re%uired that the correspondence φ(Q) 6 ψ (Q) is itself continuous and differentiable. 0n e-ample of a differentiable manifold is a twodimensional spherical surface lie the 7arths surface, that can be covered by two dimensional plane charts (even though no single chart can cover the entire globe and some points, different depending on the type of correspondence used, are left out8 for e-ample, the 9ercator pro"ection e-cludes the poles). The fact that there is no correspondence able to transport by itself the whole manifold into ℝ n is not an heavy limitation when, as often oc# curs, only single points are left out. 2ne correspondence can suffice in these cases to describe almost completely the space. /n practical terms, we can say that there is a correspondence φ: M ℝ n , meaning now with M the space deprived of some points.; ; /n any case the limitation that until now has brought us to consider vectors emerging from a single point is still valid: each vector or tensor is related to the point where it is defined and cannot be operated* with vectors or tensors defined at different points, e-cept they are infinitely close. This limitation comes from the fact that we cannot presume to freely transport vectors (and tensors) in any space as it is usually done in the plane or in spaces ℝ n . h 0s already mentioned, the set of tensors of ran k defined at a h k
point P is itself a vector space of dimension n . /n particular, the already nown vector space of dimension n of the vectors defined in a point P is called the tangent space8 the similar for covectors is named the cotangent space.
correspondence φ . 4.1 Coordinate systems
>iven a ndimensional differentiable manifold M , defining a certain correspondence φ: M ℝ n means tagging* each point P of M with n numbers x = 5, =,... n that identify it uni%uely. The ntuple e-presses the coordinates of P (another way, punctual, to designate the correspondence φ is to write P ↔ { x } ). 'ince the manifold is differentiable, the correspondence is continuous and transforms each neighborhood of P in a neighborhood of { x }. The numbers that e-press the coordinates may be lengths, angles or otherwise. We note that a coordinate system does not presuppose any definition of distance between two points of the manifold. 4.2 Coordinate lines and surfaces
/n the neighborhood of a point P whose coordinates are x 5 , x = ,... x n no other points can have the same coordinates, but there are points that have some coordinates e%ual to P (provided that at least one is different). The points that share with P only a single coordinate forms a (hyper) surface in n5 dimensions. These coordinate surfaces passing through P are n in number, one for each coordinate which remains fi-ed. The points that share with P n5 coordinates and differ only in one, lets call it x , align on a line along which only x changes. n coordinate lines of this ind emerge from P . 0 coordinate line originates from the intersection of n5 coordinate (hyper)surfaces (the intersection leaves free one coordinate only). Thus, at each point P of the manifold n coordinate surfaces and n coordinate lines intersect. The representation comes easy in ?<: 0t any point P three coordinate surfaces intersect8 on each of them the value of a coordinate remains constant. These surfaces, intersecting two by two, generate ? coordinate lines8 on each of them two coordinates are constant and only the third varies. The coordinate surfaces are labeled by the coordinate that remains constant, the coordinate lines by the coordinate that varies, as 4?
shown in the figure. x
?
x x
=
5
= const
= const P ?
x = const x
x
=
5
+oughly speaing: defining a coordinate system in a manifold means to draw an infinitely dense ndimensional grid of coordinate lines within the space (in general not straight nor intersecting at right angle). 4.3 Coordinate bases
0n opportunity that arises after defined a coordinate system is to lin the bases of vectors and covectors to the coordinates themselves. The idea is to associate to the coordinates a basis of vectors consisting of n basisvectors, each of which is tangent to one of the n coordinate lines that intersect in the point. x
=
P ' d x
x
5
P
l
>iven a generic point P of coordinate x , infinite parametric lines ;; pass through it8 let l be one of them. 9oving along l by an infinitesimal the point moves from P to P' while the parameter ; 0 parametric line in a ndimensional manifold is defined by a system of n e%uations x = x s with parameter s (@ ! 5, =, ... n). To each value of s corresponds a point of the line. 4A
increases from s to s ! ds and the coordinates of the point move from x5 , x = , ... x n to x5 dx 5 , x = dx = , ... x n dx n . The displacement from P to P' is then d x d x
comp
dx
, or:
= dx e .
A.5
This e%uation wors as a definition of a vector basis { e } in P , in such a manner that each basisvector is tangent to a coordinate line. ote that d x is a vector (inasmuch it is independent of the coordinate system which we refer to). 0 basis of vectors defined in this way, related to the coordinates, is called a coordinate vector basis . 2f course it is a basis among other comp ones, but the easiest to use because only here d x dx C ;; /t is worth noting that in general it depends upon the point P . 1 0 further relation between d x and its components dx is: dx
= e , d x 〉
A.=
(it is nothing but the rule e".#.#$, according to which we get the components by applying the vector to the dual basis)8 its blocwise representation in Tmosaic metaphor loos lie:
e
d x
≡
≡
e e
dx
dx
The basis { e } is the coordinate covector basis, dual to the previ# ously introduced coordinate vector basis. 1 We aim now to lin even this covector basis to the coordinates. Det us consider a scalar function of the point f defined on the manifold (at least along l ) as a function of the coordinates. /ts variation from the initial point P along a path d x is given by its total differential: df =
f μ μ dx x
A.?
; /f the basis is not that defined by e".%.#, we can still decompose d x along the coordinate lines x , but dx is no longer a component along a basisvector. 4E
f are the components of a covector xμ because their product by the components dx of the vector d x We note that the derivatives
gives the scalar df (in other words, e".%.& can be interpreted as an heterogeneous scalar product).; ;
f this covector, whose components x μ
comp ̃ Det us denote by ∇ f →
in a coordinate basis are the partial derivatives of the function f and therefore it can be identified with the gradient of f itself. /n symbols, in a coordinate basis: comp
̃ ̃ → ≡ ∇ grad or, in short notation
μ ≡
x μ
A.A
xμ
for partial derivatives: comp
A.E
ote that the gradient is a covector, not a vector. The total differential (e".%.&) can now be written in vector form: d f
̃ f ( d x⃗ ) = ∇ ̃ f , d x⃗ 〉 = ∇
/f as scalar function f we tae one of the coordinates x dx
e
;;;;we get:
x , d x = 〉
By comparison with the already nown dx
⇒
x =
=e , d x 〉 (e".%.) ⇒
in a coordinate basis
A.4
This means that in coordinate bases the basiscovector e coincides with the gradient of the coordinate x and is therefore oriented in the direction of the faster variation of the coordinate itself, which is that of the normal to the coordinate surface x = const. /n summary, provided we use coordinate bases: • the basisvectors e are tangent to coordinate lines • the basiscovectors e are normal to coordinate surfaces
; ote that, as we as dx to be a component of d x , we assume to wor in a coordinate basis. ;; /t means taing as line l the coordinate line x . 44
as shown in the following figures that illustrate the ?< case: x
?
e ? e P
e = =
e 5 x
x
x
=
!
x
?
t s n o c
5
= const
P
e
5
e
=
?
x = const
5
When the coordinate system is orthogonal ( ⇒ the coordinate lines, in general not straight, intersect at each point at right angle) basis vectors and basiscovectors have the same direction.; ; /f the coordinates are not orthogonal the two sets of vectors and covectors are differently oriented. $ovectors, as well as vectors, can be represented by arrows, although we must eep in mind that they are entities of a different ind (for instance, vectors and covectors are not directly summable).;; ;; 4.4 Coordinate bases and non-coordinate bases
/n ?< $artesian coordinates it is usual to write the infinitesimal displacement as: ⃗ =⃗i dx F ⃗ ( dy F ⃗k d) d x or, labeling by e the *nit vectors: d x
= e5 dx 5 e = dx =e ? dx ? = e dx
A.G
which is precisely the condition of coordinate basis d x = e dx (e".%.#+ with e = e .
( , ⃗k , are a coordinate basis. This means that unit vectors e , or ⃗i , ⃗ ( , ⃗k , are in fact everywhere oriented as coordinate lines, ectors ⃗i , ⃗ and this is true not only punctually, but in the whole space ℝ ? . /t is not always the case. 3or e-ample, in plane polar coordinates (I, J) ; /n general, however, they will not match in length or magnitude. ;; 'ome authors shy away from this representation, that is entirely lawful with the warning of the diversity of ind. 4G
d r is e-pressed as: d x
= e d e d
A.K
and this can be identified with the coordinate basis condition = e dx provided we tae as a basis: d x e
= e , e = e
A.L
Here the basisvector e includes* , its no longer a unit vector and also varies from point to point. $onversely, the basis of unit vectors { e , e } which is currently used in ector 0nalysis for polar coordinates is not a coordinate basis. 0n e-ample of a noncoordinate basis in ?< $artesian coordinates is obtained by applying a AEM rotation to unit vectors ⃗i , ⃗ ( on the horiNontal plane, leaving ⃗k unchanged. 7-pressing ⃗i , ⃗( in terms of the new rotated vectors we ⃗i ' , ⃗( ' : get for d x d x ⃗
= ! = ( ⃗i ' "⃗ (' ) dx F
! =
=
=
( ⃗i ' F ⃗( ' ) dy # ⃗k d) ,
A.5O
and this relation matches the coordinate bases condition ( e".%.#) only by taing as basis vectors: ! = =
( ⃗i ' " ⃗( ' ) ,
! = =
( ⃗i ' F⃗ ( ' ) , ⃗k
that is nothing but the old coordinate basis {⃗i , ⃗( , ⃗ k } . /t goes without saying that the rotated basis {⃗i ' ,⃗ (' , ⃗k } is not a coordinate basis (as we clearly see from e".%.#$). This e-ample shows that, as already said, only in a coordinate basis d x
comp
dx
holds good. ot otherwiseC
1 0s an e-ample we can deduce the dual coordinate basis of covectors e 〉 = % in both the following cases: from the duality relation $ e , /n ?< $artesian coordinates it is easy to see that the vector coordinate basis {⃗ e & }≡{⃗i , ⃗ ( , ⃗k } coincides with the covector μ coordinate basis { ẽ }≡{̃i , (̃ ,̃ k } obtained by duality ;;
u
;
⃗i and ̃i (in general ⃗e & and ẽ μ ) remain distinct entities, although super imposable. /n $artesian they have the same e-pansion by components and their ParrowsP coincide, but under change of coordinates they return to differ. 4K
/ndeed, it is already nown that in $artesian the coordinate basis is made of the unit vectors ⃗i , ⃗( , ⃗ k . They can be written in terms of components:
⃗i = ( 5 , O , O ) , ⃗ ( =( O ,5 , O) , ⃗k = ( O , O ,5 ) The coordinate covector basis will necessarily be:
̃ = (O , O , 5 ) ̃i = ( 5 , O , O ) , ̃( = ( O , 5 , O ) , k because only this way you can get:
$ ̃i , ⃗i 〉 = 5 , $ ̃( , ⃗i 〉 = O , $ k̃ , ⃗i 〉 = O ,
$ ̃i , ⃗k 〉 = O , $ ̃( , ⃗k 〉 = O , $ k̃ , ⃗k 〉 = 5 ,
$ ̃i , ⃗( 〉 = O , $ ̃( , ⃗ ( 〉 = 5 , $ k̃ , ⃗( 〉 = O ,
as re%uired by the condition of duality. v /n
polar coordinates, on the contrary, the vector coordinate basis {e⃗& }≡ { ⃗e ' ,⃗ e } does not match with the covector basis deduced from & ' duality condition {ẽ }≡ { ẽ , ẽ } . i/ndeed: unit vectors in polar coordinates are by definition: e ' =( 5 , O ) , e = (O , 5) . Thus, from e".%.-: e⃗'
,
e⃗
= ( O , ' )
'
= (a ,b ) , ẽ = (c , d ) . 3rom $ e , e 〉 = % ⇒ ' 5 = $ ẽ , e⃗' 〉 = ( a ,b ) * ( 5 ,O )= a ' ' ⇒ =( 5 , O ) e ̃ O = $ ẽ ,⃗ e 〉= ( a ,b ) *( O , ')= b ' ⇒ b=O
Det be
⇒
= ( 5 , O )
ẽ
O = $ ẽ ,⃗ e' 〉= ( c , d ) * ( 5 , O) = c 5
5 = $ ẽ , e⃗ 〉 = ( c ,d ) *( O , ' )= d ' ⇒ d = '
5
⇒ ẽ =( O , ' )
ote that the dual switch tensor G , even if already defined, has no role in the calculation of a basis from its dual. 4.5 Change of the coordinate system
The change of coordinates, by itself, does not re%uest the change of the bases, but we must do it if we want to continue woring in coordinate basis in the new system as well. 0s usual, the change of basis will be done by means of a transformation matri- + ! , +μ& ' which depends upon both the old { x } and new { x ' } coordinate 4L
systems (an ape- ' mars the new ones). But which Q must we use for a given transformation of coordinatesR To find the actual form of Q we impose to vector basis a twofold condition: that the starting basis is a coordinate basis (first e%uality) and that the arrival basis is a coordinated basis, too (second e%uality):
