Gravitational thermodynamics of causal diamonds in (A)dS

Abstract

A maximally symmetric causal diamond is a solution to Einstein's equation with a cosmological constant. We establish a Smarr formula for such diamonds and a "first law" for variations to nearby solutions. The latter relates the variations of the bounding area, spatial volume of the maximal slice, cosmological constant, and matter Hamiltonian. The total Hamiltonian is the generator of evolution along the conformal Killing vector that preserves the diamond. To interpret the first law as a thermodynamic relation, it appears necessary to attribute a negative temperature to the diamond, as has been previously suggested for the special case of the static patch of de Sitter spacetime. With quantum corrections included, for small diamonds we recover the "entanglement equilibrium" result that the generalized entropy is stationary at the maximally symmetric vacuum at fixed volume, and we reformulate this as the stationarity of free conformal energy with the volume not fixed. * jacobson@umd.edu † m.r.visser@uva.nl E Conformal transformation from Rindler space to causal diamond 55 F Acceleration of the conformal Killing flow 56 (3) Rindler horizon ⇣ ⇠ i < l a t e x i t s h a 1 _ b a s e 6 4 = " Z S b F s 6 J e H r C 0 A r l k C Y O j u o 0 P K O 0 = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B i y U R Q Y 9 F L x 4 r 2 g 9 o Y 9 l s J + 3 S z S b s b o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g o r q 2 v r G 8 X N 0 t b 2 z u 5 e e f + g q e N U M W y w W M S q H < l a t e x i t s h a 1 _ b a s e 6 4 = " A w 7 J B 7 u Q b N k z 9 S A 4 r 0 1 X c s L h w / I = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E q M e i F 4 8 V 7 Q e 0 s W y 2 m 3 b p Z h N 2 J 0 I J / Q l e P C j i 1 V / k z X / j t s 1 B W x 8 M P N 6 b Y W Z e k E h h 0 H W / n c L a + s b m V n G 7 t L O 7 t 3 9 Q P j x q m T Q c z S i C t k k h r T 9 d w E / Y x q F E z y a a m X G p 5 Q N q Z D 3 r V U 0 Y g b P 5 u f O i V n V h m Q M N a 2 F J K 5 + n s i o 5 E x k y i w n R H F k V n 2 Z u J / X j f F 8 M r P h E p S 5 I o t F o W p J B i T 2 d 9 k I D R n K C e W U K a F v Z W w E d W U o U 2 n Z E P w l l 9 e J a 2 L q u d W v b v L S v 0 6 j 6 M I J 3 A K 5 + B B D e p w C w 1 o A o M h P M M r v D n S e X H e n Y 9 F a 8 H J Z 4 7 h D 5 z P H / G + j Y 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A w 7 J B 7 u Q b N k z 9 S A 4 r 0 1 X c s L h w / I = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E q M e i F 4 8 V 7 Q e 0 s W y 2 m 3 b p Z h N 2 J 0 I J / Q l e P C j i 1 V / k z X / j t s 1 B W x 8 M P N 6 b Y W Z e k E h h 0 H W / n c L a + s b m V n G 7 t L O 7 t 3 9 Q P j x q m T Q c z S i C t k k h r T 9 d w E / Y x q F E z y a a m X G p 5 Q N q Z D 3 r V U 0 Y g b P 5 u f O i V n V h m Q M N a 2 F J K 5 + n s i o 5 E x k y i w n R H F k V n 2 Z u J / X j f F 8 M r P h E p S 5 I o t F o W p J B i T 2 d 9 k I D R n K C e W U K a F v Z W w E d W U o U 2 n Z E P w l l 9 e J a 2 L q u d W v b v L S v 0 6 j 6 ...

Aspects of the first law of complexity

“…Hence we may conclude that our results will agree with those arising from the analysis of coherent states in both [125] and [126].…”

“…Similar independent discussions have appeared in [125,126] relating the volume variation to an integral of the matter stress tensor perturbation, as in (3.116), even though their derivations are different. We show below their results are equivalent to our explicit volume extremisation.…”

“…(3.165) 23 In particular, the definition of the stress energy tensor in [126] differs by a factor ε 2 /16πG N from ours. Furthermore, they use coordinates where global AdS is…”

“…Recall that in section 3.3.3, we commented on the relation of our results for δC V with similar variations studied in [125,126]. Here, we would like to consider the possible connection of our first law of complexity (2.25) with the second law in [25].…”

“…Translating the results of [126] to the AdS coordinates (3.5) and taking the large volume limit yields ζ WDW | t Σ =0 = cos ρ ∂ t with κ = 1 (3.163) (see [126] for the details). Substituting the explicit expressions into (3.158) we thus obtain…”