Quintic Quasitopological Gravity
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1 Quintic Quasitopological Gravity Luis Guajardo 1 in collaboration with Adolfo Cisterna 2 Mokthar Hassaïne 1 Julio Oliva 3 1 Universidad de Talca 2 Universidad Central de Chile 3 Universidad de Concepción November 7, 2017
2 Table of Contents 1 Introduction 2 The Theory 3 Some Properties 4 Final Comments
3 Quasitopological Gravity Context: AdS/CFT Correspondence Higher curvature theories have received some attention. Lovelock s Theory as a natural extension of General Relativity. The curvature invariants of k th order in Lovelock theory don t contribute to the field equations if D 2k. Contrary to this last property, new theories like Quasitopological Gravities appears.
4 Quasitopological Gravity First result on 2010 [J Oliva, S. Ray: arxiv.org/ ], introducing a cubic interaction in D = 5: L 3 = 7 6 Rab cdr ce bf R df ae R cd ab R be cd R a e 1 2 R ab cd R a cr b d Ra br b cr c a 1 2 RRa br b a R3. (1) On spacetimes with spherical/planar/hyperbolic symmetry, the theory has second order field equations. Among others, this theory has the following properties: The trace of the field equations is proportional to the Lagrangian. Birkhoff s Theorem. Interaction with GR and Gauss-Bonnet terms lead to an asymptotically AdS black hole.
5 Quasitopological Gravity Goals of the speech: To present a theory in D = 5 which is fifth order in curvature, but with second order field equations on spherical/planar/hyperbolic spacetimes. To give some properties: Birkhoff s Theorem. No ghosts on AdS.
6 Table of Contents 1 Introduction 2 The Theory 3 Some Properties 4 Final Comments
7 The Theory Here we are considering the following gravity theory: I [g µν ] = d 5 x [ ] R 2Λ 5 g 16πG + a k L k k=2 where L 2 stands for the Gauss-Bonnet combination (2) L 2 = R 2 4R ab R ab + R abcd R abcd, L 3 was defined on (1). The quartic quasitopological term L 4 can be written as 1 L 4 = [ 7080R pqbs R a u p b R v w a u Rqvsw 234Rpqbs R au pq R vw au R bsvw 1237 (R pqbs ) 2 R pqbs R pq R bsau R v bs p Rauvq 6912Rpq R bs R a u p q R abus 7152R pq R bs R au pb Rauqs + 308R pq R pqr bsau R bsau + 298R 2 R pqbs R pqbs R pq R bs R a b Rpsqa 115R4 912RR pq R bs R pbqs R pq R b p R s q R bs 4256RR pq R b p R qb R 2 R pq ] R pq
8 The Theory The new quintic Quasitopological combination is L 5 = A 1 RR a b R b c R c d R d a + A 2 RR a b R b a R cd ef R ef cd + A 4 R a b R b a R c d R d e R e c + A 5 R a b R b c R c a R de fg R fg de + A 7 R a b R b d R c f R de cg R fg ae + A 8 R a b R b c R cd ae R ef + A 10 R a b R b c R cd eg R ef + A 13 R a c R b d R cd ef R ef ah R gh df gh R gh ab + A 11 R a c R b d R + A 14 R a c R b d R + A 3 RR a c R b d R gh R gh df cd ef R ef ab + A 6 R a b R b d R c f R de ag R fg ce cd ab R ef gh R gh ef cd eg R ef ah R gh bf + A 9 R a b R b c R + A 12 R a c R b d R + A 15 R a c R b e R cd ef R ef gh R gh ad cd ae R ef gh R gh bf cd af R ef gh R gh bd + A 16 R a b R bc ad R de fh R fg R hi ci eg + A 17 R a b R bc de R de cf R fg R hi hi ag + A 18 R a b R bc df R de ac R fg R hi hi eg + A 19 R a b R bc df R de ah R fg R hi ei cg + A 20 R a b R bc df R de gh R fg R hi ei ac + A 21 R ab cd R cd eg R ef ai R gh R ij fj bh + A 22 R ab ce R cd af R ef gi R gh R ij bj dh + A 23R ab ce R cd ag R ef bi R gh R ij fj dh + A 24R ab ce R cd fg R ef hi R gh R ij aj bd, (3)
9 The Theory And the coefficients that define the new quintic quasi-topological interaction in (3) are: A 1 = , A 2 = , A 3 = , A 4 = , A 5 = , A 6 = , A 7 = , A 8 = , A 9 = , A 10 = , A 11 = , A 12 = , A 13 = , A 14 = , A 15 = , A 16 = , A 17 = , A 18 = , A 19 = , A 20 = , A 21 = , A 22 = , A 23 = , A 24 =
10 The Theory The quasitopological gravities are defined up to the addition of the corresponding Euler densities. A geometric interpretation of the quasitopological theories remains as an open problem.
