Shape Dynamic Black Holes and Horizons

Transcription

1 Shape Dynamic. 1/1 Shape Dynamic and Horizons Gabriel Herczeg Physics, University of California, Davis May 9, 2014

2 Shape Dynamic. 2/1 Asymptotically Flat Boundary Conditions Asymptotically Flat Boundary Conditions Before we begin discussing black hole solutions, it will be useful to review the formulation of shape dynamics for asymptotically flat boundary conditions.

3 Shape Dynamic. 2/1 Asymptotically Flat Boundary Conditions Asymptotically Flat Boundary Conditions Before we begin discussing black hole solutions, it will be useful to review the formulation of shape dynamics for asymptotically flat boundary conditions. The Good News : In the asymptotically flat formulation of shape dynamics, the constraints and equations of motion are local. Constraints: H a (ξ a ) = d 3 xπ ab L ξ g ab (1) D(ρ) = d 3 xρg ab π ab (2)

4 Shape Dynamic. 2/1 Asymptotically Flat Boundary Conditions Asymptotically Flat Boundary Conditions Before we begin discussing black hole solutions, it will be useful to review the formulation of shape dynamics for asymptotically flat boundary conditions. The Good News : In the asymptotically flat formulation of shape dynamics, the constraints and equations of motion are local. Constraints: H a (ξ a ) = d 3 xπ ab L ξ g ab (1) D(ρ) = d 3 xρg ab π ab (2) Note that there are no spatial averages in the conformal constraint D(ρ).

5 Shape Dynamic. 3/1 Asymptotically Flat Boundary Conditions Asymptotically Flat Boundary Conditions The Bad News : Just as in canonical general relativity, the presence of the boundary at spatial infinity requires that we add non-local boundary terms to the Hamiltonian in order for the variational problem to be well-posed.

6 Shape Dynamic. 3/1 Asymptotically Flat Boundary Conditions Asymptotically Flat Boundary Conditions The Bad News : Just as in canonical general relativity, the presence of the boundary at spatial infinity requires that we add non-local boundary terms to the Hamiltonian in order for the variational problem to be well-posed. These boundary terms are essentially a consequence of the fact that when calculating Poisson brackets, one cannot discard integrals of total derivatives when a boundary is present.

7 Shape Dynamic. 3/1 Asymptotically Flat Boundary Conditions Asymptotically Flat Boundary Conditions The Bad News : Just as in canonical general relativity, the presence of the boundary at spatial infinity requires that we add non-local boundary terms to the Hamiltonian in order for the variational problem to be well-posed. These boundary terms are essentially a consequence of the fact that when calculating Poisson brackets, one cannot discard integrals of total derivatives when a boundary is present. In particular, at the level of the linking theory, we must add a boundary term to the scalar constraint: t φ S(N) t φ (S(N) + B(N)). (3)

8 Shape Dynamic. 4/1 Asymptotically Flat Boundary Conditions The Hamiltonian When we perform phase space reduction to obtain shape dynamics from the linking theory, we find that the evolution Hamiltonian resides entirely on the boundary: t φ (S(N) + B(N)) t φ0 B(N 0 ) Where N 0 is a solution of the lapse-fixing equation: e 4φ 0 ( 2 N 0 + 2g ij φ 0,i N 0,j ) N 0e 12φ 0 G ijklπ ij π kl g = 0 and φ 0 satisfies the Lichnerowicz-York equation: Where Ω = e φ0. 2 Ω + R 8 Ω 1 8 πij π ij Ω 7 = 0.

9 Shape Dynamic. 5/1 Asymptotically Flat Boundary Conditions Maximal Slicing and Weyl Invariance The slicing condition we impose on the linking theory in the asymptotically flat case differs from the spatially closed case.

10 Shape Dynamic. 5/1 Asymptotically Flat Boundary Conditions Maximal Slicing and Weyl Invariance The slicing condition we impose on the linking theory in the asymptotically flat case differs from the spatially closed case. In this case, we have the conformal constraint D(ρ) = d 3 xρg ij π ij

11 Shape Dynamic. 5/1 Asymptotically Flat Boundary Conditions Maximal Slicing and Weyl Invariance The slicing condition we impose on the linking theory in the asymptotically flat case differs from the spatially closed case. In this case, we have the conformal constraint D(ρ) = d 3 xρg ij π ij Since the conformal constraint contains no g π term for a spatially non-compact manifold, we have maximal slicing rather than CMC slicing.

12 Shape Dynamic. 5/1 Asymptotically Flat Boundary Conditions Maximal Slicing and Weyl Invariance The slicing condition we impose on the linking theory in the asymptotically flat case differs from the spatially closed case. In this case, we have the conformal constraint D(ρ) = d 3 xρg ij π ij Since the conformal constraint contains no g π term for a spatially non-compact manifold, we have maximal slicing rather than CMC slicing. When viewed as a first class constraint, this slicing condition generates unrestricted spatial Weyl transformations they need not be volume preserving (there is no finite volume to preserve.)

13 Shape Dynamic. 6/1 Asymptotically Flat Boundary Conditions Boundary Conditions In order to determine the boundary terms, and hence the evolution Hamiltonian, it is necessary to impose boundary conditions of the fields at spatial infinity.

14 Shape Dynamic. 6/1 Asymptotically Flat Boundary Conditions Boundary Conditions In order to determine the boundary terms, and hence the evolution Hamiltonian, it is necessary to impose boundary conditions of the fields at spatial infinity. I will not go into the details of the boundary conditions at spatial infinity, but will assume that falloff of the fields is compatible with a suitable notion of asymptotic flatness.

15 Shape Dynamic. 6/1 Asymptotically Flat Boundary Conditions Boundary Conditions In order to determine the boundary terms, and hence the evolution Hamiltonian, it is necessary to impose boundary conditions of the fields at spatial infinity. I will not go into the details of the boundary conditions at spatial infinity, but will assume that falloff of the fields is compatible with a suitable notion of asymptotic flatness. We will however, discuss in detail the boundary conditions we must impose on the event horizon of a black hole if it is to be treated as a spatial boundary.

16 Shape Dynamic. 6/1 Asymptotically Flat Boundary Conditions Boundary Conditions In order to determine the boundary terms, and hence the evolution Hamiltonian, it is necessary to impose boundary conditions of the fields at spatial infinity. I will not go into the details of the boundary conditions at spatial infinity, but will assume that falloff of the fields is compatible with a suitable notion of asymptotic flatness. We will however, discuss in detail the boundary conditions we must impose on the event horizon of a black hole if it is to be treated as a spatial boundary. In what follows, we review the simple asymptotically flat black hole solutions for general relativity, and describe their corresponding solutions in shape dynamics.

