PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS

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1 PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS To strt on tensor clculus, we need to define differentition on mnifold.a good question to sk is if the prtil derivtive of tensor tensor on mnifold? X = ( x x c = xd x c x d x b Xb ) ( x x b Xb = xd 2 x x c x d x b Xb + xd x X b x c x b x d = x x d x b x c X b + 2 x x d x d x b x c Xb This will not be tensor s the second term involves second derivtives of the new coordinte system; unless this term is removed or somehow cncelled out, this will not trnsform in tensoril mnner. From clculus we know tht the derivtive is limiting process of two function evlutions t different points f f(x + h) f(x) (x) = lim. h 0 h We re going to consider similr expression for tensors in nlogy to this [X ] P [X ] Q lim δn 0 δn where δu indictes the seprtion between P nd Q. There is problem, however, s the coordinte trnsformtions t these different points will in generl be different, since these objects re evluted t points: X P = [ x x b ] P Xb P, nd, ) X Q = [ x x b ]Q Xb Q. In order to define tensoril derivtive, we must devise clever wy to drg tensor t one plce to nother. 1. The Lie Derivtive To mke this ide work, we need to describe how the coordintes of point s it is moved in the mnifold, this is known s n ctive trnsformtion. To describe how the coordintes chnge, we generlize the ide of curve on mnifold to congruence of curves, which is collection of curves tht cover the mnifold, nd connect points on the mnifold together. Any congruences of smooth curves, x (u) : R M, going through point on the mnifold, p M, cn lwys be described by vector field, X (p) : M T p M, on the mnifold: dx = X (x b (u)) 1

2 2 PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS Figure 1. A congruence of curves on mnifold M With these curves, we will be ble to move tensor t one position P to nother t Q. Given prticulr vector-field X ssocited with congruence of curves, consider x x = x + δux (x) we will tret this s shift from position P to Q, thus the vector X my be seen s P Q for infinitesimlly close points P nd Q. Computing the first derivtives, x x b = δ b + δu b X Let us consider the effect of contrvrint rnk 1 tensor (or vector field) under the process of drgging T (x) T ( x) Figure 2. A vector T being drgged long curve generted by X from P to Q on mnifold We will lso hve second tensor t Q given by the tensor found from the ctive coordinte chnge, T ( x) - where we note tht T ( x) nd T ( x) re distinct contrvrint vectors. Since T ( x) is tensor, (1) T ( x) = x x b T b (x) = (δ b + δu b X )T b (x) = T (x) + [ b X T b (x)]δu;

3 PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS 3 for the tensor found by the ctive coordinte chnge, we ssume δu is smll nd pply Tylor expnsion bout the coordintes for P (2) T ( x) = T (x b + δux b ) = T (x) + δux b δ b T + O(δu 2 ) The Lie derivtive is then defined s the difference of (1) nd (2) divided by δu: X T = T ( x) lim ( x) δu 0 δu = X b b T T b b X Notice tht the Lie derivtive of vector field T in the direction of X reces to the Lie brcket, however the Lie derivtive cn be extended to more generl tensors of type (p, q), T. 2. Properties of the Lie Derivtive The Lie derivtive in the direction of X stisfies the following properties Linerity: X (λy + µz ) = λ X Y + µ X Z where λ, µ R. Leibniz: X Y Z+ bc ) = ( X Y )Z bc + Y ( X Z bc ) Commutes with Contrction: δ b X T b = XT Using these properties nd our knowledge of the Lie derivtive of contrvrint vector field we my derive expressions for more generl tensors. As two simple exmples, we hve: The Lie Derivtive of covrint vector field, X Y = X b b Y + Y b X b. The Lie Derivtive of sclr field, x φ = X φ. As one more exmple we consider the Lie derivtive of type (1,1) tensor Exmple 2.1. Given type (1,1) tensor, T b, contrct with two rbitrry vectors Y b nd Z to proce sclr T b Xb Y nd pply Lie derivtive in the direction of X X (T by b Z ) = X (T b)y b Z + T b X (Y b )Z + T by b X (Z ). = X (T b)y b Z + T b[x e e Y b Y e e X b ]Z + T by b [X e e Z + Z e X e ] Next suppose tht Y b = δ b d nd Z = δ c for some fixed vlue of d nd 1, then this expression becomes X T c d = X (T c d) + T b[ d X b ] + T by b [ X c ] Solving for X (T b ) gives the finl expression X (T c d) = X T c d + T c b d X b T c by b X c Remrk 2.2. Notice tht the Lie derivtive is type preserving, tht is, the Lie derivtive of type (r,s) tensor is nother type (r,s) tensor. See P.72 of the textbook for the definition of the Lie derivtive of n rbitrry type (r,s) tensor. The Lie derivtive is very useful tool, especilly when mnifold does not come equipped with metric. In the context of Riemnnin geometry, the Lie derivtive llows one to define specil clss of trnsformtions clled isometries. These hve the property tht distnce reltionships re unchnged under their ctive trnsformtions. 1 Tht is the vectors Y nd Z re ligned with bsis vector of the tngent nd cotngent spce respectively

