Covariant Energy-Momentum Conservation in General Relativity with Cosmological Constant

Transcription

1 Energy Momentum in GR with Cosmologicl Constnt Covrint Energy-Momentum Conservtion in Generl Reltivity with Cosmologicl Constnt by Philip E. Gibbs Abstrct A covrint formul for conserved currents of energy, momentum nd ngulr-momentum is derived from generl form of Noether s theorem pplied directly to the Einstein- Hilbert ction of clssicl generl reltivity. Energy conservtion in closed nd flt bigbng cosmologies including cosmic rdition nd drk energy is discussed s specil cses. Specil cre is tken to distinguish between kinemtic nd dynmic expressions. e-mil: philip.gibbs@pobox.com 7 Stnley Med, Bristol, BS 0EF, UK

2 Energy Momentum in GR with Cosmologicl Constnt Is Energy Conserved in Generl Reltivity? This my seem strnge question to be sking in 00 considering tht conservtion of energy in generl reltivity ws estblished by Einstein himself in 96, the yer fter the theory ws published []. Yet there remins sour tste to Einstein s pseudo-tensor formultion becuse of its lck of covrince. Modern ttempts to improve on it cn only relly be sid to be successful in specific cses such s symptoticlly flt spce-times nd different forms of pseudo-tensor hve only dded to the mbiguity. To exemplify the problem it is helpful to sk gin question which is often posed by students. Where does the energy of the Cosmic Bckground Rdition go when it is redshifted? Ech photon crries quntum of energy which slowly decreses s the universe expnds. Ignoring interctions with mtter which re irrelevnt on the cosmologicl scle, the totl number of photons remins constnt. The totl energy in the rdition must therefore decrese, where does it go? To void objections let us ssume tht the universe is closed so tht the totl number of photons is finite. The obvious nswer is tht the energy is trnsferred to the grvittionl field which hs n incresing (negtive) totl energy. But to justify this response we must be ble to provide generl formul for energy which cn be pplied to this cse. The pseudo-tensor is not sufficient becuse it is co-ordinte dependent nd it is not possible to define co-ordinte system over whole sptil hypersurfce of the closed universe if its topology is tht of three-sphere. How cn we then be sure tht we hve not missed lek of energy t the inevitble co-ordinte singulrity where the formultion breks down? It my well be possible to stop the gp with some mthemticl slog, but survey of most of the populr textbooks on generl reltivity does not turn up such n nlysis. Those which use modern differentil geometry shy wy from the pseudo-tensor ltogether nd often leve generl energy-momentum conservtion outside the scope of the book. This is surprising becuse there is n elementry covrint wy to tret energy nd momentum conservtion s I will show in this pper. Aside from the exmple I hve given, the problem is importnt becuse of other cses such s energy-momentum trnsferred by grvittionl wves nd the contribution of grvittionl self energy to mss which hs been verified in tests of the strong equivlence principle by lser rnging of the moon. An interesting twist to the problem of totl energy in closed universe is tht it comes out to be exctly zero rther thn n rbitrry constnt. This fct is often quoted when people sk where ll the energy cme from to set off the big bng, the nswer: There is none! Agin, it would be useful to hve tidy wy of proving this. Sometimes the null totl energy is tken s n indiction tht energy conservtion in generl reltivity is n empty sttement. This is not true becuse the energy is only zero dynmiclly, not kinemticlly. I.e. it is only zero when the grvittionl field equtions re stisfied. To see this we must provide the right formul for energy which cn be pplied to kinemtics. Another question similr to the one bout cosmic rdition hs become more relevnt since the discovery of drk energy which requires cosmologicl constnt in the field equtions. This provides constnt density of vcuum energy so s the universe expnds the totl drk energy in n expnding volume of spce increses in proportion to the volume. Where does this energy come from? The nswer is tht it comes from the grvittionl field which gives

3 Energy Momentum in GR with Cosmologicl Constnt negtive contribution of energy tht increses in mgnitude with the rte of expnsion. To understnd this fully we must include the cosmologicl constnt in our equtions. Energy nd Noether s Theorem Let us turn now to the grvittionl field equtions, R Rg g 8 GT By the Binchi Identities, the divergence of the left hnd side vnishes nd implies, T ; 0 If we were doing specil reltivity the divergence of the energy-momentum stress tensor would be dequte to estblish energy-momentum conservtion, but in generl reltivity it is not unless there is killing vector field K α, which genertes n isometry of the metric, x g K ( K K T ; K ; ; ) 0 0 When we hve such divergenceless vector current for energy flow, we cn derive conservtion becuse, v ; g g x nd therefore the current cn be integrted over closed spce-like hypersurfce of constnt time co-ordinte to give conserved Energy, E g 0 d 3 x When we hve no Killing field nd we try this with the energy-momentum tensor we get, g T g ; gt x gt x 0 The second term, which cnnot be integrted, signifies flow of energy between mtter nd the grvittionl field. A term representing contribution to the energy momentum tensor 3

