Black Hole Entropy in the presence of Chern-Simons terms

Transcription

1 Black Hole Entropy in the presence of Chern-Simons terms Yuji Tachikawa School of Natural Sciences, Institute for Advanced Study Dec 25, 2006, Yukawa Inst. based on hep-th/ , to appear in Class. Quant. Grav. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 1 / 35

2 Contents 1. Introduction 2. Black Hole Spacetime 3. Black Hole Spacetime in Higher Derivative Gravity 4. Derivation of the First law 5. Summary Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 2 / 35

3 Black Holes Thermodynamics Classical General Relativity leads to κ δa 2π 4G N = δm ΩδJ Semiclassical analysis identifies Very natural to identify T H = κ 2π. S = and look for statistical explanation. A 4G N Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 3 / 35

4 String theory Extremal charged black hole D-branes Excitations can be counted, account for S = A 4G N [Strominger-Vafa], [Maldacena-Strominger-Witten] Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 4 / 35

5 String theory Extremal charged black hole D-branes Excitations can be counted, account for S = A 4G N + [Strominger-Vafa], [Maldacena-Strominger-Witten] Also predicts subleading corrections. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 4 / 35

6 Corrections to the area law Einstein-Hilbert L = g R 16πG N Area law S = A 4G N Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 5 / 35

7 Corrections to the area law Einstein-Hilbert corrected: L = R g( + c 16πG N 2 R2 + ) Area law accordingly modified : S = A + 8πc R rtrt + 4G N hor Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 5 / 35

8 Corrections to the area law Einstein-Hilbert corrected: L = R g( + c 16πG N 2 R2 + ) Area law accordingly modified : S = A + 8πc R rtrt + 4G N hor Wald s formula: δl S = 2π g ɛ ab ɛ cd hor δr abcd Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 5 / 35

9 String theory works Two way to calculate corrections: Microscopic d.o.f. on the brane Macroscopic Wald s formula Completely agrees! [de Wit et al.][ooguri-strominger-vafa] Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 6 / 35

10 String theory works Two way to calculate corrections: Microscopic d.o.f. on the brane Macroscopic Wald s formula Completely agrees! In four dimensions. [de Wit et al.][ooguri-strominger-vafa] Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 6 / 35

11 Odd dimensions Black rings in 5d. Entropy correction: Microscopic done Macroscopic not yet Why? Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 7 / 35

12 Odd dimensions Black rings in 5d. Entropy correction: Microscopic done Macroscopic not yet Why? Wald s formula isn t applicable to gravitational Chern-Simons: F tr Γ R Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 7 / 35

13 Odd dimensions Black rings in 5d. Entropy correction: Microscopic done Macroscopic not yet Why? Wald s formula isn t applicable to gravitational Chern-Simons: F tr Γ R Our aim today Entropy correction from grav. CS. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 7 / 35

14 Contents 1. Introduction 2. Black Hole Spacetime 3. Black Hole Spacetime in Higher Derivative Gravity 4. Derivation of the First law 5. Summary Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 8 / 35

15 Contents 1. Introduction 2. Black Hole Spacetime 3. Black Hole Spacetime in Higher Derivative Gravity 4. Derivation of the First law 5. Summary Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 9 / 35

16 Terminology Killing Horizon 2 ξ =0 n ξ Killing Horizon : ξ 2 = 0. Binormal ɛ ab = ξ a n b ξ b n a Surface gravity a ξ b = κɛ ab + Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 10 / 35

17 Terminology Bifurcate Horizon 2 ξ =0 n Bifurcation surface ξ=0 ξ Bifurcation Surface ξ = 0. Bifurcate horizon : A pair of horizons which pass BS. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 11 / 35

18 Hawking radiation Bifurcate Horizon 2 ξ =0 n Bifurcation surface ξ=0 ξ Free field radiates at T H = κ/2π. Unruh effect near B.S. Depends only on bg metric, Not quantum gravity per se. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 12 / 35

19 Bifurcation surface Horizon Bifurcate Horizon 2 ξ =0 Bifurcation surface ξ=0 Horizon cross section at finite t is the b.s. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 13 / 35

