Irreducible forms for the metric variations of the action terms of sixth-order gravity and approximated stress-energy tensor
aa r X i v : . [ g r- q c ] A ug Irreducible forms for the metric variations of theaction terms of sixth-order gravity andapproximated stress-energy tensor
Yves D´ecanini and Antoine Folacci
UMR CNRS 6134 SPE,Equipe Physique Semi-Classique (et) de la Mati`ere Condens´ee,Universit´e de Corse, Facult´e des Sciences, Boˆıte Postale 52, 20250 Corte, FranceE-mail: [email protected] , [email protected] Abstract.
We provide irreducible expressions for the metric variations of thegravitational action terms constructed from the 17 curvature invariants of order sixin derivatives of the metric tensor i.e. from the geometrical terms appearing in thediagonal heat-kernel or Gilkey-DeWitt coefficient a . We then express, for a fourdimensional spacetime, the approximated stress-energy tensor constructed fromthe renormalized DeWitt-Schwinger effective action associated with a massivescalar field. We also construct, for higher dimensional spacetimes, the infinitecounterterms of order six in derivatives of the metric tensor appearing in theleft hand side of Einstein equations as well as the contribution associated withthe cubic Lovelock gravitational action. In an appendix, we provide a list ofgeometrical relations we have used and which are more generally helpful forcalculations in two-loop quantum gravity in a four dimensional background orfor calculations in one-loop quantum gravity in higher dimensional background.We also obtain the approximated stress-energy tensors associated with a massivespinor field and a massive vector field propagating in a four dimensionalbackground.PACS numbers: 04.62.+v
1. Introduction
Calculations which must be carried out in field theories defined on curved spacetimesare in general highly non-trivial. This is more particularly true in the contextof renormalization of quantum fields and of quantum gravity but, even at theclassical level, analogous difficulties can appear, for example, in the context of theradiation reaction problem of gravitational wave theory. This is mainly due to thesystematic occurrence, in these calculations, of Riemann polynomials (i.e., polynomialsformed from the Riemann tensor by covariant differentiation, multiplication andcontraction) whose complexity, degree and number rapidly increase with the precisionof the approximations needed or with the dimension of the gravitational backgroundconsidered. The results of these calculations may be moreover darkened because of thenon-uniqueness of their final forms what moreover complicates the comparison betweenthe works completed by different authors: Indeed, the symmetries of the Riemanntensor as well as Bianchi identities are not used in a uniform manner and monomialsformed from the Riemann tensor may be linearly dependent in non-trivial ways. In a etric variations in higher-derivative gravity and stress-energy tensor a and therefore in the unrenormalized or renormalized DeWitt-Schwingereffective action [7, 8, 9, 10, 11, 12] associated with massive fields propagating in curvedspacetime. As a consequence, our results will permit us to unambiguously express,in Section 3, for a four dimensional spacetime, the approximated stress-energy tensorconstructed from the renormalized effective action associated with a massive scalarfield as well as to construct, for an arbitrary six dimensional spacetime, the infinitecounterterms of order six in derivatives of the metric tensor which appear in theleft hand side of the bare Einstein equations. In a brief conclusion (Section 4), weshall consider some possible immediate prolongations of our present work. Finally,in an appendix, we shall provide a list of geometrical relations we have used (theserelations are more generally helpful for calculations in two-loop quantum gravity in afour dimensional background or for calculations in one-loop quantum gravity in higherdimensional background) and we shall briefly extend the result obtained in Section3 by providing, in the large mass limit, simplified expressions for the approximatedstress-energy tensors associated with a massive spinor field and a massive vector fieldpropagating in a four dimensional background.All our results are obtained by using the geometrical conventions of Hawking andEllis [13] as far as the definitions of the scalar curvature R , the Ricci tensor R pq andthe Riemann tensor R pqrs are concerned and the commutation of covariant derivativesin the form T p...q... ; sr − T p...q... ; rs = + R ptrs T t...q... + . . . − R tqrs T p...t... − . . . (1.1)We use furthermore the FKWC-notations R rs,q and R r { λ ... } : R rs,q denotes the spaceof Riemann polynomials of rank r (number of free indices), order s (number ofdifferentiation of the metric tensor) and degree q (number of factors ∇ q R ...... ) while R r { λ ... } denotes the space of Riemann polynomials of rank r spanned by contractionsof products of the type ∇ λ R ...... . We refer to the FKWC-article [1] for more precisionson these notations and rigor on the subject. etric variations in higher-derivative gravity and stress-energy tensor
2. Irreducible forms for the metric variations of the action termsconstructed from 17 curvature invariants of order six in derivatives of themetric tensor and for the cubic Lovelock tensor
In this section, we consider spacetime as an arbitrary D -dimensional pseudo-Riemannian manifold ( M , g ab ) and we assume that its dimension D is sufficientlyhigh in order to avoid any degeneracy of the Riemann tensor as well as anytopological constraint associated with Euler-Gauss-Bonnet-Lovelock densities andnumbers. Later, we shall drop this strong hypothesis. The most general expression of a gravitational lagrangian of order six in derivativesof the metric tensor is obtained by expanding it on the FKWC-basis for Riemannpolynomials of order six and rank zero. This basis consists of the 17 following elements[1]: R , : (cid:3)(cid:3) R R { , } : R (cid:3) R R ; pq R pq R pq (cid:3) R pq R pq ; rs R prqs R { , } : R ; p R ; p R pq ; r R pq ; r R pq ; r R pr ; q R pqrs ; t R pqrs ; t R , : R RR pq R pq R pq R pr R qr R pq R rs R prqs RR pqrs R pqrs R pq R prst R qrst R pqrs R pquv R rsuv R prqs R p qu v R rusv . (2.1)This basis is a natural one and is often used on this form in