4 4 PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS While the Lie derivtive llows us to define the derivtive of tensor, it is not quite wht we wnt. This definition requires the choice of vector field X in order to evlute the definition of the Lie deritive of tensor field T. Furthermore the type preserving nture mens this is not quite nlogous to pplying - lthough we my choose coordintes os tht it ppers this wy. We will see tht the covrint derivtive will hve the properties we wnt. However in order to discuss this derivtive, we must define new structure on our mnifold, clled connection which will llow us to prllel trnsport vectors. 3. Prllel Trnsport As thought experiment, consider n rrow being moved long closed pth strting nd ending t the sme point p, with the rrow pointing in some initil fixed direction for the plne nd sphere respectively. Figure 3. Moving fixed rrow long rectngle in the plne R 2 On the plne, we my simply consider rectngle of length m nd n. Moving the rrow long this closed pth, there will be no chnge in the direction of the rrow. For the sphere we will choose tringle s the closed pth for the rrow 2. Figure 4. Moving fixed rrow long tringle on the sphere S 2 2 Wht hppens if the rrow trvels long tringle in the plne, nd is there ny difference to rectngle?

5 PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS 5 Choosing the initil point p to be the North pole, the rrow trvels to the equtor, then to second point long the equtor, nd then bck to the North pole. Despite trvelling in the sme mnner s the rrow in the plne, the strting nd ending vector on the sphere no longer line-up! This chnge is cused by the bsis vectors on the sphere chnging from point to point. While we used congruences of curves to trnsport tensors to define the Lie derivtive, this is not the only wy to move tensors. Suppose we re given two points P = P (x) nd Q = Q(x + δx), we would like to define one-to-one mp between the tngent spces T p (M) nd T Q (M), with the following properties: When P = Q the mp must rece to the identity. The trnsformtion rule is liner 3 We will cll the corresponding vectors in T Q (M) prllel. Denoting vector t P by X (x) we define the trnsported vector t Q s X (x) + δx (x). Linerity ensures tht we cn write the secon term s δx = W bx b where W b depends on the choice of P nd Q. Both requirements cn be stisfied by setting 4 W b = Γ bcδx c since lim δx 0 W b = 0 for, b [1, n] ensuring the identity mp when P = Q. For the trnsported vector t Q, X (x) + δx (x) = (δ b Γ bcδx c )X b the Γ bc must depend only on P, while the seprtion to Q is expressed through δx c. For n n-dimensionl mnifold Γ bc corresponds to n3 functions. The Γ bc re clled connection coefficients nd tken together these define connection. For given mnifold there re mny choices of connection. 4. The Covrint Derivtive To define derivtive we will lso need to consider vector field evluted t Q = x + δx, using Tylor expnsion X (x + δx) = X (x) + δx b b X + O(δx 2 ) Given X (x + δx) nd the trnsported vector X (x) + δx (x) we now define the covrint derivtive: c X 1 [ = lim X δx 0 δx c (x + δx) (X (x) + δx (x)) ]. As this is sum of vectors evluted t Q, this is n cceptble tensoril definition. Tking the limit δx = 0 this becomes c X = c X + Γ bcx b. Remrk 4.1. The covrint derivtive of vector X is lso written s X ;c or x c. 3 This is the simplest trnsformtion we cn pick, nd it will ensure our mp is one-to-one 4 The negtive sign is included to follow nottionl convention

