141SPECIALRELATIVITYANDFLATSPACETIMEIn a 4-dimensional spacetime there will be 4k+I basis tensors in all. In component notationwethen write our arbitrary tensor asT = T..e(m) @..@e() @(o) ...(n) .(1.48)Alternatively,we could define the components by acting the tensor on basis vectors and dualvectors:- = T((u),.., (x), (),..,e(u) .Tμ1*k,(1.49)You can check for yourself, using (1.33) and so forth, that these equations all hang togetherproperly.As with vectors, we will usually take the shortcut of denoting the tensor T by its com-ponents T+-v.--. The action of the tensors on a set of vectors and dual vectors followsthe pattern established in (1.35):T(w),., (), ),., v)) - T.-w) .. )i.. . .(1.50)The order of the indices is obviously important, since the tensor need not act in the same wayon its various arguments. Finally, the transformation of tensor components under Lorentztransformations can be derived by applying what we already know about the transformationof basis vectors and dual vectors.The answer is just what you would expect from indexplacement,T = A".. AA.. T k(1.51)v.nThus, each upper index gets transformed like a vector, and each lower index gets transformedlike a dual vector.Although we have defined tensors as linear maps from sets of vectors and tangent vectorsto R, there is nothing that forces us to act on a full collection of arguments. Thus, a (1, 1)tensor also acts as a map from vectors to vectors:(1.52)T:V→TVYou can check foryourself thatT,visavector (i.e.obeys the vector transformation law).Similarly, we can act one tensor on (all or part of) another tensor to obtain a third tensor.For example,UW= THP.S°pv(1.53)is a perfectly good (1,1) tensorYou may be concerned that this introduction to tensors has been somewhat too brief.given the esoteric nature of the material. In fact, the notion of tensors does not require agreat deal of effort to master; it's just a matter of keeping the indices straight, and the rulesfor manipulating them are very natural. Indeed, a number of books like to define tensors as
1 SPECIAL RELATIVITY AND FLAT SPACETIME 14 In a 4-dimensional spacetime there will be 4k+l basis tensors in all. In component notation we then write our arbitrary tensor as T = T µ1···µk ν1···νl eˆ(µ1) ⊗ · · · ⊗ eˆ(µk) ⊗ ˆθ (ν1) ⊗ · · · ⊗ ˆθ (νl) . (1.48) Alternatively, we could define the components by acting the tensor on basis vectors and dual vectors: T µ1···µk ν1···νl = T( ˆθ (µ1) , . . ., ˆθ (µk) , eˆ(ν1) , . . ., eˆ(νl)) . (1.49) You can check for yourself, using (1.33) and so forth, that these equations all hang together properly. As with vectors, we will usually take the shortcut of denoting the tensor T by its components T µ1···µk ν1···νl . The action of the tensors on a set of vectors and dual vectors follows the pattern established in (1.35): T(ω (1), . . ., ω(k) , V (1), . . ., V (l) ) = T µ1···µk ν1···νlω (1) µ1 · · · ω (k) µk V (1)ν1 · · · V (l)νl . (1.50) The order of the indices is obviously important, since the tensor need not act in the same way on its various arguments. Finally, the transformation of tensor components under Lorentz transformations can be derived by applying what we already know about the transformation of basis vectors and dual vectors. The answer is just what you would expect from index placement, T µ ′ 1 ···µ ′ k ν ′ 1 ···ν ′ l = Λµ ′ 1 µ1 · · ·Λ µ ′ k µkΛν ′ 1 ν1 · · ·Λν ′ l νlT µ1···µk ν1···νl . (1.51) Thus, each upper index gets transformed like a vector, and each lower index gets transformed like a dual vector. Although we have defined tensors as linear maps from sets of vectors and tangent vectors to R, there is nothing that forces us to act on a full collection of arguments. Thus, a (1, 1) tensor also acts as a map from vectors to vectors: T µ ν : V ν → T µ νV ν . (1.52) You can check for yourself that T µ νV ν is a vector (i.e. obeys the vector transformation law). Similarly, we can act one tensor on (all or part of) another tensor to obtain a third tensor. For example, U µ ν = T µρ σS σ ρν (1.53) is a perfectly good (1, 1) tensor. You may be concerned that this introduction to tensors has been somewhat too brief, given the esoteric nature of the material. In fact, the notion of tensors does not require a great deal of effort to master; it’s just a matter of keeping the indices straight, and the rules for manipulating them are very natural. Indeed, a number of books like to define tensors as