literature. However,it should be noted that certain authors prefer to use the scalar monomial R pqrs (cid:3) R pqrs instead of the scalar monomial R pq ; rs R prqs . This is the case of Gilkey in Refs. [5, 6].This choice is only a matter of taste because these two terms appear equally inthe calculations carried out in field theories defined on curved spacetimes and theelimination of one of them can be achieved by using the identity (5.4). It should bealso noted that in these calculations, other Riemann monomials of order six and rankzero such as R pq R pr ; qr , R prqs R pquv R uvrs , R rspq R puqv R rusv and R prqs R puqv R r sv u aresystematically encountered. They can be eliminated by using the geometrical identities(5.3) and (5.5 a )-(5.5 c ) which permit us to expand them on the FKWC-basis (2.1). The functional derivatives with respect to the metric tensor of an action termconstructed from a gravitational lagrangian of order six is obtained by expandingit on the FKWC-basis for Riemann polynomials of order six and rank two. This basisconsists of the 42 following elements [1]: R , : ( (cid:3) R ) ; ab (cid:3)(cid:3) R ab R { , } : RR ; ab ( (cid:3) R ) R ab R ; pa R pb R (cid:3) R ab R pa (cid:3) R pb R pq R pq ; ab R pq R pa ; bq R pq R ab ; pq R ; pq R paqb ( (cid:3) R pq ) R paqb R pq ; ra R rqpb R p ; qra R pqrb R pqrs R pqrs ; ab R { , } : R ; a R ; b R ; p R pa ; b R ; p R ; pab R pq ; a R pq ; b R pq ; a R bp ; q R pa ; q R ; qpb R pa ; q R qb ; p R pq ; r R rqpa ; b R pq ; r R paqb ; r etric variations in higher-derivative gravity and stress-energy tensor R pqrs ; a R pqrs ; b R pqra ; s R ; spqrb R , : R R ab RR pa R pb R pq R pq R ab R pq R pa R qb RR pq R paqb R pr R qr R paqb R pq R ra R rqpb RR pqra R pqrb R ab R pqrs R pqrs R pa R qrsp R qrsb R pq R rspa R rsqb R pq R prqs R rasb R pq R prsa R qrsb R pqrs R pqta R trs b R prqs R tpqa R trsb R pqrs R pqrt R s ta b . (2.2)Here, it should be noted that we have slightly modified the FKWC-basis of Ref. [1]:i) We have replaced the term R pq R pa ; qb proposed in Ref. [1] by the term R pq R pa ; bq . We think it is more interesting to work with the latter which directlyreduces to an element of the scalar basis (2.1) by contraction on the free indices a and b . In fact, these two terms are linked by the geometrical identity (5.7) so it is easy toreturn to the original FKWC-basis.ii) We have replaced the terms R pq ; ra R prqb , R pq ; r R prqa ; b and R pq R ra R prqb proposed in Ref. [1] by their opposites respectively given by R pq ; ra R rqpb , R pq ; r R rqpa ; b and R pq R ra R rqpb . This choice has been done for obvious mnemotechnic reasons.In the calculations carried out in field theories defined on curved spacetimes aswell as in quantum gravity and more particularly in the calculations we shall achievein the present section, other Riemann monomials of order six and rank two which arenot in the FKWC-basis (2.2) are systematically encountered. They can be eliminated(i.e., expanded on the FKWC-basis (2.2)) more or less trivially from the geometricalidentities (5.6),(5.7),(5.8),(5.9), (5.10 a )-(5.10 a ),(5.11),(5.12),(5.13),(5.14) and (5.15 a )-(5.15 e ) we have obtained and displayed in the Appendix. From the 17 elements of the FKWC-basis (2.1), we can construct 17 action termswhich permit us to express the most general gravitational action constructed from alagrangian of order six in derivatives of the metric tensor. For a general spacetime( M , g ab ) of arbitrary dimension D (with D sufficiently high), these 17 action terms areindependent but if we assume that the considered spacetime has no boundary (i.e., if ∂ M = ∅ ), some of these terms are linked together due to Stokes’s theorem. From nowon, we shall work under this hypothesis and therefore consider that, for any vectorfield V , we have Z M d D x √− g V p ; p = 0 . (2.3)By using integration by part, the contracted Bianchi identities (5.2 b ) and (5.2 c )as well as the geometrical identities (5.3) and (5.4) and Stokes’s theorem in the form(2.3), it is easy to prove that only the ten action terms Z M d D x √− g R (cid:3) R (2.4) Z M d D x √− g R pq (cid:3) R pq (2.5) Z M d D x √− g R (2.6) etric variations in higher-derivative gravity and stress-energy tensor Z M d D x √− g RR pq R pq (2.7) Z M d D x √− g R pq R pr R qr (2.8) Z M d D x √− g R pq R rs R prqs (2.9) Z M d D x √− g RR pqrs R pqrs (2.10) Z M d D x √− g R pq R prst R qrst (2.11) Z M d D x √− g R pqrs R pquv R rsuv (2.12) Z M d D x √− g R prqs R p qu v R rusv (2.13)are independent and that one of the remaining action terms vanishes while the sixother ones can be expressed in terms of some of the previous ones: Z M d D x √− g (cid:3)(cid:3) R = 0 (2.14) Z M d D x √− g R ; pq R pq = Z M d D x √− g (cid:18) R (cid:3) R (cid:19) (2.15) Z M d D x √− g R pq ; rs R prqs = Z M d D x √− g (cid:18) − R (cid:3) R + R pq (cid:3) R pq − R pq R pr R qr + R pq R rs R prqs (cid:19) (2.16) Z M d D x √− g R ; p R ; p = Z M d D x √− g ( − R (cid:3) R ) (2.17) Z M d D x √− g R pq ; r R pq ; r = Z M d D x √− g ( − R pq (cid:3) R pq ) (2.18) Z M d D x √− g R pq ; r R pr ; q = Z M d D x √− g (cid:18) − R (cid:3) R − R pq R pr R qr + R pq R rs R prqs (cid:19) (2.19) Z M d D x √− g R pqrs ; t R pqrs ; t = Z M d D x √− g (cid:0) R (cid:3) R − R pq (cid:3) R pq + 4 R pq R pr R qr − R pq R rs R prqs − R pq R prst R qrst + R pqrs R pquv R rsuv + 4 R prqs R p qu v R rusv (cid:1) . (2.20) The functional derivatives with respect to the metric tensor of the ten independentaction terms (2.4)-(2.13) can be obtained by using the behaviour of these action terms etric variations in higher-derivative gravity and stress-energy tensor g µν → g µν + δg µν (2.21)of the metric tensor. From the corresponding variations of the geometrical tensors √− g , R , (cid:3) R , R ab , (cid:3) R ab and R abcd given in Subsection (5.2) of the Appendix (see(5.18 a )-(5.18 h )) or in Ref. [14] and after tedious calculations, we have expanded thefunctional derivatives of the ten action terms (2.4)-(2.13) on the FKWC-basis (2.1)and (2.2) and we have obtained the following results: H { , } (1) ab ≡ √− g δδg ab Z M d D x √− g R (cid:3) R = 2 ( (cid:3) R ) ; ab − (cid:3) R ) R ab + R ; a R ; b + g ab [ − (cid:3)(cid:3) R − (1 / R ; p R ; p ] , (2.22) H { , } (3) ab ≡ √− g δδg ab Z M d D x √− g R pq (cid:3) R pq = ( (cid:3) R ) ; ab − (cid:3)(cid:3) R ab + R ; p ( a R pb ) − R pq R pq ;( ab ) + 6 R pq R p ( a ; b ) q − (cid:3) R pq ) R paqb + 4 R pq ; r ( a R | rqp | b ) + 3 R ; p R p ( a ; b ) − R pq ; a R pq ; b + 2 R pq ;( a R b ) p ; q + 4 R pq ; r R rqp ( a ; b ) − R pq R pa R qb + 4 R pr R qr R paqb + 2 R pq R r ( a R | rqp | b ) + g ab [ − (1 / (cid:3)(cid:3) R − R ; pq R pq + R pq (cid:3) R pq + 2 R pq ; rs R prqs − (1 / R ; p R ; p + (5 / R pq ; r R pq ; r − R pq ; r R pr ; q − R pq R pr R qr + 2 R pq R rs R prqs ] , (2.23) H (6 , ab ≡ √− g δδg ab Z M d D x √− g R = 6 RR ; ab + 6 R ; a R ; b − R R ab + g ab [ − R (cid:3) R − R ; p R ; p + (1 / R ] , (2.24) H (6 , ab ≡ √− g δδg ab Z M d D x √− g RR pq R pq = RR ; ab − ( (cid:3) R ) R ab + 2 R ; p ( a R pb ) − R (cid:3) R ab + 2 R pq R pq ;( ab ) + R ; a R ; b + 2 R ; p R p ( a ; b ) − R ; p R ; pab + 2 R pq ; a R pq ; b − R pq R pq R ab − RR pq R paqb + g ab [ − (1 / R (cid:3) R − R ; pq R pq − R pq (cid:3) R pq − R ; p R ; p − R pq ; r R pq ; r + (1 / RR pq R pq ] , (2.25) H (6 , ab ≡ √− g δδg ab Z M d D x √− g R pq R pr R qr etric variations in higher-derivative gravity and stress-energy tensor
7= (3 / R ; p ( a R pb ) − R p ( a (cid:3) R pb ) + 3 R pq R p ( a ; b ) q + (3 / R ; p R p ( a ; b ) + 3 R pq ;( a R b ) p ; q − R pa ; q R ; qpb + 3 R pq R r ( a R | rqp | b ) + g ab [ − (3 / R ; pq R pq − (3 / R ; p R ; p − (3 / R pq ; r R pr ; q − R pq R pr R qr + (3 / R pq R rs R prqs ] , (2.26) H (6 , ab ≡ √− g δδg ab Z M d D x √− g R pq R rs R prqs = − (1 /
2) ( (cid:3) R ) R ab + R ; p ( a R pb ) + R pq R pq ;( ab ) − R pq R ab ; pq − ( (cid:3) R pq ) R paqb − R pq ; r ( a R | rqp | b ) + (1 / R ; a R ; b − R ; p R ; pab + 2 R pq ; a R pq ; b − R pq ;( a R b ) p ; q + R pa ; q R qb ; p − R pq ; r R rqp ( a ; b ) − R pq ; r R paqb ; r + R pq R pa R qb − R pr R qr R paqb + 2 R pq R r ( a R | rqp | b ) + 2 R pq R rspa R rsqb − R pq R prsa R qrsb + g ab [(1 / R ; pq R pq − R pq (cid:3) R pq − R pq ; rs R prqs − R pq ; r R pq ; r + 2 R pq ; r R pr ; q + R pq R pr R qr − (1 / R pq R rs R prqs ] , (2.27) H (6 , ab ≡ √− g δδg ab Z M d D x √− g RR pqrs R pqrs = 2 RR ; ab − R (cid:3) R ab − R ; pq R paqb + 2 R pqrs R pqrs ;( ab ) + 8 R ; p R p ( a ; b ) − R ; p R ; pab + 2 R pqrs ; a R pqrs ; b + 4 RR pa R pb − RR pq R paqb − RR pqra R pqrb − R ab R pqrs R pqrs + g ab [ − R pq ; rs R prqs − R pqrs ; t R pqrs ; t + (1 / RR pqrs R pqrs − R pq R prst R qrst + 2 R pqrs R pquv R rsuv + 8 R prqs R p qu v R rusv ] , (2.28) H (6 , ab ≡ √− g δδg ab Z M d D x √− g R pq R prst R qrst = R ; p ( a R pb ) − R p ( a (cid:3) R pb ) + 2 R pq R p ( a ; b ) q − R pq R ab ; pq − R ; pq R paqb + 4 R p ; qr ( a R | pqr | b ) + (1 / R pqrs R pqrs ;( ab ) + R ; p R p ( a ; b ) − R ; p R ; pab + 2 R pq ;( a R b ) p ; q − R pa ; q R ; qpb + 2 R pa ; q R qb ; p − R pq ; r R rqp ( a ; b ) − R pq ; r R paqb ; r + (1 / R pqrs ; a R pqrs ; b − R pqra ; s R ; spqrb + 2 R pq R pa R qb etric variations in higher-derivative gravity and stress-energy tensor − R pr R qr R paqb + 2 R pq R r ( a R | rqp | b ) − R p ( a R qrs | p | R | qrs | b ) + 2 R pq R prqs R rasb − R pq R prsa R qrsb + R pqrs R pqta R trs b + 4 R prqs R tpqa R trsb − R pqrs R pqrt R s ta b + g ab [ − R pq ; rs R prqs − R pq ; r R pq ; r + R pq ; r R pr ; q − (1 / R pqrs ; t R pqrs ; t + (1 / R pqrs R pquv R rsuv + R prqs R p qu v R rusv ] , (2.29) H (6 , ab ≡ √− g δδg ab Z M d D x √− g R pqrs R pquv R rsuv = 24 R p ; qr ( a R | pqr | b ) − R pa ; q R ; qpb + 12 R pa ; q R qb ; p + 3 R pqrs ; a R pqrs ; b − R pqra ; s R ; spqrb − R pq R rspa R rsqb + 12 R prqs R tpqa R trsb + g ab [(1 / R pqrs R pquv R rsuv ] , (2.30) H (6 , ab ≡ √− g δδg ab Z M d D x √− g R prqs R p qu v R rusv = (3 / R ; pq R paqb − (cid:3) R pq ) R paqb − R pq ; r ( a R | rqp | b ) − (3 / R pqrs R pqrs ;( ab ) + 3 R pq ; a R pq ; b − R pq ;( a R b ) p ; q + 3 R pa ; q R qb ; p + 6 R pq ; r R rqp ( a ; b ) + (3 / R pqrs ; a R pqrs ; b + 3 R pr R qr R paqb + 3 R p ( a R qrs | p | R | qrs | b ) + (3 / R pq R rspa R rsqb − R pq R prqs R rasb − R pq R prsa R qrsb − (3 / R pqrs R pqta R trs b − R prqs R tpqa R trsb + (3 / R pqrs R pqrt R s ta b + g ab [(1 / R prqs R p qu v R rusv ] . (2.31)It should be here noted that the ten geometrical tensors (2.22)-(2.31) are automaticallyconserved due to the invariance of the actions (2.4)-(2.13) under spacetimediffeomorphisms.From the relations (2.14)-(2.20) and by using (2.22)-(2.31), we can now directlyobtain the functional derivatives with respect to the metric tensor of the sevenremaining action terms. We obtain seven conserved tensors given by H (6 , ab ≡ √− g δδg ab Z M d D x √− g (cid:3)(cid:3) R = 0 , (2.32) H { , } (2) ab ≡ √− g δδg ab Z M d D x √− g R ; pq R pq = 12 H { , } (1) ab (2.33 a )= ( (cid:3) R ) ; ab − ( (cid:3) R ) R ab + (1 / R ; a R ; b + g ab [ − (cid:3)(cid:3) R − (1 / R ; p R ; p ] , (2.33 b ) etric variations in higher-derivative gravity and stress-energy tensor H { , } (4) ab ≡ √− g δδg ab Z M d D x √− g R pq ; rs R prqs = − H { , } (1) ab + H { , } (3) ab − H (6 , ab + H (6 , ab (2.34 a )= (1 /
2) ( (cid:3) R ) ; ab − (cid:3)(cid:3) R ab + (1 / R ; p ( a R pb ) + 3 R p ( a (cid:3) R pb ) − R pq R pq ;( ab ) + 3 R pq R p ( a ; b ) q − R pq R ab ; pq − (cid:3) R pq ) R paqb + 2 R pq ; r ( a R | rqp | b ) + (3 / R ; p R p ( a ; b ) − R ; p R ; pab + R pq ; a R pq ; b − R pq ;( a R b ) p ; q + 3 R pa ; q R ; qpb + R pa ; q R qb ; p + 2 R pq ; r R rqp ( a ; b ) − R pq ; r R paqb ; r − R pq R pa R qb + 2 R pr R qr R paqb + R pq R r ( a R | rqp | b ) + 2 R pq R rspa R rsqb − R pq R prsa R qrsb + g ab [ R pq ; rs R prqs + (1 / R pq ; r R pq ; r − (1 / R pq ; r R pr ; q ] , (2.34 b ) H { , } (1) ab ≡ √− g δδg ab Z M d D x √− g R ; p R ; p = − H { , } (1) ab (2.35 a )= − (cid:3) R ) ; ab + 2 ( (cid:3) R ) R ab − R ; a R ; b + g ab [2 (cid:3)(cid:3) R + (1 / R ; p R ; p ] , (2.35 b ) H { , } (2) ab ≡ √− g δδg ab Z M d D x √− g R pq ; r R pq ; r = − H { , } (3) ab (2.36 a )= − ( (cid:3) R ) ; ab + (cid:3)(cid:3) R ab − R ; p ( a R pb ) + 2 R pq R pq ;( ab ) − R pq R p ( a ; b ) q + 2 ( (cid:3) R pq ) R paqb − R pq ; r ( a R | rqp | b ) − R ; p R p ( a ; b ) + R pq ; a R pq ; b − R pq ;( a R b ) p ; q − R pq ; r R rqp ( a ; b ) + 2 R pq R pa R qb − R pr R qr R paqb − R pq R r ( a R | rqp | b ) + g ab [(1 / (cid:3)(cid:3) R + 2 R ; pq R pq − R pq (cid:3) R pq − R pq ; rs R prqs + (1 / R ; p R ; p − (5 / R pq ; r R pq ; r + 4 R pq ; r R pr ; q + 2 R pq R pr R qr − R pq R rs R prqs ] , (2.36 b ) H { , } (3) ab ≡ √− g δδg ab Z M d D x √− g R pq ; r R pr ; q = − H { , } (1) ab − H (6 , ab + H (6 , ab (2.37 a ) etric variations in higher-derivative gravity and stress-energy tensor − (1 /
