Spherically symmetric loop quantum gravity

Transcription

1 Spherically symmetric loop quantum gravity Martin Bojowald The Pennsylvania State University Institute for Gravitation and the Cosmos University Park, PA Spherical symmetry p.1

2 Reduction in loop quantum gravity Models necessary to simplify, understand and develop dynamics of full theory. But: Beware of artefacts. For reliable results, constructions must be ensured to work equally well in many different models. Key issues for dynamics: Difference equations: underlying discreteness, step-size affected by lattice refinement, directly captured only in inhomogeneous situations. Specific form important for acceptable behavior even of homogeneous models stable in semiclassical regimes). Effective equations: intuitive geometrical pictures, but quantum back-reaction important in strong quantum regimes e.g. near singularities). Spherical symmetry p.

3 Kantowski Sachs Schwarzschild interior metric ds = 1 M ) dt + r 1 )dr + r dω 1 M r homogeneous for r < M. Corresponding invariant connection/triad SU)-gauge fixed): A i aτ i dx a E a i τ i x a = cτ 3 dx + bτ dϑ bτ 1 sin ϑdϕ + τ 3 cos ϑdϕ = p c τ 3 sin ϑ x + p bτ sin ϑ ϑ p bτ 1 ϕ Spatial metric x = t for interior): ds = p b p c dx + p c dω On shell: p c = 0 at singularity, p b = 0 at horizon. Spherical symmetry p.3

4 Phase space b, c, p b, p c ) R 4, with { b, p b } = γg/l 0, { c, p c } = γg/l 0 after choosing x,ϑ,ϕ)-integration region of size L 0 4π. Rescale b,c) := b,l o c), p b,p c ) := L o p b, p c ) for coordinate independent variables but L 0 -dependent). Co-triad: a c = p b / p c = L 0 ã c, a b = p c = p c. Note: L 0 or V 0 in isotropic models) key culprit for possible artefacts in minisuperspace constructions. Only coordinate and L 0 -independent variables are b, a b and c/a c. Can quantum corrections only depend on the curvature scale c/a c, or possibly just on the spatial scale a c as suggested by inverse triad corrections)? Here, role of quantum reduction becomes important. Spherical symmetry p.4

5 Holonomy-flux algebra τl0 h x τ) A) = exp 0 h µ) ϑ A) = exp µ generate states µ,τ = ĥµ) eigenstates: 0 dx cτ 3 = cos τc + τ 3 sin τc dϑ bτ = cos µb + τ sin µb ϑ ĥτ) x 0,0 with c,b 0,0 = 1. Triad ˆp b µ,τ = 1 γl P µ µ,τ, ˆp c µ,τ = γl P τ µ,τ Completes kinematical setting. [A. Ashtekar, MB: CQG 3 006) 391] As symmetric states, µ,τ can be realized as invariant distributions. Key properties of reduction enter at dynamical level: Hamiltonian constraint and lattice refinement. Spherical symmetry p.5

6 Ĥ δ) = Hamiltonian constraint Standard construction: = i γ 3 δ 3 l P tr IJK +γ δ τ 3 ĥ δ) ǫ IJK ĥ δ) I ĥ δ) x, ˆV ) ] x [ĥδ) 1 4i γ 3 δ 3 l 8sin δb P sin δb + 4sin δb ˆV cos δb cos δb J ĥδ) 1 I sin δc ĥ δ) 1 J ĥ δ) K [ĥδ) 1 K, ˆV ] δc cos ) δb cos ˆV sin δb cos δb + γ δ ) sin δc ˆV cos δc Real parameter δ expected to be constant in pure minisuperspace setting. δc cos ˆV sin δc ) ) Spherical symmetry p.6

7 Difference equation Derive explicit action of Ĥδ), transform to triad representation. Resulting difference equation non-singular: evolves across τ = 0 p c = 0, singularity) with C + µ,τ)ψ µ+δ,τ+δ ψ µ δ,τ+δ ) +C 0 µ,τ) µ + δ)ψ µ+4δ,τ 1 + γ δ )µψ µ,τ ) +µ δ)ψ µ 4δ,τ +C µ,τ)ψ µ δ,τ δ ψ µ+δ,τ δ ) = 0 C ± µ,τ) = δ τ ± δ + τ ) C 0 µ,τ) = τ + δ τ δ Spherical symmetry p.7

8 Instability With constant δ, difference equation unstable in large region of minisuperspace µ > 4τ): Only exponential rather than oscillating solutions even in supposedly semiclassical regions. [J. Rosen, J.-H. Jung, G. Khanna: CQG 3 006) 7075] Seen based on analysis of difference equation von Neumann stability); also indicated by tree-level equation holonomized ): H δ) 4 = γ 3 δ 3 8sin δb δb δc δc cos sin cos pc G + 4sin δb ) δb cos + γ δ pb p c easier to analyze numerically. [D.-W. Chiou, L. Modesto, K. Vandersloot] Incomplete effective equation, but reliable for inferring problems in semiclassical regimes with weak quantum back-reaction). Near singularity: state parameters essential.) ) Spherical symmetry p.8