151 SPECIALRELATIVITYANDFLATSPACETIMEcollections of numbers transforming according to (1.51). While this is operationally useful, ittends to obscure the deeper meaning of tensors as geometrical entities with a life independentofany chosen coordinate system.There is, however,one subtlety which wehaveglossed over.The notions of dual vectors and tensors and bases and linear maps belong to the realm oflinear algebra, and are appropriate whenever we have an abstract vector space at hand. Inthe case of interest to us we have not just a vector space, but a vector space at each point inspacetime.More often than not we are interested in tensor fields, which can be thought ofas tensor-valued functions on spacetime. Fortunately, none of the manipulations we definedabovereally care whether wearedealingwithasinglevector space or acollection ofvectorspaces, one for each event. We will be able to get away with simply calling things functionsof r when appropriate. However, you should keep straight the logical independence of thenotions we have introduced and their specific application to spacetime and relativity.Nowlet's turn to some examples of tensors.First we consider theprevious example ofcolumn vectors and their duals, row vectors. In this system a (1, 1) tensor is simply a matrix,M'.Its action on a pair (w,V) is given by usual matrix multiplication:(M)M'2MinV1V2M2nM?1M?2=wiM'i,vi.(1.54)M(w, V)= (wiw2... wn)MniM"z:MIf you like, feel free to think of tensors as“"matrices with an arbitrary number of indices."In spacetime, we have already seen some examples of tensors without calling them that.The most familiar example of a (o,2) tensor is the metric, nuw. The action of the metric ontwo vectors is so useful that it gets its own name, the inner product (or dot product):(1.55)n(V,W)=nuV"WV=V.W.Just as with the conventional Euclidean dot product, we will refer to two vectors whose dotproduct vanishes as orthogonal. Since the dot product is a scalar, it is left invariant underLorentz transformations; therefore the basis vectors of any Cartesian inertial frame, whichare chosen to be orthogonal by definition, are still orthogonal after a Lorentz transformation(despite the“scissoring together"we noticed earlier).The norm of a vector is defined to beinner product of the vector with itself; unlike in Euclidean space, this number is not positivedefinite:<0, Vμ is timelikeif nwVuVis=O, Vμis lightlikeor null(>0,Vμis spacelike
1 SPECIAL RELATIVITY AND FLAT SPACETIME 15 collections of numbers transforming according to (1.51). While this is operationally useful, it tends to obscure the deeper meaning of tensors as geometrical entities with a life independent of any chosen coordinate system. There is, however, one subtlety which we have glossed over. The notions of dual vectors and tensors and bases and linear maps belong to the realm of linear algebra, and are appropriate whenever we have an abstract vector space at hand. In the case of interest to us we have not just a vector space, but a vector space at each point in spacetime. More often than not we are interested in tensor fields, which can be thought of as tensor-valued functions on spacetime. Fortunately, none of the manipulations we defined above really care whether we are dealing with a single vector space or a collection of vector spaces, one for each event. We will be able to get away with simply calling things functions of x µ when appropriate. However, you should keep straight the logical independence of the notions we have introduced and their specific application to spacetime and relativity. Now let’s turn to some examples of tensors. First we consider the previous example of column vectors and their duals, row vectors. In this system a (1, 1) tensor is simply a matrix, Mi j . Its action on a pair (ω, V ) is given by usual matrix multiplication: M(ω, V ) = (ω1 ω2 · · · ωn) M1 1 M1 2 · · · M1 n M2 1 M2 2 · · · M2 n · · · · · · · · · · · · · · · · · · Mn 1 Mn 2 · · · Mn n V 1 V 2 · · · V n = ωiMi jV j . (1.54) If you like, feel free to think of tensors as “matrices with an arbitrary number of indices.” In spacetime, we have already seen some examples of tensors without calling them that. The most familiar example of a (0, 2) tensor is the metric, ηµν . The action of the metric on two vectors is so useful that it gets its own name, the inner product (or dot product): η(V, W) = ηµνV µWν = V · W . (1.55) Just as with the conventional Euclidean dot product, we will refer to two vectors whose dot product vanishes as orthogonal. Since the dot product is a scalar, it is left invariant under Lorentz transformations; therefore the basis vectors of any Cartesian inertial frame, which are chosen to be orthogonal by definition, are still orthogonal after a Lorentz transformation (despite the “scissoring together” we noticed earlier). The norm of a vector is defined to be inner product of the vector with itself; unlike in Euclidean space, this number is not positive definite: if ηµνV µV ν is < 0 , V µ is timelike = 0 , V µ is lightlike or null > 0 , V µ is spacelike
