Second law of black-hole thermodynamics in Lovelock theories of gravity

Second law of black-hole thermodynamics in Lovelock theories of gravity Nilay Kundu YITP, Kyoto Reference : 62.04024 ( JHEP 706 (207) 090 ) With : Sayantani Bhattacharyya, Felix Haehl, R. Loganayagam, Mukund Rangamani

2nd law of BH thermodynamics in Lovelock Theories of gravity

2nd law of BH thermodynamics in Lovelock Theories of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory

2nd law of BH thermodynamics in Lovelock Theories of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory Why Higher derivative theory of gravity?

2nd law of BH thermodynamics in Lovelock Theories of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory Why Higher derivative theory of gravity? I = 6 G N g R + L matter + L HD - Low energy limit of any UV-complete theory of quantum gravity is expected to generate higher derivative corrections to the leading two-derivative Einstein-Hilbert action with matter - Particular form of the higher derivative correction depends on the particular UV completion, however there are limitations of a fully consistent quantum theory of gravity

2nd law of BH thermodynamics in Lovelock Theories of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory Why Higher derivative theory of gravity? I = 6 G N g R + L matter + L HD - We need to go beyond classical Einstein s theory of general relativity - String theory - a prominent consistent candidate for a UV-complete theory of gravity - It also has limitations ===> time dependent processes!!

2nd law of BH thermodynamics in Lovelock Theories of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory I = 6 G N g R + L matter + L HD Is there any general principle to constrain the low energy behavior of the effective theory of gravity? - Dynamical black holes==> They are theoretical laboratories for understanding quantum nature of gravity!! - One such general principle is 2nd law of black hole thermodynamics : which we can test on solutions of low energy effective theory of gravity.

2nd law of BH thermodynamics in Lovelock Theories of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory I = 6 G N g R + L matter + L HD

2nd law of BH thermodynamics in Lovelock Theories of gravity A general principle to constrain low energy effective theory of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory I = 6 G N g R + L matter + L HD What is the statement of 2nd law? - This is obviously a statement beyond equilibrium Eq ) Eq 2, Total Entropy Eq2 Total Entropy Eq - This is a non-local statement ==> Depends only on initial and final end points of the time evolution, two equilibrium points. - We can ensure this by constructing a local entropy function - fn. of the state variables - that is (a) monotonically increasing under time evolution (b) reduces to familiar notion of equilibrium values at the two end points

2nd law of BH thermodynamics in Lovelock Theories of gravity A general principle to constrain low energy effective theory of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory I = 6 G N g R + L matter + L HD What is the statement of 2nd law in gravity? Eq ) Eq 2, Total Entropy Eq2 Total Entropy Eq Equlibrium configuration ==> Metric with a killing horizon Equilibrium entropy ==> BH entropy on Killing horizon metric ) metric 2, BH Entropy metric2 BH Entropy metric - There is an entropy functional interpolating bw/ two equilibrium metric ==> a local version of the 2nd law

2nd law of BH thermodynamics in Lovelock Theories of gravity A general principle to constrain low energy effective theory of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory I = 6 G N g R + L matter + L HD BH Entropy metric2 BH Entropy metric What happens for Einstein s general relativity? - BH Entropy ==> Area of the black hole horizon - st law and 2nd law (Hawking s area theorem) are both known to be satisfied

2nd law of BH thermodynamics in Lovelock Theories of gravity A general principle to constrain low energy effective theory of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory I = 6 G N g R + L matter + L HD BH Entropy metric2 BH Entropy metric What happens for Einstein s general relativity? - BH Entropy ==> Area of the black hole horizon - st law and 2nd law (Hawking s area theorem) are both known to be satisfied What happens Beyond Einstein s general relativity? S Wald = 2 H L R abcd ab cd - Wald Entropy satisfies st law for any higher derivative correction to GR - No general proof of 2nd law