2) ( (cid:3) R ) ; ab − (1 / R ; p ( a R pb ) + 3 R p ( a (cid:3) R pb ) + R pq R pq ;( ab ) − R pq R p ( a ; b ) q − R pq R ab ; pq − ( (cid:3) R pq ) R paqb − R pq ; r ( a R | rqp | b ) − (3 / R ; p R p ( a ; b ) − R ; p R ; pab + 2 R pq ; a R pq ; b − R pq ;( a R b ) p ; q + 3 R pa ; q R ; qpb + R pa ; q R qb ; p − R pq ; r R rqp ( a ; b ) − R pq ; r R paqb ; r + R pq R pa R qb − R pr R qr R paqb − R pq R r ( a R | rqp | b ) + 2 R pq R rspa R rsqb − R pq R prsa R qrsb + g ab [(1 / (cid:3)(cid:3) R + 2 R ; pq R pq − R pq (cid:3) R pq − R pq ; rs R prqs + (1 / R ; p R ; p − R pq ; r R pq ; r + (7 / R pq ; r R pr ; q + 2 R pq R pr R qr − R pq R rs R prqs ] , (2.37 b ) H { , } (4) ab ≡ √− g δδg ab Z M d D x √− g R pqrs ; t R pqrs ; t = H { , } (1) ab − H { , } (3) ab + 4 H (6 , ab − H (6 , ab − H (6 , ab + H (6 , ab + 4 H (6 , ab (2.38 a )= − H { , } (4) ab − H (6 , ab + H (6 , ab + 4 H (6 , ab (2.38 b )= − (cid:3) R ) ; ab + 4 (cid:3)(cid:3) R ab − R ; p ( a R pb ) − R p ( a (cid:3) R pb ) + 4 R pq R pq ;( ab ) − R pq R p ( a ; b ) q + 12 R pq R ab ; pq + 8 R ; pq R paqb − R pq ; r ( a R | rqp | b ) + 16 R p ; qr ( a R | pqr | b ) − R pqrs R pqrs ;( ab ) − R ; p R p ( a ; b ) + 6 R ; p R ; pab + 8 R pq ; a R pq ; b − R pq ;( a R b ) p ; q − R pa ; q R ; qpb + 16 R pa ; q R qb ; p + 24 R pq ; r R rqp ( a ; b ) + 12 R pq ; r R paqb ; r + 5 R pqrs ; a R pqrs ; b − R pqra ; s R ; spqrb + 8 R pr R qr R paqb − R pq R r ( a R | rqp | b ) + 16 R p ( a R qrs | p | R | qrs | b ) − R pq R rspa R rsqb − R pq R prqs R rasb − R pqrs R pqta R trs b − R prqs R tpqa R trsb + 8 R pqrs R pqrt R s ta b + g ab [(1 / R pqrs ; t R pqrs ; t ] . (2.38 c )It should be noted that the last six expressions can be also obtained by consideringthe variation of the action terms on the left hand side of (2.15)-(2.20) from (5.20 a )-(5.20 e ). Of course, the corresponding redundant calculations are tedious ones but wehave also achieved them in order to check the validity and the internal coherence ofour results.Finally, it seems to us interesting to provide also the functional derivative of thealternative action term constructed from the gravitational lagrangian R pqrs (cid:3) R pqrs .From (5.4) and (2.16) and by using (2.22)-(2.31) or directly by using integration byparts, we obtain for this conserved tensor etric variations in higher-derivative gravity and stress-energy tensor H { , } (alt) ab ≡ √− g δδg ab Z M d D x √− g R pqrs (cid:3) R pqrs = − H { , } (1) ab + 4 H { , } (3) ab − H (6 , ab + 4 H (6 , ab + 2 H (6 , ab − H (6 , ab − H (6 , ab (2.39 a )= 4 H { , } (4) ab + 2 H (6 , ab − H (6 , ab − H (6 , ab (2.39 b )= − H { , } (4) ab . (2.39 c )It appears as the opposite of H { , } (4) ab and its explicit expression can be obtaineddirectly from (2.38 c ). In 1971, Lovelock found the most general symmetric and conserved tensor whichis quasi-linear in the second derivatives of the metric tensor and does not containhigher derivatives. It therefore generalizes the Einstein tensor R ab − (1 / Rg ab (seeRefs. [2, 3] for the original discussion but also the paper by Deruelle and Madore [15]for a historical and physical presentation of this subject). Lovelock found moreoverthat this tensor can be obtained by functional derivation with respect to the metrictensor of an action constructed from a lagrangian which is the sum of dimensionallyextended Euler densities. The Lovelock gravitational theory is an appealing one beingfree of ghosts [16, 17] and is today more particularly considered in the context of stringtheory and brane cosmology.The Lovelock lagrangian L L reads L L = X n ≥ c k L ( n ) (2.40)where the c k are real arbitrary coefficients while L ( n ) for n ≥ n in the Riemann tensor (or the Euler-Gauss-Bonnet-Lovelock invariant of order n ) given by L ( n ) = 12 n δ p q ......p n q n r s ......r n s n R r s p q . . . . . . R r n s n p n q n . (2.41)Here δ p q ......p n q n r s ......r n s n denotes the generalized Kronecker symbol which is totallyantisymmetric in its upper and lower indices and which can be considered as the n × n normalized determinant δ p q ......p n q n r s ......r n s n = X σ ∈ Π n sign( σ ) δ p σ ( r ) δ q σ ( s ) . . . . . . δ p n σ ( r n ) δ q n σ ( s n ) . (2.42)We have L (0) = 1 by convention, L (1) = R and L (2) which reduces to the Gauss-Bonnetdensity, i.e. L (2) = R pqrs R pqrs − R pq R pq + R . (2.43)The part of the Lovelock lagrangian which is cubic in the Riemann tensor is explicitlygiven by L (3) = R − RR pq R pq + 16 R pq R pr R qr + 24 R pq R rs R prqs + 3 RR pqrs R pqrs − R pq R prst R qrst + 4 R pqrs R pquv R rsuv − R prqs R p qu v R rusv (2.44) etric variations in higher-derivative gravity and stress-energy tensor S (3) = Z M d D x p − g ( x ) L (3) ( x ) , (2.45)the cubic part of the Lovelock tensor G (3) ab = 1 √− g δS (3) δg ab . (2.46)With our previous notations, we have G (3) ab = H (6 , ab − H (6 , ab + 16 H (6 , ab + 24 H (6 , ab + 3 H (6 , ab − H (6 , ab + 4 H (6 , ab − H (6 , ab (2.47)and more explicitly, by using (2.22)-(2.31), we obtain G (3) ab = − R R ab + 12 RR pa R pb + 12 R pq R pq R ab − R pq R pa R qb + 12 RR pq R paqb − R pr R qr R paqb + 48 R pq R r ( a R | rqp | b ) − RR pqra R pqrb − R ab R pqrs R pqrs + 24 R p ( a R qrs | p | R | qrs | b ) + 12 R pq R rspa R rsqb − R pq R prqs R rasb + 24 R pq R prsa R qrsb − R pqrs R pqta R trs b + 24 R prqs R tpqa R trsb + 12 R pqrs R pqrt R s ta b + g ab [(1 / L (3) ] . (2.48)This result is not a new one. In fact, it has been obtained by M¨uller-Hoissen in Ref. [18]much more directly. Indeed, in order to functionally derive the Lovelock lagrangian, itis not necessary to functionally derive independently all the geometrical terms whichcompound it as we have done. Here, we recover this result mainly in order to check ourprevious calculations. It should be however noted that the comparison of (2.48) withthe result of M¨uller-Hoissen is not immediate: Indeed, in Ref. [18], the FKWC-basisis not used. But, by using (5.15 c ), it is easy to put the result of M¨uller-Hoissen in ourirreducible form (2.48).
