A Vectors and Tensors in General Relativity

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1 1 A Vectors nd Tensors in Generl Reltivity A.1 Vectors, tensors, nd the volume element The metric of spcetime cn lwys be written s ds 2 = g µν dx µ dx ν µ=0 ν=0 g µν dx µ dx ν. (1) We introduce Einstein s summtion rule: there is sum over repeted indices (tht is, we don t bother to write down the summtion sign in this cse). Greek (spcetime) indices go over the vlues 0, Ltin (spce) indices over the vlues 1, i.e., g ij dx i dx j i=1 j=1 g ijdx i dx j. The objects g µν re the components of the metric tensor. They hve, in principle, the dimension of distnce squred. In prctice one often ssigns the dimension of distnce (or time) to some coordintes, nd then the corresponding components of the metric tensor re dimensionless. These coordinte distnces re then converted to proper ( rel or physicl ) distnces with the metric tensor. The components of the metric tensor form symmetric 4 4 mtrix. Exmple 1. The metric tensor for 2-sphere (discussed bove s n exmple of curved 2D spce) hs the components [g ij ] = [ ] sin 2. (2) ϑ Exmple 2. The metric tensor for Minkowski spce hs the components g µν = () in Crtesin coordintes, nd g µν = r r 2 sin 2 ϑ (4) in sphericl coordintes. Exmple. The Robertson-Wlker metric, which we discuss in Chpter, hs components g µν = Kr r 2 0. (5) r 2 sin 2 ϑ Note tht the metric tensor components in the bove exmples lwys formed digonl mtrix. This is the cse when the coordinte system is orthogonl. The vectors which occur nturlly in reltivity re four-vectors, with four components, e.g., the four-velocity. The vlues of the components depend on the bsis {e α } used. Note tht the index of the bsis vector does not refer to component, but specifies which one of the four bsis vectors is in question. The components of the bsis vectors in the bsis they define re, of course, (e α ) β = δ β α. (6)

2 A VECTORS AND TENSORS IN GENERAL RELATIVITY 2 Given coordinte system, we hve two bses (lso clled frmes) nturlly ssocited with it, the coordinte bsis nd the corresponding normlized bsis. If the coordinte system is orthogonl, the ltter is n orthonorml bsis. When we use the coordintes to define the components of vector, like the 4-velocity in Chpter 2, the components nturlly come out in the coordinte bsis. The bsis vectors of coordinte bsis re prllel to coordinte lines, nd their length represents the distnce from chnging the vlue of the coordinte by one unit. For exmple, if we move long the coordinte x 1 so tht it chnges by dx 1, the distnce trveled is ds = g 11 dx 1 dx 1 = g 11 dx 1. The length of the bsis vector e 1 is thus g 11. Since in the coordinte bsis the bsis vectors usully re not unit vectors, the numericl vlues of the components give the wrong impression of the mgnitude of the vector. Therefore we my wnt to convert them to the normlized bsis eˆα ( 1 gαα ) eα. (7) (It is customry to denote the normlized bsis with ht over the index, when both bses re used. In the bove eqution there is no sum over the index α, since it ppers only once on the left hnd side.) For four-vector w we hve where w = w α e α = wˆα eˆα, (8) wˆα g αα w α. (9) For exmple, the components of the coordinte velocity of mssive body, v i = dx i /dt could be greter thn one; the physicl velocity, i.e., the velocity mesured by n observer who is t rest in the comoving coordinte system, is 1 vî = g ii dx i / g 00 dx 0, (10) with components lwys smller thn one. The volume of region of spce (given by some rnge in the sptil coordintes x 1, x 2, x ) is given by V = dv = det[g ij ]dx 1 dx 2 dx (11) V V where dv det[g ij ]dx 1 dx 2 dx is the volume element. Here det[g ij ] is the determinnt of the submtrix of the metric tensor components corresponding to the sptil coordintes. For n orthogonl coordinte system, the volume element is dv = g 11 dx 1 g 22 dx 2 g dx. (12) The metric tensor is used for tking sclr (dot) products of four-vectors, The (squred) norm of four-vector w is w u g αβ u α w β. (1) w w g αβ w α w β. (14) Exercise: Show tht the norm of the four-velocity is lwys 1. 1 When g 00 = 1, this simplifies to g iidx i /dt.