17 Shape Dynamic. 7/1

18 Shape Dynamic. 8/1 Spherical Symmetry Schwarzschild Birkhoff s Theorem: The unique spherically symmetric solution to Einstein s equations in vacuum is the Schwarzschild metric: ( ds 2 = 1 2m ) ( dt m ) 1 dr 2 2 s + r ( s dθ 2 + sin 2 θdφ 2) (4) r s r s Karl Schwarzschild Penrose diagram for Schwarzschild

19 Shape Dynamic. 9/1 Spherical Symmetry Kruskal Coordinates The solution written in terms of Schwarzschild coordinates has a coordinate singularity on the event horizon at r = 2m.

20 Shape Dynamic. 9/1 Spherical Symmetry Kruskal Coordinates The solution written in terms of Schwarzschild coordinates has a coordinate singularity on the event horizon at r = 2m. In order to extend past the horizon, we need to transform to coordinates that are well-defined everywhere, e.g. Kruskal-Szekeres coordinates:

21 Shape Dynamic. 9/1 Spherical Symmetry Kruskal Coordinates The solution written in terms of Schwarzschild coordinates has a coordinate singularity on the event horizon at r = 2m. In order to extend past the horizon, we need to transform to coordinates that are well-defined everywhere, e.g. Kruskal-Szekeres coordinates: ds 2 = 32m3 e r/2m ( dv 2 + du 2) + r 2 ( dθ 2 + sin 2 θdφ 2) r where r is defined implicitly by V 2 U 2 = ( 1 r ) e r/2m 2m

22 Shape Dynamic. 9/1 Spherical Symmetry Kruskal Coordinates The solution written in terms of Schwarzschild coordinates has a coordinate singularity on the event horizon at r = 2m. In order to extend past the horizon, we need to transform to coordinates that are well-defined everywhere, e.g. Kruskal-Szekeres coordinates: ds 2 = 32m3 e r/2m ( dv 2 + du 2) + r 2 ( dθ 2 + sin 2 θdφ 2) r where r is defined implicitly by V 2 U 2 = ( 1 r ) e r/2m 2m The solution written in this form is complete. The entire spacetime is covered by this one coordinate patch.

23 Shape Dynamic. 10/1 Spherical Symmetry Isotropic Coordinates It can be shown that the ADM decomposition of the schwarzschild metric is the unique, spherically symmetric, asymptotically flat solution of shape dynamics outside the event horizon.

24 Shape Dynamic. 10/1 Spherical Symmetry Isotropic Coordinates It can be shown that the ADM decomposition of the schwarzschild metric is the unique, spherically symmetric, asymptotically flat solution of shape dynamics outside the event horizon. Is there a spatial diffeomorphism that locally takes (4) to a complete solution?

25 Shape Dynamic. 10/1 Spherical Symmetry Isotropic Coordinates It can be shown that the ADM decomposition of the schwarzschild metric is the unique, spherically symmetric, asymptotically flat solution of shape dynamics outside the event horizon. Is there a spatial diffeomorphism that locally takes (4) to a complete solution? Yes! The isotropic line element provides the Birkhoff theorem for shape dynamics (Gomes, arxiv: [gr-qc]). ( 1 m ) 2 ( ds 2 2r = 1 + m dt m ) 4 (dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) ) (5) 2r 2r

26 Shape Dynamic. 10/1 Spherical Symmetry Isotropic Coordinates It can be shown that the ADM decomposition of the schwarzschild metric is the unique, spherically symmetric, asymptotically flat solution of shape dynamics outside the event horizon. Is there a spatial diffeomorphism that locally takes (4) to a complete solution? Yes! The isotropic line element provides the Birkhoff theorem for shape dynamics (Gomes, arxiv: [gr-qc]). ( 1 m ) 2 ( ds 2 2r = 1 + m dt m ) 4 (dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) ) (5) 2r 2r There are a few things worth noting about this solution...

27 Shape Dynamic. 11/1 Spherical Symmetry The line isotropic line element is well known in the context of GR as a representation of the Schwarzschild metric outside of the event horizon.

28 Shape Dynamic. 11/1 Spherical Symmetry The line isotropic line element is well known in the context of GR as a representation of the Schwarzschild metric outside of the event horizon. The novel feature is that, in analogy with the Kruskal-Szekeres coordinates in GR, this is the complete solution for shape dynamics, valid both outside, and within the horizon.

29 Shape Dynamic. 11/1 Spherical Symmetry The line isotropic line element is well known in the context of GR as a representation of the Schwarzschild metric outside of the event horizon. The novel feature is that, in analogy with the Kruskal-Szekeres coordinates in GR, this is the complete solution for shape dynamics, valid both outside, and within the horizon. An infalling observer would take infinite proper time to reach r = 0. This is physically different than the Schwarzschild spacetime!

30 Shape Dynamic. 12/1 Spherical Symmetry Shape Dynamic are Wormholes The solution (5) represents a wormhole.

31 Shape Dynamic. 12/1 Spherical Symmetry Shape Dynamic are Wormholes The solution (5) represents a wormhole. This can be seen by noting that the transformation r m 2 /4r leaves the form of the line element (5) invariant.

32 Shape Dynamic. 12/1 Spherical Symmetry Shape Dynamic are Wormholes The solution (5) represents a wormhole. This can be seen by noting that the transformation r m 2 /4r leaves the form of the line element (5) invariant. = no curvature singularity at r = 0!

33 Shape Dynamic. 13/1 Spherical Symmetry The Skeptical Listener By now the skeptical listener can barely contain himself...

34 Shape Dynamic. 13/1 Spherical Symmetry The Skeptical Listener By now the skeptical listener can barely contain himself... But Gabe! He shouts.

35 Shape Dynamic. 13/1 Spherical Symmetry The Skeptical Listener By now the skeptical listener can barely contain himself... But Gabe! He shouts. Isn t it true that isotropic coordinates break down at the event horizon? I m afraid this whole approach is hopelessley naïve...

36 Shape Dynamic. 14/1 Spherical Symmetry A Well-Practiced Reply It is true that isotropic coordinates break down at the event horizon from the viewpoint of general relativity: det(g (4) ) = ( 1 m 2r ) 2 ( 1 + m 2r ) 10 r 4 sin 2 θ Vanishes on the horizon! Spacetime metric is degenerate!

37 Shape Dynamic. 14/1 Spherical Symmetry A Well-Practiced Reply It is true that isotropic coordinates break down at the event horizon from the viewpoint of general relativity: det(g (4) ) = ( 1 m 2r ) 2 ( 1 + m 2r ) 10 r 4 sin 2 θ Vanishes on the horizon! Spacetime metric is degenerate! However, from the viewpoint of shape dynamics, it is the conformal spatial geometry, not the space-time geometry, which is considered fundamental. det(g) = ( 1 + m 2r ) 12 r 4 sin 2 θ Non-zero on horizon. Spatial metric is invertible.