⃗ d x
= e⃗μ d xμ = e⃗&' d x &'
A.55
The generic new coordinate x ' will be related to all the old ones ' ' 5 = n 5 = n x , x , ... x by a function x = x x , x , ... x , i.e. & '
x = x whose total differential is:
& '
( ... , x μ , ... )
x & ' μ dx = dx x μ & '
'ubstituting into the double e%uality e".%.##:
x & ' μ d x ⃗ = e⃗μ dx = e⃗& ' μ dx x μ
⇒
x & ' e⃗μ = e⃗& ' xμ
A.5=
0 comparison with the change of basisvectors e".&.& (inverse, from ne to old ): ' e = . e ' leads to the conclusion that:
x & ' A.5? + = μ x is the matri- element of . complying with the definition & ' μ
⇒ . '
e".&./ that we were looing for.;; 'ince the 0acobian matrix of the transformation is defined as:
0
/
, -
x 5 ' x 5 ' x 5 ' 0 5 = x x x n x = ' x = ' x = ' & ' x x 5 x = x n = x μ 1 1 x n ' x n ' 0 x n ' x 5 x = x n
; 2r e%ually, as in the case, element of its transpose. GO
, -
A.5A
we can identify:
. ≡ 0
A.5E
1 The coordinate transformation has thus as related matrix the Sacobian matri- . ≡ 0 (whose elements are the partial derivatives of the new coordinates with respect to the old ones). /t states, together with its inverse and transpose, how the old coordinate bases transform into the new coordinate bases, and conse%uently how the components of vectors and tensors transform. 0fter a coordinate transformation we have to transform bases and components by means of the related matri- . ≡ 0 (and its inverse transpose) to continue woring in a coordinate basis. 1 0s always, to avoid confusion, it is more practical to start from the transformation formulas and tae care to balance the inde-es8 the inde-es of . will suggest the proper partial derivatives to be taen. xμ μ μ 3or e-ample, P & ' = + & ' P μ ⇒ use the matri- + & ' = & ' . x ote that the upper lower position of the inde- mared ' in partial derivatives agrees with the upper lower position on the symbol . . 4.6 Contravariant and covariant tensors
We summariNe the transformation laws for components of vectors and covectors in a coordinate basis: & ' x μ (contra2ariant scheme) • vector components: 1 = 1 x μ xμ P μ (co2ariant scheme) • covector components: P & ' = & ' x & '
/n the traditional formulation of Tensor $alculus by components*, the two transformation schemes above are assumed as definitions of contra2ariant 2ector and co2ariant 2ector respectively. The e%uivalence with the terminology used so far is: • vector 6 • covector (or 5form*) 6
contravariant vector covariant vector
5 3or a tensor of higher ran, for e-ample = , the following law of
transformation of the components applies: G5
2 ' 3 μ ' & '
x 2 ' xμ x & 2 = 2 μ ' & ' 3 μ & x x x
A.54
which is "ust a different way of writing e".&.#$ in case of coordinate basis (in that case e".%.#& holds). The generaliNation of e".&.#$ to tensors of any ran is obvious. h
/n general, a tensor of ran k is told contravariant of order (! ran) h and covariant of order k . Traditional te-ts begin here, using the transformation laws under coordinate transformation of components; ;as a definition: a tensor is defined as a multidimensional entity whose components transform according to e-ample e".%.#4 . 4. !ffine tensors
/n conformity with the traditional definition, tensors are those %uantit# ies that transform according to the contravariant or covariant schemes under any admissible coordinate transformation. /f we restrict the class of the allowed coordinate transformations, the number of %uant# ities that underlie the schemes of tensorial* (contravariant covari# ant) transformation increases. /n particular, if we restrict to consider linear transformations, we identify the class of affine tensors, wider than that of tensors without further %ualification. The coordinate transformations to which we restrict in this case have ' ' ' the form x = a x where a are purely numerical coefficients. These coefficients can be arranged in a matri- , a - that e-presses the law of coordinate transformation. But also the transformation matri- for vector components, i.e. the Sacobian matri- 0 given by '
x &' is made in this case by the same numerical coefficients + = xμ & ' x& ' = because here is a μ . xμ & ' μ
'o, for affine tensors, the law of coordinate transformation and the transformation law for vector components coincide and are both e-pressed by the same matri- . (the components of covectors transform as usual by the transposed inverse of . ). ; 0lthough the traditional by components* approach avoids speaing about bases, it is implied that there e alays ork in a coordinate basis (both old and new). G=
4." Cartesian tensors
/f we further restrict to linear orthogonal transformations, we identify the class of $artesian tensors (wider than affine tensors). 0 linear orthogonal transformation is represented by an orthogonal ' matri-.;; 0lso for $artesian tensors the same matri- , a - rules both the coordinate transformation and the transformation of vector components. 9oreover, since an orthogonal matri- coincides with its ' transposed inverse, the same matri- , a - transforms as well covector components. 'o, as vectors and covectors transform in the same way, they are the same thing: the distinction between vectors and covectors (or contravariant tensors and covariant tensors) falls in the case of $artesian tensors. 4.# $agnitude of vectors
the scalar 31 3 such We define magnit*de (or norm+ of a vector 1 that: A.5G 31 3= = 1 * 1 = G 1 , 1 ote that this definition, given by means of a scalar product, depends on the switch G that has been chosen. 4.1% &istance and metric tensor
/n a manifold with a coordinate system x a distance ds from a point P to a point P' shifted by d x can be defined as: ds
=
= 3d x3 = = G d x , d x
A.5K
/n fact, what we define is an infinitesimal distance, also called arc element*. 0fter defined a distance the manifold becomes a metric space. 'ince its definition is given by the scalar product, the distance depends on the G we use. /f we want the distance to be defined in a certain way, it follows that G must be chosen appropriately. O We will denote g the tensor = we pic out among the possible
switches G to get the desired definition of distance. ; 0 matri- is orthogonal when its rows are orthogonal vectors (i.e. scalar product in pairs ! O), and so its columns. G?
The distance ds given by the tensor e%uation: ds
=
= g d x , d x
A.5L
is an invariant scalar and does not depend on the coordinate system. 0t this point, the tensor g defines the geometry of the manifold, i.e. its metric properties, based on the definition of distance between two points. 3or this reason g is called metric tensor , or metric*. 1 /f (and only if) we use coordinate bases, the distance can be written: ds
=
= g 2 4 dx 2 dx 4
A.=O
/ndeed, (only) in a coordinate basis is d x = e dx
, hence:
ds = = g d x e 4 dx 4 = g e2 , e 4 dx 2 dx 4 = , d x = g e2 dx2 ,
= g 2 4 dx2 dx 4 4.11 uclidean distance
/f the manifold is the usual 7uclidean space ℝ ? , using a $artesian coordinate system, the distance is classically defined: ds = 5 dx
or: ds
=
=
dy = d) =
A.=5
= dx5 = dx == dx ?=
A.==
which e-press the Uythagoras theorem and coincides with the form (e".%.$) that applies in coordinate bases ds
=
= dx5 = dx == dx ?= = g 2 4 dx 2 dx 4
provided we tae: g 67 =
{
5 for 6 ! 7 O for 6 8 7
otice that in this case g = g
} -1
i.e.
g=
, 5 O O
O 5 O
24
, namely g 2 4 = g
O O 5
.
A.=?
= diag 5 A.=A
4.12 (enerali)ed distances
The definition of distance given by e".%.$ is more general than the 7uclidean case and includes any bilinear form in dx5 , dx = , ... dx n such as: g 55dx 5 dx 5 g 5=dx 5 dx = g 5?dx 5 dx ?... g =5 dx =dx 5 g == dx = dx =... g nn dx n dx n A.=E GA
where g 2 4 are sub"ect to few restrictions. /nasmuch they build up the matri- that represents the metric tensor g, we must at least re%uire that the matri- is symmetric and det , g 2 4 -8O such that g is invertible and
g
-1
e-ists.
ote, however, that g is symmetriNable in any case: for e-ample, if it were g 5=8 g =5 in the bilinear form e".%.5, "ust taing g ' 5= = g ' =5 = g 5= g =5 9 = nothing would change in the definition of distance. 9etric spaces where the distance is defined by e".%.#- with any g , provided symmetric and invertible, are called Riemann spaces. 1 >iven a manifold, its metric properties are fully described by the metric tensor g associated to it. 0s a tensor, g does not depend on the coordinate system imposed onto the manifold: g does not change if the coordinate system changes. However, we do not now how to represent g by itself: its representation is only possible in terms of components g 2 4 and they do depend upon the particular coordinate system (and therefore upon coordinate basis). 3or a given g we then have different matrices , g 2 4 - , one for each coordinate system, which mean the same g, i.e. the same metric properties for the manifold. 3or instance, for the =< 7uclidean plane, the metric tensor g, which e-presses the usual 7uclidean metric properties is represented by
, g 2 4 - = , 5O
O 5
-
, -
5 O in $artesian coordinates and by , g 2' 4 ' - = = O
in
polar coordinates. 2f course, there are countless other , g ' ' - which can be obtained from the previous ones by coordinate transformation and subse%uent change of basis by means of the related matri- . . 2' . 4 '
2 4 (remind that , g 2 4 - , g ' ' - , i.e g ' ' = . ' . ' g 2 4 ). /n other words, the same 7uclidean metric can be e-pressed by all the , g ' ' - attainable by coordinate transformation from the $artesian
, g 2 4 - = , 5O
1 'ummary:
O 5
-.
{
geometry of manifold coordinate system
GE
}
: g ⇒ ⇒ g ⇒
4.13 Tensors and not -
The contra covariant transformation laws (e-emplified by e".%.#4 ) provide a criterion for determining whether a %uantity is a tensor. /f the %uantity changes (in fact its components change) according to those schemes, the %uantity is a tensor. 3or e-ample, the transformation for (components of) d x (e".%.##):
x & ' μ dx = dx x μ & '
confirms that the infinitesimal displacement vector is a contravariant tensor of the first order. /nstead, the position* or radiusvector* x is not a tensor because its components do not transform according to the law of coordinate transformation (which in general has nothing to do with the contra covariant schemes). 1 either is a tensor the derivative of (components of) a vector. /n fact, given a vector that transforms its components: μ
1
& '
= 1
x μ & ' x
A.=4
its derivatives transform (according to DeibniN rule ; ; and using the chain rule*) as: = μ 1 μ 1 & ' x μ xμ 1 & ' x ; ' xμ & ' & ' x 2 = 2 2 2 & ' # 1 & ' = ; ' & ' # 1 x x x x x x x x x 2 x & '
A.=G which is not the tensor transformation scheme (it would be for a 5 tensor 5 without the additional term in = ). This conclusion is not surprising because partial derivatives are not independent of the coordinates. 4.14 Covariant derivative
1 /n general, the derivative of a vector (which is itself a vector) is not simply the derivative of the components, because even the basis vectors vary from point to point in the manifold:
1 ≡ = = 1 e 1 e 1 1 e x ; DeibniN rule for derivative of a product: ( f V g )' ! f '6 g ! f 6 g' G4
A.=K
2f the two terms, the first describes the variability of the components of the vector, the second the variability of the basisvectors along the coordinate lines x . e , the partial derivative of a basisvector, We observe that although not a tensor (see 0ppendi-) is a vector in a geometric sense (because it represents the change of the basis vector e along the infinitesimal arc dx on the coordinate line x ) and therefore can be e-panded on the basis of vectors. < , we can write: e = Hence, setting
< = < e But in reality < is already labeled with lower inde-es < = < e , so that: e = < e
A.=L
, , hence A.?O
The vectors < are n= in number, one for each pair , , with n components < each (for ! 5,=,... n). The coefficients < are called Christoffel symbols or connection coefficients8 they are in total n? coefficients. 7".%.&$ is e%uivalent to: comp
e
<
A.?5
The $hristoffel symbols < are therefore the components of the vectors partial derivative of a basis vector* on the basis of the vectors themselves namely: The recipe* of the partial derivative of a basis vector uses as ingredients* the same basis vectors while $hristoffel symbols represent their amounts. /nserting e".%.&$, e".%.8 becomes: 1 = e 1 < e 1 =
in the last right term , are dummy summation inde-es: they can therefore be freely changed (and interchanged). /nterchanging , :
= e 1 < e 1 = e 1 < 1 GG
/n short:
?