4 Energy Momentum in GR with Cosmologicl Constnt from the grvittionl field is required. In deriving it Albert Einstein got some help from Emmy Noether who provided generl theorem relting conservtion lws to symmetries of Lgrngins []. Noether s theorem could be pplied to the Hilbert-Einstein ction for grvittion fter n integrtion by prts designed to remove second derivtives of the metric tensor from the ction. This leds to Einstein s non-covrint energy-momentum pseudotensor for which, x ( gt t ) 0 A concise exposition of the derivtion cn be found in Dirc s book on Generl Reltivity [3]. After further mnipultions which re described in the book of Lndu nd Lifshitz [4], version of the pseudo-tensor which is symmetric in the two indices cn be found. This symmetry is importnt if conservtion of ngulr momentum is lso to be demonstrted, but it mkes the nlysis even more complicted nd less well motivted since Noether s theorem is not used directly. Mny other forms of the pseudo-tensor re possible such s the symmetric one given by Weinberg [5]. Recently, Bk, Cngemi nd ckiw derived symmetric pseudotensor using Noether s theorem nd provided useful comprison with other pseudo tensors [6]. Their result is still not covrint nd is presented in form which is divergenceless even kinemticlly. The full formul for the energy which shows the contribution to energy from different sources, nd which is conserved dynmiclly but not kineticlly, is often simplified using the field equtions. With so mny different versions of energy-momentum conservtion tht do not give exctly the sme nswer, one might sk wht energy relly is. The different formultions ll differ by quntity which is kinemticlly conserved (perhps even before the simplifiction using the field equtions) nd which is symptoticlly negligible in the wek field pproximtion to grvittionl wves. Even so, we might hve expected unique concept of energy. The correct nswer to the question Wht is energy? is tht the correct energy is tht which is given by Noether s theorem. This is still not unique since the Lgrngin is only unique up to term which is perfect divergence. This feture is wht is used to get rid of the terms with second derivtives in the metric tensor before deriving the energy eqution, but it lso destroys covrince. Even with covrince other forms of the ction re known but t lest it might be worthwhile hving the correct expression for energy for the trditionl Einstein- Hilbert ction. S R ( g L 6G M ) d 4 x This cn be done using form of Noether s theorem which works with second derivtives in the Lgrngin. 4

5 Energy Momentum in GR with Cosmologicl Constnt Consider generl ction of the form, S L (,,,, ) 4 d x re ll the components of the field vribles including the metric tensor. Applying vrition principle, the Euler-Lgrnge equtions re, L L L,, 0 A continuous symmetry of the ction which might include co-ordinte chnges s well s field trnsformtions would hve generl infinitesiml form, nd invrince implies tht, x k L L L L k L,, ( ),, Combining results we finlly derive the conserved Noether current, I I L L L k L v,,,,, g ; 0 I 0 5

6 Energy Momentum in GR with Cosmologicl Constnt The Covrint Formultion The next step is to pply this form of Noether s theorem directly to the Einstein-Hilbert ction. The ction is invrint under diffeomorphisms (co-ordinte trnsformtions). Infinitesiml diffeomorphisms re generted by x k g g ( k k ) i ; ; i ( k ) re ll the field components other thn the metric tensor. The nswer will not tke the form of n energy-momentum tensor, it will be contrvrint vector current which depends on the contrvrint trnsport vector field k α nd by Noether s theorem it will be conserved for ny choice of k α. This is nturl consequence of the fct tht in generl reltivity the symmetry is diffeomorphism invrince rther thn globl Lorentz invrince. ψ i The derivtion is quite long but will nevertheless by given in some detil for the grvittionl prt. Our strting point is the Lgrngin density, L g ( R ) 6 G L M It is ssumed tht the mtter Lgrngin contins no higher thn first covrint derivtives in field vribles, nd trnsforms s sclr-density under diffeomorphisms In tht cse the mtter field contribution to Noether current is given by, G M M The Riemnn curvture tensor is given by, R R R k T ( ) ( ) ( ) R ( g g g g ),,,, ( ) R g g ( g g g ) R R g g,,, 6

