Quantum Extension of Kruskal Black Holes

Transcription

1 Quantum Extension of Kruskal Black Holes Javier Olmedo Penn State University & Louisina State University In collaboration with Abhay Ashtekar and Parampreet Singh arxiv: , arxiv: ILQGS, 09/10/2018

2 Motivation Symmetry reduced models (FRW, Bianchi, Schwarzschild, Kerr,...) allow us to carry out concrete calculations (local and global properties of the spacetime geometry, primordial power spectrum, Hawking radiation and black hole evaporation,...). Within loop quantum gravity, it takes lots of effort to obtain similar reduced results... and efforts have begun (Alesci, Cianfrani, Dapor, Liegener, Pawłowski,...). The (loop) quantization of symmetry reduced models of GR is simpler and has provided very useful lessons about the physics and mathematics applicable into the full theory (singularity resolution, semiclassical sectors, potential predictions,...). The most fruitful example is loop quantum cosmology. Rigorous quantization of various cosmological spacetimes based on a difference (discrete) quantum Hamiltonian operator. 1/21

3 Motivation Spherically symmetric spacetimes in vacuum provide the simplest models for black holes. They have important applications in quantum gravity. For instance: Singularity resolution and its role in the understanding of information loss paradox (LQG viewpoint). Other approaches argue that BH singularities may persist. A lot of literature in the past years focused on the application of LQC ideas (based on the analogs of µ o and µ schemes) to the interior of black holes. But detailed predictions have physically undesirable features. Other alternatives (adopting an Abelian constraint) need to be further explored (Bojowald, Brahma, Campiglia, Corichi, Gambini, O, Pullin, Saeed,...). 2/21

4 The model 3/21 Main ideas: We focus on spacetimes with spherical symmetry in vacuum (black holes) using homogeneous slicings (trivial constraint algebra) and adopt loop quantum cosmology techniques. We limit ourselves to effective descriptions (classical evolution equations modified to incorporate quantum geometry corrections). What is new: Extension to the exterior region. Judicious choices of plaquettes to define the curvature operator in the Hamiltonian constraint (in the spirit of improved dynamics proposals). Consequences: We provide the quantum extension of the macroscopic Kruskal black holes. The singular regions are replaced by regular transition surfaces that separate trapped and anti-trapped regions. There, curvature reaches universal upper bounds. Away from the Planck regime, the spacetime metric (and then the curvature) is well approximated by the classical theory. ADM masses are the same.

5 Kinematical setting Topology of homogeneous Cauchy slices is R S 2. The fiducial metric q ab on Σ is q ab dx a dx b = dx 2 + r 2 o (dθ2 + sin 2 θdφ 2 ), x (, ), r o = const. (1) Reduced connection and triad can be expressed as A i a τ i dx a = c τ 3 dx + b r o τ 2 dθ b r o τ 1 sin θ dφ + τ 3 cos θ dφ, (2) E a i τ i a = p c r 2 o τ 3 sin θ x + p b r o τ 2 sin θ θ p b r o τ 1 φ. (3) The symplectic structure (fiducial cell x [0, L o]) takes the form Ω = Lor2 o ( ) 2d b d pb + d c d p c. (4) 2Gγ New suitable variables b = r o b, c = Lo c, p b = L or o p b and p c = r2 o pc satisfying the Poisson brackets: {c, p c} = 2Gγ, {b, p b } = Gγ. (5) These variables, under the transformation r o βr o, are invariant. Under a rescaling of fiducial length L o the combinations c/l o and p b /L o, and b, p c are invariant. The spacetime metric is g ab dx a dx b ds 2 = Nt 2 dt2 + p2 b p clo 2 dx 2 + p c(dθ 2 + sin 2 θdφ 2 ), (6) 4/21

6 Classical dynamics of the interior region Classical dynamics in the interior dictated by the Hamiltonian constraint H cl [N cl ] = 1 ( ) ) 2c p c + (b + γ2 p b, N cl = γ b 1 p 1/2 c. (7) 2Gγ b The solutions to the dynamical equations and Hamiltonian constraint (well adapted to Schwarzschild geometry) are b(t cl ) = γ ( e T cl 1 ) 1/2, pb (T cl ) = 2mL 0 e T cl ( e T cl 1 ) 1/2, (8) where < T cl 0 and With τ = 2me T cl we get c(t cl ) = γlo 4m e 2T cl, p c(t cl ) = 4m 2 e 2T cl, (9) cp c L oγ = m = 1 2L oγ ds 2 = ( 2m τ 1 ) 1dτ 2 + (b + γ2 b ) p b. (10) ( 2m τ 1 ) dx 2 + τ 2 (dθ 2 + sin 2 θdφ 2 ). (11) Singularity located at τ = 0 (or T cl = ). Horizon located at τ = 2m (or T cl = 0). 5/21