6 6 PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS The requirement tht c X must be tensor will constrin how the Γ bc trnsform. To see this consider the covrint derivtive of X in new coordinte system c X = x x b x d x c ( dx b + Γ b edx e ) = c X + Γ bc X b Expnding out c X we my derive n eqution for the Γ bc : Γ bc = x x e x f x d x b x c Γd ef xd x e 2 x x b x c x d x e Equivlently, since d ( x x c x c x b ) = d (δ b ) = 0 Γ bc = x x e x f x d x b x c Γd ef + x 2 x d x d x b x c Anyt set of functions tht trnsform ccording to this trnsformtion lw is clled n ffine connection. Remrk 4.2. The second term compenstes for the dditionl components ssocited with the prtil derivtives under coordinte chnges. 5. Properties of the Covrint Derivtive The covrint derivtive hs some useful properties, just like the Lie derivtive is liner. stisfies the Liebniz property. commutes with contrction. Using these properties we my show tht For ny sclr field φ, φ = φ The covrint derivtive of covrint vector field is X b = A X b Γ c bx c A generl type (r,s) tensor hs the covrint derivtive c T 1,...,r = c T 1,...,r + Γ 1 dc T d,...,r Γ r dc T 1,...,d Γ d bct 1,...,r d,...,b s... Γ d bct 1,...,r b 1,...,d While Γ bc is not tensor, nd neither is sum of two connections, the difference of two connections is tensor of type (1,2). If the mnifold is equipped with metric g b so tht one my form line element, then there is unique connection stisfying two properties c g b = 0 nd tht this connection is torsion-free, tht is the torsion tensor vnishes. For the moment we will focus on the second condition, nd stte without proof tht the torsion-free condition is equivlent to T bc = Γ bc Γ cb = 0 For vnishing torsion, this implies tht the connection coefficients must be symmetric in the lower indices Γ bc = Γ cb With the covrint derivtive defined we my extend the definition of the totl d derivtive,, to consider the bsolute derivtive. For ny congruence of curves

7 PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS 7 x (u) defined by the tngent vector X (x(u)) through the first order differentil eqution dx (u) = X (x(u)) the bsolute derivtive of tensor T 1,...,r is D 1,...,r (Tb Du 1,...,b s ) = X T 1,...,r = X c c T 1,...,r The definition of prllel propgtion my be expressed simply s the vnishing of the bsolute derivtive D 1,...,r (Tb Du 1,...,b s ) = 0 Further, for vector field λ (u) long the curves x (u) D Du (λ ) = dxb bλ = dxb ( bλ + Γ cbλ c ) = dxb λ x b + Γ cbλ c dxb = dλ + Γ cbλ c dxb. This shows the reltionship between the totl nd bsolute derivtive. 6. Geodesics In words, we cn describe geodesic s the shortest distnce between two points in stright line While we hve not relly exmined distnce on mnfiold, we cn still consider the ide of stright line in mnifold. For stright line, the tngent vector long the curve must be trnsported to be copy of itself multiplied by function β(u) expnding this out we find D Du ( dx ) = β(u) dx d 2 x 2 + dx b dx c Γ bc = β(u)dx this defines n ffine geodesic. If we cn find prmetriztion, s, to mke β(u) vnish then this becomes ( ) D dx = 0, Ds ds or equivlently d 2 x 2 + dx b dx c Γ bc = 0. Such prmeter s is clled n ffine prmeter. Any trnsformtion s s = s + b will mintin this differentil eqution with s s ; we cll such mp, n ffine trnsformtion Geodesics will ply fundmentl role in Generl Reltivity. They define the pths tht freely flling objects tke, s well s photons.