20 Holonomy at B.S. At the B.S., c ɛ ab = κ 1 c a ξ b = κ 1 R abcd ξ d = 0. holonomy reduces to SO(D 1, 1) SO(1, 1) N SO(D 2) Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 14 / 35

21 Black Hole Thermodynamics Zeroth law κ constant on the horizon. First law κ δa 2π 4G N = δm ΩδJ Second law A increases with time. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 15 / 35

22 Contents 1. Introduction 2. Black Hole Spacetime 3. Black Hole Spacetime in Higher Derivative Gravity 4. Derivation of the First law 5. Summary Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 16 / 35

23 Higher derivative corrections RG perspective Planck suppressed terms R 2, R 4 etc. Coefficients calculable in string compactifications Affects black hole solutions. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 17 / 35

24 Conceptual problems Null dir. of g ab maximal propagation speed. e.g. Field redefinition g ab g ab + cr ab + c Rg ab + changes the light cone Physics should be invariant! Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 18 / 35

25 Bifurcate Horizon in Higher derivative gravity Define the horizon using the light cone of g ab. Suppose it has Killing, bifurcate horizon : under g ab g ab + h ab, Bifurcation surface invariant, defined by ξ a = 0 Bifurcate horizon ξ a ξ a g ab ξ a ξ b + h ab ξ a ξ b. 2nd term invariant under ξ zero by evaluating at B.S. Hawking temperature Evaluate κ 2 = a ξ b 2 at B.S. Christoffel drops off because ξ a = 0. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 19 / 35

26 Black Hole Thermo. in Higher Derivative Gravity Zeroth law κ constant on the horizon : assumption First law κ S = δm ΩδJ, 2π S given by Wald s formula Second law Difficult to establish Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 20 / 35

27 Wald s formula Black Hole Entropy S = 2π hor δl g ɛ ab ɛ cd δr abcd for L = L(g ab, R abcd, e R abcd, ; φ, ) It does not include gravitational Chern-Simons or Green-Schwarz type coupling δl = tr Γ R 2n 1 δl = 1 2 H 2 with H = db + tr Γ R. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 21 / 35

28 Contents 1. Introduction 2. Black Hole Spacetime 3. Black Hole Spacetime in Higher Derivative Gravity 4. Derivation of the First law 5. Summary Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 22 / 35

29 Notation D space-time dimension L(φ) Lagrangian density as D-form φ collective symbol for the fields. g µν, A µ,... ξ Lie derivative by a vector field ξ, ξ ω = (dι ξ + ι ξ d)ω. δ ξ Variation under diffeo. e.g. where (U ξ ) a b = bξ a. δ ξ Γ a b = ξγ a b + d(u ξ) a b Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 23 / 35

30 Diffeomorphism Invariance Assumption by Wald: Cannot incorporate the CS. δ ξ L(φ) = ξ L(φ). Today: δ ξ L(φ) = ξ L(φ) + dξ ξ. Almost verbatim transcript of Wald s original. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 24 / 35

31 Covariant Hamiltonian Method EOM E φ and symplectic potential Θ via δl = E φ δφ + dθ(φ, δφ) Θ pairs of coordinate and momenta L(φ) = 1 dφ dφ 2 Θ = dφ δφ symplectic form given by Ω(φ, δ 1 φ, δ 2 φ) = δ 1 Θ(φ, δ 2 φ) δ 2 Θ(φ, δ 1 φ). Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 25 / 35

32 Lemma j : a form constructed from fields φ an external field ξ suppose j is closed on-shell for any ξ. It is then exact on-shell. i.e. : equality on-shell. dj ξ 0 j ξ dq ξ. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 26 / 35

33 Noether s theorem Symmetry leads to conserved current : j ξ = Θ(φ, δ ξ φ) ι ξ L Ξ ξ satisfies The lemma implies j ξ dq ξ, dj ξ 0 j ξ C C Q ξ. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 27 / 35

34 Noether s theorem Symmetry leads to conserved current : j ξ = Θ(φ, δ ξ φ) ι ξ L Ξ ξ satisfies The lemma implies j ξ dq ξ, i.e. Global symmetry Gauge symmetry dj ξ 0 j ξ C C Q ξ. Conserved charges Gauss law Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 27 / 35