1 = e 1 < 1
A.?=
1 ;
We define covariant derivative of a vector with respect to x :
1 ;
/ 1 < 1
A.??
(we introduce here the subscript ; to denote the co2ariant deri2ati2e). The covariant derivative 1 ; thus differs from the respective partial derivative for a corrective term in < . &nlie the ordinary derivative the covariant derivative is in nature a tensor: the n= covariant derivatives of e".%.&& are the components of a tensor, as we will see (in 0ppendi- a chec calculation in extenso). 1 The derivative of a vector is written in terms of covariant derivative (from e".%.&) as: 1 = e 1 ; A.?A The (partial) derivative of a vector is thus a vector whose scalar components are the n covariant derivatives of the vector. 1 /n the same way we treat the case of the derivative of a covector:
9 ≡ = = 9 e 9 e 9 9 e x
A.?E
0s before, the term e describes the variability of the basis covector e along the coordinate line x and, though not a covector in tensorial sense, can be e-panded on the basis of covectors. 'etting
μ ẽ & = D̃ , we write: ̃ = D >̃ e D
>
̃ is already labeled with an upper But in reality D inde-, then: & & > D̃μ = Dμ > ẽ hence: μ ẽ & = D&μ > ẽ > which, substituted into e".%.&5, gives:
μ̃ 9 = ẽ & μ 9 # Dμ& > ẽ > 9 = &
&
GK
and a lower
A.?4
, in the last term: =̃ e & μ 9 # Dμ> & ẽ & 9 = ẽ & ( μ 9 # Dμ> & 9 ) A.?G interchanging the dummy inde-es &
>
&
>
1 What is the relationship between @ and DR They are simply opposite in sign: > > Dμ & ="@μ & A.?K /t can be shown by calculating separately the derivatives of both e 〉 = % . right and left members of the duality condition $ e , 'ince % = O or 5 (however a constant), then A % = O . 2n the other hand: A $ e , e 〉 = $ A e , e 〉 $ e , A e 〉
= $ Dμ; > ẽ > ⃗, e 〉 # $ ẽ μ , @>;& e⃗ 〉 &
>
= Dμ;> $ ẽ > ⃗, e 〉 # @ >; & $ ẽ μ , e⃗ 〉 &
>
= Dμ;> B>& # @ >; & Bμ> = Dμ; & # @μ;& μ
μ
Hence O = D ; & # @ ; &
⇒ e".%.&8, %.e.d.
?
Therefore:
e = "< e
A.?L
and from e".%.&/ :
9 = e 9 " < 9
A.AO
9 ;
We define covariant derivative of a covector with respect to x : 9& ;μ /
μ 9&" @μ> & 9>
A.A5
which is analogous to e".%.&&. 1 0fter this position, the derivative of a covector is written (from e".%.%$): 9 = e 9 ; A.A= The (partial) derivative of a covector has as scalar components n covariant derivatives of the covector.
GL
4.15 The gradient
̃ at *or+ ∇
is a covector whose We have already stated that the gradient ∇ components in coordinate bases are the partial derivatives with respect to the coordinates (e".%.%, e".%.5): comp grad ≡
= e
namely That
A.A?
is a tensor has been deduced from e".%.& in case of ∇
applied to f , but it is in general true, confirmed by the fact ∇ x that the chain rule* applied to its components ' = ' x
coincides with the covariant transformation scheme e".%.#4 . 0s a tensor, various tensor operation can be performed by applying it to other tensors. We will e-amine the cases: • outer product with vectors, covectors or tensors • inner product with vector or tensor u Tensor
outer ,roduct
⇒ gradient ⇒ divergence
C vector ∇
5 The result is the tensor 5 gradient of vector . n coordinate bases:
that is to say;;
C1 = e C1 1 = e 1 ∇
1 comes from e".%.&. 'ubstituting: 1 = e e 1 < 1 ∇ 1 may be written as e-pansion: 2n the other hand, ∇
1 = ∇ 1 e e ∇
A.AA
A.AE
A.A4
and by comparison between the last two we see that
∇ 1 = 1 < 1
A.AG
which is e%uivalent to e".%.&& in an alternative notation: ; /t is convenient to thin of the symbol C as a connective between tensors. /n μ ⃗ Tmosaic it means to "u-tapose and past the two blocs ẽ μ and 1 KO
1 ≡ 1 ;
A.AK
7".%.%/ represents the co2ariant deri2ati2e of 2ector , %ualifying it as ̃⃗ 1 ( ≡ ∇ ̃ C 1 ) ⃗ . component of the tensor gradient of vector ∇ The covariant derivatives of a vector are the n= components of the C 1 (more often written 1 ). tensor gradient of vector*
/n 0ppendi- we show that the covariant derivatives 1 ; transform according to the tensor contra covariant scheme 1 e-emplified by e".%.#4 and this confirms that the gradient of vector of which they are components is indeed a tensor. 1 The partial derivative of a vector (e".%.&%) may be written in the new as: notation in terms of 1 = e 1
A.AL
comp 1 ∇ 1
A.EO
namely:
1 What we have seen about gradient tensor and derivative of a vector can be summariNed in the setch: covariant derivative
gradient
1
n= components
n
1 & & > = F @ 1 μ> xμ
1 n
components
derivative of component
components
⃗ 1 ≡ μ x
μ⃗ 1 partial derivative
1 When in particular the vector is a basisvector the setch above reduces to: =
̃ e⃗& ∇
n components
@μ>& n
n
μ⃗ e & K5
according to e".%.%%, e".%.&#:
̃ ⃗e & =̃ e μ ⃗e & =̃ e ⃗e > @μ & . ∇ μ
μ
>
A.E5
Terminology
We reserve the term “covariant derivative” to the components which together form the “gradient tensor”:
1 ∇
?
comp
∇ 1
?
gradient tensor
1 ? < 1
=
covariant derivative
?
partial derivative
Christoffel symbol
Notation
The covariant derivative of 1
1
with respect to x or
1 ;
Not to be confused with the partial derivative
1
or
is symbolized
1 & x 2
, indicated as
1 ,
The lower inde after the , means partial derivative The lower inde after the ; means covariant derivative
v Tensor
outer ,roduct
C covector ∇
O The result is the = tensor gradient of covector . n coordinate bases:
or
C 9 = e C 9 9 = e 9 ∇
A.E=
We go on as in the case of gradient of a vector: comes from e".%.%$8 replacing: 9
9 = e e 9 " < 9 ∇
A.E?
and by comparison with the generic definition we get:
9 = ∇ 9 e e ∇
A.EA
∇ 9 = 9 " < 9
A.EE
which is e%uivalent to e".%.%# with the alternative notation: ∇ 9 ≡ 9 ; K=
A.E4
7".%.55 represents the co2ariant deri2ati2e of co2ector and %ualifies it 9 C 9 ( ≡ ) . as a component of the tensor gradient of covector 7".%.55 is the analogous of e".%.%/ , which was stated for vectors. 1 The partial derivative of a covector ( e".%.%) may be written in the as: new notation in terms of
= e 9 9
A.EG
namely: comp
9 ∇ 9
A.EK
1 3or the gradient and the derivative of a covector a setch lie that shown for vectors applies, with few obvious changes. Mnemo
To write the covariant derivative of vectors and covectors: ! we had better writing the X before the component, not vice versa ! "st step:
9 = 9 <
or
9 = 9 "<
#ad$ust the indees of X according to those of the " st member% ! &nd step: paste a dummy inde to X and to the component 9 that follows so as to balance the upper ' lower indees:
. . . . . . . .
< A 9A
Tensor outer ,roduct
̃ T ) is a ̃ C T (or ∇ ∇
. . . . . . .
or
C tensor
"
h k
h tensor, the gradient of tensor. k 5
By the same procedure of previous cases we can find that, for e-ample = 27 D for a ran 5 tensor T = 3 D e⃗2⃗ e7 ẽ is:
̃ T = ∇ μ 3 D2 7 e⃗μ e 2 ⃗e 7 e D namely: ∇ ⃗ ̃ comp ̃ ∇ T → ∇ μ3 D27 = μ3 D27 # @2μ ; 3 D;7 # @7μ; 3 D2 ; " @μ; D 3 27 ; A.EL h /n general, the covariant derivative of a tensor k is obtained by
attaching to the partial derivative h terms in X with positive sign and k terms in X with negative sign. Treat the inde-es individually, one after the other in succession.
K?
Mnemo
To write the covariant derivative of tensors, for eample ! after the term in
∇ 3 E2 4
:
consider the indees of 3 one by one:
∇ 3 2 , then complete 3 2 2 A4 with its remaining indees E4 : < A < A 3 E 4 ! write the second term in X as it wer e 3 , then complete 3 4 4 2A 2 with its remaining indees E : < A < A 3 E ! write the third X as it were 3 E , then complete 3 A A 24 24 with its remaining indees : "< E "< E 3 A ! write the first term in X as it were
Considerations on the form of com,onents of tensor
̃ is a tensor and μ are its n 2nce stated that the gradient ∇ ̃ T is a tensor raned r F5 if T has ran components, it follows that ∇ r and that μT are its component at first level, but not in general its scalar components: they are scalar for r ! O, i.e. T Y f scalar, but even for r =5, μ T are vectors and assume a form more comple- than the simple partial derivative (e".%.&), while for r Z 5 they are tensors. otice that in any case μT are n in number while the scalar ̃ T ought to be n r!5 when T has ran r . The scalar components of ∇ ̃ T are in fact the n r F5 covariant components of the gradient ∇ derivatives. They have a form that depends upon the ran r of the tensor, since μ applied to T operates (see e".%.8) the derivatives of each basisvector or covector of T as well as those of the components.
where [ [ [ [
A.4O
∇ μ are the scalar components which consist of:
μ if applied to a scalar μ F @ ... if applied to (a component of) a vector 1 μ"@ ... if applied to (a component of) a covector P μF various terms in @ ... according to the ran of tensor T
The components covariant derivative ∇ then tae a variable ar# ̃ applies. rangement* depending on the ob"ect to which ∇
KA
x nner
,roduct
* vector 1 = ∇ 1 = 1 ; = di2 1 ∇
A.45
/t is the scalar divergence of a vector .
applied as inner product to a vector loos similar to a ote that ∇ covariant derivative: 1 = 1 < A 1 A
A.4=
but with one (repeated) inde- only, and is actually a number. +oughly: The divergence of a vector di2 1 loos lie a covariant derivative, e-cept for a single repeated inde-: 1 or 1 ; To "ustify the presence of additional terms in < it is appropriate to thin of the divergence ∇ 1 as a covariant derivative 1 followed by an inde- contraction = (more precisely it is a contraction of the gradient):
̃ C 1 ⃗ = ∇ & 1 μ ẽ & C e⃗μ ∇
contraction
&=μ
∇ μ 1 μ
A.4?