2nd law of BH thermodynamics in Lovelock Theories of gravity A general principle to constrain low energy effective theory of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory I = 6 G N g R + L matter + L HD BH Entropy metric2 BH Entropy metric No general proof of 2nd law beyond GR for Wald entropy BIG GOAL - To prove 2nd law for dynamical BH soln. in general Higher derivative theory of gravity ==> A local version construct an entropy functional

2nd law of BH thermodynamics in Lovelock Theories of gravity A general principle to constrain low energy effective theory of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory I = 6 G N g R + L matter + L HD BH Entropy metric2 BH Entropy metric No general proof of 2nd law beyond GR for Wald entropy BIG GOAL - To prove 2nd law for dynamical BH soln. in general Higher derivative theory of gravity ==> A local version construct an entropy functional OR - To find a concrete counter example such that we can rule out theories demanding 2nd law

2nd law of BH thermodynamics in Lovelock Theories of gravity A general principle to constrain low energy effective theory of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory I = 6 G N g R + L matter + L HD BH Entropy metric2 BH Entropy metric No general proof of 2nd law beyond GR for Wald entropy BIG GOAL - To prove 2nd law for dynamical BH soln. in general Higher derivative theory of gravity ==> A local version construct an entropy functional OR - To find a concrete counter example such that we can rule out theories demanding 2nd law What we achieved : - A small step!! == > We checked things in one simple model of higher derivative gravity ==> Lovelock theory

I = 6 G N g R + L matter + L HD S Wald = 2 H L R abcd ab cd Our Aim : Wald Entropy metric2 Wald Entropy metric - Constructing a local entropy function that is (a) monotonically increasing under time evolution (b) reduces to Wald entropy at the two equilibrium end points of the time evolution. - Modification of Wald entropy away from equilibrium Let us organize things a little better : - Higher derivative terms in the action comes with a characteristic length scale - We have dimension-less coupling - We start with an initial equilibrium configuration : A stationary black hole - We then perturb it slightly - Perturbations are denoted by two parameters ==> (a) amplitude a, (b) frequency w - The entropy function S total should have knowledge about these three parameters ==> (a) amplitude a, (b) frequency w, and, (c) coupling

I = 6 G N g R + L matter + L HD S Wald = 2 H L R abcd ab cd Our Aim : S total [a,!, ] Wald Entropy metric2 Wald Entropy metric - Constructing a local entropy function that is (a) monotonically increasing under time evolution (b) reduces to Wald entropy at the two equilibrium end points of the time evolution. - Modification of Wald entropy away from equilibrium What is known so far in literature : - For any coupling, but with a=0 : Wald entropy may be constructed as the desired entropy function - For f(r) theories, in finite range of coupling, but arbitrary a and w, the entropy function can be constructed - For small amplitude expansion (a << ), considering 4-derivative theories of gravity the entropy function can be constructed - In the context of holographic EE, particular correction to Wald entropy has been constructed, but again in small amplitude expansion (a << )

I = 6 G N g R + L matter + L HD S Wald = 2 H L R abcd ab cd Our Aim : S total [a,!, ] Wald Entropy metric2 Wald Entropy metric - Constructing a local entropy function that is (a) monotonically increasing under time evolution (b) reduces to Wald entropy at the two equilibrium end points of the time evolution. - Modification of Wald entropy away from equilibrium We work in a different expansion : - In our work we aim to construct the entropy function in the frequency expansion, but for arbitrary amplitude,!l s - We would work perturbatively in higher derivative interactions, treating the correction to Einstein s GR in a gradient expansion. - The small parameter is the dimensionless number!l s with arbitrary amplitude away from the equilibrium.