7 Extension to the exterior We keep a homogeneous slicing, i.e. time-like hypersurfaces with ( + +) 3-metric. The connection and triad are now SU(1, 1) valued (instead of SU(2)). or equivalently: A i a τ i dx a = c L o τ 3 dx + b τ 2 dθ b τ 1 sin θ dφ + τ 3 cos θ dφ, (12) E a i τ i a = p c τ 3 sin θ x + p b L o τ 2 sin θ θ p b L o τ 1 φ, (13) τ 1 iτ 1, τ 2 iτ 2, τ 3 τ 3 ; b i b, p b i p b ; c c, p c p c (14) The Poisson brackets are now given by: The spacetime metric is { c, p c} = 2Gγ, { b, p b } = Gγ. (15) g ab dx a dx b d s 2 = p2 b p clo 2 dx 2 + Ñτ 2 dτ 2 + p c(dθ 2 + sin 2 θdφ 2 ). (16) 6/21

8 Classical dynamics of the exterior region Classical dynamics in the exterior dictated by the Hamiltonian constraint H cl [Ñ cl ] = 1 ( ) ) 2 c p c + ( b + γ2 1/2 p b, Ñ = γ b 1 p c. (17) 2Gγ b Solutions to the dynamical equations and Hamiltonian constraint are given by b(tcl ) = ±γ ( 1 e T cl ) 1/2, pb (T cl ) = 2mL 0 e T cl ( 1 e T cl ) 1/2, (18) c(t cl ) = γlo 4m e 2T cl, p c(t cl ) = 4m 2 e 2T cl, (19) where 0 T cl < and c p c L oγ = m = 1 2L oγ ( b + γ2 b ) p b. (20) The exterior and interior regions match smoothly across the horizon (T cl = 0). Spatial infinity is located at T cl =. With τ = 2me T cl we get ( ds 2 = 1 2m τ ) ( dx m τ ) 1dτ 2 + τ 2 (dθ 2 + sin 2 θdφ 2 ). (21) 7/21

9 Classical Penrose diagram of the Kruskal spacetime τ = 0 i + III J + II I i 0 IV J τ = 0 i 8/21

10 Quantum corrected dynamics for the interior Quantum corrections to the classical theory. i) H[N] contains curvature F i ab of the gravitational connection Ai a. ii) F i ab can be obtained from (h /A ), where h is a loop of holonomies of A i a and A the area enclosed by (we choose it to be equal to the LQG area gap = 4 3πγl 2 Pl ). iii) curvature operators ˆF θ,φ, ˆF x,θ, ˆF x,φ are determined by the fractional lengths of the links in these plaquettes: δ c for the x-directional link and δ b for links in the 2-spheres. LQC effective dynamics dictated by H eff [N] = 1 [2 sin(δcc) ( sin(δb b) p c + + γ 2 δb ) p b ], N = γ p1/2 c δ b 2Gγ δ c δ b sin(δ b b) sin(δ b b). (22) The solutions to the dynamical equations, provided that δ b and δ c are chosen to be appropriate constants of motion, take the form cos ( δ b b(t) ) = b o tanh tan ( 12 ( b ot + 2 tanh 1 ( 1 bo ) )), p b (T) = 2 sin(δc c(t)) sin(δ b b(t)) δc δ b pc(t) sin 2, (23) (δ b b(t)) δ b 2 +γ 2 ( ) δc c(t) = γloδc 2 8m e 2T, p c(t) = 4m 2( e 2T + γ2 L o 2 δ2 ) c 64m 2 e 2T, b o = (1 + γ 2 δb 2 )1/2, (24) where [ sin(δc c) ] p c = m = 1 [ sin(δb b) + γ 2 δb ] pb (25) γl oδ c 2 δ b sin(δ b b) γl o 9/21

11 Causal structure The spacetime has the Killing vector field X a a = / x that becomes null when p b = 0 (horizons at T = 0 and T = 4/b o tanh 1 (1/b o)). The expansions ( θ ± ) of the null normal vectors l ± a to the 2-spheres vanish at ṗ c = 0, i.e. at T T = 1 2 ln γloδc. Each solution has one and only one transition surface T that separates a 8m trapped region in the past (BH region) and an anti-trapped region to the future (WH region). The entire geometry is smooth since the transition surface T replaces the classical singularity by a regular highcurvature region. Here p c(t T ) = 1 2 γ (Loδc). In summary, the geometry consists of a boundary T between a trapped region (BH type) in the past and an antitrapped region (WH type) to the future. 10/21