38 Shape Dynamic. 15/1 Spherical Symmetry No Space-time on Horizon The implications of the above analysis are startling.

39 Shape Dynamic. 15/1 Spherical Symmetry No Space-time on Horizon The implications of the above analysis are startling. While the solution seems well behaved on the horizon, it does not form a spacetime there.

40 Shape Dynamic. 15/1 Spherical Symmetry No Space-time on Horizon The implications of the above analysis are startling. While the solution seems well behaved on the horizon, it does not form a spacetime there. More on this later...

41 Shape Dynamic. 15/1 Spherical Symmetry No Space-time on Horizon The implications of the above analysis are startling. While the solution seems well behaved on the horizon, it does not form a spacetime there. More on this later... Many features of the spherically symmetric solution are preserved in extending to the rotating solution, which we will consider in what follows.

42 Shape Dynamic. 16/1 Axisymmetry Axisymmetric Solutions H. Gomes, G. Herczeg arxiv: [gr-qc]

43 Shape Dynamic. 17/1 Axisymmetry The Stationary, Axisymmetric Line Element We begin our consideration of rotating black holes by analyzing the stationary, axisymmetric line element: ds 2 = (N 2 ΩΨξ 2 )dt 2 + Ω[(dx 1 ) 2 + (dx 2 ) 2 + Ψdφ 2 ] + 2ΩΨξdφdt (6)

44 Shape Dynamic. 17/1 Axisymmetry The Stationary, Axisymmetric Line Element We begin our consideration of rotating black holes by analyzing the stationary, axisymmetric line element: ds 2 = (N 2 ΩΨξ 2 )dt 2 + Ω[(dx 1 ) 2 + (dx 2 ) 2 + Ψdφ 2 ] + 2ΩΨξdφdt (6) Every stationary, axisymmetric solution of Einstein s equations can (locally) be put in the above form (e.g. Bergamini, Viaggiu 2003).

45 Shape Dynamic. 17/1 Axisymmetry The Stationary, Axisymmetric Line Element We begin our consideration of rotating black holes by analyzing the stationary, axisymmetric line element: ds 2 = (N 2 ΩΨξ 2 )dt 2 + Ω[(dx 1 ) 2 + (dx 2 ) 2 + Ψdφ 2 ] + 2ΩΨξdφdt (6) Every stationary, axisymmetric solution of Einstein s equations can (locally) be put in the above form (e.g. Bergamini, Viaggiu 2003). We will show that the ADM decomposition of this line element is maximally sliced. We can map it directly onto a shape dynamics solution!

46 Shape Dynamic. 17/1 Axisymmetry The Stationary, Axisymmetric Line Element We begin our consideration of rotating black holes by analyzing the stationary, axisymmetric line element: ds 2 = (N 2 ΩΨξ 2 )dt 2 + Ω[(dx 1 ) 2 + (dx 2 ) 2 + Ψdφ 2 ] + 2ΩΨξdφdt (6) Every stationary, axisymmetric solution of Einstein s equations can (locally) be put in the above form (e.g. Bergamini, Viaggiu 2003). We will show that the ADM decomposition of this line element is maximally sliced. We can map it directly onto a shape dynamics solution! Generic local equivalence of GR and shape dynamics = most general local form of the shape dynamics solution.

47 Shape Dynamic. 18/1 Axisymmetry A Quick Calculation A quick calculation shows that the ADM decomposition of (6) is maximally sliced:

48 Shape Dynamic. 18/1 Axisymmetry A Quick Calculation A quick calculation shows that the ADM decomposition of (6) is maximally sliced: Hamilton s equation for g ij : ġ ij = 2N(Ω 3 Ψ) 1/2 (π ij 1 2 πg ij) + L ξ g ij (7) where L ξ g ij denotes the Lie derivative of the spatial metric along the shift vector. Since the solution is axisymmetric L ξ g ij = i ξg φj + j ξg iφ = 2ΩΨδ φ (iξ,j) (8)

49 Shape Dynamic. 18/1 Axisymmetry A Quick Calculation A quick calculation shows that the ADM decomposition of (6) is maximally sliced: Hamilton s equation for g ij : ġ ij = 2N(Ω 3 Ψ) 1/2 (π ij 1 2 πg ij) + L ξ g ij (7) where L ξ g ij denotes the Lie derivative of the spatial metric along the shift vector. Since the solution is axisymmetric L ξ g ij = i ξg φj + j ξg iφ = 2ΩΨδ φ (iξ,j) (8) Putting (8) into (7), and using the fact that the solution is stationary, we find 2N(Ω 3 Ψ) 1/2 (π ij 1 2 πg ij) + 2ΩΨδ φ (iξ,j) = 0 (9)

50 Shape Dynamic. 19/1 Axisymmetry A Quick Calculation Contracting the last equation with g ij yields: 2N(Ω 3 Ψ) 1/2 (π 3 2 π) + 2ξ,ig iφ = N(Ω 3 Ψ) 1/2 π + 2(ΩΨ) 1 ξ,φ = 0 (10)

51 Shape Dynamic. 19/1 Axisymmetry A Quick Calculation Contracting the last equation with g ij yields: 2N(Ω 3 Ψ) 1/2 (π 3 2 π) + 2ξ,ig iφ = N(Ω 3 Ψ) 1/2 π + 2(ΩΨ) 1 ξ,φ = 0 (10) Noting that ξ is independent of φ, we find that N(Ω 3 Ψ) 1/2 π = 0. This proves that the line element (6) is maximally sliced.

52 Shape Dynamic. 20/1 Axisymmetry The Kerr Spacetime Now that we have established this boring lemma, we can use it for exciting things!

53 Shape Dynamic. 20/1 Axisymmetry The Kerr Spacetime Now that we have established this boring lemma, we can use it for exciting things! We should be able to express a stationary, axisymmetric general relativistic black hole in the form (6).

54 Shape Dynamic. 20/1 Axisymmetry The Kerr Spacetime Now that we have established this boring lemma, we can use it for exciting things! We should be able to express a stationary, axisymmetric general relativistic black hole in the form (6). If we can, this should give us a local expression for the corresponding solution in shape dynamics!