11 Table of Contents 1 Introduction 2 The Theory 3 Some Properties 4 Final Comments
12 Some Properties: Birkhoff s Theorem Claim: For generic values of the couplings a k, the spherically (planar or hyperbolic) symmetric solution is static and it is determined by a quintic polynomial equation. The proof is done through the Reduced Action approach, evaluating the Lagrangian on the metric ds 2 = f (t, r)b 2 (t, r)dt 2 +2m(t, r)b(t, r) dtdr+ dr 2 f (t, r) +r 2 dσ 2 γ. Here dσ γ denotes the line element of a Euclidean 3d manifold of constant curvature γ {±1, 0}.
13 Some Properties: Birkhoff s Theorem For convenience, we define h(t, r) = f (t, r) γ. The variation of the reduced action with respect to h, b, m, and a posteriori gauge fixing m(t, r) = 0 leads to: 0 = ( 24r 6 h(t, r)a 2 6r 4 h(t, r) 2 a 3 4r 2 h(t, r) 3 a 4 + 5h(t, r) 4 a 5 + 6r 8 b(t, r) ) r 0 = h(t, r) 5 r 5 a 5 h(t, r) 4 r 3 a 4 2h(t, r) 3 r 1 a 3 12h(t, r) 2 ra 2 + r 5 Λ + 6r 3 h(t, r) + µ(t)r (5) 0 = ( 24r 6 h(t, r)a 2 6r 4 h(t, r) 2 a 3 4r 2 h(t, r) 3 a 4 + 5h(t, r) 4 a 5 + 6r 8 h(t, r) ) t (4) (6)
14 Some Properties: Birkhoff s Theorem The fact that the values a k are generic induces that h(t, r) b(t, r) = = 0, hence µ(t) is in fact constant. t r From this, b(t) can be absorbed by a time reparametrization, which means that the metric now reads: ds 2 = f (r)dt 2 + dr 2 f (r) + r 2 dσ 2 γ, Now it s easy to see that the solution is static.
15 Some Properties: No-ghosts around AdS Claim: Around maximally symmetric backgrounds, quintic quasitopological gravities lead to the same propagator that G.R, with an effective Newton s constant which depends on the values of the couplings a k. Fast-linearization procedure for gravity theories involving contractions of the Riemman tensor, ie, L(R αβρσ, g µν ) around maximally symmetric backgrounds. [P. Bueno, P.Cano: arxiv.org/ ].
16 Some Properties: No-ghosts around AdS The linearized field equations are written in terms of values a, b, c, e, which depend on the theory under consideration. Their method consists in the evaluation of the Lagrangian on a deformed curvature that depends on two parameters, (α, χ). The values a, b, c, e can be obtained by taking specific derivatives and evaluations on this effective action.
17 Some Properties: No-ghosts around AdS In the quintic quasitopological gravity case, we assume that the maximally symmetric solution has a dressed constant curvature, λ, which is fixed by the polynomial P[λ] := a 5 λ 5 + 6a 4 λ 4 72a 3 λ a 2 λ (Λ λ) = 0 Scaling λ λ 6 for simplicity, the linearized equations read dp[λ] dλ G L µν = 0, where G L µν is the linearized Einstein tensor.
18 Table of Contents 1 Introduction 2 The Theory 3 Some Properties 4 Final Comments
19 Final Comments Assuming that (5) has a solution f (r) with a single zero located at r = r h, the Black Hole thermodynamical properties can be analyzed. In this case we have: T = 1 2πr h ( 3a5 γ 5 2a 4 γ 4 r 2 h + 2a 3γ 3 r 4 h + 6γr 8 h 2Λr 10 h 5a 5 γ 4 + 4a 4 γ 3 rh 2 6a 3γ 2 rh a 2γrh 6 + 6r h 8 ( ) S = Vol(Σ γ ) 4πrh πγa 2 r h + 12πγ2 a 3 rh 2 8πγ3 a 4 3rh 3 2πγ4 a 5 rh 5 M = Vol(Σ ( γ) γ 5 a 5 2 rh 6 + γ4 a 4 rh 4 2γa ) 3 rh γ 2 a 2 + 6γrh 2 Λrh 4 satisfying the 1st Law of Thermodynamics, dm = T ds. )
20 Final Comments Quasitopological Gravities shares with its Lovelock counterpart a lot of properties. For mention a few: 2nd order field equations, although QTG have shown to have this property on spherically/planar/hyperbolic spacetimes. The asymptotic behavior allowed by Wheeler s polynomial coincides with that of General Relativity. Birkhoff s Theorem.
21 Final Comments THANKS FOR YOUR ATTENTION!
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