9 Lattice refinement [MB, D. Cartin, G. Khanna: PRD ) ] Reason for instability: Wave function highly oscillating in semiclassical regimes; discrete lattice must be sufficiently small. In Hamiltonian constraint, holonomies with constant δ do not take into account necessary refinement of discrete structure as geometry grows. Cannot be seen purely in minisuperspace setting extra input required. Holonomies depend on vertex numbers N I determining discreteness: h lx 0) x = expl x 0 cτ 3 ) = expl x 0L 1 0 cτ 3) = expcτ 3 /N x ), h lϑ 0) ϑ = expl ϑ 0bτ ) = expbτ /N ϑ ) for holonomies along edges of coordinate length l x 0, lϑ 0 : N x = L 0 /l x 0 edges of length lx 0 in interval of size L 0. Spherical symmetry p.9

10 Improvised uniqueness Discrete versus classical evolving geometry: N x λ)v x λ) = p bλ) pc λ) L 0, N ϑ λ)v ϑ λ) = p c λ) Allows more freedom since two free functions in discrete geometry for each free function of classical geometry: Number of sites N as well as sizes v change in internal) time λ. Precise behavior needed to model full dynamics. Possible assumption: v x λ) constant, then N x ã c L 0 = a c and only c/a c appears in holonomies: improv[is]ed dynamics. Apparently unique, but based on assumption. Gives only one special case, ruled out by failure to capture near-horizon dynamics properly instabilities since N x small near p b = 0). Spherical symmetry p.10

11 Lattice refining difference equation General refinement schemes to be considered; gives difference equation whose step-sizes 1/N I vary: C + µ,τ) ψ µ+δnϑ µ,τ) 1,τ+δN x µ,τ) 1 ) ψ µ δnϑ µ,τ) 1,τ+δN x µ,τ) 1 +C 0 µ,τ) µ + δn ϑ µ,τ) 1 )ψ µ+4δnϑ µ,τ) 1,τ 1 + γ δ N ϑ µ,τ) )µψ µ,τ +µ δn ϑ µ,τ) 1 )ψ µ 4δNϑ µ,τ) 1,τ +C µ,τ) ψ µ δnϑ µ,τ) 1,τ δn x µ,τ) 1 ψ µ+δnϑ µ,τ) 1,τ δn x µ,τ) 1 ) = 0 ) Spherical symmetry p.11

12 Less symmetry Lattice refinement of any kind non-constant δ) goes beyond minisuperspace models. Try to embed homogeneous models in less symmetric ones, such as spherical symmetry. Advantages: Triad representation still available; explicit inhomogeneity; different types of singularities can be studied. Spherical symmetry p.1

13 Spherically symmetric gravity A i adx a τ i = A x τ 3 dx + A ϕ e 1 iπτ Λ A ϕe 1 iπτ dϑ + A ϕ Λ A ϕ sin ϑdϕ +τ 3 cos ϑdϕ E a i x a τ i = E x τ 3 sin ϑ x + Eϕ e 1 iπτ Λ ϕ E e 1 iπτ sinϑ ϑ + Eϕ Λ ϕ E ϕ with U1)-gauge theory A x,e x ) on 1-dimensional Σ, A ϕ :Σ R, P ϕ = E ϕ cos α using cos α := Λ A ϕ Λ E ϕ ) e iβ :Σ U1), P β = A ϕ E ϕ sin α in Λ A ϕ = cosβ)τ 1 + sinβ)τ ). Gauss constraint: x E x + A ϕ E ϕ Λ A ϕ Λ E ϕ = x E x + P β = 0. Thus sin α = 0 if homogeneous Kantowski Sachs) but new freedom in general spherical symmetry. Complicated relation between momenta and triad components if cos α is free. Will make volume in terms of fluxes complicated. Spherical symmetry p.13

14 Basic variables and smearing A simple canonical transformation A ϕ A ϕ cos α = γk ϕ P ϕ E ϕ gives densitized triad components as momenta. Use extrinsic curvature component K ϕ instead of connection component A ϕ, but keep U1)-connection A x for now). Still allows natural smearing in 1-dimensional model: ) 1 h I A x ) = exp i A x x)dx, F v E x ) = E x v) I h v K ϕ ) = expiγk ϕ v)), F I E ϕ ) = dxe φ x) I h v β) = expiβv)), F v P β ) = dxp β x) I Spherical symmetry p.14