55 Shape Dynamic. 20/1 Axisymmetry The Kerr Spacetime Now that we have established this boring lemma, we can use it for exciting things! We should be able to express a stationary, axisymmetric general relativistic black hole in the form (6). If we can, this should give us a local expression for the corresponding solution in shape dynamics! In the Boyer-Lindquist coordinates, the Kerr metric takes the form: ds 2 = Σ where ( dt a sin 2 θdφ ) 2 + sin 2 θ Σ ( (r 2 BL + a2 )dφ adt ) 2 + Σ dr 2 BL + Σdθ2 (11) = r 2 BL 2mr BL + a 2, Σ = r 2 BL + a 2 cos 2 θ.

56 Shape Dynamic. 21/1 Axisymmetry Prolate Spheroidal Coordinates Next, we would like to put this solution in the form (6) which we know is maximally sliced.

57 Shape Dynamic. 21/1 Axisymmetry Prolate Spheroidal Coordinates Next, we would like to put this solution in the form (6) which we know is maximally sliced. Change to prolate spheroidal coordinates: r BL = m 2 a 2 cosh µ + m (12)

58 Shape Dynamic. 21/1 Axisymmetry Prolate Spheroidal Coordinates Next, we would like to put this solution in the form (6) which we know is maximally sliced. Change to prolate spheroidal coordinates: In these coordinates, the line element reads: where r BL = m 2 a 2 cosh µ + m (12) ds 2 = λ 1 (dt ωdφ) 2 + λ[m 2 e 2γ (dµ 2 + dθ 2 ) + s 2 dφ 2 ] (13) s = mp sinh µ sin θ e 2γ = p 2 cosh 2 µ + q 2 cos 2 [ θ 1 (14) ] ω = e 2γ 2a sin 2 θ(p cosh µ + 1) [ ] λ = e 2γ (p cosh µ + 1) 2 + q 2 cos 2 θ

59 Shape Dynamic. 22/1 Axisymmetry Properties The ADM decomposition of this line element is the rotating black hole solution for shape dynamics.

60 Shape Dynamic. 22/1 Axisymmetry Properties The ADM decomposition of this line element is the rotating black hole solution for shape dynamics. It has some interesting properties.

61 Shape Dynamic. 22/1 Axisymmetry Properties The ADM decomposition of this line element is the rotating black hole solution for shape dynamics. It has some interesting properties. Like the spherically symmetric case, it is a wormhole solution.

62 Shape Dynamic. 22/1 Axisymmetry Properties The ADM decomposition of this line element is the rotating black hole solution for shape dynamics. It has some interesting properties. Like the spherically symmetric case, it is a wormhole solution. It has a reflection symmetry under µ µ

63 Shape Dynamic. 22/1 Axisymmetry Properties The ADM decomposition of this line element is the rotating black hole solution for shape dynamics. It has some interesting properties. Like the spherically symmetric case, it is a wormhole solution. It has a reflection symmetry under µ µ It appears to be free of physical singularities. More on this later...

64 Shape Dynamic. 22/1 Axisymmetry Properties The ADM decomposition of this line element is the rotating black hole solution for shape dynamics. It has some interesting properties. Like the spherically symmetric case, it is a wormhole solution. It has a reflection symmetry under µ µ It appears to be free of physical singularities. More on this later... Unlike the Kerr solution, it is free of inner horizons and closed timelike curves.

65 Shape Dynamic. 22/1 Axisymmetry Properties The ADM decomposition of this line element is the rotating black hole solution for shape dynamics. It has some interesting properties. Like the spherically symmetric case, it is a wormhole solution. It has a reflection symmetry under µ µ It appears to be free of physical singularities. More on this later... Unlike the Kerr solution, it is free of inner horizons and closed timelike curves. Like the spherically symmetric case, is does not form a space-time at the horizon.

66 Shape Dynamic. 23/1 Conformal Regularity of the Horizon Singularity at the Horizon? Since this is a wormhole-type solution, we do not expect a central curvature singularity.

67 Shape Dynamic. 23/1 Conformal Regularity of the Horizon Singularity at the Horizon? Since this is a wormhole-type solution, we do not expect a central curvature singularity. Rather, we might expect to find a singularity at the throat of the wormhole (the horizon), as we find in analogous wormhole solutions of GR.

68 Shape Dynamic. 23/1 Conformal Regularity of the Horizon Singularity at the Horizon? Since this is a wormhole-type solution, we do not expect a central curvature singularity. Rather, we might expect to find a singularity at the throat of the wormhole (the horizon), as we find in analogous wormhole solutions of GR. This does not appear to be the case.

69 Shape Dynamic. 24/1 Conformal Regularity of the Horizon Change of Conformal Frame In order to simplify calculations, we can make a change of conformal frame, so that the spatial metric has only one functional degree of freedom, Ψ: ( ) g ij = m 2 δ µ i δ µ j + δi θ δj θ + Ψδ φ i δ φ j (15)

70 Shape Dynamic. 24/1 Conformal Regularity of the Horizon Change of Conformal Frame In order to simplify calculations, we can make a change of conformal frame, so that the spatial metric has only one functional degree of freedom, Ψ: ( ) g ij = m 2 δ µ i δ µ j + δi θ δj θ + Ψδ φ i δ φ j (15) Since Weyl transformations of the spatial metric are pure gauge from the point of view of shape dynamics, this should not effect any of our conclusions.

71 Shape Dynamic. 25/1 Conformal Regularity of the Horizon The Cotton-York Tensor Like the Weyl tensor in higher dimensions, the cotton tensor C ijk contains all of the information about the local conformal structure of a three dimensional Riemannian manifold: C ijk := k (R ij 1 ) ( 4 Rg ij j R ik 1 ) 4 Rg ik (16)

72 Shape Dynamic. 25/1 Conformal Regularity of the Horizon The Cotton-York Tensor Like the Weyl tensor in higher dimensions, the cotton tensor C ijk contains all of the information about the local conformal structure of a three dimensional Riemannian manifold: C ijk := k (R ij 1 ) ( 4 Rg ij j R ik 1 ) 4 Rg ik (16) The rank-two Cotton-York tensor, C ij, can be defined by it s relation to the Cotton tensor: C ij := 1 2 g mj ɛ ikl C mkl (17)

73 Shape Dynamic. 25/1 Conformal Regularity of the Horizon The Cotton-York Tensor Like the Weyl tensor in higher dimensions, the cotton tensor C ijk contains all of the information about the local conformal structure of a three dimensional Riemannian manifold: C ijk := k (R ij 1 ) ( 4 Rg ij j R ik 1 ) 4 Rg ik (16) The rank-two Cotton-York tensor, C ij, can be defined by it s relation to the Cotton tensor: C ij := 1 2 g mj ɛ ikl C mkl (17) It is traceless, transverse, and transforms homogeneously under Weyl transformations.