161 SPECIALRELATIVITYANDFLATSPACETIME(A vector can have zero norm without being the zero vector.) You will notice that theterminology is the same as that which we earlier used to classify the relationship betweentwo points in spacetime; it's no accident, of course, and we will go into more detail later.Another tensor is the Kronecker delta S, of type (1,1), which you already know thecomponents of. Related to this and the metric is the inverse metric n, a type (2,0)tensor defined as the inverse of the metric:(1.56)nnvp=npn=In fact, as you can check, the inverse metric has exactly the same components as the metricitself. (This is only true in flat space in Cartesian coordinates, and will fail to hold in moregeneral situations.) There is also the Levi-Civita tensor,a (0,4) tensor:(+1if μvpo isanevenpermutationof0123(1.57)-1if μvpois an oddpermutationof0123Euvp40otherwise.Here, a“permutation of 0123" is an ordering of the numbers 0, 1, 2,3 which can be obtainedby starting with 0123 and exchanging two of the digits; an even permutation is obtained byan even number of such exchanges, and an odd permutation is obtained by an odd number.Thus, for example, E0321 =-1.It is a remarkable property of the above tensors - the metric, the inverse metric, theKronecker delta, and the Levi-Civita tensor-that, even though they all transform accordingtothetensortransformationlaw(1.5l),theircomponentsremainunchangedinanyCartesiancoordinate system in flat spacetime. In some sense this makes them bad examples of tensors,since most tensors do not have this property. In fact, even these tensors do not have thisproperty oncewegotomoregeneral coordinate systems,withthe single exception of theKronecker delta.This tensor has exactly the same components in any coordinate systemin any spacetime. This makes sense from the definition of a tensor as a linear map; theKronecker tensor can be thought of as the identity map from vectors to vectors (or fromdual vectors to dual vectors), which clearly must have the same components regardless ofcoordinatesystem.The other tensors (the metric,its inverse, and the Levi-Civitatensor)characterize the structure of spacetime, and all depend on the metric. We shall thereforehave to treat them more carefully when we drop our assumption of flat spacetime.Amore typical example of a tensor is the electromagnetic field strength tensor.Weall know that the electromagnetic fields are made up of the electric field vector E, and themagnetic field vector Bi. (Remember that we use Latin indices for spacelike components1,2,3.)Actually these are only“"vectors"under rotations in space, not under the full Lorentz
1 SPECIAL RELATIVITY AND FLAT SPACETIME 16 (A vector can have zero norm without being the zero vector.) You will notice that the terminology is the same as that which we earlier used to classify the relationship between two points in spacetime; it’s no accident, of course, and we will go into more detail later. Another tensor is the Kronecker delta δ µ ν , of type (1, 1), which you already know the components of. Related to this and the metric is the inverse metric η µν, a type (2, 0) tensor defined as the inverse of the metric: η µν ηνρ = ηρν η νµ = δ ρ µ . (1.56) In fact, as you can check, the inverse metric has exactly the same components as the metric itself. (This is only true in flat space in Cartesian coordinates, and will fail to hold in more general situations.) There is also the Levi-Civita tensor, a (0, 4) tensor: ǫµνρσ = +1 if µνρσ is an even permutation of 0123 −1 if µνρσ is an odd permutation of 0123 0 otherwise . (1.57) Here, a “permutation of 0123” is an ordering of the numbers 0, 1, 2, 3 which can be obtained by starting with 0123 and exchanging two of the digits; an even permutation is obtained by an even number of such exchanges, and an odd permutation is obtained by an odd number. Thus, for example, ǫ0321 = −1. It is a remarkable property of the above tensors – the metric, the inverse metric, the Kronecker delta, and the Levi-Civita tensor – that, even though they all transform according to the tensor transformation law (1.51), their components remain unchanged in any Cartesian coordinate system in flat spacetime. In some sense this makes them bad examples of tensors, since most tensors do not have this property. In fact, even these tensors do not have this property once we go to more general coordinate systems, with the single exception of the Kronecker delta. This tensor has exactly the same components in any coordinate system in any spacetime. This makes sense from the definition of a tensor as a linear map; the Kronecker tensor can be thought of as the identity map from vectors to vectors (or from dual vectors to dual vectors), which clearly must have the same components regardless of coordinate system. The other tensors (the metric, its inverse, and the Levi-Civita tensor) characterize the structure of spacetime, and all depend on the metric. We shall therefore have to treat them more carefully when we drop our assumption of flat spacetime. A more typical example of a tensor is the electromagnetic field strength tensor. We all know that the electromagnetic fields are made up of the electric field vector Ei and the magnetic field vector Bi . (Remember that we use Latin indices for spacelike components 1,2,3.) Actually these are only “vectors” under rotations in space, not under the full Lorentz