3. Applications: Renormalization in the effective action and stress-energytensor a In this section, we consider a massive scalar field Φ propagating on the D -dimensionalcurved spacetime ( M , g ab ) and obeying the wave equation (cid:0) (cid:3) − m − ξR (cid:1) Φ = 0 . (3.1)Here m is the mass of the scalar field while ξ is a dimensionless factor which accountsfor the possible coupling between this field and the gravitational background. Theassociated DeWitt-Schwinger effective action W [8, 9, 10, 11, 12], which contains all theinformation on the ultraviolet behaviour of the quantum theory, may be representedby the asymptotic series [12] W = Z M d D x p − g ( x ) × " π ) D/ Z + ∞ d ( is )( is ) D/ e − im s + ∞ X k =0 a k ( x )( is ) k . (3.2) etric variations in higher-derivative gravity and stress-energy tensor a k ( x ) are the diagonal DeWitt coefficients and we have for the four first ones a = 1 , (3.3 a ) a = − ( ξ − / R, (3.3 b ) a = − (1 /
6) ( ξ − / (cid:3) R + (1 /
2) ( ξ − / R − (1 / R pq R pq + (1 / R pqrs R pqrs (3.3 c )and a = − (1 /
60) ( ξ − / (cid:3)(cid:3) R + (1 /
6) ( ξ − /
6) ( ξ − / R (cid:3) R − (1 /
90) ( ξ − / R ; pq R pq − (1 / R pq (cid:3) R pq + (1 / R pq ; rs R prqs + (1 /
12) [ ξ − (2 / ξ + 17 / R ; p R ; p − (1 / R pq ; r R pq ; r − (1 / R pq ; r R pr ; q + (1 / R pqrs ; t R pqrs ; t − (1 /
6) ( ξ − / R + (1 / ξ − / RR pq R pq + (1 / R pq R pr R qr − (1 / R pq R rs R prqs − (1 / ξ − / RR pqrs R pqrs + (1 / R pq R prst R qrst − (4 / R pqrs R pquv R rsuv − (22 / R prqs R p qu v R rusv . (3.3 d )The results we have obtained in the previous section are therefore helpful in orderto understand those of the physical aspects of the scalar field theory which are moreparticularly associated with the coefficient a since it is of sixth order in the derivativesof the metric tensor. In the following subsections, we shall focus our attention on twoof them. To be more precise, it is important to recall that the effective action (3.2)is divergent at the lower limit of the integral over s for all the positive values of thedimension D . By considering the dimensionality D of spacetime as a complex number,the effective action can be regularized by analytic continuation and its divergent partcan be extracted coherently. In a four dimensional background, the divergent part ofthe effective action is proportional to [8, 9, 10, 11, 12] Z M d x p − g ( x ) (cid:2) a ( x ) − m a ( x ) + ( m / a ( x ) (cid:3) (3.4)while its regularized part is proportional, in the large mass limit, to [10, 11] Z M d x p − g ( x ) a ( x ) . (3.5)In a six dimensional background, the divergent part of the effective action isproportional to Z M d x p − g ( x ) (cid:2) a ( x ) − m a ( x ) + ( m / a ( x ) − ( m / a ( x ) (cid:3) . (3.6)In both cases, it should be noted that the global (or integrated) Gilkey-DeWittcoefficient Z M d D x p − g ( x ) a ( x ) (3.7)plays a central role. It is therefore necessary to have at our disposal its explicitexpression as well as the expressions of the three first global (or integrated) DeWitt etric variations in higher-derivative gravity and stress-energy tensor d ) and(2.14)-(2.20), we can easily obtained them. We have Z M d D x √− g a = Z M d D x √− g, (3.8) Z M d D x √− g a = Z M d D x √− g [ − ( ξ − / R ] (3.9)and Z M d D x √− g a = Z M d D x √− g (cid:2) (1 /
2) ( ξ − / R − (1 / R pq R pq + (1 / R pqrs R pqrs ] (3.10)and Z M d D x √− g a = Z M d D x √− g (cid:0) [(1 / ξ − (1 / ξ + 1 / R (cid:3) R +(1 / R pq (cid:3) R pq − (1 /
6) ( ξ − / R + (1 / ξ − / RR pq R pq − (4 / R pq R pr R qr + (1 / R pq R rs R prqs − (1 / ξ − / RR pqrs R pqrs + (1 / R pq R prst R qrst +(17 / R pqrs R pquv R rsuv − (1 / R prqs R p qu v R rusv (cid:1) . (3.11) In the large mass limit of the quantized scalar field, the renormalized effective actionin four dimensions reduces to [10, 11] W ren = 132 π m Z M d x p − g ( x ) a ( x ) (3.12)and from (3.11) we can then write the effective action in the form (see also Ref. [10, 11]) W ren = 1192 π m Z M d x √− g (cid:0) [(1 / ξ − (1 / ξ + 1 / R (cid:3) R +(1 / R pq (cid:3) R pq − ( ξ − / R + (1 /
30) ( ξ − / RR pq R pq − (8 / R pq R pr R qr + (2 / R pq R rs R prqs − (1 /
30) ( ξ − / RR pqrs R pqrs + (1 / R pq R prst R qrst +(17 / R pqrs R pquv R rsuv − (1 / R prqs R p qu v R rusv (cid:1) . (3.13)By functional derivation of the effective action (3.13), we obtain an approximationfor the expectation value of the stress-energy tensor associated with the scalar field.With the notations introduced in Section 2, we can write h T ab i ren = 2 √− g δW ren δg ab etric variations in higher-derivative gravity and stress-energy tensor
15= 196 π m (cid:16) [(1 / ξ − (1 / ξ + 1 / H { , } (1) ab +(1 / H { , } (3) ab − ( ξ − / H (6 , ab + (1 /
30) ( ξ − / H (6 , ab − (8 / H (6 , ab + (2 / H (6 , ab − (1 /
30) ( ξ − / H (6 , ab +(1 / H (6 , ab + (17 / H (6 , ab − (1 / H (6 , ab (cid:17) (3.14)and from the relations (2.22)-(2.31), we have then explicitly(96 π m ) h T ab i ren = [ ξ − (2 / ξ + 3 /
70] ( (cid:3) R ) ; ab − (1 / (cid:3)(cid:3) R ab − ξ − / ξ − (1 / ξ + 1 / RR ; ab − ( ξ − / ξ − /
5) ( (cid:3) R ) R ab + (1 / ξ − / R ; p ( a R pb ) + (1 / ξ − / R (cid:3) R ab + (1 / R p ( a (cid:3) R pb ) + (1 / ξ − / R pq R pq ;( ab ) + (2 / R pq R p ( a ; b ) q − (1 / R pq R ab ; pq + (2 / ξ − / R ; pq R paqb − (1 / (cid:3) R pq ) R paqb + (4 / R pq ; r ( a R | rqp | b ) + (2 / R p ; qr ( a R | pqr | b ) − (1 / ξ − / R pqrs R pqrs ;( ab ) − ξ − / ξ − / R ; a R ; b − (1 / ξ − / R ; p R p ( a ; b ) + (1 / ξ − / R ; p R ; pab + (1 / ξ − / R pq ; a R pq ; b − (1 / R pa ; q R ; qpb + (1 / R pa ; q R qb ; p − (1 / R pq ; r R rqp ( a ; b ) − (1 / R pq ; r R paqb ; r − (1 / ξ − / R pqrs ; a R pqrs ; b − (1 / R pqra ; s R ; spqrb + 3( ξ − / R R ab − (2 / ξ − / RR pa R pb − (1 / ξ − / R pq R pq R ab − (2 / R pq R pa R qb + (1 / ξ − / RR pq R paqb + (1 / R pr R qr R paqb + (1 / R pq R r ( a R | rqp | b ) + (1 / ξ − / RR pqra R pqrb + (1 / ξ − / R ab R pqrs R pqrs − (4 / R p ( a R qrs | p | R | qrs | b ) − (2 / R pq R rspa R rsqb + (4 / R pq R prqs R rasb − (1 / R pq R prsa R qrsb + (2 / R pqrs R pqta R trs b + (4 / R prqs R tpqa R trsb − (2 / R pqrs R pqrt R s ta b + g ab [[ − ξ + (2 / ξ − / (cid:3)(cid:3) R + 6( ξ − / ξ − (1 / ξ + 1 / R (cid:3) R − (1 / ξ − / R ; pq R pq − (1 / ξ − / R pq (cid:3) R pq + (4 / ξ − / R pq ; rs R prqs + 6[ ξ − (13 / ξ + (17 / ξ − / R ; p R ; p − (1 / ξ − / R pq ; r R pq ; r − (1 / R pq ; r R pr ; q + (1 / ξ − / R pqrs ; t R pqrs ; t − (1 / ξ − / R + (1 / ξ − / RR pq R pq + (1 / R pq R pr R qr etric variations in higher-derivative gravity and stress-energy tensor − (1 / R pq R rs R prqs − (1 / ξ − / RR pqrs R pqrs + (2 / ξ − / R pq R prst R qrst − (1 / ξ − / R pqrs R pquv R rsuv − (4 / ξ − / R prqs R p qu v R rusv ] . (3.15)We have therefore expressed the approximated