which results in a scalar product, that is a scalar. This definition of divergence is a direct generaliNation of the divergence as nown from ector 0nalysis, where it is defined in $artesian coordinates as the sum of the partial derivatives of the components of the vector. /n fact, in a $artesian frame, it is = ∇ 1 = 1 provided the coefficients < are null. di2 1 y nner
,roduct
* tensor
h k
/f there are more than one upper inde-, various divergences can be defined, one for each upper inde-. 3or each inde- of the tensor, both upper or lower, including the one involved, an ade%uate term in < must be added. 3or e-ample, the divergence with respect to \ inde- of 5 = 27 D the ran ( 5) tensor T = 3 D e⃗2⃗ e7̃ e is the ( 5) tensor:
̃ ( T ) = ∇ 7 3 D2 7 e⃗2 ẽ D namely: ∇ ̃ ( T ) comp ∇ → ∇ 7 3 D2 7 = 7 3 D2 7 F @27 ; 3 D;7 F @ 77; 3 D2 ; "@7; D 3 2; 7 A.4A KE
/t comes to the contraction of
̃ T ( e".%.5- ) for μ=7 . ∇
4.16 (radient of some fundamental tensors
/t is interesting to calculate the gradient of some fundamental tensors: the identity tensor and the metric tensor. 1
I ∇
comp
∇ % 24 = % 24 < 2 A % A4 " < A 4 %2A = = O <2 4 " < 2 4 = O
A.4E
as might be e-pected. 1
comp ∇ g ∇ g 2 4 = g 2 4 "
A.44
/ndeed (e"..&/ , e".%.&$):
g 2 4 = e2 * e4 = ? e 2 * e4 e2 * ? e4 = =< A 2 eA
=< A 4 eA
= < A 2 eA * e4 e2 *
Diewise:
g =O ∇
A.4G
g-1 = O
A.4K 24
/t can be obtained lie e".%.44 , passing through ∇ g and g 2 4 and then considering that e 2 ="< 2 A e A (e".%.&-). The last e".%.4/ , e".%.48 are often respectively written as:
∇ ; g 2 7 = O and:
μ&
∇ ; g = O
or or
g 2 7 ; ; = O μ&
g
;;
=O
A.4L
4.1 Covariant derivative and inde/ raising 0 lo*ering
The covariant derivative and the raising or lowering of the inde-es are operations that commute (! you can reverse the order in which they tae place). 3or e-ample: g 2 3 2 4 ; A = 3 4 ; A A.GO ;
g = O all over the space does not mean that g is unvarying, but that it is punctually stationary (lie a function in its ma-imum or minimum) in each point. K4
(on the left, covariant derivative followed by raising the inde-8 on the right, raised inde- followed by covariant derivative). /ndeed:
2
3 4 ; A = g 3 2 4
A = g 2 A 3 2 4 g 2 3 2 4 A = g 2 3 2 4 A ;
;
;
?
;
=O
having used the e".%.4-. /f is thus allowed to raiselower inde-es under covariant derivative*. 4.1" Christoffel symbols
$hristoffel symbols < give account of the variability of basis vectors from one point to the other of the manifold. /n addition, they are what mae the difference between covariant derivatives and ordinary derivatives: G < = O : = ∇ /f (and only if) all $hristoffel symbols < are null, ordinary derivatives and covariant derivatives coincide. 1 /n the usual ?< 7uclidean space described by a $artesian coordinate system, basisvectors do not vary from point to point and conse%uently G < = O everywhere (so that covariant derivatives coincide with partial derivatives). That is no more true when the same space is described by a spherical coordinate system ' , H , because in this case the basisvectors e⃗' ,⃗ e H ,⃗ e are functions of the point (as we saw for the plane polar coordinates) and this implies that at least some < are ] O. Sust this fact maes sure that the < cannot be the components of a 5 hypothetical tensor = as might seem: the hypothetical tensor Γ
would be null in $artesian (since all its components are Nero) and not null in spherical coordinates, against the invariance of a tensor under change of coordinates. The $hristoffel symbols are rather a set of n? coefficients depending on ? inde-es but do not form any tensor. However, observing the way = < e , e".%.- , each < related in which it was introduced < to a fi-ed pair of lower inde-es is a vector whose components are mared by the upper inde- ;. /t is therefore lawful (only for this inde-) the raising lowering by means of g: KG
g A < = < A
A.G5
1 0n important property of the $hristoffel symbols is to be symmetric in a coordinate basis with respect to the e-change of lower inde-es:
< = <
A.G=
.To prove this we operate a change of coordinates and a conse%uent transformation of coordinate basis by the appropriate related matri-. Det us e-press the old < in terms of the new
x ; ' basis (using for this purpose the matri- + = & ) : x ; ' &
< e / e = . A ' eA ' = . A ' eA ' . A ' eA ' = x ; ' x ; ' = x ; ' x ; ' x > ' = μ & e⃗; ' # & μ e⃗; ' = μ & e⃗; ' # & μ > ' e⃗; ' x x x x x x x '
where we have used the chain rule to insert x . 'ince this e-pression is symmetric with respect to the inde-es @,. ^, the same result is obtained by calculating < e / e on the new basis, then < = < . 1 0n important relationship lins < to g and shows that in coordinate bases the coefficients < are functions of g and its first derivatives only :
A.G?
(note the cyclicity of inde-es @, \, _ within the parenthesis). That comes out from:
g=O ⇒ ∇ g = g " < g " < g = O ∇ 24 24 2 4 4 2 ⇒ (using the comma instead of as derivative symbol) :
g 2 4 , = < 2 g 4 < 4 g 2
A.GA
$yclically rotating the three inde-es _, \, @ we can write two other similar e%uations for g 4 , 2 = .... and g 2 , 4 = .... By summing member to member: 5 st ` =nd F ?rd e%uation and using the symmetry properties of the lower inde-es e".%./ (which applies in a KK
coordinate basis) we get: g 2 4 , " g 4 , 2 g 2 , 4
= = g 2 < 4
A2 A 9ultiplying both members by g A2 and since g g 2 = % : ;;
@;μ7 = 5 g ;2 ( g 2 7 ,μ " g 7μ , 2 # g μ 2 , 7) , c.v. = This result has as important conse%uence:;; ;; G g 2 4 = constant
:
G < = O
A.GE
/f the components g 2 4 of the metric tensor g are constant, all the coefficients < are Nero, and vice versa. This is especially true in $artesian coordinate systems. /n general we will %ualify (incorrectly) flat* a coordinate system where G < = O . ;;;;;; 4.1# Covariant derivative and invariance of tensor euations
0 tensor e%uation contains only tensors, thus it may contain covariant derivatives but not ordinary derivatives (which are not tensors). 3or g = O or its componentwise e%uivalent g 2 4 = O instance, (which actually stands for n? e%uations) are tensor e%uations8 it is not the case of e".%./&. 0 strategy often used to obtain tensor e%uations is called comma goes to semi
: I a coordinate
4.2% T-mosaic re,resentation of gradient divergence and covariant derivative
is the covector ∇
∇ e
that applies to scalars, vectors or tensors.
̃ f ∇
1 >radient of scalar:
∇ ! f
e
e e
C1 ∇
1 >radient of vector:
∇ 1 e
∇ f
e
∇ 1 e
= 1 e e ?
covariant derivative
1 >radient of covector:
C P ∇
∇ P e
e
∇ P
e e
= ∇ P e e ?
covariant derivative
1 >radient of tensor, e.g. g:
Cg ∇
∇ g e
LO
e e
∇ g
e e e
1
di2 9
∇ e
9 ∇
∇
e
9
1
9
∇ 9
(with respect to inde- =):
∇ e
T ∇
e
e
e
∇
3
e
∇ μ 3 &μ >
3
e
e
L5
e
5 Curved manifolds 5.1 Symptoms of curvature
Consider a spherical surface within the usual 3D Euclidean space. The spherical surface is a manifold 2D. A 3D observer, or O3D, which has an outside view, sees the curvature of the spherical surface in the third dimension. A 2D observer O2D, who lives on the spherical surface has no outside view. O3D sees the light rays propagate on the spherical surface along maimum circle arcs! for O2D they will be straight lines, by definition "they are the only possible inertial tra#ectories, run in absence of forces$. %ow can O2D reali&e that his space is curved' (y studying the geometry in its local contet O2D will discover the common properties of the 2D Euclidean geometry, the usual plane geometry. )or eample, he finds that the ratio of the circumference to diameter is a constant * + for all circles and that the sum of the inner angles of any triangle is -/. 0n fact, in its environment, i.e. in the small portion of the spherical surface on which he lives, his two1 dimensional space is locally flat, as for us is flat the surface of a water pool on the Earths surface "the observer O3D sees this neighborhood practically lying on the plane tangent to the spherical surface in the point where O2D is located$. %owever, when O2D comes to consider very large circles, he reali&es that the circumference diameter ratio is no more a constant and it is smaller and smaller than + as the circle enlarges, and that the sum of the inner angles of a triangle is variable but always 4 -/. O2D can deduce from these facts that his space is curved, though not capable of understanding the third dimension. The curvature of a space is thus an intrinsic property of the space itself and there is no need of an outside view to describe it. 0n other words, it is not necessary for a n1dimensional space to be considered embedded in another n 5 dimensional space to reveal the curvature. 6 Another circumstance that allows O2D to discover the curvature of his space is that, carrying a vector parallel to itself on a large enough closed loop, the returned vector is no longer parallel " * does not overlap$ to the initial vector. 0n a flat space such as the Euclidean one vectors can be transported 72
parallel to itself without difficulty, so that they can be often considered delocali&ed. )or eample, to measure the relative velocity of two particles far apart, we may imagine to carry the vector v 2 parallel to itself from the second particle to the first so as to ma8e their origins v . coincide in order to carry out the subtraction v 2 − (ut what does it mean a parallel transport in a curved space' 0n a curved space, this epression has no precise meaning. %owever, O2D can thin8 to perform a parallel transport of a vector by a step1by1 step strategy, ta8ing advantage from the fact that in the neighborhood of each point the space loo8s substantially flat.
9et l be the curve along which we want to parallel1transport the "foot . At first the observer O2D and the vector are in P ! of the$ vector V the vector is transported parallel to itself to P 2 , a point of the "infinitesimal$ neighborhood of P . Then O2D goes to P 2 and parallel transports the vector to P 3 that belongs to the neighborhood of P 2 , and so on. 0n this way the transport is always within a neighborhood in which the space is flat and the notion of parallel transport is unambiguous: vector in P vector in P 2 ! vector in P 2 vector in P 3 , etc. (ut the ;uestion is: is vector in P vector in P n ' That is: does the parallelism which wor8s locally step1by1step by infinitesimal amounts hold globally as well' To answer this ;uestion we must transport the vector along a closed path: only if the vector that returns from parallel transport is superimposable onto the initial vector we can conclude for global parallelism. 0n fact it is not so, at least for large enough circuits: #ust ta8e for eample the transport A-B-C-A along the meridians and the e;uator in 73
the picture: C
B
A
The mismatch between the initial vector and the one that returns after the parallel transport can give a measure of the degree of curvature of the space "this measure is fully accessible to the two1dimensional observer O2D$. To do so we must give first a more precise mathematical definition of parallel transport of a vector along a line. 5.2 Derivative of a scalar or vector along a line
⃗= U
⇒ u
μ
dx ⃗ U = i.e. U d τ μ
comp
d ⃗ x d τ
=.
μ
dx d τ
in a coordinate basis.
derivative of a scalar f along a line
)rom the total differential of f : df =
f we immediately dx x
deduce the definition of derivative along a line: 7>
d f d
f d x = x d
=.2
but, since for a scalar f partial and covariant derivatives identify, in a coordinate base is: d f d
f , U 〉 = U f ≡ U ∇ f = 〈 ∇
=.3
?oughly spea8ing: the derivative of f along the line can be seen as a @ projection o r @component of the gradient in direction of the tangent vector. 0t is a scalar. 6 The derivative of a scalar along a line is a generali&ation of the partial "or covariant ∇ $ derivative and reduces to it along coordinated lines.
⃗ e⃗̄μ ⇒ U ̄ is the B10ndeed: along a coordinate line x it is U only non&ero component! the summation on implicit in U ∇ f "eq..!$ thus collapses to the single 1th term. μ
oreover, if is the arc, then d τ≡ d x
̄μ
̄μ
⇒ U =
μ
d x̄ d τ
= .