3 A VECTORS AND TENSORS IN GENERAL RELATIVITY For n orthonorml bsis we hve We shll use the short-hnd nottion eˆ0 eˆ0 = 1 eˆ0 e ĵ = 0 eî eĵ = δ ij. (15) eˆα eˆβ = η αβ, (16) where the symbol η αβ is like the Kronecker symbol δ αβ, except tht η 00 = 1. A.2 Contrvrint nd covrint components We sometimes write the index s subscript, sometimes s superscript. This hs precise mening in generl reltivity. This is explined in this smll-print (indicting stuff not relly needed in this course) subsection. The component w α of four-vector is clled contrvrint component. We define the corresponding covrint component s The norm is now simply In prticulr, for the 4-velocity we lwys hve w α g αβ w β. (17) w w = w α w α. (18) u µ u µ = g µν u µ u ν = ds2 = 1. (19) dτ2 We defined the metric tensor through its covrint components (Eq. 1). We now define the corresponding covrint components g αβ s the inverse mtrix of the mtrix [g αβ ], Now g αβ g βγ = δ γ α. (20) g αβ w β = g αβ g βγ w γ = δ α γw γ = w α. (21) The metric tensor cn be used to lower nd rise indices. For tensors, A β α = g αγ A γβ A αβ = g αγ g βδ A γδ A αβ = g αγ g βδ A γδ. (22) Note tht the mixed components A β α A β α, unless the tensor is symmetric, in which cse we cn write A β α. For n orthonorml bsis, gˆαˆβ = gˆαˆβ = η αβ, (2) nd the covrint nd contrvrint components of vectors nd tensors hve the sme vlues, except tht the rising or lowering of the time index 0 chnges the sign. These orthonorml components re lso clled physicl components, since they hve the right mgnitude. Note tht the symbols δ αβ nd η αβ re not tensors, nd the loction of their index crries no mening.

4 A VECTORS AND TENSORS IN GENERAL RELATIVITY 4 A. Einstein eqution From the first nd second prtil derivtives of the metric tensor, g µν / x σ, 2 g µν /( x σ x τ ), (24) one cn form vrious curvture tensors. These re the Riemnn tensor R µ νρσ, the Ricci tensor R µν R α µαν, nd the Einstein tensor G µν = R µν 1 2 g µνr, where R is the Ricci sclr g αβ R βα, lso clled the sclr curvture (not to be confused with the scle fctor R(t) of the Robertson Wlker metric). We shll not discuss these curvture tensors in this course. The only purpose of mentioning them here is to be ble to show the generl form of the Einstein eqution, before we go to the much simpler specific cse of the Friedmnn models. In Newton s theory the source of grvity is mss, in the cse of continuous mtter, the mss density ρ. According to Newton, the grvittionl field g N is given by the eqution 2 Φ = g N = 4πGρ (25) Here Φ is the grvittionl potentil. In Einstein s theory, the source of spcetime curvture is the energy-momentum tensor, lso clled the stress-energy tensor, or, for short, the energy tensor T µν. The energy tensor crries the informtion on energy density, momentum density, pressure, nd stress. The energy tensor of frictionless continuous mtter ( perfect fluid) is T µν = (ρ+p)u µ u ν +pg µν, (26) where ρ is the energy density nd p is the pressurein the rest frme of the fluid. In cosmology we cn usully ssume tht the energy tensor hs the perfect fluid form. T 00 is the energy density in the coordinte frme. (T i0 gives the momentum density, which is equl to the energy flux T 0i. T ij gives the flux of momentum i-component in j-direction.) We cn now give the generl form of the Einstein eqution, G µν = 8πGT µν. (27) This is the lw of grvity ccording to Einstein. Compring to Newton (Eq. 25) we see tht the mss density ρ hsbeen replced by T µν, nd 2 Φhs been replced by the Einstein tensor G µν, which is short wy of writing complicted expression contining first nd second derivtives of g µν. Thus the grvittionl potentil is replced by the 10 components of g µν in Einstein s theory. In the cse of wek grvittionl field, the metric is close to the Minkowski metric, nd we cn write, e.g., g 00 = 1 2Φ (28) (in suitble coordintes), where Φ is smll. The Einstein eqution for g 00 becomes then 2 Φ = 4πG(ρ+p). (29) Compring this to Eq. (25) we see tht the density ρ hs been replced by ρ+p. For reltivistic mtter, where p cn be of the sme order of mgnitude thn ρ this is n importnt modifiction to the lw of grvity. For nonreltivistic mtter, where the prticle velocities re v 1, we hve p ρ, nd we get Newton s eqution.

