The geometry of black hole entropy John Dougherty UC San Diego - PowerPoint PPT Presentation

The geometry of black hole entropy John Dougherty UC San Diego March 13, 2015 John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 1 / 26

Introduction The laws of BHT 0 κ is constant on the horizon 1 1 δ M = 8 π κ δ A + Ω δ J 2 δ A ≥ 0 in any process 3 κ = 0 not achievable by any process John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 2 / 26

Introduction The laws of BHT 1 1 δ M = 8 π κ δ A + Ω δ J 2 δ A ≥ 0 in any process Puzzles 1 The δ acting on M and J represents a perturbation of a quantity at spatial infinity; δ A is a perturbation at the horizon. How do they relate? (Curiel 2014) 2 How do the δ A in the first and second laws relate? John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 3 / 26

Introduction Wald entropy Iyer and Wald (1994) propose a definition of entropy that will help us answer these. The Wald entropy applies to any diffeomorphism covariant Lagrangian field theory of the form � � ◦ ◦ ◦ ◦ ◦ ◦ ◦ L g ab , ∇ a 1 g ab , . . . , ∇ a k ) g ab , ψ, ∇ a 1 ψ, . . . , ∇ a l ) ψ, γ ∇ ( a 1 · · · ∇ ( a 1 · · · On this definition, the perturbations do not act at spatial infinity and the horizon, but globally. John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 4 / 26

Introduction Wald entropy Two steps 1 Show that � � ◦ ◦ ◦ ◦ ◦ ◦ ◦ L g ab , ∇ a 1 g ab , . . . , ∇ ( a 1 · · · ∇ a k ) g ab , ψ, ∇ a 1 ψ, . . . , ∇ ( a 1 · · · ∇ a l ) ψ, γ may be writtenmay be written � � L g ab , ∇ a 1 R bcde , . . . , ∇ ( a 1 · · · ∇ a m ) R bcde , ψ, ∇ a 1 ψ, . . . , ∇ ( a 1 · · · ∇ a l ) ψ with R bcde the Riemann tensor of g ab . 2 Construct differential forms satisfying the first law using our new L . John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 5 / 26

Variational calculus Let M be some spacetime, and consider a bundle π : E → M over it. John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 6 / 26

Variational calculus The 1-jet bundle π 1 : J 1 E → M adds data about first derivatives. John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 7 / 26

Variational calculus The 2-jet bundle π 2 : J 2 E → M adds data about second derivatives. John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 8 / 26

Variational calculus The ( ∞ -)jet bundle π ∞ : JE → M is the inverse limit JE = lim − J n E . ← JE π ∞ π ∞ n +1 n π n +2 π n π n +1 n +1 J n +1 E n − 1 n J n E · · · · · · Above some point p ∈ M , JE has all possible Taylor series around p . It gives all the ways a section of E could look over an infinitesimal region. For any section φ of E , there is a section j ∞ φ of E that assigns to p ∈ M the Taylor expansion of φ about p . John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 9 / 26

Variational calculus Variational bicomplex The variational bicomplex Ω ∗ , ∗ ( JE ) of E is the de Rham complex of differential forms on JE with the exterior derivative d + δ . For a Lagrangian L , the global first variational formula δ L = E + d Θ is an equation of ( n , 1)-forms on JE . Wald’s locally constructed forms Ω ∗ loc ( M ) are the image of the pullback along ( id , j ∞ ) → M × Γ( JE ) ev e ∞ : M × Γ( E ) − − − − − → JE John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 10 / 26

Variational calculus Global equation of first variation (Zuckerman, 1987) Theorem For any ( n , 0)-form L , there is an ( n , 1)-form E and an ( n − 1, 1)-form Θ such that δ L = E + d Θ John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 11 / 26

Gauss–Bonnet Example: The Gauss–Bonnet theorem (Anderson 1989, xxii–xxv) The Gauss–Bonnet theorem: � K dA = 2 πχ ( X ) M for a 2D Riemannian X with Gaussian curvature K . Take E = R 2 × R 3 over M = R 2 , restrict attention to local, regularly parametrized surfaces. δ L = d η , so the LHS vanishes. John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 12 / 26

