7.6 The Riemann curvature tensor

Transcription

1 7.6. The Riemnn curvture tensor The Riemnn curvture tensor Before we egin with the derivtion of the Riemnn curvture tensor, rief discussion of the concept of curvture ppers pproprite. Mthemticins discriminte etween two different types of curvture. These re the intrinsic Gussin curvture nd the exterior curvture. Intrinsic curvture is detectle y the inhitnts of surfce, s well s y outside oservers. Extrinsic curvture, on the other hnd, relies on the existence of higher dimensionl spce in which lower-dimensionl spce is emedded, nd cn e detected only y outside oservers. In the cse of generl reltivity, curvture is defined intrinsiclly, so tht one does not need to ssume tht the Universe is emedded in higher dimensionl spce. The mgnitude of intrinsic curvture is x 1 =+ δ B 2 x = e 1 x 1 = A e 2 C x 2 =+ δ D Figure 7.2: Smll closed loop in Riemnnin mnifold. The chnge δv of fourvector V prllel trnsported from A B C D A is used to determine the curvture of the mnifold. expressed in terms of the Riemnn curvture tensor, which will e derived next [3]. For this purpose we consider smll closed loop in continuous spce s illustrted in figure 7.2, know s Riemnnin mnifold. Prllel trnsport of four-vector V µ long this loop is then given y e β V ;β = 0, where e β denotes tngent vector see eqution 7.43 pointing long the coordinte line x β. We egin with the identity V B V A B A dv = B A V x 1 dx From the prllel trnsport eqution one otins for β = 1 the condition e 1 V ;1 = 0, from which if follows tht V ;1 = 0. Reclling 7.25, this mens tht V,1 = µ1v µ so tht we rrive for the chnge of V from A B t V B = V A initil +δ For the segment B C we egin with the identity V C V B C B dv = µ1 V µ x 2 = dx C B V x 2 dx

2 54 Chpter 7. The Einstein Eqution As efore, we use the prllel trnsport eqution to otin for β = 2 the condition e 2 V ;2 = 0 so tht now V ;2 = 0. Reclling 7.25, this mens tht V,2 = µ2 V µ so tht we rrive for the chnge of V from B C t V C = V B +δ Similrly, one otins for the segments C D nd D A nd V D = V C+ V A finl = V D+ +δ µ2 V µ x 1 =+δ dx δ µ1 V µ x 2 =+δ dx1, 7.52 µ2 V µ x 1 = dx The net chnge in V is otined from δv V A finl A A initil s δv = +δ + +δ µ2 V µ +δ x 1 = dx2 µ2 V µ x 1 =+δ dx2 µ1 V µ +δ x 2 =+δ dx1 µ1 V µ x 2 = dx Eqution7.54 cn e rewritten further y performing Tylor expnsions of the second nd third integrnds on the right-hnd-side of µ2 V µ x 1 =+δ µ2v µ + µ2 V µ x 1 = x 1 µ1 V µ x 2 =+δ µ1 V µ + µ1 V µ x 2 = x 2 x 1 = x 2 = δ, 7.55 δ Sustituting these two expressions into 7.54 leds to δv = +δ µ2 V µ x 1 δ δx 2 + x 1 = +δ µ1 V µ x 2 x 2 = δ δx Crrying out the prtil derivtives in 7.57 leds, fter some extr lger, to µ2 V µ = x 1 µ2,1 V µ + ν2 ν µ1 V µ, 7.58 µ1 V µ = x µ1,2v µ ν1 ν µ2v µ

3 7.7. Properties of the Riemnn curvture tensor 55 Upon sustituting 7.58 nd 7.59 into 7.57, one rrives for 7.57 t δv = δ δ µ1,2 µ2,1 + ν2 ν µ1 ν1 ν µ2 V µ The indices 1 nd 2 pper ecuse the pth in 7.2 ws chosen to go long coordinte lines x 1 nd x 2. Using generl coordinte lines x σ nd x λ, insted of x 1 nd x 2, one otins for 7.60 for the chnge in orienttion of n ritrry four-vector [3], δv = δ δ µσ,λ µλ,σ + νλ ν µσ νσ ν µλ V µ The expression in round rckets in 7.61 is the Riemnn curvture tensor, R µλσ µσ,λ µλ,σ + νλ ν µσ νσ ν µλ, 7.62 which is rnk-4 tensor chrcterized y 4 4 = 256 components. All these components vnish, R µλσ = 0, if the Riemnnin mnifold is flt. For the curved spce-time geometry of generl reltivity one thus hs R µλσ 0. For the ske of completeness, we lso write down the expression for ll four vector components of δv. In the ltter cse eqution 7.61 ecomes δ V = δv 0,δV 1,δV 2,δV 3 = δv e = δ δ R µλσ V µ e σ e λ e Properties of the Riemnn curvture tensor The Riemnn curvture tensor possesses severl intriguing properties [3, 7], which re very useful when using this tensor to derive equtions for tensors of lower rnk. Let us egin with R βµν = βν,µ βµ,ν + σµ σ βν σν σ βµ In locl inertil frme we hve µν = 0 so tht the Riemnn tensor ecomes R βµν = βν,µ βµ,ν The prtil derivtives of the Christoffel symols in 7.65 re given y βν,µ = 1 2 gδ g δβ,νµ +g δν,βµ g βν,δµ Sustituting 7.66 into 7.65 leds for the Riemnn tensor to R βµν = 1 2 gδ g δβ,νµ +g δν,βµ g βν,δµ g δβ,µν g δµ,βν +g βµ,δν = 1 2 gδ g δν,βµ g δµ,βν +g βµ,δν g βν,δµ 7.67