)or these reasons, eq..! reduces to: d f d
= U ∇ f [ no sum ] = ∇ f """"
=.>
;.e.d. 6 (ecause of its relationship with the covariant derivative and the gradient, the derivative of a scalar along a line is also denoted with: d f d
≡ ∇ U f
=.=
6 #q..! and eq.. enable us to write symbolically: d d
= ∇ U = U = U ∇
=.F
Gote that along the coordinate line x , provided is the arc: e dx e dx d x ≝ U = = [ no sum ] = e d d dx The indication Hno sumI offs the summation convention for repeated indees in the case.
7=
v
along a line derivative of a vector V
0n analogy with the result #ust found for the scalar f "eq..!$ we define along a line l with tangent vector U as the derivative of a vector V the $projection% or $component% of the gradient on the tangent vector : d V V , ≝ 〈 ∇ U 〉 =.J d The derivative of a vector along a line is a vector, since tensor. Epanding on the bases: d V d
V is a ∇
V ,U 〉 = 〈∇ V e e ,U e 〉 = ∇ V U 〈 e , e 〉 e = = 〈 ∇
= U ∇ V e =
&V d
≝
d V d &V d
#q..' can be written: after set:
comp
&V e d
=.-
&V d
≝ U ∇ V
=.7 =.
d V &V is the (1component of the derivative of vector and is d d referred to as covariant derivative a)ong )ine, epanded:
&V d
V V dx V U = ≝ U ∇ V = U V = x d x
=
dx d
d V 5 V U = =. d A new derivative symbol has been introduced since the components of d V d V are not simply d d
, but have the more comple form eq..**.
6 Another notation, the most widely used in practice, for the derivative of a vector along a line is the following:
d V d
≡ ∇ U V 7F
=.2
"that recalls the definition @pro#ection of the gradient
V on U $. ∇ V
6 9et us recapitulate the various notations for the derivative of a vector
d V V alon along g a line line.. (esi (eside de the the defi defini niti tion on eq..+ d
V , ≝ 〈 ∇ U U 〉 were
appended eq..*, eq..*, eq.., eq.., eq..*, eq..*, summari&ed in:
d V d
&V d
≡ ∇ U V
comp
≡ U ∇ V
=.3
≡ U ∇
=.>
or, symbolically: d d
≡ ∇ U
comp
& d
6 The relationship relationship between the derivative derivative of a vector vector along a line and the covariant derivative arises when the line l is is a coordinate line x .
B 0n this case U = provided is the arc and the summation on eq../$ for the derivative of a collapses, as already seen " eq../$ scalar! then the chain eq..' changes eq..' changes into:
d V d namely:
= U ∇ V e = ∇ V e d V d
=.=
comp
∇ V
The derivative of a vector vector along a coordinate line x has as componK ents the n covariant derivatives corresponding to the bloc8ed inde . 6 Comparing the two cases of derivative along a line for a scalar and a vector we see "eq..0 " eq..0 , eq..*/$ eq..*/$ that in both cases the derivative is d = ∇ U , but design designate ated d by but the the oper operat ator or ∇ U has has a diff differ eren entt d meaning in the two cases: •
∇ U = U ∇
•
∇ U U ∇ for a vector.
for a scalar
comp
6 The derivative along a line of scalars and vectors are sometimes called led @absolute derivat vatives or "more properly$ ly$ @ directiona) covariant derivatives. derivatives . 7J
5.3 T-mosaic representation of derivatives along a line u
Derivative along along a line of a scalar: scalar:
∇ f ∇ f df d
f f , U 〉 = 〈 ∇
e
=
L
e
=
U ∇ f =
U
U
∇ U f
=
"scalar$
v
Derivative along along a line of a vector:
e
∇ d V V d
V , U U 〉 = = 〈 ∇
e
e
∇ V
V
e
L
e
e
=
e
U ∇ V
U
U
e
=
&V d
≡
∇ U V "vector$
7-
=
Mnemo
Derivative along a line of a vector: the previous block diagram can help recalling the right formulas. In particular, writing ∇ U V recalls the ∇ V disposition of the symbols in the block:
U Also note that:
d V V V , contain the sign and are vectors ▪ ≡ 〈 ∇ U 〉 〉 ≡ ∇ U V d &V ▪ ≡ U ∇ V are scalar components ( without sign d If a single thing is to keep in mind choose this: ▪ directional covariant derivative ≡ U ∇ : applies to both a scalar and vector components
5.4 arallel transport along a line of a vector
a vector defined 9et V defined in each each point of a parametric line x . ovi oving ng from from the the poin pointt P to the the poin pointt P the vector vector undergoes an increase:
d V V 1 2 V = V d
=.F
0n case the first degree term ter m is missing, namely:
= V 1 2 V
!
d V = d
=.J
is parallel transported . we say that the vector V ?oughly: transporting its origin along the line for an amount the vector undergoes only a variation of the 2 nd order "and thus 8eeps itself almost unchanged$. is defined locally, in its own flat The parallel transport of a vector V local system: over a finite length the parallelism between initial and transported vector is no longer maintained . x = d 6 0n each point of the line let a tangent vector U be defined. d Me denote by 1" x$ x$ a ;uantity of the same order of x of x.. 0n this case it is matter of a ;uantity of the same order of 23 τ42, i.e. of the second order with respect to 3 τ. 77
a)ong U can be epressed by The para))e) transport condition of V one of the following forms: V = , 〈 ∇ , ∇ U V U 〉 = , U ∇ V =
=.-
all e;uivalent to the early definition because "eq..+ , eq..*!$:
comp d V V , U 〉 U ∇ V . ∇ U V = = 〈 ∇ d to be transported can have any orientation with respect The vector V
is transported parallel to Mhen it happens that the tangent vector U itself along a line, the line is a geodesic. 5eodesic condition for a line is then: d U = d
along the line
=.7
E;uivalent to eq..* are the following statements:
= ∇ U U
,
U , U 〉= 〈 ∇
or componentwise:
=.2 =.2
% μ d U " d 2 x " " % μ " dx dx U ∇ μ U = , #$μ % U U = , #$μ % = 2 d τ d τ d τ d τ which come respectively from eq..*, eq..+ , eq..*, eq..** "and eq..*$. All the eq..*, eq.., eq..* are e;uivalent and represent the eq6ations of the geodesic. Nome properties of geodesics are the following: μ
"
6 Along a geodesic the tangent vector is constant in magnitude. B.0ndeed:
d g & ' U &U ' = U ∇ g &' U &U ' = d
= U g & ' U & ∇ U ' g & ' U ' ∇ U &U & U ' ∇ g & ' = ' & because U ∇ U = U ∇ U = is the parallel transport The converse is not true: the constancy in magnitude of the tangent vector is not a sufficient condition for a line to be a geodesic.
along the geodesic and ∇ g & ' = "eq./.0+ $. condition of U & ' , (2 = const , ;.e.d. %ence: g &' U U = const ⇒ g U U ≝ (U 6 A geodesic parameteri&ed by a parameter is still a geodesic when re1parameteri&ed with an @affine parameter s li8e s = a b "two parameters are called affine if lin8ed by a linear relationship$. 6 The geodesics are the @straightest possible lines in a curved space, and can be thought of as inertial tra#ectories of particles not sub#ect to eternal forces. Onder these conditions a particle transports its velocity vector parallel to itself "constant in magnitude and tangent to the path$. A curve must have precise characteristics to be a geodesic: it must be a path that re;uires no acceleration. )or eample on a 2D spherical surface the geodesics are the greatest circles, li8e the meridians and the e;uator! not so the parallels: they can be traveled only if a constant acceleration directed along the meridian is in action. B Me add some comments to clarify a situation that is not immediately intuitive. An aircraft flying along a parallel with speed v tangent to the parallel itself does not parallel1transport this vector along its at path. A ;uestion arises about how would evolve a vector # first coincident with v , but for which we re;uire to be parallel transported all along the trip. )or this purpose lets imagine to cut from the spherical surface a thin ribbon containing the path, in this case the whole Earths parallel. This ribbon can be flattened and layed down on a plane! on this plane the path is an arc of a circumference "whose radius is the generatri of the cone whose ais coincides with the Earths ais and is tangent to the spherical surface on the parallel$! the length of the arc will be that of the Earths parallel. initially 9ets now parallel1transport on the plane the vector # tangent to the arc of circumference until the end of the flattened arc "wor8ing on a plan, there is no ambiguity about the meaning fin be the vector at the end of the of parallel transport$. 9et # transport. Gow we imagine bringing bac8 to its original place on A line may be parametri&ed in more than one way. )or eample, a parabola can 2 3 F be parametri&ed by x = t ! 7 = t or x = t ! 7 = t .
the spherical surface the ribbon with the various subse;uent drawn on. 0n that way we positions ta8en by the vector # achieve a graphical representation of the evolution of the vector parallel transported along the Earths parallel. # The steps of the procedure are eplained in the figure below. After having traveled a full circle of parallel transport on the has undergone a Earths parallel of latitude & the vector # cloc8wise rotation by an angle ) =
2 * r cos & r + tg &
to # fin . # Pnly for & = "along the e;uator$ we get
= 2 * sin & , from
) = after a turn.
The tric8 to flatten on the plane the tape containing the tra#ectory wor8s whatever is the initial orientation of the vector you want to parallel transport "along the @flattened path the vector may be moved parallel to itself even if it is out of the plane$.
5." ositive and negative curvature
Me have so far considered the case of a 2D space @spherical surface with constant curvature only. There are, however, other possible types of curved spaces and other types of curvature. 0n fact, the curvature 2
can be positive "e.g. a spherical space$ or negative "hyperbolic space$. 0n a space with positive curvature the sum of the internal angles of a triangle is 4- /, the ratio of circumference to diameter is Q + and two parallel lines end up with meeting each other! in a space with negative curvature the sum of the internal angles of a triangle is Q-/, the ratio of circumference to diameter is 4 + and two parallel lines end up to diverge. 6 A space with negative curvature or hyperbolic curvature is less easily representable than a space with positive curvature or spherical. An intuitive approach is to imagine a thin flat metallic plate "a$ that is unevenly heated. The thermal dilatation will be greater in the points where the higher is the temperature.
a$
flat space
b$
space with positive curvature
c$
space with negative curvature
0f the plate is heated more in the central part the termal dilating will be greater at the center and the foil will sag dome1shaped "b$.