74 Shape Dynamic. 25/1 Conformal Regularity of the Horizon The Cotton-York Tensor Like the Weyl tensor in higher dimensions, the cotton tensor C ijk contains all of the information about the local conformal structure of a three dimensional Riemannian manifold: C ijk := k (R ij 1 ) ( 4 Rg ij j R ik 1 ) 4 Rg ik (16) The rank-two Cotton-York tensor, C ij, can be defined by it s relation to the Cotton tensor: C ij := 1 2 g mj ɛ ikl C mkl (17) It is traceless, transverse, and transforms homogeneously under Weyl transformations. No singularities in the spatial conformal geometry manifest themselves in the scalar (density) C ij C ij.

75 Shape Dynamic. 26/1 Conformal Regularity of the Horizon Cotton-York Squared Using the reduced form of the metric, we can express C ij C ij in terms of Ψ and it s coordinate derivatives:

76 Shape Dynamic. 26/1 Conformal Regularity of the Horizon Cotton-York Squared Using the reduced form of the metric, we can express C ij C ij in terms of Ψ and it s coordinate derivatives: C 2 := C ij C ij = 2Ψ [ (C µφ ) 2 + (C θφ ) 2] = 1 4Ψ 2 [Ψ,µΨ,µµ + Ψ,θ Ψ,θθ + (Ψ,µ + Ψ,θ )Ψ,µθ ] 1 4Ψ (Ψ,µµµ + Ψ,µµθ + Ψ,µθθ + Ψ,θθθ ). (18)

77 Shape Dynamic. 26/1 Conformal Regularity of the Horizon Cotton-York Squared Using the reduced form of the metric, we can express C ij C ij in terms of Ψ and it s coordinate derivatives: C 2 := C ij C ij = 2Ψ [ (C µφ ) 2 + (C θφ ) 2] = 1 4Ψ 2 [Ψ,µΨ,µµ + Ψ,θ Ψ,θθ + (Ψ,µ + Ψ,θ )Ψ,µθ ] 1 4Ψ (Ψ,µµµ + Ψ,µµθ + Ψ,µθθ + Ψ,θθθ ). (18) Note that since Ψ is an even function of µ, any odd number of µ derivatives acting on Ψ will be zero when evaluated on the horizon. Taking this into account we can put (18) in the simplified form C 2 (0, θ) = [ 1 4Ψ 2 Ψ,θΨ,θθ 1 4Ψ (Ψ,µµθ + Ψ,θθθ )] µ=0 (19)

78 Shape Dynamic. 27/1 Conformal Regularity of the Horizon Cotton-York Squared At the horizon, we have Ψ(0, θ) = 4a 2 sin θ m 2 (4 + q 2 cos 2 θ) 2 (20) which is nonzero except on the axis of rotation θ = {0, π}. We will have to be careful with these points.

79 Shape Dynamic. 27/1 Conformal Regularity of the Horizon Cotton-York Squared At the horizon, we have Ψ(0, θ) = 4a 2 sin θ m 2 (4 + q 2 cos 2 θ) 2 (20) which is nonzero except on the axis of rotation θ = {0, π}. We will have to be careful with these points. A tedious but straightforward calculation yields the other ingredients of C 2 (0, θ): Ψ,θ (0, θ) = 28 a 2 q 2 sin 2 θ cos θ m 2 (4 + q 2 cos 2 θ) 3 [ ] 2 8 Ψ,θθ (0, θ) = a 2 q 2 sin 2 θ(sin 2 θ cos 2 θ) 6a2 q 4 sin 3 θ cos 2 θ m 2 (4 + q 2 cos 2 θ) q 2 cos 2 θ Ψ,θθθ (0, θ) = 29 a 2 q 2 sin 2 θ cos θ [ ] q 4 sin 2 θ cos 2 θ q 2 (sin 2 θ cos 2 θ) + 2(4 + q 2 cos 2 θ) m 2 (4 + q 2 cos 2 θ) 4 Ψ,θµµ (0, θ) = 4 sin θ cos θ a 2 q 2 m 2 sin θ cos θ (4 + q 2 cos 2 θ) 4 (21)

80 Shape Dynamic. 28/1 Conformal Regularity of the Horizon None of these functions can diverge for any values of θ.

81 Shape Dynamic. 28/1 Conformal Regularity of the Horizon None of these functions can diverge for any values of θ. If we insert these functions back into C 2 (0, θ), we see that C 2 (0, 0) = C 2 (0, π) = 0. We do not see any singularities in the conformal structure manifesting themselves in the Cotton-York tensor at the horizon.

82 Shape Dynamic. 28/1 Conformal Regularity of the Horizon None of these functions can diverge for any values of θ. If we insert these functions back into C 2 (0, θ), we see that C 2 (0, 0) = C 2 (0, π) = 0. We do not see any singularities in the conformal structure manifesting themselves in the Cotton-York tensor at the horizon. This is not conclusive proof that there are no singularities on the horizon.

83 Shape Dynamic. 28/1 Conformal Regularity of the Horizon None of these functions can diverge for any values of θ. If we insert these functions back into C 2 (0, θ), we see that C 2 (0, 0) = C 2 (0, π) = 0. We do not see any singularities in the conformal structure manifesting themselves in the Cotton-York tensor at the horizon. This is not conclusive proof that there are no singularities on the horizon. This is a heuristic argument in the same vein as showing that e.g., the Kretschmann invariant K = R abcd R abcd remains finite for a GR solution.

84 Shape Dynamic. 28/1 Conformal Regularity of the Horizon None of these functions can diverge for any values of θ. If we insert these functions back into C 2 (0, θ), we see that C 2 (0, 0) = C 2 (0, π) = 0. We do not see any singularities in the conformal structure manifesting themselves in the Cotton-York tensor at the horizon. This is not conclusive proof that there are no singularities on the horizon. This is a heuristic argument in the same vein as showing that e.g., the Kretschmann invariant K = R abcd R abcd remains finite for a GR solution. In order to complete the proof, one would need to identify a complete set of conformal-diffeo invariants and repeat the analysis for each one.

85 Shape Dynamic. 29/1 Spherically Symmetric Limit Zero Angular Momentum Limit If we take the limit as a := J/m 0, we should reobtain the spherically symmetric solution.