expectation value of the stress-energytensor associated with the scalar field on the FKWC-basis described in Section 2. Wehave a final expression which is simplified and without any ambiguities. It should benoted that in recent articles [19, 20, 21], Matyjasek has calculated this stress-energytensor but, being only interested by the result in particular spacetimes, he has notobtained a general simplified result valid in an arbitrary background.To conclude this subsection, we would like to emphasize some possible othersimplifications coming from “topological” and geometrical constraints associated withthe four dimensional nature of spacetime. In that special case, it is well-known thatthe Euler number Z M d x p − g ( x ) L (2) ( x ) (3.16)where L (2) is given by (2.43) is a topological invariant. As a consequence, its metricvariation vanishes identically and we have H (4 , ab − H (4 , ab + H (4 , ab = 0 (3.17)with H (4 , ab ≡ √− g δδg ab Z M d x √− g R = 2 R ; ab − RR ab + g ab [ − (cid:3) R + (1 / R ] , (3.18) H (4 , ab ≡ √− g δδg ab Z M d x √− g R pq R pq = R ; ab − (cid:3) R ab − R pq R paqb + g ab [ − (1 / (cid:3) R + (1 / R pq R pq ] , (3.19) H (4 , ab ≡ √− g δδg ab Z M d x √− g R pqrs R pqrs = 2 R ; ab − (cid:3) R ab + 4 R pa R pb − R pq R paqb − R pqra R pqrb + g ab [(1 / R pqrs R pqrs ] , (3.20)or more explicitly, − RR ab + 4 R pa R pb + 4 R pq R paqb − R pqra R pqrb + 12 g ab ( R pqrs R pqrs − R pq R pq + R ) = 0 . (3.21)Moreover, in four dimensions, we have the Xu’s geometrical identity (see Ref. [22] aswell as the introduction of Ref. [23] for a simplest derivation and a clear interpretation) R − RR pq R pq + 8 R pq R pr R qr + 8 R pq R rs R prqs + RR pqrs R pqrs − R pq R prst R qrst = 0 (3.22) etric variations in higher-derivative gravity and stress-energy tensor H (6 , ab − H (6 , ab + 8 H (6 , ab + 8 H (6 , ab + H (6 , ab − H (6 , ab = 0 . (3.23)Finally, in four dimensions, the cubic Lovelock lagrangian L (3) given by (2.44) aswell as its metric variation, i.e. the cubic Lovelock tensor G (3) ab given by (2.48), vanishidentically. As a consequence, we have with (3.17) or (3.21), (3.22) and (3.23), L (3) = 0and G (3) ab = 0 five new geometrical relations which could permit us to fully simplify theexpression (3.15) of the approximated stress-energy tensor. We leave this task to theinterested reader because the choice of the (scalar and tensorial) Riemann monomialsto be eliminated is a matter of taste and depends on the problem treated as well as onthe gravitational background considered. We remark however that in the expression(3.14) of this stress-tensor, it would be certainly interesting to drop two of the tenterms by using (3.23) as well as G (3) ab = 0 in the form H (6 , ab − H (6 , ab + 16 H (6 , ab + 24 H (6 , ab + 3 H (6 , ab − H (6 , ab + 4 H (6 , ab − H (6 , ab = 0 . (3.24) As we have noted in Subsection (3.1), the divergent part of the effective actionassociated with the scalar field is proportional to (3.6) in a six dimensional background.It can be removed by renormalization of the Newton’s gravitational constant andof the cosmological constant and by adding to the Einstein-Hilbert gravitationallagrangian three counterterms of order four ( R , R pq R pq and R pqrs R pqrs ) in orderto eliminate the divergences associated with the DeWitt coefficient a (see (3.10)) aswell as ten counterterms of order six ( R (cid:3) R , R pq (cid:3) R pq , R , RR pq R pq , R pq R pr R qr , R pq R rs R prqs , RR pqrs R pqrs , R pq R prst R qrst , R pqrs R pquv R rsuv , R prqs R p qu v R rusv ) inorder to eliminate the divergences associated with the Gilkey-DeWitt coefficient a (see (3.11)). These last ten counterterms induce in the bare Einstein equations acorrection of sixth order in the derivative of the metric tensor which is of the form α H { , } (1) ab + α H { , } (3) ab + α H (6 , ab + α H (6 , ab + α H (6 , ab + α H (6 , ab + α H (6 , ab + α H (6 , ab + α H (6 , ab + α H (6 , ab (3.25)with the coefficients α i containing terms in 1 / ( D −
6) and so diverging in the physicaldimension limit. Furthermore, because in six dimensions the Euler number Z M d x p − g ( x ) L (3) ( x ) (3.26)where L (3) is given by (2.44) is a topological invariant, its metric variation G (3) ab givenby (2.48) vanishes identically. We then have here again the constraint (3.24) whichcould permit us to eliminate one of the ten contributions of order six in the bareEinstein equations. etric variations in higher-derivative gravity and stress-energy tensor
4. Concluding remarks
The metric variations of the gravitational action terms constructed from the curvatureinvariants of order six in derivatives of the metric tensor have been already consideredby numerous authors (see, for example, Refs. [24, 25, 19, 20, 21, 26, 27]). However, inthe works achieved until now, all the possible simplifications due to the symmetriesof the Riemann tensor as well as to Bianchi identities have not been systematicallydone. As a consequence, there did not exist until now, in literature, explicit irreducibleformulas for these functional derivatives as it was the case for the functional derivativeswith respect to the metric tensor of the action terms constructed from the gravitationallagrangians (cid:3) R , R , R pq R pq and R pqrs R pqrs , i.e. from the curvature invariants oforder four in derivatives of the metric tensor (see, for example, Ref. [9]). In the presentpaper, by using the results obtained by Fulling, King, Wybourne and Cummings [1]based on group theoretical considerations, we have solved unambiguously this problem,filling up a void in quantum field theory in curved spacetime. We have been able tothen discuss some aspects of quantization linked to the DeWitt-Schwinger effectiveaction (simplified form for the renormalized effective action in four dimensions andsimplified forms for the infinite counterterms of order six in the derivatives of themetric tensor which must appear in the left hand side of the bare Einstein equationsin six dimensions).We think that our results could be helpful not only in four dimensions but alsoin treating some aspects of the quantum physics of extra spatial dimensions which iscurrently exploding under the impulsion of string theory. In a near future, these resultscould be more particularly interesting i) in order to understand the back reactionproblem (in the large mass limit of the quantized fields) for a wide class of metrics [28]and ii) in order to discuss, from a general point of view, the Hadamard renormalizationof the stress-energy tensor in an arbitrary D -dimensional spacetime [29] and to studyprecisely its ambiguities. Acknowledgments
We would like to thank Jerzy Matyjasek for kind correspondence during last summer.We are furthermore grateful to Rosalind Fiamma for help with the English.