3
0f the plate is more heated in the periphery, the termal dilation will be greater in the outer band and the plate will @embar8 assuming a wavy shape "c$. The simplest case of negative curvature "c$ is shown, in which the curved surface has the shape of a hyperbolic paraboloid "the form often ta8en by fried potato chipsR$. 5.# $lat and curved manifold
The metric tensor g completely characteri&es the geometry of the manifold, in particular its being flat or curved. Gow, either g is constant all over the manifold "i.e. unvarying$ or varies with the point P . (y definition: g = g , P ! curved manifold =.22 and conversely: g constant ! flat manifold =.23 Nince g is a tensor, it is independent of the coordinate system, even if its representation by components g depends upon. Then g may be constant all over the manifold despite in some system its components g μ" are P 1dependent! only if g μ" are P 1dependent in a)) systems we are sure that g is not constant. Pnly in that case we can say that g is P 1dependent, or g = g , P - . According to what said above, componentwise translations of eq., eq.!, are respectively: S if some g μ" is P 1dependent in a)) coordinate systems, then the manifold is curved "and viceversa$ S if a)) g μ" are constant in some coordinate system, then the manifold is flat "and viceversa$ Mhat is discriminating is the eistence of a reference system where all g μ " are independent of the point P , that is to say that all elements of the matri [ g μ" ] are numerical constants. 0f that system eists, the manifold is flat, otherwise it is curved. 0t will be seen later on apropos of ?iemanns tensor. %ere and below we consider only positive definite metrics "for indefinite metrics eq.. applies only to the left . and eq..! only to the right ⇒ $. 0t is useful to 8eep in mind as a good eample the polar coordinates ", U$, where g // = + 0 despite the flatness of the plane. >
0n practice, given a g μ " P 1dependent, 1dependent, it is not easy to determine if the manifold is flat or curved: you should find a coordinate system where all g μ "= numerical constants, or eclude the possibility to find one. 6 Gote that in a flat manifold 1 $ = due to eq./.+!. eq./.+!. 6 Due to a theorem of matri calculus, any symmetric matri "such as [ g μ " ] with with consta constant nt coeffi coefficie cients nts$$ can can be put into into the canoni canonical cal form diag "V$ by means of an other appropriate matri 2 . ?emind that the matri tri 3 is rela relatted to a certa rtain coordi rdinate transf transform ormati ation on "it is not the the transformation, but it comes downR$. 0n fact, to reduce the metric to canonical form, we have to apply a coor coordi dina nate te trans transfo forma rmati tion on to the the mani manifo fold ld so that that "the "the trans transpo pose sed d inverse of$ its Wacobian matri 3 & ' ma8es the metric canonical:
3 & ' 3 ' ' g = g &' ' ' ' = diag 4
=.2>
)or a flat space, being being the metric independent independent of the point, point, the matri 3 hold holdss glo globall bally y, for for the the manif anifol old d as a whol wholee. %enc %encee, a flat flat manifold admits a coordinate system in which the metric tensor at each point has the canonical form g ] = diag 5 [ g
=.2=
which can be achieved by a proper matri 3 valid for the whole manifold. This property can be ta8en as a definition of flatness: 0s flat a manifold that admits a coordinate system where g can be everywhere epressed epressed in canonical form diag 5 g ] = diag the 0n parti particu cula larr, if [ g the mani manifo fold ld is the the Eucl Euclid idea ean n spac space! e! if [ g ] = diag 5 with a single X the manifold is a in8ows8i space. The occurrences of 5 and X in the main diagonal, or their sum called @signature, is a characteristic of the given space: transforming the coordinates the various 5 and X can interchange their places on the diagonal but do not change in number " 87)vester9s theorem$. theorem$. %ereaf %ereafter ter by 6 we will will refe referr to the the canon nonica ical metri etricc diag "V$, "V$, regardless of the number of 5 and X in the main diagonal. Conversely, for curved manifolds there is no coordinate system in Me mai main n here here nume numeri ricc mat matri rice ces. s. =
which the metric tensor has the canonical form diag "V$ diag "V$ on the whole manifold. Pnly p6nct6a))7 Pnly p6nct6a))7 we we can do this: given any point P , we can turn the metric to the canonical form in this point by means of a certain matri 3 , but that wor8s only for that point "as well as the coor coordi dina nate te tran transf sfor orma matio tion n from from whic which h it come comes$ s$.. The The coord coordina inate te transformation that has led to the matri 3 good for P for P does does not wor8 for all other points of the manifold: to put the metric into canonical form here too, you have to use other transformations and other matrices. 5.% $lat local system
0ndeed, for each point of a curved manifold, we can do better: not only putting the metric punctually in canonical form, but also ma8ing its first derivatives to vanish at that point. 0n that way the "canonical$ metric is stationary in the point! that is, it is valid, ecept for infinitesimals infinitesimals of higher higher order, order, also )oca))7 in )oca))7 in the neighborhood of the point. This ensures that the metric is flat in the neighborhood of the point "a metric that is the canonical 6 in a single point is not enough to guarantee the flatness: it is necessary that this metric remains stationary in the neighborhood. 0n other words, the flatness is a concept that cannot be defined punctually, but locally in a neighborhood$. All that is formali&ed in the following important theorem. 5.& 'ocal flatness t(eorem
[
g P 9 = 6 P 1 x
− x 2 ]
=.2F
where P 9 x is any point of the neighborhood of P x . The term 1H...I is for an infinitesimal of the second order with respect to the displacement from P x to P 9 x , that we have improperly denoted for simplicity by x − x .
The The metr metric ic is in any any case case stat statio iona nary ry in each each poin point: t: ∇ g = , eq./.0+. Analogy with a function f x that is stationary in a maimum or minimum. 0n those points f 9 = and for small shifts to the right or left f x varies only for infinitesimals of the second order with respect to the shift. Nee Nee no note on eq..*0 on eq..*0 . F
The above e;uation is merely a Taylor series epansion around the
g μ " point P , missing of the first order term , x − x 8 , P - because x 8
8
the first derivative is supposed to be null in P in P . Me epl eplic icit itly ly obse observ rvee that that the the neig neighb hbor orho hood od of P con conta tain inin ing g P9 belongs to the manifold, not to the tangent space. Pn the other hand 6 is the "global$ flat metric of the tangent space in P . . ?oug ?oughl hly y, the the theo theore rem m mean meanss that that the the spac spacee is loca locall lly y flat flat in the the neighborhood of each point and there it can be confused with the tangent space to less than a second1order infinitesimal. The analogy is with the Earths surface, that can be considered locally flat in the neighborhood of every point. 6 The proof of the theorem is a semi1constructive one: it can be shown that there is a coordinate transformation whose related matri ma8es the metric canonical in P in P with with all its first derivatives null! then well try to rebuild the coordinate transformation and its related matri "or at least the initial terms of the series epansions of its inverse$. Me report here a scheme of proof. &
6
'
g ' ' P 9 = 3 ' 3 ' g & ' P 9
=.2J
'
"denoted 3 & the transformati transformation on matri 3 "see eq.!.+ , eq.!.'$, eq.!.'$, to transform g & ' we have to use twice its transposed inverse$. 0n the latter both g both g and and 3 are meant to be calculated in P9 . 0n a differentiable manifold, what happens in P9 can can be approimated by what happens in P and and a Taylor series. Me epand in Taylor series around P around P both both members of eq..+ , first the left: 9 '
9 '
g ' ' P 9 = g ' ' P x − x g ' ' , 9 ' P
x 9 ' − x 9 ' x ' − x ' ' g ' ' , 9 ' ' P .... 2
and then the right one: %ence %encefo fort rth h we wil willl ofte often n use use the the com comma ma ) to denote partial derivatives. J
=.2-
3 & ' 3 ' ' g & ' P 9 = 3 & ' 3 ' ' g & ' P x 9 ' − x 9 '
& ' 3 3 ' ' g & ' P 9 '
x
2
x 9 ' − x 9 ' x ' − x ' 9 ' ' 3 & ' 3 ' ' g & ' P .... 2 x x
=.27
Gow lets e;ual order by order the right terms of eq..' and eq..:
g ' ' P
0$
= 3 & ' 3 ' ' g & ' P
00$ g ' ' , 9 ' P =
000$
=.3
& ' 3 3 ' g & ' P ' 9 '
x
2 & ' g ' ' , 9 ' ' P = 9 ' ' 3 ' 3 ' g & ' P x x
=.3
=.32
%ereafter all terms are calculated in P and for that we omit this specification below.
x & ?ecalling that 3 = ' and carrying out the derivatives of product x & '
we get: 0$ g ' '
00$
x & x ' = ' ' g & ' x x
=.33
2 x & x ' x & 2 x ' x & x ' g ' ' , 9 ' = 9 ' ' ' g & ' ' 9 ' ' g & ' ' ' g & ' , 9 ' x x x x x x x x 3
000$ g ' ' , 9 ' ' =
&
'
x x 9 ' ' ' ' g & ' ... x x x x
=.3>
=.3=
The right side members of these e;uations contain terms such as g & ' , 9 ' x : that can be rewritten, using the chain rule, as g & ' , 9 ' = g & ' , : 9 ' . x The typology of the right side member terms thus reduces to: • 8nown terms: the g & ' and their derivatives with respect to the old
coordinates, such as g & ' , : • un8nowns to be determined: the derivatives of the old coordinates with respect to the new ones. -
"Mell see that #ust the derivatives of various order of old coordinates with respect to new ones allow us to rebuild the matri 3 related to the transformation and the transformation itself.$ 6 Gow lets perform the counting of e;uations and un8nowns for the case n * >, the most interesting for
>; > symmetric tensor g ' ' . The g & ' are 8nown.
To get g ' ' = 6 ' ' ⇒ we have to put = or V each of the independent elements of g ' ' "in the right side member$. Conse;uently, among the F un8nown elements of matri 3 , F can be arbitrarily assigned, while the other are determined by e;uations 0 "eq..!!$. "The presence of si degrees of freedom means that there is a multiplicity of transformations and hence of matrices 3 able to ma8e canonical the metric$. No have been assigned or calculated values for all F first derivatives. 00$ There are > e;uations " g ' ' has independent elements! :9 can
2 x & ta8e > values$. The independent second derivatives li8e 9 ' ' x x
are > "> values for Y at numerator! different pairs of Z, at denominator$. All the g & ' and first derivatives are already 8nown. %ence, we can set * all the > g ' ' , 9 ' at first member and determine all the > second derivatives as a conse;uence. g ' ' to derive with respect to the different pairs generated by > numbers$.
000$ There are e;uations " independent elements
3 x& [nown the other factors, - third derivatives li8e ' 9 ' ' are x x x to be determined "the 3 indees at denominator give 2 combinaK tions, the inde at numerator > other$. Gow, we cannot find a set of values for the - third derivatives such that all the g ' ' , 9 ' ' vanish! 2 among them remain non&ero. 0t is a matter of combinations "in the case of > items$ with repeats, different for at least one element. 7
Me have thus shown that, for a generic point P , by assigning appropriate values to the derivatives of various order, it is possible to get "more than$ a metric g ' ' such that in this point: S g ' ' = 6 ' ' , the metric of flat manifold S all its first derivatives are null: 1 g ' ' , 9 ' = S some second derivatives are non&ero: < some g ' ' , 9 ' ' 6 This metric 6 ' ' ;ith
'
around P
x & x x L x P 9 = x P x − x 9 ' P x &
&
'
&
9 '
# 3
' & x '
L
& μ ' , P 9 -
2
9 '
2
&
x > ' % ' % ' > ' , x − x -, x − x - > ' % ' , P - #... 2 x x
2 &μ ' =2 # , x − x - > ' , P - # x 2 3 & ' 9 ' 9 ' ' ' x − x x − x 9 ' ' P ...* 2 x x & μ ' , P -
> '
> '
9 ' 9 ' x & 2 x& = ' P x − x 9 ' ' P x x x 9 ' ' ' 3 x& 9 ' x − x x − x 9 ' ' ' P ... 2 x x x
only derivatives of old coordinates with respect to the new ones appear as coefficients.
Pnce 8nown its inverse, the direct transformation x ' x & and the ' related matri 3 = 3 & that induce the canonical metric in P are in principle implicitly determined, ;.e.d.
[
]
6 0t is worth noting that also in the flat local system the strategy @comma goes to semico)on is applicable. 0ndeed, eq./.+! ensures that even in the flat local system is 1 = and therefore covariant derivatives ∇ and ordinary derivatives coincide. B That enables us to get some results in a straightforward way. )or eample, as in the flat local system "li8e in all flat manifolds$ g & ' , = and g & ' , = g & ' ; , it follows g &' ; ≡ ∇ g &' = , g = , true in general. which is the well18nown result ∇ 0n the following we will use again this strategy. 5.1* +iemann tensor
\arallel transporting a vector along a closed line in a curved manifold, due to the the final vector differs from the initial by an amount V depends on the path, but it can be used curvature. This amount V as a measure of the curvature in a point if calculated along a closed infinitesimal path around the point.
&
x '
B
x
C
&
&
x
'
A
along the circuit ABC&A fin"" and we parallel1transport the vector V The loop ABC&A fin always closes: in fact it lies on the surface or @hypersurface in which the n-2 not involved coordinates are constant.