4 56 Chpter 7. The Einstein Eqution Multiplying 7.67 with g λ leds to the totlly covrint Riemnn tensor R βµν = 1 2 δ δ g δν,βµ g δµ,βν +g βµ,δν g βν,δµ = 1 2 gν,βµ g µ,βν +g βµ,ν g βν,µ Inspection of eqution 7.68 revels tht R βµν possesses the following symmetries, nd R βµν = R βµν = R βνµ = R µνβ 7.69 R βµν +R νβµ +R µνβ = Equtions 7.69 nd 7.70 re vlid tensor equtions, lthough derived in locl inertil frme, they re vlid in ll coordinte systems. We cn use the two identities 7.69 nd 7.70 to reduce the numer of independent components of R βµν from 256 to Einstein s generl reltivistic field eqution In this section we re looking for generl reltivistic version of the clssicl lw of grvity, 2 U = 4πǫ, where U denoted the grvittionl potentil generted y given mss energy density distriution, ǫ. In the frmework of generl reltivity, the source of curvture is mss energy, which enters in the energy momentum tensor derived in section 7.1. This suggests the replcement ǫ T µν ǫ,p in the clssicl eqution of grvity. The energy-momentum tensor is of rnk 2. We re therefore looking for rnk-2 tensor tht replces the clssicl grvittionl potentil in 2 U = 4πǫ. This tensor, known s the Einstein tensor G µν, must ccount, ccording to the foundtions of generl reltivity, for the curvture produced y given mss-energy distriution. It ppers thus nturl to go ck to the Riemnn curvture tensor nd try to uild rnk-2 tensor from it. In doing so, the fct tht energy-momentum is conserved loclly T µν ;ν = 0 is used s very importnt guiding principle since it demnds tht the tensor we re serching for oeys G µν ;ν = 0 s well. We egin y differentiting 7.68 with respect to x λ, which gives R βµν,λ = 1 2 gν,βµλ g µ,βνλ +g βµ,νλ g βν,µλ Next we mke use of the fct tht g β = g β nd tht prtil derivtives commute. We then otin from 7.71 the reltion R βµν,λ +R βλµ,ν +R βνλ,µ = 0, 7.72

5 7.8. Einstein s generl reltivistic field eqution 57 which is vlid in locl inertil frme. Therefore, in generl reference frme we will hve to del with the reltions R βµν;λ +R βλµ;ν +R βνλ;µ = 0, 7.73 which re known s the Binchi identities [3,24]. They re vlid in ny coordinte system. The Ricci tensor, R β, which is rnk-2 tensor, is otined from the Riemnn curvture tensor upon contrction of R µ µβ on the first nd third indices, R β = R µ µβ = R β Einstein s first guess for the field eqution of generl reltivity ws R µν T µν. This reltion is however not comptile with energy-momentum conservtion, T µν ;ν = 0, nd therefore does not qulify s cndidte for the field eqution of generl reltivity. Next, let us ssume tht the Ricci tensor is one of the uilding locks of which the Einstein tensor is mde of, nd tht R µν cn e supplemented such tht energymomentum is conserved. To find these supplementl terms, we introduce the Ricci sclr sclr curvture, R, given y simple contrction, R R β β = g β R β = g β R µ µβ = g β g µν R µνβ A rnk-2 tensor, which oeys G µν ;ν = 0, is then otined y multiplying the Binchi identities 7.73 with the metric tensor ccording to g β R βµν;λ +R βλµ;ν +R βνλ;µ = 0, 7.76 ndymkinguseofthefctthtthecovrintderivtiveofthemetrictensorvnishes, g β;µ = 0. ThisfollowsfromthereltionV ;β = g µ V µ ;β ndthelinerityofv ;β, which mens tht V ;β cn lso e written s V ;β = g µ V µ ;β = g µ;β V µ +g µ V µ ;β. Since g β;µ = g β ;µ we cn tke g β in nd out of covrint derivtives t will. Bering this in mind, we get from 7.76 R µ βµν;λ +R µ βλµ;ν +R µ βνλ;µ = Using R µ βλµ;ν = R µ βµλ;ν eqution 7.77 cn e written s R βν;λ R µ βµλ;ν +R µ βνλ;µ = 0, 7.78 which is known s the contrcted Binchi identity [3,24]. Using R µ βµλ;ν = R βλ;ν nd contrcting second time of the indices β nd ν gives from which it follows tht g βν R βν;λ R βλ;ν +R µ βνλ;µ = 0, 7.79 R ν ν;λ R ν λ;ν +R µν νλ;µ =