5. Appendix
In the present subsection of the Appendix we provide some geometrical identities whichpermit us to eliminate “alternative” Riemann monomials of order six by expressingthem in terms of elements of the FKWC-basis. These relations are more generallyuseful for calculations in two-loop quantum gravity in a four dimensional backgroundor for calculations in one-loop quantum gravity in higher dimensional background. Allthese relations can be derived more or less trivially from the “symmetry” properties ofthe Ricci and the Riemann tensors (pair symmetry, antisymmetry, cyclic symmetry) R ab = R ba , (5.1 a ) R abcd = R cdab , (5.1 b ) etric variations in higher-derivative gravity and stress-energy tensor R abcd = − R bacd and R abcd = − R abdc , (5.1 c ) R abcd + R adbc + R acdb = 0 , (5.1 d )from the Bianchi identity and its consequences obtained by contraction of index pairs R abcd ; e + R abec ; d + R abde ; c = 0 (5.2 a ) R ; dabcd = R ac ; b − R bc ; a (5.2 b ) R ; bab = (1 / R ; a (5.2 c )as well as from the commutation of covariant derivatives in the form (1.1).It is of course possible to derive numerous scalar relations between scalar Riemannmonomials of order six but, in fact, it seems to us that only five of these relations arereally important: R pq R pr ; qr = 12 R ; pq R pq + R pq R pr R qr − R pq R rs R prqs (5.3)and R pqrs (cid:3) R pqrs = 4 R pq ; rs R prqs + 2 R pq R prst R qrst − R pqrs R pquv R rsuv − R prqs R p qu v R rusv (5.4)and R prqs R pquv R uvrs = 12 R pqrs R pquv R rsuv , (5.5 a ) R rspq R puqv R rusv = 14 R pqrs R pquv R rsuv , (5.5 b ) R prqs R puqv R r sv u = − R pqrs R pquv R rsuv + R prqs R p qu v R rusv . (5.5 c )Similarly, among all the tensorial relations between Riemann monomials of ordersix and rank two, we have chosen to retain more particularly the 15 following ones: (cid:3) ( R ; ab ) = ( (cid:3) R ) ; ab + 2 R ; p ( a R pb ) − R ; pq R paqb − R ; p R ; pab + 2 R ; p R p ( a ; b ) (5.6)and R pq R pa ; qb = R pq R pa ; bq + R pr R qr R paqb + R pq R ra R rqpb (5.7)and R p ; qra R rqpb = 12 R sa R pqrs R pqrb − R pq R rspa R rsqb (5.8)and R pq ; r R raqb ; p = R pq ; r R rqpb ; a + R pq ; r R paqb ; r (5.9)and R pqrs ; a R pqrb ; s = 12 R pqrs ; a R pqrs ; b , (5.10 a ) R prsa ; q R qsrb ; p = 14 R pqrs ; a R pqrs ; b (5.10 b ) etric variations in higher-derivative gravity and stress-energy tensor R pqrs R pqra ; bs = 12 R pqrs R pqrs ; ab − R pqrs R pqrt R s ta b + 12 R pqrs R pqta R trs b + 2 R prqs R tpqa R trsb (5.11)and R pqra (cid:3) R pqrb = − R pq ; rb R rqpa − R p ; qrb R pqra + R pb R qrsp R qrsa + R pq R rspa R rsqb − R pqrs R pqta R trs b − R prqs R tpqa R trsb (5.12)and R pq (cid:3) R paqb = R pq R pq ;( ab ) − R pq R p ( a ; b ) q + R pq R ab ; pq − R pq R r ( a R | rqp | b ) − R pq R rspa R rsqb − R pq R prqs R rasb + 2 R pq R prsa R qrsb (5.13)and R pq R prsa R qsrb = R pq R prsa R qrsb − R pq R rspa R rsqb (5.14)and R pqrs R tpqa R trsb = 14 R pqrs R pqta R trs b , (5.15 a ) R pqrs R pqta R trsb = − R pqrs R pqta R trs b , (5.15 b ) R prqs R pqta R trs b = 12 R pqrs R pqta R trs b , (5.15 c ) R prqs R tpqa R tsrb = R prqs R tpqa R trsb − R pqrs R pqta R trs b , (5.15 d ) R prqs R pqta R tsrb = 14 R pqrs R pqta R trs b . (5.15 e )There exists in particular a lot of other relations involving terms cubic in the Riemanntensor which are useful in calculations but they can be obtained trivially from the fiveprevious ones. In this short subsection of the Appendix, we provide a list of relations describingthe behaviour of some important geometrical tensors in an elementary variation ofthe metric tensor. These relations are useful to obtain, in Section 2, the functionalderivatives with respect to the metric tensor of the action terms constructed from the17 scalar Riemann monomials of order six. Apart from two of them which we haveestablished, these relations can be found in Ref. [14] but we prefer to collect them i)in order to avoid the reader having to read this reference and ii) because in certaincases we have adopted a more practical notation.In the elementary variation g ab → g ab + h ab (5.16) etric variations in higher-derivative gravity and stress-energy tensor g ab → g ab + δg ab , (5.17 a ) √− g → √− g + δ ( √− g ) , (5.17 b )Γ cab → Γ cab + δ Γ cab , (5.17 c ) R → R + δR, (5.17 d ) (cid:3) R → (cid:3) R + δ ( (cid:3) R ) , (5.17 e ) R ab → R ab + δR ab , (5.17 f ) (cid:3) R ab → (cid:3) R ab + δ ( (cid:3) R ab ) , (5.17 g ) R abcd → R abcd + δR abcd , (5.17 h )with δg ab = − g pa g qb h pq , (5.18 a ) δ ( √− g ) = 12 √− g h with h ≡ g pq h pq , (5.18 b ) δ Γ cab = 12 (cid:16) − h ; cab + h ca ; b + h cb ; a (cid:17) , (5.18 c ) δR = h pq ; pq − (cid:3) h − R pq h pq , (5.18 d ) δ ( (cid:3) R ) = (cid:3) ( h pq ; pq ) − (cid:3)(cid:3) h − R pq ; r h pq ; r − R pq (cid:3) h pq − ( (cid:3) R pq ) h pq − R ; pq h pq − R ; p h pq ; q + 12 R ; p h ; p , (5.18 e ) δR ab = 12 ( h pa ; bp + h pb ; ap − h ; ab − (cid:3) h ab ) , (5.18 f ) δ ( (cid:3) R ab ) = 12 [ (cid:3) ( h pa ; bp ) + (cid:3) ( h pb ; ap ) − (cid:3) ( h ; ab ) − (cid:3)(cid:3) h ab ] − R ab ; p h pq ; q + 12 R ab ; p h ; p − R ab ; pq h pq − R ap ( (cid:3) h pb + h pq ; bq − h ; pqbq ) − R bp ( (cid:3) h pa + h pq ; aq − h ; pqaq ) − R ap ; q ( h p ; qb − h q ; pb + h pq ; b ) − R bp ; q ( h p ; qa − h q ; pa + h pq ; a ) , (5.18 g ) δR abcd = 12 ( h ab ; dc − h ab ; cd + h ad ; bc − h ac ; bd − h bd ; ac + h bc ; ad )+ R pbcd h ap . (5.18 h )In the elementary variation (5.16) of the metric tensor, we have moreover R ; a → R ; a + δ ( R ; a ) (5.19 a ) R ; ab → R ; ab + δ ( R ; ab ) (5.19 b ) R ab ; c → R ab ; c + δ ( R ab ; c ) (5.19 c ) R ab ; cd → R ab ; cd + δ ( R ab ; cd ) (5.19 d ) R abcd ; e → R abcd ; e + δ ( R abcd ; e ) (5.19 e )with δ ( R ; a ) = ( δR ) ; a (5.20 a ) δ ( R ; ab ) = ( δR ) ; ab − R ; p ( δ Γ pab ) (5.20 b ) δ ( R ab ; c ) = ( δR ab ) ; c − R pa ( δ Γ pbc ) − R pb ( δ Γ pac ) , (5.20 c ) δ ( R ab ; cd ) = ( δR ab ) ; cd − ( δ Γ pac ) ; d R pb − ( δ Γ pbc ) ; d R pa − ( δ Γ pcd ) R ab ; p etric variations in higher-derivative gravity and