I = 6 G N g R + L matter + L HD S Wald = 2 H L R abcd ab cd Our Aim : S total [a,!, ] Wald Entropy metric2 Wald Entropy metric - Constructing a local entropy function that is (a) monotonically increasing under time evolution (b) reduces to Wald entropy at the two equilibrium end points of the time evolution. - Modification of Wald entropy away from equilibrium We work in a different expansion : - In our work we aim to construct the entropy function in the frequency expansion, but for arbitrary amplitude,!l s - In other words, we allow for arbitrary time evolution away from equilibrium, as long as this time evolution is sensibly captured by the low-energy effective action. - Geometrically, We allow fluctuations of BH horizon with at the horizon is small compared to the curvature scales. - We assume that the Classical gravity description is valid ==> no loop correction etc. enter in the game.

I = 6 G N g R + L matter + L HD S Wald = 2 H L R abcd ab cd Our Aim : S total [a,!, ] Wald Entropy metric2 Wald Entropy metric - Constructing a local entropy function that is (a) monotonically increasing under time evolution (b) reduces to Wald entropy at the two equilibrium end points of the time evolution. - Modification of Wald entropy away from equilibrium Our analysis :,!l s, a > 0 Question?? Naive answer If two derivative gravity, i.e. Einstein s GR, dominates at the horizon, is it possible at all that we need to modify the entropy away from equilibrium? Though the leading area contribution is large, it s variation may be anomalously small and contribution from higher derivative terms may overcome it.

The action for the Lovelock theory I = 4 L m = g R + X m=2 m `2m 2 s L m + L matter! µ µ m m m m R µ R m µm m m. m = dimensionless numbers, coupling l s = Some scale at which the higher derivative terms become important We restrict to the Gauss-Bonnet theory, m=2 I = 4 - We are neglecting the matter part, as it will not play any role in our analysis. Or we need to impose NEC. g R + 2 l 2s L 2 L 2 =L GB = R 2 R µ R µ + R µ R µ

The action for the Lovelock theory I = 4 L m = g R + X m=2 m `2m 2 s L m + L matter! µ µ m m m m R µ R m µm m m. m = dimensionless numbers, coupling l s = Some scale at which the higher derivative terms become important We restrict to the Gauss-Bonnet theory, m=2 - We start with one equilibrium/stationary metric of a BH with a regular horizon, by fixing a coordinate chart ds 2 =2dv dr I = 4 - We are neglecting the matter part, as it will not play any role in our analysis. Or we need to impose NEC. f(r, v, x) dv 2 +2k A (r, v, x) dv dx A + h AB (r, v, x) dx A dx B f(r, v, x) H + = k A (r, v, x) H + = @ r f(r, v, x) H + =0 g R + 2 l 2s L 2 L 2 =L GB = R 2 R µ R µ + R µ R µ The null hypersurface of the horizon H + is the locus r =0 a Spatial section (const. v slices) of H + = nely parametrized null generator of H + = @ v

ds 2 =2dv dr f(r, v, x) dv 2 +2k A (r, v, x) dv dx A + h AB (r, v, x) dx A dx B f(r, v, x) H + = k A (r, v, x) H + = @ r f(r, v, x) H + =0 The geometry has a horizon, a null hypersurface at r=0 H + The coordinates on the horizon = {v, x A } The coordinates on the constant v-slices of horizon ) {x A } SchematicsofHorizon coordinates A - generator r - generator Away from the horizon the coordinate r ==> Affinely parametrized along null geodesics piercing the horizon at an angle (@ v, @ r ) H + =, (@ r, @ A ) H + =0 v - generator r= 0 surface Horizon

We are working with Gauss-Bonnet theory I = 4 g R + 2 l 2s L 2 L 2 =L GB = R 2 R µ R µ + R µ R µ We want to construct an entropy functional Condition : @ v S total 0 Condition 2 : S total reduces to Wald entropy S Wald in equilibrium The Wald entropy needs to be modified away from equilibrium S total = S Wald + S cor S cor equilibrium =0

