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 ...

We study several aspects of higher-order gravities constructed from general contractions of the Riemann tensor and the metric in arbitrary dimensions. First, we use the fast-linearization procedure presented in arXiv:1607.06463 to obtain the equations satisfied by the metric perturbation modes on a maximally symmetric background in the presence of matter and to classify $\mathcal{L}($Riemann$)$ theories according to their spectrum. Then, we linearize all theories up to quartic order in curvature and use this result to construct quartic versions of Einsteinian cubic gravity (ECG). In addition, we show that the most general cubic gravity constructed in a dimension-independent way and which does not propagate the ghost-like spin-2 mode (but can propagate the scalar) is a linear combination of $f($Lovelock$)$ invariants, plus the ECG term, plus a New ghost-free gravity term. Next, we construct the generalized Newton potential and the Post-Newtonian parameter $\gamma$ for general $\mathcal{L}($Riemann$)$ gravities in arbitrary dimensions, unveiling some interesting differences with respect to the four-dimensional case. We also study the emission and propagation of gravitational radiation from sources for these theories in four dimensions, providing a generalized formula for the power emitted. Finally, we review Wald's formalism for general $\mathcal{L}($Riemann$)$ theories and construct new explicit expressions for the relevant quantities involved. Many examples illustrate our calculations.Comment: 83 pages, 3 figures, 4 tables; v2: minor modifications to match published version, references adde

We present measurements of the radial gravitational acceleration around isolated galaxies, comparing the expected gravitational acceleration given the baryonic matter (gbar) with the observed gravitational acceleration (gobs), using weak lensing measurements from the fourth data release of the Kilo-Degree Survey (KiDS-1000). These measurements extend the radial acceleration relation (RAR), traditionally measured using galaxy rotation curves, by 2 decades in gobs into the low-acceleration regime beyond the outskirts of the observable galaxy. We compare our RAR measurements to the predictions of two modified gravity (MG) theories: modified Newtonian dynamics and Verlinde’s emergent gravity (EG). We find that the measured relation between gobs and gbar agrees well with the MG predictions. In addition, we find a difference of at least 6σ between the RARs of early- and late-type galaxies (split by Sérsic index and u − r colour) with the same stellar mass. Current MG theories involve a gravity modification that is independent of other galaxy properties, which would be unable to explain this behaviour, although the EG theory is still limited to spherically symmetric static mass models. The difference might be explained if only the early-type galaxies have significant (Mgas ≈ M⋆) circumgalactic gaseous haloes. The observed behaviour is also expected in Λ-cold dark matter (ΛCDM) models where the galaxy-to-halo mass relation depends on the galaxy formation history. We find that MICE, a ΛCDM simulation with hybrid halo occupation distribution modelling and abundance matching, reproduces the observed RAR but significantly differs from BAHAMAS, a hydrodynamical cosmological galaxy formation simulation. Our results are sensitive to the amount of circumgalactic gas; current observational constraints indicate that the resulting corrections are likely moderate. Measurements of the lensing RAR with future cosmological surveys (such as Euclid) will be able to further distinguish between MG and ΛCDM models if systematic uncertainties in the baryonic mass distribution around galaxies are reduced.

We show that the linearized higher derivative gravitational field equations are equivalent to an equilibrium condition on the entanglement entropy of small spherical regions in vacuum. This extends Jacobson's recent derivation of the Einstein equation using entanglement to include general higher derivative corrections. The corrections are naturally associated with the subleading divergences in the entanglement entropy, which take the form of a Wald entropy evaluated on the entangling surface. Variations of this Wald entropy are related to the field equations through an identity for causal diamonds in maximally symmetric spacetimes, which we derive for arbitrary higher derivative theories. If the variations are taken holding fixed a geometric functional that we call the generalized volume, the identity becomes an equivalence between the linearized constraints and the entanglement equilibrium condition. We note that the fully nonlinear higher curvature equations cannot be derived from the linearized equations applied to small balls, in contrast to the situation encountered in Einstein gravity. The generalized volume is a novel result of this work, and we speculate on its thermodynamic role in the first law of causal diamond mechanics, as well as its possible application to holographic complexity.

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