"of course, the construction could be repeated using all possible couples of coordinates with & , ' = ,2,... n $. e ' , coordinated Tangent to the coordinates are the basis1vectors e& , to x& , x ' .
d V re;uires that = or, componentK The parallel transport of V d &
wise U ∇ & V = "eq..*!4, which along a segment of coordinate & line x reduces to ∇ & V = "eq..*4. (ut:
V V ? & V = ⇒ = ⇒ = − V V =.3F & & & & x x
'
and a similar relation is obtained for the transport along x . (ecause of the transport along the line segment AB, the components of undergo an increase: the vector V
V " & " " % & V , B - = V , A - # x V V x , A @ = − $ @ , &% , AB x & , AB =.3J "
"
where " AB$ means @calculated in an intermediate point between A and B. Along each of the four segments of the path there the changes are:
= V A − & V AB x &
= V B − ' V BC x '
= V C & V C& x&
AB :
V B
BC :
V C
C& :
V &
&A fin :
V A fin = V &
"in the last two segments
V A fin − V A
' V &A
x '
x & , x ' are negative in sign$. Numming:
= ' V &A x ' − ' V BC x ' & V C& x & − & V AB x &
[
] [
= ' V &A − ' V BC x '
2
V & C&
]
− & V AB x &
=−
& ' ' & V x x V x x ' & x& x ' "in the last step we have used once again the theorem of the finite increase! now we mean the two terms in .. be calculated in an intermediate point between " BC $ and " &A$ and in an intermediate point between " AB$ and "C&$ respectively, dropping to indicate it$
=−
V & ' V ' & x x V x x & , ' & x & x '
V ' , &
'
V = − V and reusing the result "eq..!0 $: & & x
=−, $"A% , & V %−$A" % $%& μ V μ @ x & @ x A 5 , $ "& % , A V % −$&" % $%Aμ V μ @ x A @ x & Osing as dummy inde instead of ] in the st and 3rd term: % μ A & =−, $"Aμ , & V μ−$"A% $ %&μ V μ @ x & @ xA 5 , $"& μ ,A V μ−$"& % $Aμ V @ x @ x
= V x & x ' − ' , & ' & & , ' − & ' Me conclude:
V = V x & x ' − ' , & & , ' ' & −& ' = ' &
=.3-
= ' & must be a tensor because the other factors are tensors as well as
V . 0ts ran8 is Net:
for the ran8 balancing: = B . 3 3
= ' & ≝ − ' , & & , ' ' & − & ' we can write: that is:
V = V x & x ' = '& P = R P , V , x , x V
7 = any covector , or even "multiplying eq../ by e $ : being P
3
=.37 =.> =.>
= e V x & x ' = & ' V
=.>2
0t is tacitly supposed a passage to the limit for x & , x ' : R and its components are related to the point P at which the infinitesimal loop degenerates. "
μ A &
R ≡ =μ A& e⃗" e7 e7 e7 , the Riemann tensor ) epresses the @proporK tionality constant between the change undergone by a parallel transported vector along an "infinitesimal$ loop and the area of the loop, and contains the whole information about the curvature. 0n general it depends upon the point: R = R" P $ .
= ⇒ the vector V overlaps 0f R = from eq../* it follows V itself after the parallel transport along the infinitesimal loop around the point P ⇒ the manifold is flat at that point. %ence: R P =
⇒ manifold flat in P
0f that applies globally the manifold is everywhere flat. 0f R , i n P ,the manifold is curved in that point, although it is possible to define a local flat system in P . 6 R is a function of ^ and its first derivatives " eq..!$, namely of g and its first and second derivatives "see eq./.+!$. 6 >n the f)at )oca) coordinate s7stem where g assumes the canonical form with &ero first derivatives, R depends only on the second derivatives of g. 9et us clarify now this dependence. (ecause in each point of the manifold in its flat local system is 1 = , the definition eq..! given for = ' & reduces to:
= ' & = − ' , & & , '
=.>3
That is e;uivalent to R applied to an incomplete list. Completing the list gives eq../*. #q../ is eq../ componentwise. e
= ' &
e e
V
e '
e e'
&
e e&
x ' x & >
*
V
as functions of g and their derivatives ' = g C − g ' , C g C , ' g C ' , and since & g C = :
)rom eq./.+! that gives the
x
2
= ' & =− g
C
2
− g ' ,C & g C , ' & g C' , &
2 g C − g & , C ' g C , & ' g C& , ' = g C g ' ,C &− g C' , &− g & , C' g C& , ' 2
C C and multiplying both members by g we get "recall that g g = and hence 0 8 $ :
= ' & = g ' , &− g ' , & − g & , ' g & , ' 2
To write, as usual, Y _ as first pair and ` as second we have to ma8e the inde echanges 8 & , μ A , A μ , & " ! rearranging the terms and by the symmetry of g & ' we get: = & ' = g & , ' g ' , & − g & , ' − g ' , & 2
=.>>
that is true in each point of the manifold in its respective flat local system "but not in general$. B 0t is worth noting that the eq..// is no longer valid in a generic system because it is not a tensor e;uation: outside the flat local system we need to retrieve the last two terms in from eq..!. Therefore in a generic coordinate system it is: D
%
D
%
= &Aμ" = , g & " , Aμ# g Aμ , & "− g &μ , A" − g A" , &μ -# g D &,$μ% $ " A−$ "% $μA 2
=.>= Mnemo
!o write down = & ' in the local flat system ( eq..// the "#ascal snail$ can be used as a mnemonic aid to suggest the pairs of the inde%es before the comma:
g & , .. g ' ,..
& '
− g & μ ,..− g A " , ..
−
&se the remaining inde%es for pairs after the comma.
=
6 0n = & ' it is usual to identify two pairs of indees: = & '
st
2nd pair pair
6 (ringing order also in the indees of eq..! we rewrite it as: E
&
&
&
%
&
%
=Aμ " = $A " , μ − $ Aμ , " #$ %μ $A " − $% " $Aμ
=.>F
6 The ?iemann tensor R provides a flatness criterion more general than that one stated by eq.. e eq..! and also valid for indefinite metrics "that can be flat even if P 1dependent$. 5.11 Symmetries of tensor R
The symmetries of = & ' can be studied in the flat local system by interchanging indees in eq..//. )or eample, echanging the indees Y_ within the first pair gives: = ' & = g ' , & g & , ' − g ' , & − g & , ' 2
whose right side member is the same as in the starting e;uation changed in sign and thus ⇒ =' & =− = & ' . 0t turns out from eq..// that R is: i$ s8ew1symmetric with respect to the echange of indees within a pair! ii$ symmetric with respect to the echange of a pair with the other! it also en#oys a property such that: iii$ the sum with cyclicity in the last 3 indees is null: i$
= ' & =− =& '
= & ' =− = & '
=.>J
ii$ = & ' = =&'
=.>-
iii$ = & ' = & ' =& ' =
=.>7
The four previous relations, although deduced in the flat local system "eq..// applies there only$ are tensor e;uations and thus valid in any reference frame. )rom i$ it follows that the components with repeated indees within the pair are null "for eample = = =' & 33 = =22 = = = $. 0ndeed, from i$: = && = − =& & ⇒ =& & = ! and so on. 6 0n the end, due to symmetries, among the n> components of = & ' nly n2 n2− + 2 are independent and "namely for n = 2! F for n * 3, 2 for n * >! ...$. F
5.12 ianc(i identity
0t s another relation lin8ing the covariant first derivatives of = & ' : = & ' ; =& ' ; =& ' ; =
=.=
"the first two indees Y _ are fied! the others ` ] rotate$. B.To get this result we place again in the flat local system of a point P and calculate the derivatives of = & ' from eq..//:
= & ' , = g & , ' g ' , & − g & , ' − g ' , & x 2
= 2 g & , ' g ' , & − g & , ' − g ' , & and similarly for = & ' , e =&' , . Numming member to member these three e;uations and ta8ing into account that in g & ' , 9 the indees can be echanged at will within the two groups Y _ , Z , the right member vanishes so that: = & ' , =&' , =& ' , = Nince in the flat local system there is no difference between derivative and covariant derivative, namely: = & ' ; = =& ' , ! = & ' ; = =& ' , ! = & ' ; = =& ' , we can write = & ' ; =& ' ; =& ' ; = which is a tensor relationship and applies in any coordinate system, ;.e.d.
5.12 +icci tensor and +icci scalar
= & ' may be contracted on indees of the same pair or on indees of different pairs. 0n the first case the result is , in the second it is significant. The following schemes illustrate some possible contractions and their results:
g &A is symmetric and the derivation order doesnt matter. &
0n fact, it is = ' to undergo a contraction. J
&'
= & ' =
F g
'
='
'&
F g
− =' & − =&&
=
− =
⇒ = = =.=
&
= & ' = = & '
F g
='
&
F g
&
=& '
='
⇒ =' = = ' = '
Pther similar cases still give as result or 5 =A " . Me define Ricci tensor the 2 tensor with components: st
rd
=A " ≝ contraction of =& A μ " with respect to and 3 inde
=.=2
6 Caution should be paid when contracting ?iemann tensor on any two indees: before contracting we must move the two indees to contract in st and 3rd position using the symmetries of = & A μ " , in order to perform anyway the contraction 13 whose result is 8nown and positive by definition. 0n such manner, due to the symmetries of = & A μ " and using schemes li8e eq..* we see that: S have sign 5 the results of contraction on indees 13 or 1> S have sign the results of contraction on indeers 1> or 213 "while contractions on indees 12 or 31> have as result$ 6 The tensor =A " is symmetric "see eq..*$. 6 (y further contraction of the ?icci tensor we get the Ricci scalar = : '
= '
F g
=
=
" = trace of = ' . "
=.=3
5.13 instein tensor
)rom (ianchi identity eq../, raising and contracting twice the inK dees to reduce ?iemann to ?icci tensors and using the commutation property between inde raising and covariant derivative: μ
μ
?emind that the trace is defined only for a mied tensor ? " as the sum ? μ of μ" elements of the main diagonal of its matri. 0nstead, the trace of ? μ" or ? is by 8μ μ" definition the trace of g ? μ " or, rispectively, of g 8μ ? . -
A"
&μ
A"
μ
g g , = &Aμ" ; % # = &A"% ; μ # =&A%μ ; " -= g , =
Aμ " ; %
# =μA" % ;μ # =μA% μ ; " -=
contracting = μ Aμ" ; % on indees 13 "sign 5$ μ and = A%μ ; " on indees 1> "sign $ : A"
g , =A " ; % # =
μ A" % ; μ
− =A% ; " -=
using symmetry = μA"% ;μ =− = Aμ" % ;μ to allow the hoo8ing and rising of inde _ : A"
μ
g , =A " ; % − = A "% ;μ − =A% ; " -= =
" ";%
− ="μ "% ;μ − =%" ; " = "μ contracting = "% ; μ on indees 13 "sign 5$
"
μ
"
= " ; % − =% ; μ − = % ; " = The first term contracts again! the 2 nd and 3rd term differ for a dummy inde and are in fact the same:
= ; − 2 = ; =
=.=>
that, than8s to the identity 2 = − = ; ≡ 2 = ; − =; , may be written as: 2 = − = ; = =.== This epression or its e;uivalent eq../ are sometimes called t;ice contracted Bianchi identities. Pperating a further rising of the indees: "%
g
, 2 =μ% −Gμ% =- ; μ =
, 2 ="μ−G "μ = -; μ = that, since
G "μ ≡ g "μ "eq../0 $, gives: , 2 ="μ− g "μ = -; μ = =
"μ
− g "μ = 2
= ;μ
=.=F
Me define Einstein tensor the 2 tensor Gwhose components are:
Gothing to do, of course, with the @dual switch earlier denoted by the same symbolR 7
5
"μ
≝ ="μ− g "μ = 2
The eq..0 can then be written: 5
"μ ;μ
=
=.=J
which shows that the tensor G has null divergence. Nince both = " μ and g " μ are symmetric under indees echange, G is symmetric too: 5 "μ = 5 μ " . The tensor G has an important role in
G is our last stop. What's all that for? Blending physics and mathematics with an overdose of intuition, one day of a hundred years ago instein wrote:
5
= ?
anticipating, according to some, a theory of the !hird Millennium to the twentieth century. !his e"uation e#presses a direct proportionality $etween the content of matter%energy represented $y the &stress%energy% momentum tensor T ' a generali(ation in the Minkowski space%time of the stress tensor and the curvature of space e#pressed $y G )it was to ensure the conservation of energy and momentum that instein needed a (ero divergence tensor like G *. In a sense, T $elongs to the realm of physics, while G is matter of mathematics: that maths we have toyed with so far. +ompati$ility with the ewton's gravitation law allows to give a value to the constant 8 and write down the fundamental e"uation of -eneral elativity in the usual form:
5
= - *>5 ? c
May $e curious to note in the margin as, in this e"uation defining the fate of the /niverse, $eside fundamental constants such as the gravitational constant - and the light speed c, peeps out the ineffa$le @ , which seems to claim in that manner its key role in the architecture of the world, although no one seems to make a great account of that )those who think of this o$servation as trivial try to imagine a /niverse in which @ is, yes, a constant, $ut different from 0.12 ....*.