15 Quantization Orthonormal basis 1-dimensional graph g, k e,k v Z, µ v R) T g,k,µ A) = Flux operators: e Eg) 1 exp ik e e ) A x dx v V g) e iγµ vk ϕ v)+ik v βv) Ê x x)fa) = iγ e δh e δa x x) f h e = 1 γ x e h e f h e such that Êx x)t g,k,µ = 1 γk ex)t g,k,µ. Similarly: Ê ϕ T g,k,µ = γ µ v T g,k,µ, I v I I ˆP β T g,k,µ = γ v I k v T g,k,µ Spherical symmetry p.15

16 Constraints Gauss constraint: E x ) + P β = 0 implies k v = 1 k e + v) k e v)) such that T g,k,µ A) = e Eg) exp 1 ik e e e v) ) A x + β )dx v V g) v e + v) exp iγµ v K ϕ v)) Now only extrinsic curvature: γk x = A x + β U1)-invariant). Diffeomorphism constraint: moves vertices. Hamiltonian constraint: H[N] = dxnx) E x 1/ 1 Γ ϕ + Kϕ)E ϕ with Γ ϕ = Ex ) E ϕ. Σ K ϕ K x E x + E x Γ ) ϕ Spherical symmetry p.16

17 Hamiltonian constraint Operator constructed from holonomy and flux operators following general scheme [T. Thiemann] and trh x h ϑ v + )h 1 x h ϑ v) 1 h ϕ {h 1 ϕ,v }) δγ K ϕ v)k x v) E x trh ϑ v)h ϕ v)h ϑ v) 1 h ϕ v) 1 h x {h 1 x,v }) δγ K ϕ v) E ϕ E x Radial edge length δ gives discretized integration measure; h x, h ϑ and h ϕ suitable SU)-holonomies. Operator changes labels through holonomies, may add new vertices to graphs as source for lattice refinement. Spherical symmetry p.17

18 Physical states Without explicit refinement: Decomposition in basis states k e Z, 0 µ v R) ψa,φ) = k, µ ψ k, µ) k k + µ µ µ A,φ) + gives rise to ĤψA,φ) = 0 as set of coupled difference equations one for each edge) Ĉ 0 k) ψ...,k,k +,...) + Ĉ R+ k) ψ...,k,k +,...) +Ĉ R k) ψ...,k,k + +,...) + Ĉ L+ k) ψ...,k,k +,...) +ĈL k) ψ...,k +,k +,...) + = 0 with difference operators Ĉ 0 and Ĉ ± acting on the µ-dependence. Non-singular: evolution across k i = 0. [PRL ) ] Bounded inverse triad. [V. Husain, O. Winkler: CQG 005) L17] Spherical symmetry p.18

19 Simpler dilaton models? [MB, J. Reyes: CQG 6 009) ] Use canonical transformation to introduce arbitrary dilaton potential e.g. higher-dimensional black holes, BF [JT]): Hamiltonian constraint K ϕe ϕ +K ϕ K x E x +E ϕ E x 1 V 1 4 Ex ) Γ ϕ where again Γ ϕ = E x ) /E ϕ. ) + E x Γ ϕ = 0 Only one term affected, which turns out to be the simplest one directly quantized in terms of flux operators. Terms containing extrinsic curvature or spatial derivatives spin connection) do not depend on the potential. Advantage: Most details of loop quantization holonomies, spatial discretization of derivatives) model independent. Disadvantage: No simplifications from alternative potentials. Spherical symmetry p.19

20 Consistent deformations Inverse triad corrections can be incorporated in anomaly-free way: H inv grav[n] = Σ dxnx)f[e x ] E x 1/ 1 Γ ϕ + K ϕ)e ϕ K ϕ K x E x + E x Γ ) ϕ together with matter Hamiltonian H inv matter[n] = Σ provided that f = νσ. dxnx)ν[e x ]H kin + σ[e x ]H grad + H pot ) Constraint algebra deformed, but remains first class. [To be compared with partially gauge-fixed version: V. Husain et al.] Spherical symmetry p.0

21 LTB models [MB, T. Harada, R. Tibrewala: PRD ) ] Alternatively, consistent formulation with partial) holonomy corrections under LTB-like condition [E x ) = g[k ϕ ]E ϕ ]. Constraint equation for R = E x, in terms of mass function Fx)): R R + Ṙ 1 γ δ Ṙ = 0 together with the evolution equation 4Ṙ R 1 γ δ Ṙ + 8RṘṘ ) = F γ δ Ṙ 1 γ δ Ṙ. Spherical lattice refinement: δ[r], more complicated analysis. Potentially naked singularities. No indication of resolution yet. Spherical symmetry p.1

22 Conclusions General features of black hole dynamics in models of loop quantum gravity available, but not unique. Consistency conditions e.g. stability) can be checked; provide conditions also for acceptable behavior in more general cases or full theory. Several models already ruled out! Spacelike singularities in spherical symmetry resolved even inhomogeneous ones), but no clear effective picture yet: quantum back-reaction still to be captured reliably. For constraints, not all corrections implemented consistently anomaly-free) yet. Spherical symmetry p.