Higher Derivative Gravity Theories

Transcript

1 Higher Derivative Gravity Theories Black Holes in AdS space-times James Mashiyane Supervisor: Prof Kevin Goldstein University of the Witwatersrand Second Mandelstam, 20 January 2018 James Mashiyane WITS) Higher Derivative Gravity November / 15

2 Table of Contents 1 introduction 2 Equations of motion 3 Linearized gravity 4 Critical Gravity In 4 Dimensions 5 Massive Gravity Theory in 4 Dimensions 6 Thermodynamics 7 Continuation James Mashiyane WITS) Higher Derivative Gravity November / 15

3 Introduction We know that the AdS-Schwarzschild black hole in d = 4 is given by ds 2 = 1 2M r and coming from + Λ ) 3 r 2 + dt M r + Λ ) 1 3 r 2 + dr 2 + r 2 dω 2 2, 1) S = gd 4 x R 2Λ), 2) James Mashiyane WITS) Higher Derivative Gravity November / 15

4 Equations of motion I = gd 4 x R 2Λ + αr µν R µν + βr 2), 3) G µν + E µν = 0, G µν = R µν 1 2 Rg µν + Λg µν, E µν = 2α R ρµνσ R ρσ + 14 ) Rρσ R ρσ g µν + 2βR + α R µν Rg µν µ ν R arxiv.org/abs/ R µν 1 ) 4 Rg µν ) + 2β g µν R µ ν R). 4) James Mashiyane WITS) Higher Derivative Gravity November / 15

5 Linearized Gravity We consider excitations around AdS 4 and expand the metric around an AdS bachbround, under AdS background, we have g µν g µν + δg µν = g µν + h µν, 5) R µνρσ = Λ 3 g µρ g νσ g µσ g νρ ), We now vary up to first order expansion. R µν = Λg µν, R = 4Λ. 6) g µν = g µν h µν + Oh 2 ), 7) James Mashiyane WITS) Higher Derivative Gravity November / 15

6 Linearized Gravity G 1) µν = R 1) µν 1 2 R1) g µν Λh µν ρ µ h νρ + ρ ν h µα h µν µ ν h ) R µν 1) = 1 2 R 1) = µ ν h µν h Λh G 1) µν + E 1) µν ) = 1 + 2Λα + 4β)) G 1) µν + α 2Λ 3 )G 1) µν 2Λ 3 R1) g µν ) + α + 2β) µ ν + g µν + Λ g µν ) R 1). 8) James Mashiyane WITS) Higher Derivative Gravity November / 15

7 Linearized Gravity Solve for h µν, there are different gauge conditions that can be used to simplify these equations to solve them analytically. Choose the gauge condition, µ h µν = ν h. 9) ) g µν G µν 1) + E µν 1) = Λh 2α + 3β) h) = 0. 10) In this gauge h describes a propagating massive scalar mode. Case 1: α = 3β, then we see that it implies, h = 0, Case 2: α 3β. James Mashiyane WITS) Higher Derivative Gravity November / 15

8 Critical Gravity Massive spin-2 modes becomes mass-less at α = 3β. R 1) = 0, G 1) δg µν + E µν ) = 3β 2 µν = 1 2 h µν + Λ 3 h µν 2Λ 3 ) 4Λ 3 1 3β ) h µν = 0 11) James Mashiyane WITS) Higher Derivative Gravity November / 15

9 Critical Gravity In 4 Dimensions h µν transverse trace-less gauge, 2Λ 3 )hm) µν = 0, 12) where h µν is a mass-less graviton, and 4Λ 3 1 ) h µν M) = 0, 13) 3β describes a massive spin-2 field. In order to have stable spin-2 modes satisfying 2Λ 3 M2) h µν = 0 in the AdS 4 background requires that M 2 0, we have taken Λ < 0, we must have 0 < β 1 2Λ. 14) James Mashiyane WITS) Higher Derivative Gravity November / 15

10 Critical Gravity in 4 Dimensions If we set β = 1 2Λ. Then the equations of motion become, ) ) Λ 2Λ 4Λ Λ 3 1 )) 2Λ ) 4Λ Λ h µν = 0 ) h µν = 0. 15) Thus we have arrived to a theory that only describes only mass-less gravitons. James Mashiyane WITS) Higher Derivative Gravity November / 15

11 Massive Gravity in 4 Dimensions Case 2 α 3β 0 = δg µν + E µν ) = α 2 2 h µν Λα ) Λβ h µν + Λ 1 + 4Λα ) Λβ h µν ) 1 + 3α + 2β) 4α + 3β) + Λ ν ν h Λ 5α + 6β 12 α + 3β + 4Λ ) α + 6β) g µν h 16) 3 James Mashiyane WITS) Higher Derivative Gravity November / 15

12 Black Hole Thermodynamics Use the Wald s formalism to calculate the black hole s entropy. Define Q µν = 2L µνρσ ρ l σ + 17) where Q µν is an anti-symmetric tensor, and the terms denoted by ellipses vanish for stationary horizons. In our case, L µνρσ = δl δr µνρσ 18) For a stationary black hole, the entropy is given by, S = 1 Q µν dσ µν, 19) T where T is the horizon temperature, H is the horizon, and l k is the horizon normal. H James Mashiyane WITS) Higher Derivative Gravity November / 15

13 Black Hole Thermodynamics L = 1 g R 2Λ + αr µν 2κ 2 R µν + βr 2), δl g = δr µνρσ 4κ 2 g µρ g νσ g µσ g νρ ) + α g 2κ 2 R ρµ g νσ R νρ g µσ ) + β g 2κ 2 Rg µρ g σν g µσ g νρ ). 20) g Q µν = 2κ 2 µ l ν ν l µ ) α g κ 2 R ρµ g µ ρ l ν R νρ ρ l µ ) β g κ 2 R µ l ν ν l µ ). 21) James Mashiyane WITS) Higher Derivative Gravity November / 15

14 Black Hole Thermodynamics On the AdS background, R µν = Λg µν, Q µν = 1 + 2αΛ + 8βΛ)Q µν Einstein, 22) thus the entropy of black holes is given by, For critical gravity, S = 1 + 2αΛ + 8βΛ) A h 4G. 23) S CG = 1 + 2βΛ) A h 4G. 24) James Mashiyane WITS) Higher Derivative Gravity November / 15

15 Continuations We need to solve the of equations of motion using numerical methods, and search for new AdS non-schwarzschild black holes We need to determine the asymptotic solutions of the full set of equations of motion, as we did before. Check if the AdS non-schwarzschild black holes obey the first law of thermodynamics. Calculate the energy of both critical gravity and massive gravity. Matching conditions for the numerical solutions and the asymptotic solutions. James Mashiyane WITS) Higher Derivative Gravity November / 15