2
/ppendi0
$ under coordinate change e by "scalar$ multiplying Me eplicit from eq./.! e = both members by e : 1 - Transformation of
〈 e , e 〉 = 〈 e , e 〉
= 〈 e , e 〉 Gow let us operate a coordinate change x H I x J ' "and coordinate bases, too$ and epress the right member in the new frame: = 〈 e , e 〉
〈
〉 〉 〉
& ' 3 ' ' e , 3 e 9 ' x = ' ' 9 ' & ' x x
=3 & '
〈
"chain rule on x& ' $
3 ' ' e , 3 e 9 ' ' ' 9 ' & '
x
=3 & ' 3 9 '
〈 〈
3 ' ' e , e 9 ' ' ' & '
x e' ' ' ' &' ' ' 9 ' =3 3 9 ' & ' 3 e' ' & ' 3 , e x x &' ' ' e' ' 9 ' & ' ' ' 9 ' =3 3 9 ' 3 3 3 3 〈 〉 , e e , e 9 ' ' ' & ' & ' x x &' ' ' e' ' 9 ' & ' ' ' 9 ' e =3 3 9 ' 3 3 3 3 , 9 ' & ' & ' ' ' x x
〈 〉 〈〉
〉
9& ' ' ' '
=3 & ' 3 9 ' 3 ' ' 9& ' ' ' ' 3 & ' 3 ' ' & ' 3 ' ' x
x & ' x 8 2 x A ' # μ A ' & ' " x x x x
& ' μ
8 > '
A ' > ' " & ' A '
&'
9 '
' ' 9 ' & ' ' '
=2 2 2 $
=3 3 3
x 2 x' ' ' ' "chain rule on x& ' $ x x x
The first term would describe a tensor transform, but the additional term leads to a different law and confirms that is not a tensor. 2
2 - Transformation of covariant derivative under coordinate change
To
V ; = V , V
lets apply a coordinate change x K I x L ' and epress the right member in the new coordinate frame:
V ; = V , V
= 3 9 ' V 9 ' "transformation of $ 3 : ' V : ' x
=3 & ' & ' 3 9 ' V 9 ' "transformation of $ 3 : ' V : ' x
& '
9 '
V 9 ' & '
=3 3
x
9 '
& '
& '
V 3
x
9 '
V 9 ' & '
=3 3
' ' 9 ' & ' ' '
9 '
& '
x
& '
V 3
9 '
3 3 3
3
& ' 3 9 '
x 2 x ' ' : ' ' ' 3 : ' V x x x
: ' V : '
x =3 3 V 3 & ' 9 ' x x x & ' 9 ' ' ' x 2 x ' ' : ' 3 3 9 ' &' ' ' V '' : ' V x x x 9 ' & ' &' V 9 ' x 2 x =3 3 9 ' & ' V & ' 9 ' x x x x & ' 9 ' ' ' x 2 x' ' : ' 3 3 9 ' &' ' ' V '' : ' V x x x &'
' '
x
' ' 9 ' & ' ' '
2 ' ' x x ' ' 3 : ' V : ' x x x
9 '
3 3 3 & '
& ' 3 9 '
V 9 ' & '
9 '
& '
or, changing some dummy indees: &'
9 '
2 x V & ' : ' x x x x & ' 9 ' ' ' x 2 x& ' : ' 3 3 9 ' &' ' ' V & 9 : ' V x x x
V 9 ' & '
=3 3
: ' x
& '
' ' ' ' x x x = : ' = ':' ' ?ecall that 3 3 = : ' x x x ' '
: '
22
=3 & ' 3 9 ' V , 9&' ' 3 & ' 3 9 ' 9& ' ' ' ' V ' ' terms in 2 =3 & ' 3 9 ' V 9, &' ' 9& ' ' ' ' V ' ' terms in 2 The two terms in 2 cancel each other because opposite in sign. To show that lets compute:
x x & ' x x & ' x x x& ' : ' = : ' & ' = : ' & ' & ' : ' = x x x x x x x x x x x x & ' 2 x x 2 x& ' = : ' & ' & ' : ' x x x x x x x = Pn the other hand: : ' : ' = , hence: x x x x & ' 2 x x 2 x & ' : ' & ' = − & ' : ' , ;.e.d. x x x x x x
The transformation for V ; is then:
& '
9 '
9 '
' '
& '
9 '
V ; = V , V = 3 3 9 ' V , & ' & ' ' ' V = 3 3 9 ' V ; & ' ⇒
⇒ V ; transforms as a "component of a$ tensor. V comp That allows us to define a tensor ∇ V ; . 3 - Non-tensoriality of basis-vectors, their derivatives and gradients Are not tensors: a$ basis1vectors ⃗e " ! b$ their derivatives μ ⃗e " ! c$ 7 ⃗e " their gradients ∇
a$ (asis1vector is a role that is given to certain vectors: within a vector space n vectors are chosen to wear the #ac8et of basis1vectors. Onder change of coordinates these vectors remain unchanged, tensorially transforming their components "according to the usual contravariant scheme eq.!./$, while their role of basis1vectors is transferred to other vectors of the vector space. ore than a law of transformation of the basis1vectors, eq.!. is the law ruling the role or #ac8et transfer. )or instance, under transformation from Cartesian coordinates to ⃗ ≡, , , - that in Cartesian plays the role of spherical, the vector V basis1vector ⃗i transforms its components "following the contravariant 23
scheme eq.!./$ and remains unchanged, but loses the role of basis1 vector which is transferred to new vectors according to eq.!. "which )oos )ie a covariant scheme$. ectors underlying basis1vectors have then a tensorial character that does not be)ong to basis-vectors as s6ch. e " are the Christoffel symbols "eq./.!, b$ Ncalar components of μ ⃗ eq./.!*$ which, as shown in Appendi , do not behave as tensors: that ecludes that μ⃗e " is a tensor. The same conclusion is reached whereas the derivatives of basic1 vectors μ ⃗e " are null in Cartesian coordinates but not in spherical ones. Nince a tensor which is null in a reference must be &ero in all, e " are not tensors. the derivatives of basis-vectors μ ⃗
7 ⃗e " have as scalar components c$ Also the gradients of basis1vectors ∇ the Christoffel symbols "eq./.*$! since they have not a tensorial character "see Appendi $, it follows that gradients of basis-vectors 7 ⃗e " are not tensors. ∇ 7 ⃗e " are &ero As above, the same conclusion is reached whereas the ∇ in Cartesian but not &ero in spherical coordinates.
⃗ ∇ 7⃗ V own Me note that this does not conflict with the fact that V , tensor character: the scalar components of both are the covariant derivatives which transform as tensors.
2>
Bibliographic references ▪ Among texts specifically devoted to Tensor Analysis the following retain a relatively soft profile: Bertschinger, E. 1999, Introduction to Tensor Calculus for General Relativity, assach!sset "nstit!te of Technology # $hysics %.9&', pag. () *ownload: http:++we.mit.ed!+edert+-+gr1.pdf "n /!st over (0 pages a systematic, well str!ct!red and clear tho!gh concise treatment of Tensor Analysis aimed to -eneral elativity. The approach to tensors is of the geometric type, the est s!ited to ac!ire the asic concepts in a systematic way. The notation conforms to standard texts of elativity. 2o !se is made of matrix calc!l!s. These notes are modeled in part on this text, possily the est among those 3nown to the a!thor. ecommended witho!t reserve4 it can e at times challenging at first approach. 5ay, *. 6. 19%%, Tensor Calculus, c-raw78ill, pag. ''% This oo3 elongs to the mythical cha!m ;!tline eries<, !t witho!t having the !alities. The approach to Tensor Analysis is the most traditional y components< =except a final chapter where the road lines of the geometric approach are piled !p in a very formal and almost incomprehensile s!mmary>. The traditional separation etween statements and prolems characteristic of cha!m?s ;!tline is here interpreted in a not very happy way: the theoretical parts are somewhat astract and formal, the prolems are mostly examples of mechanisms not s!itale to clarify the concepts. "n fact, the concept!al part is the great asent in this oo3 =which yo! can read almost to the end witho!t having clear the very f!ndamental invariance properties of vectors and tensors s!/ect to coordinate transformations@>. Extensive !se of matrix calc!l!s =to which a far too ins!fficient review chapter is devoted>. The second half of the oo3 is dedicated to topics of interest for *ifferential -eometry !t marginal for -eneral elativity. The notations !sed are not always standard. This text is not recommended as an introd!ction to the mathematics of -eneral elativity, to which it is rather misleading. elf7st!dy ris3y. ;n the contrary it is !sef!l for cons!ltation of certain s!/ects, when the same are somewhat 3nown. oveloc3, *., !nd, 8. 19%9, Tensors, Differential Forms and Variational Principles, *over $!lication, pag. (&& Text with a somewhat mathematical setting, !t still accessile and attentive to the concept!al propositions. The approach is that traditional y components< 4 are first introd!ced affine tensors and then general tensors according to a =!estionale> policy of grad!alism. The topics that are preliminary to -eneral elativity do not exceed one third of the oo3. "t may e !sef!l as a s!mmary and reference for some single s!/ects.
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Cleisch, *.A. '01', A Student's Guide to Vectors and Tensors, 6amridge Dniversity $ress, pag. 1(( A very friendly introd!ction to Fector and Tensor Analysis, !nderstandale even witho!t special prere!isites, neverthless with a good completeness =!ntil the introd!ction of the iemann tensor, !t witho!t going into c!rved spaces>. The approach to tensor is traditional. Bea!tif!l ill!strations, many examples from physics, many calc!lations carried o!t in f!ll, together with a speech that gives the impression of proceeding methodically and safely, witho!t /!mps and witho!t leaving ehind dar3 spots, ma3e this oo3 an excellent a!todidactict tool, also accessile to a good high school st!dent. ▪ Among the texts of -eneral elativity =->, the following contains a s!fficiently practicale introd!ction to Tensor Analysis carried on y a geometrical approach: ch!tG, B. C. 19%, A First Course in General Relativity, 6amridge Dniversity $ress, pag. (H& 6lassic introd!ctory text to -eneral elativity, it contains a !ite systematic and complete disc!ssion, altho!gh fragmented, of tensors. At least in the first part of the oo3 =the one that concerns !s here>, the disco!rse is caref!lly arg!ed, even metic!lo!s, with all the explanations of the case and only few occasional disconnection =even if the impression is sometimes that of a it c!mersome machinery<>. This is a good introd!ction to the topic which has the only defect of flattening< significantly concepts and res!lts witho!t giving a different emphasis depending on their importance. Cor that it wo!ld e etter dealing with it with some landmar3 already ac!ired. "t has een !sed in vario!s circ!mstances as a reference while writing these notes. *!nsy, $. 5. . '000 An Introduction to Tensors and Relativity, Dniversity of 6ape Town, o!th Africa, pag. 1(0 *ownload: http:++www.mth.!ct.ac.Ga+omei+gr+ =$ format> This is a schematic !t well7made s!mmary that follows the text of ch!tG. Iith few variations, some simplifications and m!ch more incisiveness. 6arroll, .. '00) Spacetime and eometry, Addison7Iesley, pag. 1( Text that has estalished itself as a leader among those recently p!lished on the s!/ect -. "t?s a fascinating example of scientific prose of conversational style: reading it se!entially yo! may have the impression of participating in a co!rse of lect!res. Tensors are introd!ced and developed closely integrated with the -. This text places itself at a level of diffic!lty somewhat higher than that of ch!tG. 6arroll, . . 199H !ecture "otes on General Relativity, Dniversity of 6alifornia anta Barara, pag. '(1 *ownload: http:++xxx.lanl.gov+$Jcache+gr7c+pdf+9H1'+9H1'019v1.pdf These are the original readings, in a even more conversational tone, from which the text !oted aove has een developed. All that is important is located here, too, 1'&