We are working with Gauss-Bonnet theory We want to construct an entropy functional I = 4 Condition : @ v S total 0 Condition 2 : S total reduces to Wald entropy g R + 2 l 2s L 2 L 2 =L GB = R 2 R µ R µ + R µ R µ S Wald in equilibrium The Wald entropy needs to be modified away from equilibrium S total = S Wald + S cor S cor equilibrium =0 How is 2nd law proved then? S final Wald S initial Wald = final initial @ v S total dv 0

I = 4 g R + 2 l 2s L 2 L 2 =L GB = R 2 4R µ R µ + R µ R µ Condition : @ v S total 0 Condition 2 : S total reduces to Wald entropy S Wald in equilibrium S total = S Wald + S cor S cor equilibrium =0 Strategy : () Define entropy density : S total = (2) Define ) @ v S total = 2 x p h d d (3) Considering! 0, as v! d d 2 x p h total ) calculate @ v, and show that @ v 0, (4) We get > 0, for all v (5) In tern we get @ v S total 0

Strategy : () Define entropy density : S total = (2) Define ) @ v S total = d d 2 x p h total d d 2 x p h (3) To obtain @ v S total 0, show that @ v 0 How does it work for Einstein GR (Hawking s area increase theorem) I = () S Wald = 2 x p h gr 4 d d (2) Einstein = 2 hab @ v h AB = K A A (3) @ v Einstein = K AB K AB R vv = K AB K AB T vv 0 (We used EOM R vv = T vv, and NEC T vv 0) What is the problem with higher derivative gravity? I = g R + L matter + L HD 6 G N @ v = K AB K AB R vv = K AB K AB T vv, R vv = T vv?? This equation will be changed due to the higher derivative term

Strategy : () Define entropy density : S total = (2) Define ) @ v S total = d d 2 x p h total d d 2 x p h (3) To obtain @ v S total 0, show that @ v 0 What is the problem with higher derivative gravity? I = g R + L matter + L HD 6 G N @ v = K AB K AB T vv, R vv = T vv?? This equation will be changed due to the higher derivative term @ v = For example we can have a situation where h AA0 h BB0 K A 0 B 0 KAB + l 2 s@ r @ v K AB T vv And @ r @ v K AB K AB l 2 s@ r @ v K AB May remain unsuppressed and violate the proof

In a perturbative amplitude expansion things do work out up to linearized order @ v S Wald = 4 I = 4 S Wald = 4 d d g (R + L HD ) d d 2 x p h + HD 2 x p h @ v (log p h)( + HD )+ @ v HD

In a perturbative amplitude expansion things do work out up to linearized order @ v S Wald = 4 I = 4 S Wald = 4 d d g (R + L HD ) d d 2 x p h + HD 2 x p h @ v (log p h)( + HD )+ @ v HD {z } = @ v = T vv + r v r v HD HD R vv + EOM vv

I = 4 Let us examine the Gauss-Bonnet case g R + 2 ls(r 2 2 4R µ R µ + R µ R µ ) S Wald = d d 2 x p h [ + 2 2 ls 2 R ind ] 4 STEP : () Obtain eq ) @ v S Wald = 2 x p h eq d d (2) Compute @ v eq and convince that @ v eq 0 not satisfied Conclusion : We need to modify Wald entropy STEP 2 : () S total = S Wald + S cor, S cor equilibrium =0 (2) Obtain total = eq + cor ) @ v S total = (3) Make sure that @ v total 0 d d 2 x p h eq + cor Conclusion : 2nd law is satisfied for Gauss-Bonnet theory

The final result for the Gauss-Bonnet case The Equilibrium quantities : I = 4 g R + 2 ls 2 (R 2 R µ R µ + R µ R µ ) S Wald = d d 2 x p h [ + 2 2 ls 2 R ind ] 4 The non-equilibrium correction : S total =S Wald + S cor S cor = d d 2 x p h cor 4 X such that, cor = n ls n @ v n 2 ls 2 h A B ls n @ v n 2 ls 2 h B A n=0 where, h A B = AA 2 BB B 2 A 2 The conditions for ==> @ v total = @ v eq + @ v cor 0 A n =2 n 2 n A n 2, for n = 2,, 0,, (for 2 = /2, 0 =, = 2), the constraint reads : A n 0 for n 2.