86 Shape Dynamic. 29/1 Spherically Symmetric Limit Zero Angular Momentum Limit If we take the limit as a := J/m 0, we should reobtain the spherically symmetric solution. In this limit, we have p := m/ m 2 a 2 = 1, q := a/ m 2 a 2 = 0. The metric functions become: s = m sinh µ cos θ, λ = (cosh µ + 1)2 sinh 2 µ ω = 0, e 2γ = sinh 2 µ (22)

87 Shape Dynamic. 29/1 Spherically Symmetric Limit Zero Angular Momentum Limit If we take the limit as a := J/m 0, we should reobtain the spherically symmetric solution. In this limit, we have p := m/ m 2 a 2 = 1, q := a/ m 2 a 2 = 0. The metric functions become: s = m sinh µ cos θ, λ = (cosh µ + 1)2 sinh 2 µ ω = 0, e 2γ = sinh 2 µ (22) These give the spherically symmetric line element written in terms of prolate spheroidal coordinates. ds 2 = sinh2 µ (cosh µ + 1) 2 dt2 + m 2 (cosh µ + 1) 2 ( dµ 2 + dθ 2 + sin 2 θdφ 2) (23)

88 Shape Dynamic. 30/1 Spherically Symmetric Limit Zero Angular Momentum Limit This is not exactly the isotropic line element, but we can put it in that form by performing the spatial diffeomorphism r = (m/2)e µ.

89 Shape Dynamic. 30/1 Spherically Symmetric Limit Zero Angular Momentum Limit This is not exactly the isotropic line element, but we can put it in that form by performing the spatial diffeomorphism r = (m/2)e µ. This diffeomorphism is globally well-defined. It is smooth everywhere in space and its inverse is smooth everywhere in space.

90 Shape Dynamic. 30/1 Spherically Symmetric Limit Zero Angular Momentum Limit This is not exactly the isotropic line element, but we can put it in that form by performing the spatial diffeomorphism r = (m/2)e µ. This diffeomorphism is globally well-defined. It is smooth everywhere in space and its inverse is smooth everywhere in space. The solutions are related by a pure gauge transformation. = They are physically equivalent.

91 Shape Dynamic. 31/1 Equivalence Principle and Firewalls Dreams of Quantum Black holes

92 Shape Dynamic. 32/1 Equivalence Principle and Firewalls Firewall Paradox Now that we have seen the essential features of classical shape dynamic black holes, we can consider a potentially interesting application to quantum gravity.

93 Shape Dynamic. 32/1 Equivalence Principle and Firewalls Firewall Paradox Now that we have seen the essential features of classical shape dynamic black holes, we can consider a potentially interesting application to quantum gravity. The so-called firewall paradox has sparked a great deal of lively debate since it was proposed by A. Almheiri, D. Marolf, J. Polchinski, J. Sully (AMPS) in 2012 (arxiv: [hep-th]).

94 Shape Dynamic. 33/1 Equivalence Principle and Firewalls Brief Review of AMPS s Argument Before we consider what shape dynamics might contribute to this conversation, let s briefly review the main results of the AMPS paper.

95 Shape Dynamic. 33/1 Equivalence Principle and Firewalls Brief Review of AMPS s Argument Before we consider what shape dynamics might contribute to this conversation, let s briefly review the main results of the AMPS paper. It is shown that the following three postulates cannot all be true:

96 Shape Dynamic. 33/1 Equivalence Principle and Firewalls Brief Review of AMPS s Argument Before we consider what shape dynamics might contribute to this conversation, let s briefly review the main results of the AMPS paper. It is shown that the following three postulates cannot all be true: (i) Hawking radiation is in a pure state.

97 Shape Dynamic. 33/1 Equivalence Principle and Firewalls Brief Review of AMPS s Argument Before we consider what shape dynamics might contribute to this conversation, let s briefly review the main results of the AMPS paper. It is shown that the following three postulates cannot all be true: (i) Hawking radiation is in a pure state. (ii) Outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations.

98 Shape Dynamic. 33/1 Equivalence Principle and Firewalls Brief Review of AMPS s Argument Before we consider what shape dynamics might contribute to this conversation, let s briefly review the main results of the AMPS paper. It is shown that the following three postulates cannot all be true: (i) Hawking radiation is in a pure state. (ii) Outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations. (iii) The infalling observer encounters nothing unusual at the horizon.

99 Shape Dynamic. 34/1 Equivalence Principle and Firewalls Brief Review of AMPS s Argument The Hawking radiation can be decomposed as a tensor product of early and late radiation. Then by postulate (i) it is in a pure state: Ψ = i ψ E i L (24) where i L is an arbitrary basis for the late radiation.

100 Shape Dynamic. 34/1 Equivalence Principle and Firewalls Brief Review of AMPS s Argument The Hawking radiation can be decomposed as a tensor product of early and late radiation. Then by postulate (i) it is in a pure state: Ψ = i ψ E i L (24) where i L is an arbitrary basis for the late radiation. Consider an outgoing Hawking mode in the late part of the radiation. Postulate (ii) permits the identification of a unique lowering operator b associated with this mode.

101 Shape Dynamic. 34/1 Equivalence Principle and Firewalls Brief Review of AMPS s Argument The Hawking radiation can be decomposed as a tensor product of early and late radiation. Then by postulate (i) it is in a pure state: Ψ = i ψ E i L (24) where i L is an arbitrary basis for the late radiation. Consider an outgoing Hawking mode in the late part of the radiation. Postulate (ii) permits the identification of a unique lowering operator b associated with this mode. Since the early and late radiation are strongly entangled, an observer making measurements on the early radiation can know the number of photons in a given mode of the late radiation (i.e., the eigenvalue of b b).

102 Shape Dynamic. 35/1 Equivalence Principle and Firewalls Brief Review of AMPS s Argument Now consider an infalling observer with corresponding set of infalling modes with lowering operators a.

103 Shape Dynamic. 35/1 Equivalence Principle and Firewalls Brief Review of AMPS s Argument Now consider an infalling observer with corresponding set of infalling modes with lowering operators a. Recall that Hawking radiation arises due to the fact that b can be expressed in terms of a and a by the Bogoliubov transformation b = 0 dω ( B(ω)a ω + C(ω)a ω ) (25) so the full state cannot be both an a-vacuum with a Ψ = 0 and a b b eigenstate.

104 Shape Dynamic. 35/1 Equivalence Principle and Firewalls Brief Review of AMPS s Argument Now consider an infalling observer with corresponding set of infalling modes with lowering operators a. Recall that Hawking radiation arises due to the fact that b can be expressed in terms of a and a by the Bogoliubov transformation b = 0 dω ( B(ω)a ω + C(ω)a ω ) (25) so the full state cannot be both an a-vacuum with a Ψ = 0 and a b b eigenstate. Thus, if Ψ is an eigenstate of b b it is not an a-vacuum, and the infalling observer will encounter high energy modes, contradicting postulate (iii)!

105 Shape Dynamic. 36/1 Equivalence Principle and Firewalls Are Firewalls a Paradox? How does (this vast oversimplification of) the firewall paradox relate to shape dynamic black holes?

