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

Transcription

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

2 2nd law of BH thermodynamics in Lovelock Theories of gravity

3 2nd law of BH thermodynamics in Lovelock Theories of gravity Higher derivative theories of gravity > Lovelock / Gauss Bonnet Theory

4 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?

5 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

6 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

7 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

8 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

9 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!!

10 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?

11 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.

12 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

13 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

14 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.

15 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

16 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

17 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

18 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

19 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

20 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

21 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

22 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

23 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

24 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

25 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

26 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

27 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

28 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

29 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 << )

30 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.

31 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.

32 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

33 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?? 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?

34 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.

35 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

36 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

37 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 (@ r ) H + =, (@ A ) H + =0 v - generator r= 0 surface Horizon

38 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 (@ r ) H + =, (@ A ) H + =0 We define these quantities K AB = vh AB r=0, K AB = rh AB r=0 ) K AB vanishes at equilibrium The extrinsic curvature on the horizon slice v - generator r= 0 surface Horizon

39 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

40 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

41 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 v S total dv 0

42 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 ) v, and show v 0, (4) We get > 0, for all v (5) In tern we v S total 0

43 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 ) v, and show v 0, (4) We get > 0, for all v (5) In tern we v S total 0

44 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 ) v, and show v 0, (4) We get > 0, for all v (5) In tern we v S total 0

45 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 v S total 0, show 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 v h AB = K A A 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)

46 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 v S total 0, show 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 v h AB = K A A 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 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

47 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 v S total 0, show v 0 What is the problem with higher derivative gravity? I = g R + L matter + L HD 6 G v = K AB K AB T vv, R vv = T vv?? This equation will be changed due to the higher derivative term

48 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 v S total 0, show v 0 What is the problem with higher derivative gravity? I = g R + L matter + L HD 6 G v = K AB K AB T vv, R vv = T vv?? This equation will be changed due to the higher derivative v = For example we can have a situation where h AA0 h BB0 K A 0 B 0 KAB + l 2 s@ v K AB T vv v K AB K AB l 2 s@ v K AB May remain unsuppressed and violate the proof

49 In a perturbative amplitude expansion things do work out up to linearized 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 v (log p h)( + HD v HD

50 In a perturbative amplitude expansion things do work out up to linearized 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 v (log p h)( + HD v HD {z } v = T vv + r v r v HD HD R vv + EOM vv

51 In a perturbative amplitude expansion things do work out up to linearized 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 v (log p h)( + HD v HD {z } v = T vv + r v r v HD HD R vv + EOM vv {z } O(a 2 v = T vv < 0 ==>2nd law is valid up to linear order in amplitude

52 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 [ ls 2 R ind ] 4 STEP : () Obtain eq v S Wald = 2 x p h eq d d (2) v eq and convince v eq 0 not satisfied Conclusion : We need to modify Wald entropy

53 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 [ ls 2 R ind ] 4 STEP : () Obtain eq v S Wald = 2 x p h eq d d (2) v eq and convince 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 v total 0 d d 2 x p h eq + cor Conclusion : 2nd law is satisfied for Gauss-Bonnet theory

54 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 [ 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 v n 2 ls 2 h A B ls v n 2 ls 2 h B A n=0 where, h A B = AA 2 BB B 2 A 2

55 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 [ 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 v n 2 ls 2 h A B ls 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.

56 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 [ ls 2 R ind ] STEP : () Obtain eq v S Wald = 2 x p h eq d d (2) v eq and convince 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 v total 0 d d 2 x p h eq + cor Conclusion : 2nd law is satisfied for Gauss-Bonnet theory

57 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 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..

58 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 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 + 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

59 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 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 + 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 - We need to modify the equilibrium Wald entropy

60 STEP 2 : () S total = S Wald + S cor, S cor equilibrium =0 (2) Obtain total = eq + cor v S total = (3) Make sure v total 0 d d 2 x p h eq + cor Question : How to decide the correction to wald v total v eq v cor v eq = T vv {z} 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 l 2 s K A v

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

62 STEP 2 : () S total = S Wald + S cor, S cor equilibrium =0 (2) Obtain total = eq + cor v S total = (3) Make sure v total 0 d d 2 x p h eq + cor Question : How to decide the correction to wald v total v eq v cor 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 l 2 s K A v + ( 2 ls) 2 2 A 2 v BB B 2 BA A A 2 v AB B 2 A 2 {z } =@ v cor S cor = d d 2 x p h ( 2 l 2 s) v cor = A A 2 BB B 2 A 2 ( 2 ls) 2 v B A 2 AB B 2 A v A A 2 BB B 2 A 2 will produce the desired B A 2 AB B 2 A 2

63 STEP 2 : () S total = S Wald + S cor, S cor equilibrium =0 (2) Obtain total = eq + cor v S total = (3) Make sure v total 0 d d 2 x p h eq + cor S cor = d d 2 x p h ( 2 l 2 s) v cor = A A 2 BB B 2 A 2 ( 2 ls) 2 v B A 2 AB B 2 A v A A 2 BB B 2 A 2 will produce the desired B A 2 AB B 2 A 2

64 STEP 2 : () S total = S Wald + S cor, S cor equilibrium =0 (2) Obtain total = eq + cor v S total = (3) Make sure v total 0 d d 2 x p h eq + cor S cor = d d 2 x p h ( 2 l 2 s) v cor = A A 2 BB B 2 A 2 ( 2 ls) 2 v B A 2 AB B 2 A v A A 2 BB B 2 A 2 will produce the desired B A 2 AB B 2 A 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 v 2 ls 2 v A A 2 BB B 2 A 2 A A 2 BB B 2 A 2 v B A 2 AB B 2 A 2 v total 0 ==> if we fix the free parameter ==> /4

65 STEP 2 : () S total = S Wald + S cor, S cor equilibrium =0 (2) Obtain total = eq + cor v S total = (3) Make sure v total 0 d d 2 x p h eq + cor S cor = d d 2 x p h ( 2 l 2 s) v cor = A A 2 BB B 2 A 2 ( 2 ls) 2 v B A 2 AB B 2 A v A A 2 BB B 2 A 2 will produce the desired B A 2 AB B 2 A 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 v 2 ls 2 v A A 2 BB B 2 A 2 A A 2 BB B 2 A 2 v B A 2 AB B 2 A 2 v total 0 ==> if we fix the free parameter ==> /4 - Also the obstruction term (T5) vanishes r A Y A =4 2 ls 2 r A r B KK AB KCK A BC h AB 2 (K2 K CD K CD ) =0 - for spherically symmetric cases it indeed vanishes..

66 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..

67 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.. : THANK YOU VERY MUCH FOR YOUR KIND ATTENTION :