Causal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim

Transcription

1 Causal Damonds M. Aghl, L. Bombell, B. Plgrm Introducton The correcton to volume of a causal nterval due to curvature of spacetme has been done by Myrhem [] and recently by Gbbons & Solodukhn [] and later by S. Khetrapal and S. Surya [3]. Myrhem has calculated the effects of curvature up to frst order on the volume of a causal nterval for 4 dmensons, however the calculatons for the boundary parts are not clear. Gbbons et al. found a general expresson for the correctons n d dmensons assumng the soluton by Myrhem to be the soluton to 4 dmensons, and then fndng the effect of curvature on two dfferent spacetmes of dmenson d and solve for the two unknown coeffcents of curvature and the result that they obtaned s n agreement wth the prevous result. However the spacetmes that they chose for ths purpose are all sphercally symmetrc and not general. Khetrapal and Surya have the calculaton of boundary term explctly wrtten, however t seems to be unnecessarly complcated. In ths paper we are attemptng to fnd the effect of curvature assumng a totally arbtrary d dmensonal spacetme and fnd the dependence on curvature. We wll see the results are n agreement wth the prevous papers. Calclulatons The lght cone n a curved spacetme s specfed such that the volume s defned by the metrc and the shape of the cone s defned by the connectons on the manfold, whch means that the connectons and the metrc are compatble. The volume occuped by past lght cone of pont q and future lght cone of pnt p n d dmensons s V g ( A(p,q where g s the determnant of metrc g µν. In small enough causal damond we know that g µν η µν + x α α g µν + xα x β α β g µν + ( where the thrd term can be wrtten n terms of Remann curvature tensor usng Remann normal coordnates xα x β α β g µν 3 xα x β R µανβ (3 wth Remann curvature tensor evaluated at the orgn of coordnates. Accordng to Sylvester s determnant theorem, If we have an nvertble matrx X, m n matrx A and an n m matrx B we have det (I m + AB det (I n + BA (4 whch can be derved usng the fact that ( ( ( ( Im m O m n Im m + A m n B n m A m n Im m O m n Im m A m n (5 B n m I n n O n m I n n B n m I n n O n m I n n + B n m A m n where O s zero matrx. The determnant of left hand sde s the multplcaton of determnant of each matrx and snce the frst and the thrd term are trangular we can fnd the overall determnant very smple and therefore (4 s proved. Now usng Sylvester s determnant theorem we can wrte det (X + AB det (X det ( I + ( X A B det (X det ( I + BX A. (6

2 Accordng to Lebnz formula, the determnant of any matrx s a polynomal from C n n to C such that t s everywhere dfferentable, whch means we can expand the determnant usng Jacob s formula ( d (det (A da det (A A, (7 dα dα therefore for ɛ n general case we have and for the specal case of A I we have det (A + ɛx det (A + tr ( A X ɛ + O ( ɛ, (8 det (I + ɛx + tr (X ɛ + O ( ɛ. (9 Based on these we can wrte det (η µν 3 xα x β R µανβ det (η det (I 3 Iη x α x β R µανβ ( 3 ηµν x α x β R µανβ + 3 xα x β R αβ. (0 Ths leads to g 6 xα x β R αβ. ( The contrbutons of curvature to the volume of causal set comes from two parts. One s just the determnant of metrc and the second s comng from the boundary of causal damonds V g ( 6 xα x β R αβ +, ( A(p,q where I I 0. The frst term can be calculated farly easly ( 6 xα x β R αβ V flat 6 R αβ x α x β. (3 Snce the boundary of ntegral s symmetrc, the odd terms wll vansh and we are remaned wth ( 6 xα x β R αβ V flat R ( x 6 6 R 00 t. (4 usng the fact that I 0 ( x d I 0 and then R 00 + d ( 6 xα x β R αβ V flat ( 6 R 00 t d r I R R, we can derve 6 (d R r. V flat V flatτ 48 (d + R dv flat τ 00 R, (5 48 (d + (d + where I 0 + denotes the upper half cone, and obvously the total contrbuton s twce the effect on the upper cone. There exsts two ways to calculate the boundary term, one s to use the perturbatons of congruence of null geodescs, and the other one s to use the null surface n general and then mpose the condtons where the latter s not very useful because unless there s sphercal symmetry, t gves one equaton and d dervatves. Usng the frst method, we wrte down the geodesc equatons d x µ + dx α dx β Γµ αβ For an unperturbed Mnkowsk spacetme Γ µ αβ 0, so we have ( x µ (λ A µ λ + B µ, A µ A 0. (6 B µ 0 (7