stress-energy tensor − ( δ Γ pac ) R pb ; d − ( δ Γ pbc ) R pa ; d − ( δ Γ pad ) R pb ; c − ( δ Γ pbd ) R pa ; c (5.20 d ) δ ( R abcd ; e ) = ( δR abcd ) ; e − ( δ Γ pae ) R pbcd − ( δ Γ pbe ) R apcd − ( δ Γ pce ) R abpd − ( δ Γ pde ) R abcp . (5.20 e ) In this last subsection of the Appendix, we briefly extend the result obtained inSubsection (3.2) by providing, in the large mass limit, simplifified expressions forthe approximated stress-energy tensors associated with a massive spinor field and amassive vector field propagating in a four dimensional background. These results havebeen obtained very easily and quickly from the formalism developed in Section 2 andthey explicitly prove the power of this formalism.In the large mass limit, the renormalized effective action of a neutral spinor fieldis given by [10, 11] W s =1 / = 1192 π m Z M d x √− g ( − (3 / R (cid:3) R +(1 / R pq (cid:3) R pq + (1 / R − (1 / RR pq R pq − (25 / R pq R pr R qr + (47 / R pq R rs R prqs − (7 / RR pqrs R pqrs + (19 / R pq R prst R qrst +(29 / R pqrs R pquv R rsuv − (1 / R prqs R p qu v R rusv (cid:1) . (5.21)By functional derivation of this effective action, we obtain for the associatedexpectation value of the stress-energy tensor the expression (here we use the notationsintroduced in Section 2) h T s =1 / ab i ren = 2 √− g δW s =1 / δg ab = 196 π m (cid:16) − (3 / H { , } (1) ab +(1 / H { , } (3) ab + (1 / H (6 , ab − (1 / H (6 , ab − (25 / H (6 , ab + (47 / H (6 , ab − (7 / H (6 , ab +(19 / H (6 , ab + (29 / H (6 , ab − (1 / H (6 , ab (cid:17) (5.22)and from the relations (2.22)-(2.31), we have then explicitly(96 π m ) h T s =1 / ab i ren = (1 /
70) ( (cid:3) R ) ; ab − (1 / (cid:3)(cid:3) R ab − (1 / RR ; ab + (1 / (cid:3) R ) R ab + (23 / R ; p ( a R pb ) + (1 / R (cid:3) R ab + (29 / R p ( a (cid:3) R pb ) − (19 / R pq R pq ;( ab ) + (61 / R pq R p ( a ; b ) q − (11 / R pq R ab ; pq − (1 / R ; pq R paqb − (17 / (cid:3) R pq ) R paqb + (13 / R pq ; r ( a R | rqp | b ) + (16 / R p ; qr ( a R | pqr | b ) + (1 / R pqrs R pqrs ;( ab ) etric variations in higher-derivative gravity and stress-energy tensor
23+ (19 / R ; p R p ( a ; b ) − (1 / R ; p R ; pab − (1 / R pq ;( a R b ) p ; q − (1 / R pa ; q R ; qpb + (3 / R pa ; q R qb ; p − (1 / R pq ; r R rqp ( a ; b ) − (11 / R pq ; r R paqb ; r + (1 / R pqrs ; a R pqrs ; b − (4 / R pqra ; s R ; spqrb − (1 / R R ab − (7 / RR pa R pb + (1 / R pq R pq R ab − (1 / R pq R pa R qb + (11 / RR pq R paqb + (13 / R pr R qr R paqb + (97 / R pq R r ( a R | rqp | b ) + (7 / RR pqra R pqrb + (7 / R ab R pqrs R pqrs − (73 / R p ( a R qrs | p | R | qrs | b ) + (19 / R pq R rspa R rsqb + (73 / R pq R prqs R rasb − (97 / R pq R prsa R qrsb + (73 / R pqrs R pqta R trs b + (239 / R prqs R tpqa R trsb − (73 / R pqrs R pqrt R s ta b + g ab [(1 / (cid:3)(cid:3) R − (1 / R (cid:3) R + (1 / R ; pq R pq + (1 / R pq (cid:3) R pq + (3 / R pq ; rs R prqs − (1 / R ; p R ; p + (3 / R pq ; r R pq ; r − (1 / R pq ; r R pr ; q + (1 / R pqrs ; t R pqrs ; t + (1 / R − (1 / RR pq R pq − (1 / R pq R pr R qr + (1 / R pq R rs R prqs − (7 / RR pqrs R pqrs + (7 / R pq R prst R qrst − (61 / R pqrs R pquv R rsuv − (43 / R prqs R p qu v R rusv ] . (5.23)In the large mass limit, the renormalized effective action of a vector field is givenby [10, 11] W s =1ren = 1192 π m Z M d x √− g ( − (27 / R (cid:3) R +(9 / R pq (cid:3) R pq − (5 / R + (31 / RR pq R pq − (52 / R pq R pr R qr − (19 / R pq R rs R prqs − (1 / RR pqrs R pqrs + (61 / R pq R prst R qrst − (67 / R pqrs R pquv R rsuv + (1 / R prqs R p qu v R rusv (cid:1) . (5.24)By functional derivation of this effective action, we obtain for the associatedexpectation value of the stress-energy tensor the expression (here we use again thenotations introduced in Section 2) h T s =1 ab i ren = 2 √− g δW s =1ren δg ab = 196 π m (cid:16) − (27 / H { , } (1) ab +(9 / H { , } (3) ab − (5 / H (6 , ab + (31 / H (6 , ab − (52 / H (6 , ab − (19 / H (6 , ab − (1 / H (6 , ab +(61 / H (6 , ab − (67 / H (6 , ab + (1 / H (6 , ab (cid:17) (5.25) etric variations in higher-derivative gravity and stress-energy tensor π m ) h T s =1 ab i ren = (9 /
70) ( (cid:3) R ) ; ab − (9 / (cid:3)(cid:3) R ab − (1 / RR ; ab − (7 /
30) ( (cid:3) R ) R ab + (13 / R ; p ( a R pb ) − (7 / R (cid:3) R ab + (337 / R p ( a (cid:3) R pb ) + (22 / R pq R pq ;( ab ) + (34 / R pq R p ( a ; b ) q − (107 / R pq R ab ; pq + (1 / R ; pq R paqb − (22 /
35) ( (cid:3) R pq ) R paqb + (46 / R pq ; r ( a R | rqp | b ) + (116 / R p ; qr ( a R | pqr | b ) − (1 / R pqrs R pqrs ;( ab ) − (1 / R ; a R ; b + (83 / R ; p R p ( a ; b ) − (41 / R ; p R ; pab + (31 / R pq ; a R pq ; b − (14 / R pq ;( a R b ) p ; q + (221 / R pa ; q R ; qpb + (113 / R pa ; q R qb ; p + (5 / R pq ; r R rqp ( a ; b ) − (107 / R pq ; r R paqb ; r − (17 / R pqrs ; a R pqrs ; b − (29 / R pqra ; s R ; spqrb + (5 / R R ab − (2 / RR pa R pb − (31 / R pq R pq R ab + (1 / R pq R pa R qb − (19 / RR pq R paqb + (33 / R pr R qr R paqb − (139 / R pq R r ( a R | rqp | b ) + (1 / RR pqra R pqrb + (1 / R ab R pqrs R pqrs − (74 / R p ( a R qrs | p | R | qrs | b ) − (5 / R pq R rspa R rsqb + (74 / R pq R prqs R rasb − (71 / R pq R prsa R qrsb + (37 / R pqrs R pqta R trs b + (97 / R prqs R tpqa R trsb − (37 / R pqrs R pqrt R s ta b + g ab [(9 / (cid:3)(cid:3) R + (19 / R (cid:3) R − (1 / R ; pq R pq − (223 / R pq (cid:3) R pq + (79 / R pq ; rs R prqs + (163 / R ; p R ; p − (17 / R pq ; r R pq ; r + (11 / R pq ; r R pr ; q + (51 / R pqrs ; t R pqrs ; t − (5 / R + (31 / RR pq R pq + (1 / R pq R pr R qr − (53 / R pq R rs R prqs − (1 / RR pqrs R pqrs + (2 / R pq R prst R qrst − (263 / R pqrs R pquv R rsuv − (106 / R prqs R p qu v R rusv ] . (5.26)It should be finally noted that it is possible to simplify (5.22) and (5.25) or (5.23)and (5.26) by using the “topological” and geometrical constraints described at the endof Subsection (3.2) which are independent of the quantum field considered. References [1] Fulling S A, King R C, Wybourne B G and Cummings C J 1992 Normal forms for tensorpolynomials: I. The Riemann tensor
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