Let us examine the Gauss-Bonnet case with some explicit expressions I = 4 g S Wald = 4 R + 2 ls(r 2 2 4R µ R µ + R µ R µ ) d d 2 x p h [ + 2 2 ls 2 R ind ] STEP : () Obtain eq ) @ v S Wald = 2 x p h eq d d (2) Compute @ v eq and convince that @ v eq 0 not satisfied Conclusion : We need to modify Wald entropy STEP 2 : () S total = S Wald + S cor, S cor equilibrium =0 (2) Obtain total = eq + cor ) @ v S total = (3) Make sure that @ v total 0 d d 2 x p h eq + cor Conclusion : 2nd law is satisfied for Gauss-Bonnet theory

Let us examine the Gauss-Bonnet case with some explicit expressions STEP : @ v eq = T vv {z} T K AB K AB {z } T 2 + 2`2s KBK A B A0 M 0 BB0 AA {z } 0 T 3 + 2`2s KB A BA @ A 2 v AB B 2 A {z 2 + r A Y A, } {z } T 4 T 5! M BB0 AA 0 > does not contain any v-derivative.. () T +T 2 0 ) with NEC (2) T 2+T3=KBK A B A0 BA B 0 0 A 0 T 3 <T2 + 2`2sM BB0 AA 0 (3) T 2+T4=KB A K B BA A + 2`2s @ A 2 v AB B 2 A 2 This can spoil the proof - For Einstein s gravity, things work out nicely.. - We neglect Term-3 compared to Term-2 - Term-4 is potentially dangerous and if ``T4 > T2 ==>.@ v eq 0

STEP 2 : () S total = S Wald + S cor, S cor equilibrium =0 (2) Obtain total = eq + cor ) @ v S total = (3) Make sure that @ v total 0 d d 2 x p h eq + cor Question : How to decide the correction to wald entropy @ v total = @ v eq + @ v cor 0 @ v eq + @ v cor {z } = T {z} vv =@ v total T + r A Y A {z } T 5 K AB K AB {z } T 2 B A 2 AB B 2 A 2 {z } T 4 + 2 l 2 s K A B @ v + ( 2 ls) 2 2 AA @ A 2 v BB B 2 BA A 2 @ A 2 v AB B 2 A 2 {z } =@ v cor

STEP 2 : () S total = S Wald + S cor, S cor equilibrium =0 (2) Obtain total = eq + cor ) @ v S total = (3) Make sure that @ v total 0 d d 2 x p h eq + cor S cor = d d 2 x p h ( 2 l 2 s) 2 @ v cor = A A 2 BB B 2 A 2 ( 2 ls) 2 2 @ v B A 2 AB B 2 A 2 @ v A A 2 BB B 2 A 2 will produce the desired B A 2 AB B 2 A 2 @ v eq + @ v cor {z } = T {z} vv =@ v total T +( 2 l 2 s) 2 ( + r A Y A {z } T 5 K A B /4) @ v 2 ls 2 2 @ v A A 2 BB B 2 A 2 A A 2 BB B 2 A 2 2 @ v B A 2 AB B 2 A 2 ) @ v total 0 ==> if we fix the free parameter ==> /4

Final comments. This can be generalized to arbitrary orders in alpha expansion and also for Lovelock families.. 2. Our construction surely works for spherically symmetric configurations.. 3. The obstruction term should have some geometric meaning.. need to be explored.. 4. This construction is also not unique.. 5. Subtle issues regarding field re-definitions and foliation dependence.. 6. This method is also indirect.. 7. Possible connections with Holographic entanglement entropy..