171SPECIALRELATIVITYANDFLATSPACETIMEgroup. In fact they are components of a (o, 2) tensor Fw, defined by0-Ei-E2-E3Ei0B3-B2Fuv =(1.58)-Fvμ.E2Bi-B30(E30B2B1From this point of view it is easy to transform the electromagnetic fields in one referenceframe to those in another, by application of (1.51).The unifyingpower of the tensorformal-ism isevident:ratherthanacollectionof twovectorswhoserelationshipandtransformationproperties are rather mysterious, we have a single tensor field to describe all of electromag-netism. (On the other hand, don't get carried away; sometimes it's more convenient to workin a single coordinate system using the electric and magnetic field vectors.)With some examples inhand we can nowbea littlemore systematic about someprop-erties of tensors. First consider the operation of contraction, which turns a (k,l) tensorinto a (k-1,l-1)tensor.Contraction proceeds by summing overoneupper and onelowerindex:SHP。= TPov .(1.59)You can check that the result is a well-defined tensor. Of course it is only permissible tocontract an upper index with a lower index (as opposed to two indices of the same type).Note also that the order of the indices matters, so that you can get different tensors bycontracting in different ways; thus,Tpa+Tupvay(1.60)in general.The metric and inverse metric can be used to raise and lower indices on tensors. Thatis, given a tensor Tapys, we can use the metric to define new tensors which we choose todenote by the same letter T:Tapug=nnTapsTP% = nuaTaBTupo= muanupnnaps,(1.61)and so forth. Notice that raising and lowering does not change the position of an indexrelative to other indices, and also that “free" indices (which are not summed over)must bethe same on both sides of an equation, while “dummy" indices (which are summed over)only appear on one side.As an example, we can turn vectors and dual vectors into eachother by raising and lowering indices:Vu=nmVuw=nwy.(1.62)
1 SPECIAL RELATIVITY AND FLAT SPACETIME 17 group. In fact they are components of a (0, 2) tensor Fµν , defined by Fµν = 0 −E1 −E2 −E3 E1 0 B3 −B2 E2 −B3 0 B1 E3 B2 −B1 0 = −Fνµ . (1.58) From this point of view it is easy to transform the electromagnetic fields in one reference frame to those in another, by application of (1.51). The unifying power of the tensor formalism is evident: rather than a collection of two vectors whose relationship and transformation properties are rather mysterious, we have a single tensor field to describe all of electromagnetism. (On the other hand, don’t get carried away; sometimes it’s more convenient to work in a single coordinate system using the electric and magnetic field vectors.) With some examples in hand we can now be a little more systematic about some properties of tensors. First consider the operation of contraction, which turns a (k, l) tensor into a (k − 1, l − 1) tensor. Contraction proceeds by summing over one upper and one lower index: S µρ σ = T µνρ σν . (1.59) You can check that the result is a well-defined tensor. Of course it is only permissible to contract an upper index with a lower index (as opposed to two indices of the same type). Note also that the order of the indices matters, so that you can get different tensors by contracting in different ways; thus, T µνρ σν 6= T µρν σν (1.60) in general. The metric and inverse metric can be used to raise and lower indices on tensors. That is, given a tensor T αβ γδ, we can use the metric to define new tensors which we choose to denote by the same letter T: T αβµ δ = η µγT αβ γδ , Tµ β γδ = ηµαT αβ γδ , Tµν ρσ = ηµαηνβη ργη σδT αβ γδ , (1.61) and so forth. Notice that raising and lowering does not change the position of an index relative to other indices, and also that “free” indices (which are not summed over) must be the same on both sides of an equation, while “dummy” indices (which are summed over) only appear on one side. As an example, we can turn vectors and dual vectors into each other by raising and lowering indices: Vµ = ηµνV ν ω µ = η µνων . (1.62)