106 Shape Dynamic. 36/1 Equivalence Principle and Firewalls Are Firewalls a Paradox? How does (this vast oversimplification of) the firewall paradox relate to shape dynamic black holes? The simplest interpetation of AMPS s argument is that the equivalence principle breaks down at the event horizon of a black hole.

107 Shape Dynamic. 36/1 Equivalence Principle and Firewalls Are Firewalls a Paradox? How does (this vast oversimplification of) the firewall paradox relate to shape dynamic black holes? The simplest interpetation of AMPS s argument is that the equivalence principle breaks down at the event horizon of a black hole. While the equivalence principle is an axiom of general relativity, it is an emergent property in shape dynamics.

108 Shape Dynamic. 36/1 Equivalence Principle and Firewalls Are Firewalls a Paradox? How does (this vast oversimplification of) the firewall paradox relate to shape dynamic black holes? The simplest interpetation of AMPS s argument is that the equivalence principle breaks down at the event horizon of a black hole. While the equivalence principle is an axiom of general relativity, it is an emergent property in shape dynamics. As we have noted, this property fails to emerge precisely at the event horizon of a shape dynamic black hole.

109 Shape Dynamic. 36/1 Equivalence Principle and Firewalls Are Firewalls a Paradox? How does (this vast oversimplification of) the firewall paradox relate to shape dynamic black holes? The simplest interpetation of AMPS s argument is that the equivalence principle breaks down at the event horizon of a black hole. While the equivalence principle is an axiom of general relativity, it is an emergent property in shape dynamics. As we have noted, this property fails to emerge precisely at the event horizon of a shape dynamic black hole. This could be a hint that shape dynamic black holes have more consistent quantum mechanical behaviour than their general relativistic cousins.

110 Shape Dynamic. 37/1 Equivalence Principle and Firewalls Prerequisites It is an exciting possibility that shape dynamic black holes could provide a resolution to the firewall paradox.

111 Shape Dynamic. 37/1 Equivalence Principle and Firewalls Prerequisites It is an exciting possibility that shape dynamic black holes could provide a resolution to the firewall paradox. Of course in order for these claims to be taken seriously, we need to understand more than just the classical behaviour of the solutions near the horizon.

112 Shape Dynamic. 37/1 Equivalence Principle and Firewalls Prerequisites It is an exciting possibility that shape dynamic black holes could provide a resolution to the firewall paradox. Of course in order for these claims to be taken seriously, we need to understand more than just the classical behaviour of the solutions near the horizon. In particular, we need to understand whether Hawking radiation works the same way in shape dynamics as it does in GR.

113 Shape Dynamic. 37/1 Equivalence Principle and Firewalls Prerequisites It is an exciting possibility that shape dynamic black holes could provide a resolution to the firewall paradox. Of course in order for these claims to be taken seriously, we need to understand more than just the classical behaviour of the solutions near the horizon. In particular, we need to understand whether Hawking radiation works the same way in shape dynamics as it does in GR. This is no simple task, especially since hawking radiation is typically identified as modes existing on I +, which is not a natural object to define in shape dynamics.

114 Shape Dynamic. 37/1 Equivalence Principle and Firewalls Prerequisites It is an exciting possibility that shape dynamic black holes could provide a resolution to the firewall paradox. Of course in order for these claims to be taken seriously, we need to understand more than just the classical behaviour of the solutions near the horizon. In particular, we need to understand whether Hawking radiation works the same way in shape dynamics as it does in GR. This is no simple task, especially since hawking radiation is typically identified as modes existing on I +, which is not a natural object to define in shape dynamics. A natural starting place for these considerations is to see what we can learn about hawking radiation from horizon thermodynamics.

115 Shape Dynamic. Thermodynamics 38/1 Thermodynamics

116 Shape Dynamic. 39/1 Thermodynamics Basics It is well known that black holes are thermodynamic objects with characteristic temperatures and entropies.

117 Shape Dynamic. 39/1 Thermodynamics Basics It is well known that black holes are thermodynamic objects with characteristic temperatures and entropies. For a stationary black hole, the temperature is given by T = κ 2π, where κ is the surface gravity of the event horizon, k a a k b = κk b (26) and k a is the stationarity killing vector k a a = t + ω φ (27) where ω is the angular velocity of the horizon.

118 Shape Dynamic. 39/1 Thermodynamics Basics It is well known that black holes are thermodynamic objects with characteristic temperatures and entropies. For a stationary black hole, the temperature is given by T = κ 2π, where κ is the surface gravity of the event horizon, k a a k b = κk b (26) and k a is the stationarity killing vector k a a = t + ω φ (27) where ω is the angular velocity of the horizon. The entropy is given simply by S = A/4, where A is the horizon area.

119 Shape Dynamic. 40/1 Thermodynamics Canonical Partition Function The connection between gravity and thermodynamics is most easily seen in the context of Euclidean quantum gravity (Gibbons, Hawking 1976).

120 Shape Dynamic. 40/1 Thermodynamics Canonical Partition Function The connection between gravity and thermodynamics is most easily seen in the context of Euclidean quantum gravity (Gibbons, Hawking 1976). The transition amplitude for a quantum field ψ is given by the path integral: ψ 2, t 2 ψ 1, t 1 = d[ψ] exp(ii[ψ]) (28)

121 Shape Dynamic. 40/1 Thermodynamics Canonical Partition Function The connection between gravity and thermodynamics is most easily seen in the context of Euclidean quantum gravity (Gibbons, Hawking 1976). The transition amplitude for a quantum field ψ is given by the path integral: ψ 2, t 2 ψ 1, t 1 = d[ψ] exp(ii[ψ]) (28) On the other hand, ψ 2, t 2 ψ 1, t 1 = ψ 2, exp[ ih(t 2 t 1 )] ψ 1,. (29)

122 Shape Dynamic. 41/1 Thermodynamics Canonical Partition Function If we set t 2 t 1 = iβ, ψ 1 = ψ 2 and sum over all ψ 1, we obtain tr[exp( βh)] = d[ ψ] exp(ii[ ψ]) (30) where the path integral is now taken over all field configurations ψ with period β in the Euclidean time τ = it.

123 Shape Dynamic. 41/1 Thermodynamics Canonical Partition Function If we set t 2 t 1 = iβ, ψ 1 = ψ 2 and sum over all ψ 1, we obtain tr[exp( βh)] = d[ ψ] exp(ii[ ψ]) (30) where the path integral is now taken over all field configurations ψ with period β in the Euclidean time τ = it. The left hand side of this equation is just the canonical partition function at inverse temperature T 1 = β. Z = tr[exp( βh)]. (31)

124 Shape Dynamic. 42/1 Thermodynamics Grand Canonical Partition Function If we want to include conserved charges C i in our analysis, then we can use the grand canonical partition function: { [ ( Z = tr exp β H )]} µ i C i (32) i where µ i is the chemical potential associated with the charge C i.