7 Energy Momentum in GR with Cosmologicl Constnt The grvittionl contribution to the current will be split up ccordingly. The fctor g psses outside the prtil derivtives but n extr term must be included when it psses through the co-ordinte derivtives. G 6G R j g g k (), ; ( k nd we cn clculte, ( k ( R ) j ; R g ( ), ) ) R g (), g g g g g ( ) R g (), ( g ), If we knew tht the nswer must be covrint, then we could use the fct tht, in locl inertil frme, the first derivtives of the metric tensor vnish t single point. This mkes ll the difficult terms in the current vnish nd the clcultion is esy to complete. j g g g ( g ) ( )( g ) ; However, covrince hs not yet been proven, so the terms should be worked out t length to be sure of the result. R R g g ( ) ( ), R g g g g, g ( ) g ( ) g ( ) j { g ( ) g } g { g ( ), g g g }( g g ) { g g g g } g, ; 7

8 Energy Momentum in GR with Cosmologicl Constnt The current cn be expressed in terms of the trnsport vector field, G 6G ( k R k k ; k ; k ; ) This is the finl covrint expression for the contribution to energy-momentum current from the grvittionl field. The dependency on the contrvrint trnsport vector k α is left explicit. Any ttempt to remove it to derive stress tensor will destroy covrince nd mke the result co-ordinte dependent. The generlity of the expression would be lost. The term with the curvture sclr depends on second derivtives of the metric field in ddition to first derivtives. This is unusul for n expression of energy or momentum. The second derivtives could be removed by dding term which is kinemticlly divergence free but only t the cost of covrince. The choice of trnsport vector field is used to distinguish between currents of energy, momentum, ngulr-momentum, etc. In the cse where it is everywhere time-like the current is interpreted s flow of energy. Energy nd momentum re therefore not unique but this is no surprise or obstcle. For given co-ordinte system the energy nd momentum which re conjugte to time nd spce co-ordintes cn be found but they do not form tensor. Using the identity, k ; k ; k R nd including the contribution from the mtter fields we get 6G ( k R k k ; k ; k R ) If we now pply the grvittionl field equtions the expression cn be simplified drmticlly. 6 G ( k ; k ; ) This cn lso be written in form without explicitly covrint derivtives, 6 G g g( k k ) g g,, The divergence of this expression is identiclly zero but it is only correct dynmiclly. The full expressions we derived before simplifying with the field equtions re kinemtic formule for the energy-momentum currents nd their divergence is not identiclly zero, but it is dynmiclly zero. They show directly the contributions to energy nd momentum from seprte prts of the Lgrngin., k T 8

9 Energy Momentum in GR with Cosmologicl Constnt These finl dynmic expressions re the Komr superpotentil [7]. When pplied to n symptoticlly flt spce-time with pproprite use of boundry conditions this leds on to the Geroch-Winicour linkge formultion [8], the Bondi Energy nd the ADM Energy [9]. Energy in Closed Cosmology We re now in position to consider the cse of energy in closed cosmology. The covrint currents we hve constructed re dynmiclly divergenceless. When integrted over closed spce-like hypersurfce to give the totl energy or momentum of the universe, the result must be zero becuse there is no boundry. As we noted in the introduction, this is hrd to prove rigorously with pseudo-tensor formultion, but with covrint currents it is simple consequence of Guss theorem. In closed universe with 3-sphere sptil topology there is no cler distinction globlly between momentum nd ngulr-momentum. They re consequences of rottions of the sptil 3-sphere bout different points. The Minkowski model of energy-momentum 4-vectors nd ngulr-momentum tensor is therefore not relevnt to closed sphericl sptil topology. For energy we must be creful bout our choice of the trnsport vector field k α. If its covrint form k α hd components which were constnt in some co-ordinte system then the current would be zero everywhere. Tht would not necessrily be incorrect but it should normlly be chosen so tht it increments the time co-ordinte uniformly. It follows tht the energy current depends criticlly on the choice of time co-ordinte s is to be expected. In homogeneous Friedmnn-Robertson-Wlker cosmology there is nturl choice of time co-ordinte, tht of the co-moving co-ordinte system. In more relistic loclly inhomogeneous cosmology the mtter is not so simple. We could define cosmologicl time t ech spce-time event to be the mximum proper-time integrted long ll time-like pths from the big-bng singulrity to the event. This is well defined provided the cosmology hs no closed time-like curves nd cn be sequentilly folited into spce-like surfces. It is esy to show tht the surfces of equl cosmologicl time re continuous spce-like surfces, except where they meet singulrities which hve finite ge. Even without singulrities such surfces re not smooth becuse co-ordinte systems which incorporte cosmologicl time re synchronous reference frmes. It is known tht they tend to develop co-ordinte singulrities in relistic spce-times (see [4]). In generl, the energy currents cn be defined reltive to smooth spce-like folition of spce-time nd time-like trnsport vector field. It is beyond the scope of this rticle to estblish when they exist. Blck-hole singulrities do not ctully cuse brekdown of energy-momentum conservtion becuse it is possible to pss spce-like surfce through blck hole which voids the singulrities. This cn be done even t lte times. The surfce would hve to bend bck in time t nerly the speed of light but cn nrrowly void becoming time-like. This cn be seen in Crter-Penrose digrm of blck-hole. Fortuntely the totl vlue of energy momentum nd ngulr momentum for n isolted object depends dynmiclly only on the choice of trnsport vector t the boundry of the surfce nd not on the surfce or choice of trnsport vector internlly. This is not true kinemticlly. Becuse of the difficulty of choosing time co-ordinte which cn be pplied universlly nd which voids ll singulrities, the interprettion of totl energy in closed universe is not perfectly cler. This is not too surprising or worrying since energy nd momentum re usully 9