5 A VECTORS AND TENSORS IN GENERAL RELATIVITY 5 When pplied to homogeneous nd isotropic universe filled with ordinry mtter, the Einstein eqution tells us tht the universe cnnot be sttic, it must either expnd or contrct. 2 When Einstein ws developing his theory, he did not believe this ws hppening in relity. He believed the universe ws sttic. Therefore he modified his eqution by dding n extr term, G µν +Λg µν = 8πGT µν. (0) The constnt Λ is clled the cosmologicl constnt. Without Λ, universe which ws momentrily sttic, would begin to collpse under its own weight. A positive Λ cts s repulsive grvity. In Einstein s model for the universe (the Einstein universe), Λ hd precisely the vlue needed to perfectly blnce the pull of ordinry grvity. This vlue is so smll tht we would not notice its effect in smll scles, e.g., in the solr system. The Einstein universe is, in fct, unstble to smll perturbtions. When Einstein herd tht the Universe ws expnding, he threw wy the cosmologicl constnt, clling it the biggest blunder in my life. In more recent times the cosmologicl constnt hs mde comebck in the form of vcuum energy. Considertions in quntum field theory suggest tht, due to vcuum fluctutions, the energy density of the vcuum should not be zero, but some constnt ρ vc. 4 The energy tensor of the vcuum would then hve the form T µν = ρ vc g µν. Thus vcuum energy hs exctly the sme effect s cosmologicl constnt with the vlue Λ = 8πGρ vc. (1) Vcuum energy is observtionlly indistinguishble from cosmologicl constnt. This is becuse in physics, we cn usully mesure only energy differences. Only grvity responds to bsolute energy density, nd there constnt energy density hs the sme effect s the cosmologicl constnt. In principle, however, they represent different ides. The cosmologicl constnt is n ddition to the left-hnd side of the Einstein eqution, modifiction of the lw of grvity, wheres vcuum energy is n ddition to the right-hnd side, contribution to the energy tensor, i.e., form of energy. A.4 Friedmnn equtions We shll now pply the Einstein eqution to the homogeneous nd isotropic cse, which leds to Friedmnn Robertson Wlker (FRW) cosmology. The metric is now the RW metric, g µν = Kr r r 2 sin 2 ϑ. (2) 2 It leds to ä < 0, which does not llow (t) = const. If we momentrily hd ȧ = 0, would immeditely begin to decrese. If you sneeze, the universe will collpse. 4 Infieldtheory, thefundmentl physiclobjects re fields, ndprticles rejust quntofthefieldoscilltions. Vcuum mens the ground stte of the system, i.e., fields hve those vlues which correspond to minimum energy. This minimum energy is usully ssumed to be zero. However, in quntum field theory, the fields cnnot sty t fixed vlues, becuse of quntum fluctutions. Thus even in the ground stte the fields fluctute round their zero-energy vlue, contributing positive energy density. This is nlogous to the zero-point energy of hrmonic oscilltor in quntum mechnics.

6 REFERENCES 6 Clculting the Einstein tensor from this metric gives Gˆ0ˆ0 Gˆ1ˆ1 = 2(ȧ2 +K) () = 1 2(2ä+ȧ2 +K) = Gˆ2ˆ2 = Gˆˆ. (4) We use here the orthonorml bsis (signified by theˆover the index). We ssume the perfect fluid form for the energy tensor T µν = (ρ+p)u µ u ν +pg µν. (5) Isotropy implies tht the fluid is t rest in the RW coordintes, so tht uˆµ = (1,0,0,0) nd (remember, gˆµˆν = η µν = dig( 1,1,1,1)) ρ T ˆµˆν = 0 p p 0. (6) p Homogeneity implies tht ρ = ρ(t), p = p(t). The Einstein eqution Gˆµˆν = 8πGT ˆµˆν becomes now Let us rerrnge this pir of equtions to 5 2(ȧ2 +K) = 8πGρ (7) (ȧ ) 2 2ä K 2 = 8πGp. (8) (ȧ ) 2 + K 2 = 8πG ρ (4) ä = 4πG (ρ+p). (44) These re the Friedmnn equtions. ( Friedmnn eqution in singulr refers to Eq. (4).) References [1] C.W. Misner, K.S. Thorne, J.A. Wheeler, Grvittion (Freemn 197). 5 Including the cosmologicl constnt Λ these equtions tke the form or, in the rerrnged form, +K) Λ = 8πGρ (9) (ȧ 2(ȧ2 ) 2 2ä K +Λ = 8πGp. (40) 2 (ȧ ) 2 + K Λ 2 ä Λ = 8πG ρ (41) = 4πG (ρ+p). (42) We shll not include Λ in these equtions. Insted, we llow for the presence of vcuum energy ρ vc, which hs the sme effect.