125 Shape Dynamic. 42/1 Thermodynamics Grand Canonical Partition Function If we want to include conserved charges C i in our analysis, then we can use the grand canonical partition function: { [ ( Z = tr exp β H )]} µ i C i (32) i where µ i is the chemical potential associated with the charge C i. This allows us to consider systems with electic charge Q and angular momentum J, by identifying the chemical potentials Φ (electrostatic potential) and ω (angular velocity).

126 Shape Dynamic. 43/1 Thermodynamics Thermodynamic Potential If one perturbs the metric g, and matter fields ψ about the classical solutions g 0, ψ 0 with periodicity β, so that g = g 0 + ĝ, ψ = ψ 0 + ˆψ (33)

127 Shape Dynamic. 43/1 Thermodynamics Thermodynamic Potential If one perturbs the metric g, and matter fields ψ about the classical solutions g 0, ψ 0 with periodicity β, so that g = g 0 + ĝ, ψ = ψ 0 + ˆψ (33) then the action can be expanded in a Taylor series about the classical solutions: I[g, ψ] = I[g 0, ψ 0 ] + I 2 [ĝ] + I 2 [ ˆψ] +... (34)

128 Shape Dynamic. 43/1 Thermodynamics Thermodynamic Potential If one perturbs the metric g, and matter fields ψ about the classical solutions g 0, ψ 0 with periodicity β, so that g = g 0 + ĝ, ψ = ψ 0 + ˆψ (33) then the action can be expanded in a Taylor series about the classical solutions: I[g, ψ] = I[g 0, ψ 0 ] + I 2 [ĝ] + I 2 [ ˆψ] +... (34) The lowest order contributions to the thermodynamic potential F are βf ln Z = ii[g 0, ψ 0 ] + ln d[ĝ] exp(ii 2 [ĝ]) + ln d[ ˆψ] exp(ii 2 [ ˆψ])

129 Shape Dynamic. 44/1 Thermodynamics Entropy From the thermodynamic potential, we can calculate all of the thermodynamic quantities associated with an ensemble of field configurations of period β.

130 Shape Dynamic. 44/1 Thermodynamics Entropy From the thermodynamic potential, we can calculate all of the thermodynamic quantities associated with an ensemble of field configurations of period β. We can also work in the classical limit by considering just the zero order contribution to F.

131 Shape Dynamic. 44/1 Thermodynamics Entropy From the thermodynamic potential, we can calculate all of the thermodynamic quantities associated with an ensemble of field configurations of period β. We can also work in the classical limit by considering just the zero order contribution to F. When this is done, one finds for the Kerr-Newman solution F = 1 (M ΦQ) (35) 2

132 Shape Dynamic. 44/1 Thermodynamics Entropy From the thermodynamic potential, we can calculate all of the thermodynamic quantities associated with an ensemble of field configurations of period β. We can also work in the classical limit by considering just the zero order contribution to F. When this is done, one finds for the Kerr-Newman solution F = 1 (M ΦQ) (35) 2 but by definition, F = M TS ΦQ ωj. (36)

133 Shape Dynamic. 44/1 Thermodynamics Entropy From the thermodynamic potential, we can calculate all of the thermodynamic quantities associated with an ensemble of field configurations of period β. We can also work in the classical limit by considering just the zero order contribution to F. When this is done, one finds for the Kerr-Newman solution F = 1 (M ΦQ) (35) 2 but by definition, F = M TS ΦQ ωj. (36) Making use of the generalized Smarr formula we find that S = A/4 as expected. 1 2 M = κ 8π A + 1 ΦQ + ωj (37) 2

134 Shape Dynamic. 45/1 Thermodynamics Entropy and Surface Action We noted earlier that in going from the Hilbert action to the ADM action, we obtained a surface term in addition to constraints. I ADM = [ dt d 3 x g ( Σ g ij π ij NH(x) ξ i H i (x) ) 1 d 2 x σ(nk + r a N,a r i ξ j πj i ) 8π Σ ]

135 Shape Dynamic. 45/1 Thermodynamics Entropy and Surface Action We noted earlier that in going from the Hilbert action to the ADM action, we obtained a surface term in addition to constraints. I ADM = [ dt d 3 x g ( Σ g ij π ij NH(x) ξ i H i (x) ) 1 d 2 x σ(nk + r a N,a r i ξ j πj i ) 8π Σ ] If there was no boundary term, the arguments of the preceding section would imply that Ĩ = βh, where Ĩ is the Euclideanized action.

136 Shape Dynamic. 45/1 Thermodynamics Entropy and Surface Action We noted earlier that in going from the Hilbert action to the ADM action, we obtained a surface term in addition to constraints. I ADM = [ dt d 3 x g ( Σ g ij π ij NH(x) ξ i H i (x) ) 1 d 2 x σ(nk + r a N,a r i ξ j πj i ) 8π Σ ] If there was no boundary term, the arguments of the preceding section would imply that Ĩ = βh, where Ĩ is the Euclideanized action. Including the boundary term, and treating the horizon as an inner boundary, we get Ĩ = βh A/4. (38)

137 Shape Dynamic. 46/1 Thermodynamics What About Shape Dynamics? So far we have just reviewed some standard results from Euclidean quantum gravity.

138 Shape Dynamic. 46/1 Thermodynamics What About Shape Dynamics? So far we have just reviewed some standard results from Euclidean quantum gravity. What does this have to do with shape dynamics?

139 Shape Dynamic. 46/1 Thermodynamics What About Shape Dynamics? So far we have just reviewed some standard results from Euclidean quantum gravity. What does this have to do with shape dynamics? Shape dynamics is constructed from a linking theory.

140 Shape Dynamic. 46/1 Thermodynamics What About Shape Dynamics? So far we have just reviewed some standard results from Euclidean quantum gravity. What does this have to do with shape dynamics? Shape dynamics is constructed from a linking theory. If we can extend these results to the linking theory and then perform phase space reduction, we should get the corresponding results for shape dynamics.

141 Shape Dynamic. 46/1 Thermodynamics What About Shape Dynamics? So far we have just reviewed some standard results from Euclidean quantum gravity. What does this have to do with shape dynamics? Shape dynamics is constructed from a linking theory. If we can extend these results to the linking theory and then perform phase space reduction, we should get the corresponding results for shape dynamics. In what follows I will present unfinished work done in collaboration with Vasudev Shyam.