6 58 Chpter 7. The Einstein Eqution Eqution 7.80 cn e written s R ;λ 2R µ λ;µ = 0. Since R ;λ = g µ λr ;µ one rrives for 7.80 t R µ λ 1 2 gµ λ R = 0, 7.81 ;µ tensor whose covrint four-derivtive vnishes. Multipliction of 7.81 with g λν finlly leds to where R µν 1 2 gµν R G µν ;µ = 0, 7.82 ;µ G µν R µν 1 2 gµν R 7.83 is the Einstein tensor. Einstein s field eqution of generl reltivity is given y equting this tensor geometry to the energy momentum tensor mss-energy of mtter [2], G µν = 8πT µν The constnt of proportionlity turns out to e equl to 8π, which is required to otin the correct clssicl limit from 7.84 see section We stress gin tht since G µν ;ν = 0 one hs T µν ;ν = 0, which expresses the locl conservtion of energymomentum. We conclude this section y noting tht the covrint derivtive of the energy momentum tensor is given y T µν ;ν = T µν,ν + µ νγ Tγν + ν νγ Tµγ Einstein s field eqution 7.84 constitutes set of 10 non-liner, prtil differentil equtions for the components of the metric tensor g µν x, given the mtter sources T µν x. The condition G µν ;ν = 0 reduces the numer of independent prtil differentil equtions from 10 down to Vcuum Einstein eqution The vcuum Einstein eqution follows from 7.84 for T µν = 0, tht is, R µν 1 2 gµν R = For comprison, the Mxwell equtions re liner prtil differentil equtions i.e., liner in the electric nd mgnetic fields, nd chrge nd current distriutions.

7 7.10. The Cosmologicl constnt 59 Multiplying 7.86 with g µν nd summing over µ nd ν leds to g µν R µν δ µ µ R = Since µν g µν R µν = R nd µ δµ µ = 4, it follows from eqution 7.87 tht R = 0 for the vcuum cse. Einstein s vcuum field eqution thus follows from 7.86 s R µν = An exmple of vcuum solution to Einstein s field eqution is the Schwrzschild solution see eqution 9.71, which descries the grvittionl field outside of sphericl non-rotting mss distriution, such s non-rotting plnet, white dwrf, neutron str, or lck hole The Cosmologicl constnt Theoreticlly one is permitted to dd to the Einstein tensor 7.83 term like Λg µν, where Λ known s the cosmologicl constnt denotes constnt. Adding such n expressionisllowedsincethecovrintderivtiveofthemetrictensorvnishes, g µν ;ν = 0 so tht Λg µν does notruin thevlidity of thekey condition7.82. With this inmind, the Einstein tensor 7.83 cn therefore e generlized to G µν = R µν 1 2 gµν R+Λ g µν, 7.89 s ws done y Einstein in order to llow for sttic Universe. Such Universe would e neither expnding nor contrcting. In 1929, however, Edwin Hule discovered tht the Universe is in fct expnding, which led Einstein to ndon the concept of finite vlue for the cosmologicl constnt, clling the introduction of this term y him one-nd--hlf decdes erlier the iggest lunder he ever mde. The sitution chnged gin in recent yers, stimulted y the oservtions of distnt type I supernove, which originte from exploding cron-oxygen CO white dwrfs. The interprettion of the redshift dt from these explosions seems to indicte tht the Universe is not expnding t decelerting rte, s ws generlly ssumed, ut expnding t n ever incresing rte ccelerting Universe. This phenomenon is lso referred to s cosmic ccelertion [25]. The force which drives the cosmic ccelertion is ttriuted to the presence of drk energy in the Universe. The concept of drk energy cn e incorported in Einstein s field eqution y moving the term which contins the cosmologicl constnt in 7.89 to the right-hnd-side of 7.84, leding to R µν 1 2 gµν R = 8πT µν +8πT µν vcuum, 7.90 with T µν vcuum = 1 8π Λgµν the energy momentum tensor of the vcuum.