35 Un-illuminating main part Define Π ξ via δ ξ Θ = ξ Θ + Π ξ. Recall δl = E φ δφ + dθ, δ ξ = ξ L + dξ ξ. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 28 / 35

36 Un-illuminating main part Define Π ξ via δ ξ Θ = ξ Θ + Π ξ. Recall δl = E φ δφ + dθ, δ ξ = ξ L + dξ ξ. Calculating δδ ξ L in two ways: dπ ξ δd Ξ ξ Π ξ δ Ξ ξ dσ ξ. δj ξ = δθ(φ, δ ξ φ) ι ξ δl δ Ξ ξ δθ(φ, δ ξ φ) δ ξ Θ(φ, δφ) + dι ξ Θ + Π ξ δ Ξ ξ Ω(φ, δφ, δ ξ φ) + d(ι ξ Θ + Σ ξ ). Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 28 / 35

37 Un-illuminating main part Define Π ξ via δ ξ Θ = ξ Θ + Π ξ. Recall δl = E φ δφ + dθ, δ ξ = ξ L + dξ ξ. Calculating δδ ξ L in two ways: dπ ξ δd Ξ ξ Π ξ δ Ξ ξ dσ ξ. δj ξ = δθ(φ, δ ξ φ) ι ξ δl δ Ξ ξ δθ(φ, δ ξ φ) δ ξ Θ(φ, δφ) + dι ξ Θ + Π ξ δ Ξ ξ Ω(φ, δφ, δ ξ φ) + d(ι ξ Θ + Σ ξ ). Thus, for C ξ with δc ξ = ι ξ Θ + Σ ξ, δdq ξ Ω(φ, δφ, δ ξφ) where Q ξ = Q ξ C ξ. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 28 / 35

38 Recap. δdq ξ Ω(φ, δφ, δ ξφ) means is the Hamiltonian generating ξ. dq ξ Let ξ = t + Ωφ where ξ Horizon generating Killing t global time translation φ angular rotation Then δ Q ξ δ Q t + Ω δ hor Q φ. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 29 / 35

39 First Law δ Q ξ δ Q t + Ω δ Q φ. hor E = Q t, J = δ Q φ. Taking the horizon at the bifurcation surface, Q ξ = κ Q ξ ξ 0, aξb ɛ ab Then, define hor hor S = 2π Q ξ ξ 0, aξb ɛ ab hor Result. κ δs = δe + ΩδJ. 2π Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 30 / 35

40 3d Chern-Simons Start from L CS = β tr(γ R 1 3 Γ 3 ) δ ξ L CS = ξ L CS d(β tr du ξ Γ ) Ξ ξ = β tr du ξ Γ Non-covariant part in Θ is β tr Γ δγ Π ξ = β tr du ξ δγ Π ξ δξ ξ 0 Σ ξ = 0 j ξ = Θ(φ, δ ξ φ) Ξ ξ + = 2β tr du ξ Γ + Q ξ = 2β tr U ξγ +. (U ξ ) a b = bξ a. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 31 / 35

41 3d Chern-Simons L CS = β tr(γ R 1 3 Γ 3 ) where Γ N = ɛ ν µγ µ νρdx ρ /2. agrees with known corrections to BTZ. Naive application of Wald 4πβ S CS = 8πβ Γ N hor hor Γ N. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 32 / 35

42 Chern-Simons in General Dimensions L CS = β tr(γ R 2m 1 + ) S CS = 8πmβ Γ N R 2m 2 N Recall hor Holonomy at the bifurcation surface Entropy correction = CS of the normal bundle! = SO(1, 1) N SO(D 2) Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 33 / 35

43 Contents 1. Introduction 2. Black Hole Spacetime 3. Black Hole Spacetime in Higher Derivative Gravity 4. Derivation of the First law 5. Summary Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 34 / 35

44 Summary Spacetime with black holes reviewed. Entropy correction from the grav. CS. Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 35 / 35

45 Summary Spacetime with black holes reviewed. Entropy correction from the grav. CS. Outlook Need application! Black rings. working on it with friends. 7d black holes. tr Γ R 3 hor idea wanted. Γ N RN 2 Yuji Tachikawa (SNS, IAS) Dec 2006, Yukawa 35 / 35