3 The A µ and B µ are defned usng the boundary condtons. Snce we are workng wth the upper lght cone, we choose A 0 so that the affne parameter s always postve, we choose λ 0 to be the tp of the cone, ths way we wll have B 0 τ/ and B 0. If we use the sphercal coordnate r d 0 (A and replace the affne d 0 (A (τ/ t whch mples parameter wth approprate functon of tme derved from (7 we get r d ( 0 A. Also we have x r A P d A whch mples that n fact A s only the angular dependence 0 (A whch here we denote wth Ω. For the perturbed case, Chrstoffel symbols n Remann normal coordnates become Γ µ αβ 3 ( R µ βɛα + Rµ ɛαβ x ɛ. (8 Usng ths equaton and symmetry propertes of Remann curvature tensor, the geodesc equaton reduces to d x µ 3 Rµ dxα dx β βαɛxɛ 0. (9 Snce we want to fnd the perturbatons to lnear order we should use the zeroth order soluton to geodesc equaton for the terms that already have a curvature term and that leads to d x µ 3 Rµ βαɛ Bɛ A α A β 0 x µ (λ c µ λ + A µ λ + B µ, (0 c µ 3 Rµ βαɛ Bɛ A α A β. ( where A µ and B µ are agan to be determned by boundary condtons. Ths equatons are for general geodescs, and we have not yet assumed that whether they are tmlke, spacelke or null. We can wrte the coeffcents n the form of taylor expanson n powers of curvature and snce here we are only takng the lnear contrbutons we can wrte A µ A µ + õ (R + O ( R B µ B µ + B µ (R + O ( R where õ & B µ represent the frst order perturbaton and A µ & B µ satsfy lm R 0 A µ A µ and lm R 0 B µ B µ. To fnd the boundary condtons we need to fnd the change n the heght of the Alexandrov set by expandng n powers of Remann curvature tensor dx h g µ µν dx ν ( dx η µ dx ν µν 3 R µανβx α x β dxµ dx ν whch we can smplfy usng the geodesc equaton that we derved above for a geodesc that starts from the orgn and extends to the tp of the cone ( h (η µν ξ µ ξ ν + η µν ξµ ξ ν η ξµ µν ξ µ ( ξ 0 τ (3 where here we used ξ µ for the coeffcent of tmelke geodesc to avod confuson. Gettng back to the geodesc equatons we can wrte ( d r ( x λ + A à λ + c A λ 3 (4 0 t c 0 λ + A 0 λ + B 0. (5 We can set λ 0 to be at the tp of the cone whch we found ts heght and we fnd B 0 ( ξ 0 τ, and snce gong from Mnkowsk spacetme to a curved spacetme s possble through a map, we can set B 0. Now we can solve equaton (5 for λ up to the frst order n curvature λ ( B 0 t + t c 0 + t Ã0. (6 3

4 Ths equaton can be used n (4 to fnd radal coordnate n terms of tme r (c 0 + c A (Ã0 t + + A Ã t + ( B 0 t. (7 The fst term of ths equaton can be smplfed usng ( c µ A µ 3 R µαβɛa α A β B ɛ A µ 0 c 0 A 0 c A c 0 c A, (8 where we have used the ant symmetrc propertes of Remann curvature tensor, and ths cancels the frst term n (7. The second term can be derved, knowng that A µ dxµ λ0 and should satsfy the null condton up to the frst order n the curvature g µν A µ A ν 0 (η µν 3 R µανβx α x β A µ A ν 0 η µν A µ Ã ν 6 R µανβa µ A ν B α B β Ã0 + A Ã 6 R µανβa µ A ν B α B β. (9 The same should hold for ξ µ wth the dfference that ξ µ s tmelke and should satsfy the tmelke geodesc normalzaton condton g µν ξ µ ξ µ (η µν 3 R µανβx α x β ξ µ ξ ν η µν ξ µ ξν (R 6 R µανβx α x β ξ µ ξ ν ξ 0 0, (30 where n the last lne we have used the Mnkowsk soluton to ξ µ and x µ for the tmelke geodesc that connects the center of Alexandrov set to the tp of the cone. Usng (8, (9 and (30 n (7 we can smplfy the equaton to ( (τ ( ( r 6 R µανβa µ A ν B α B β t + t 6 R 0j0A A j τ (τ t + t. (3 Now everythng s ready to do the ntegraton I I 0 d d 6 R0j0A A j t+ t x dt dω drr d, I 0 whch after ntegraton over r and expanson to the frst order n curvature becomes d d x dt dω t d ( + d ( d 6 R 0j0Ω Ω j τ. (3 The frst term n parentheses gves the volume of flat Alexandrov set, and the second term can be smplfed by symmetry, knowng that ntegraton s over a unt sphere We know that Ω A x Mnk. r Mnk. the fact that d V flat + d R 00 3 dt x Mnk. τ/t and therefore dω ( Ω d R 00 R 00 we can conclude that c ( d τ 3d dω ( Ω. (33 d ( dω wth Ω. Fnally usng d+ R00 V flat 4 R 00τ, (34 where c d s the volume of a flat d dmensonal unt ball. The overall result of the calculaton usng (5 can be wrtten as d V V flat ( + 4 (d + R 00τ d 4 (d + (d + Rτ, (35 whch s n agreement wth the results of Myrhem and Gbbons et al. 4

5 References [] J. Myrhem, Statstcal Geometry, CERN preprnt TH.5 38-CERN (978. [] G. W. Gbbons and N. Solodukhn, The Geometry of Small Causal Damonds, arxv:hep-th/ v3 (05. [3] S. Khetrapal and S. Surya, Boundary Term Contrbuton to the Volume of a Small Causal Damond, arxv:.069v (0. 5