181SPECIALRELATIVITYANDFLATSPACETIMEThis explains why the gradient in three-dimensional fat Euclidean space is usually thoughtof as an ordinary vector, even though we have seen that it arises as a dual vector; in Euclideanspace (where themetric isdiagonal with all entries+1)a dual vector is turned intoa vectorwith preciselythe samecomponents when weraiseits index.You maythen wonder whywehave belabored the distinction at all. One simple reason, of course, is that in a Lorentzianspacetime the components are not equal:w = (-wo,w1,w2,w3) .(1.63)In a curved spacetime, where the form of the metric is generally more complicated, the dif-ference is rather more dramatic. But there is a deeper reason, namely that tensors generallyhave a“natural"definition which is independent of the metric.Even though we will alwayshavea metric available,itishelpful to be aware of thelogical status of eachmathematicalobject we introduce. The gradient, and its action on vectors, is perfectly well defined re-gardless of any metric, whereas the “"gradient with upper indices" is not. (As an example,we will eventually want to take variations of functionals with respect to the metric, and willtherefore have to know exactly how the functional depends on the metric, something that iseasily obscured by the index notation.)Continuing our compilation of tensor jargon, we refer to a tensor as symmetric in anyof its indices if it is unchanged under exchange of those indices. Thus, if(1.64)Suvp=Supwe say that Sμwp is symmetric in its first two indices, while if(1.65)Sp=upy=Se=Sup=Se=Se，we say that Sμvp is symmetric in all three of its indices. Similarly, a tensor is antisym-metric (or “skew-symmetric") in any of its indices if it changes sign when those indices areexchanged; thus,(1.66)Awp=-Apvμmeans that Auwye is antisymmetric in its first and third indices (or just “antisymmetric in μand p"). If a tensor is (anti-) symmetric in all of its indices, we refer to it as simply (anti-)symmetric (sometimes with the redundant modifier"completely").As examples, the metricNu and the inverse metric nu are symmetric, while the Levi-Civita tensor ewpa and theelectromagnetic field strength tensor Fwy are antisymmetric. (Check for yourself that if youraiseorlower asetof indiceswhicharesymmetricorantisymmetric,theyremainthatway.Notice that it makes no sense to exchange upper and lower indices with each other, so don'tsuccumb to the temptation to think of the Kronecker delta g as symmetric. On the otherhand, the fact that lowering an index on g gives a symmetric tensor (in fact, the metric)
1 SPECIAL RELATIVITY AND FLAT SPACETIME 18 This explains why the gradient in three-dimensional flat Euclidean space is usually thought of as an ordinary vector, even though we have seen that it arises as a dual vector; in Euclidean space (where the metric is diagonal with all entries +1) a dual vector is turned into a vector with precisely the same components when we raise its index. You may then wonder why we have belabored the distinction at all. One simple reason, of course, is that in a Lorentzian spacetime the components are not equal: ω µ = (−ω0, ω1, ω2, ω3) . (1.63) In a curved spacetime, where the form of the metric is generally more complicated, the difference is rather more dramatic. But there is a deeper reason, namely that tensors generally have a “natural” definition which is independent of the metric. Even though we will always have a metric available, it is helpful to be aware of the logical status of each mathematical object we introduce. The gradient, and its action on vectors, is perfectly well defined regardless of any metric, whereas the “gradient with upper indices” is not. (As an example, we will eventually want to take variations of functionals with respect to the metric, and will therefore have to know exactly how the functional depends on the metric, something that is easily obscured by the index notation.) Continuing our compilation of tensor jargon, we refer to a tensor as symmetric in any of its indices if it is unchanged under exchange of those indices. Thus, if Sµνρ = Sνµρ , (1.64) we say that Sµνρ is symmetric in its first two indices, while if Sµνρ = Sµρν = Sρµν = Sνµρ = Sνρµ = Sρνµ , (1.65) we say that Sµνρ is symmetric in all three of its indices. Similarly, a tensor is antisymmetric (or “skew-symmetric”) in any of its indices if it changes sign when those indices are exchanged; thus, Aµνρ = −Aρνµ (1.66) means that Aµνρ is antisymmetric in its first and third indices (or just “antisymmetric in µ and ρ”). If a tensor is (anti-) symmetric in all of its indices, we refer to it as simply (anti-) symmetric (sometimes with the redundant modifier “completely”). As examples, the metric ηµν and the inverse metric η µν are symmetric, while the Levi-Civita tensor ǫµνρσ and the electromagnetic field strength tensor Fµν are antisymmetric. (Check for yourself that if you raise or lower a set of indices which are symmetric or antisymmetric, they remain that way.) Notice that it makes no sense to exchange upper and lower indices with each other, so don’t succumb to the temptation to think of the Kronecker delta δ α β as symmetric. On the other hand, the fact that lowering an index on δ α β gives a symmetric tensor (in fact, the metric)