10 Energy Momentum in GR with Cosmologicl Constnt considered to be reltive, nd the whole universe cnnot be considered reltive to nything else. Nevertheless, there re covrintly conserved energy nd momentum currents whose totl integrtes to zero. This is not trivil result since it is not true kinemticlly if we use the correct kinemtic definition of the currents which is derived directly from Noether s theorem without using the field equtions to simplify. Remember tht it is only this kinemtic form which shows explicitly the contribution to the currents from individul fields. Energy Conservtion in Flt Cosmology with drk Energy According to the most recent precision mesurements the observble universe is very close to flt FLRW stndrd cosmology with dominnt cosmologicl constnt t the present epoch. We cn ssume tht mtter including drk mtter is pproximted by cold mssive gs of prticles. Cosmic rdition is in the form of smooth photon field tht decreses in energy s it is redshifted in line with the expnsion of spce. The metric for this cosmology is given by ds ( t) ( dx dy dz ) c dt (t) is n expnsion fctor normlised to be t the current epoch. The contribution from the Komr Superpotentil will be zero s cn be seen by direct computtion or by observing tht the only conserved current density field tht is homogenous nd isotropic is zero. The remining terms in the time component energy current density cn be clculted nd integrted by multiplying by volume of spce V = (t) 3 The eqution tht results is equivlent to the well known Friedmnn eqution E Mc 8 G 3 3 8G d dt 0 E is the totl energy in the expnding spce of volume V M is the mss of cold mtter in the sme volume. P is the energy density of cosmic rdition t the present epoch. The first term is the constnt mount of energy of cold drk mtter in the expnding volume of spce. The second term is the decresing energy in the cosmic bckground rdition. The third term is the drk energy tht increses in proportion to the expnding volume. The finl term in this expression is the energy in the grvittionl field in the expnding volume V. It is negtive nd increses in mgnitude s the expnsion fctor nd rte of expnsion increse. The rte of expnsion will be such s to mke the grvittionl energy negte ll the other contributions to the energy by the grvittionl field equtions. This justifies the conclusion tht energy loss from cosmic rdition nd energy gin from drk energy is blnced by chnges in the energy of the grvittionl field nd tht this is not trivil result becuse it depends on the grvittionl field equtions to be true. 0

11 Energy Momentum in GR with Cosmologicl Constnt References [] A. Einstein, Preussische Akdemie der Wissenschften, (96), p. [] E. Noether, Goett. Nchr Ges Wiss.,, (98), pp37, 35 [3] P.A.M. Dirc, Generl Theory of Reltivity, ohn Wiley & Sons (975) [4] L.D. Lndu nd E.M. Lifshitz, The Clssicl Theory of Fields, Pergmon Press (95) [5] S. Weinberg, Grvittion nd Cosmology, ohn Wiley & Sons (97) [6] D. Bk, D. Cngemi, R. ckiw, Energy Momentum Conservtion in Generl Reltivity, hep-th/93005 (993), Phys Rev D49, (994), Errtum-ibid D5, 3753 (995) [7] A. Komr, Covrint Conservtion Lws in Generl Reltivity, Phys Rev, (959) [8] R.P. Geroch,. Winicour Linkges in Generl Reltivity,. Mth Phys,, (98) [9] R. Arnowitt, S. Deser, C.W. Misner, The Dynmics of Generl Reltivity in Grvittion: An Introduction to Current Reserch ed L. Witten, Wiley (96)