Gauss–Bonnet What gives? We’re missing the global aspects of the problem. The problem should be invariant under Euclidean motions in R 3 and 1 orientation-preserving diffeomorphisms of the base space. If we consider only equivariant forms, then we no longer have δ L = d η . 2 This leads us to the global first variational formula 3 δ L = E + d Θ Lesson: the global first variational formula δ L = E + d Θ incorporates covariance, and encodes global information about the bundle of interest. John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 13 / 26

Iyer and Wald step 1 Iyer and Wald step 1 A Lagrangian is determined by a function L : JE → R . IW’s result: there is a bijection between functions JE g × M JE ψ × M E ◦ γ → R and functions JE g × M JE R × M JE ψ → R such that L uses the R abcd of g ab . John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 14 / 26

Iyer and Wald step 1 Proof Consider the function u : J 2 E g → E g × M E R u : ( p , g µν , g µν,λ , g µν,λσ ) �→ (( p , g µν ) , ( p , R µνλσ )) u splits the projection JE g × M JE R → JE g (i.e., pr 1 ◦ u = id ), so u ◦ pr 1 pr 1 JE g × M JE R JE g × M JE R JE g id L L ′ R is an absolute coequalizer. John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 15 / 26

Iyer and Wald step 1 Diffeomorphism invariance We’ve shown that L can always be rewritten � � ◦ ◦ ◦ ◦ ◦ ◦ ◦ L g ab , ∇ a 1 R bcde , . . . , ∇ ( a 1 · · · ∇ a m ) R bcde , ψ, ∇ a 1 ψ, . . . , ∇ ( a 1 · · · ∇ a l ) ψ, γ But what about the background fields? IW claim that they drop out when we demand covariance, but I disagree. John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 16 / 26

Iyer and Wald step 1 (Assumptions about) general covariance General covariance: Diff ( M )-equivariance Background field: a section of a bundle with a trivial Diff ( M )-action So by general covariance of L , we must have ∂ L ◦ γ = 0 L ξ ◦ ∂ γ for ξ the infinitesimal generator of the diffeomorphism. For a background field, ξ generates id ; i.e., ξ = 0. John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 17 / 26

Iyer and Wald step 1 Step 1 summary 1 The variational bicomplex allows for a bit more precision about the geometric objects involved and the role of general covariance. 2 There is a simple bijective correspondence between covariant Lagrangians L : JE g × M JE ψ → R and covariant Lagrangians L ′ : JE g × M JE R × M JE ψ → R which factor through u . 3 Assuming the definition of general covariance just given, we cannot eliminate dependence on background fields. John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 18 / 26

BH entropy as Noether charge The first law Recall that for any Lagrangian L , there are E and Θ satisfying δ L = E + d Θ The ambiguities in E and Θ are well understood. E suffices to pick out the solutions to the equations of motion, but Θ is needed to characterize conserved quantities, like δ Θ (Noether, 1918). IW define black hole entropy, and derive the first law, by considering a decomposition of Θ. John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 19 / 26

BH entropy as Noether charge Θ as a current Consider the purely geometric sector: no matter fields, no background fields, and fix the Einstein–Hilbert Lagrangian. δ Θ is a conserved current, called the Crnkovi´ c–Witten current, Ashtekar–Bombelli–Koul current, or universal current. Pick some spatial slice Σ of M , and define � ω Σ = δ Θ Σ On shell, ω Σ depends only on the homology class of Σ. Alternatively: don’t integrate, then δ Θ defines a cohomology class in H 4+1 ( M × S , R ), for S ⊆ Γ( JE g × M JE R ) the solution set. John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 20 / 26

BH entropy as Noether charge Decomposing Θ Now we break the rewriting symmetry. On JE g × M JE R we have δ L = E g δ g + E R δ R + d Θ For a spatial slice Σ, we define � S Σ = 2 π E R d Σ Σ John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 21 / 26

BH entropy as Noether charge If ξ is a stationary Killing field, then it is also a symmetry of L . Noether’s theorems give conserved charge Q in terms of δ Θ that’s closed on shell. Using ξ to decompose Q gives: � 1 � 0 = dQ = − d δ M + d 4 κ δ ( E R d Σ) + d (Ω δ J ) Integrating over Σ and applying Stokes’ theorem gives the first law: δ M = 1 8 πκ δ S Σ + Ω δ J Takeaway: this derivation of the first law is determined by ξ , the cohomology class of Σ, and E R . John Dougherty (UC San Diego) The geometry of black hole entropy March 13, 2015 22 / 26