12 Determination of δ b and δ c The quantum parameter δ b has the interpretation of the fractional length of each link constituting the plaquette within the θ-φ 2-spheres, and δ c, of the fractional length of the links in the x-direction within the plaquettes in the θ-x and φ-x planes in a fiducial cell. Our proposal: Choose the plaquettes to lie on the transition surface where the space-time has largest curvature. One then has: 2π δ cδ b p b T =, (26) (Physical (fractional) area of the annulus around equators) 4π δ 2 b pc T =. (27) (Physical (fractional) area of the sphere) The solutions can be written in a closed form for macroscopic BHs (m l Pl ): ( ) 1/3, δ b = Loδc = 1 ( γ 2 ) 1/3. (28) 2πγ 2 m 2 4π 2 m 11/21

13 Physical consequences Several curvature invariants are bounded above, reaching a maximum value at the transition surface that is mass independent, namely ( ( C abcd C abcd T = 1024π2 3γ O ) 1 ( ( 3 m 2 ln ), m2 K T = 768π2 γ O ) 1 ) 3 m 2 ln m2. (29) K T Curvature invariant for m = /21

14 Physical consequences Quantum correction can be seen as an effective fluid with a stress-energy tensor T ab characterized by the energy density ρ and the radial and tangential pressures p x and p, respectively. One can easily see that, at the transition surface, the strong energy condition is violated (T ab 1 2 g abt)t a T b T < 0. (30) 10 2 ρ ( ρ) 10 5 p ( p ) T Curvature invariant for m = /21

15 Physical consequences A puzzling aspect arises from considerations of the Komar mass at the horizons (T = 0 and T = 4/b o tanh 1 (1/b o)) with respect to the Killing vector field X = X a a = x: K[S] := 1 ɛ cd ab cx d ds ab, (31) 8πG S ( ) K[S 2 ] K[S 1 ] = 2 Tab (T/2)g ab X a dv b. (32) Σ 14/21 The 3-surface integral is large and negative because of properties of T ab in the interior. But it is also delicately balanced to make the volume integral precisely equal to 2K[S 1 ] (in the large m limit). Therefore, K[S 2 ] = K[S 1 ]. (33) Since ADM mass refers to the unit future pointing asymptotic time-like Killing vector field, this is the sufficient condition for r B = r W when m 2. This shows that there are highly non-trivial constraints that our effective geometry satisfies: singularity resolution and (nearly) unit amplification factor for the mass in the transition from the trapped to the anti-trapped region.

16 Comparison with other proposals i) Constant µ 0 schemes (physical results depend on fiducial cell) δ b ii) Improved dynamics µ schemes (BV scheme) δ b l Pl, L oδ c. (34) p c, L oδ c iii) Schemes based on dimensional arguments (CS scheme) iv) Our proposal (AOS scheme) δ b pc (p b /L o). (35) 2m, Loδc. (36) ( ) 1/3, δ b = Loδc = 1 ( γ 2 ) 1/3. (37) 2πγ 2 m 2 4π 2 m 15/21

17 Comparison with other proposals BV AOS CS Class pc(t ) T Comparison for m = In the classical theory, p c decreases steadily corresponding to the monotonic decrease in the radius of the round 2-spheres. In CS and AOS, it undergoes precisely one bounce, with a trapped region to the past of the bounce and anti-trapped to the future. In BV, it undergoes several bounces. The anti-trapped region after the first bounce is very short lived. After the second bounce, this µ scheme cannot be trusted because its underlying assumptions are violated. 16/21

18 Comparison with other proposals pb(t )/Lo BV AOS CS Class T Comparison for m = This triad component signals the emergence of a black or white hole type horizon when it vanishes. The white hole type horizon emerges much much later in the CS approach (large mass amplification in the CS approach) than in AOS one (no amplification in the large m limit). The BV approach becomes unreliable after T 12. Besides, very near the black hole type horizon, the BV dynamics deviates from the classical theory. The AOS dynamics is indistinguishable from classical dynamics near this horizon. 17/21

19 Extension to the exterior We follow the same ideas discussed in the classical theory (homogeneous slicing and SU(1, 1) valued triad and connection) The dynamics in the exterior is dictated by H eff [Ñ] = 1 2Gγ [2 sin(δ c c) ( p c + sinh(δ b b) + γ2 ) ] δ b p b, Ñ = γ p1/2 c δ b. (38) δ c b) b) δ b sinh(δ b sinh(δ b The solutions to the dynamical equations take the form cosh ( δ b b(t) ) = bo tanh tan ( 1 2 ( bot + 2 tanh 1 ( 1 bo ) )), p b (T) = 2 sin(δ c c(t)) b(t)) sinh(δ b pc(t) δ c δ b γ 2 sinh2 b(t)) (δ b δ 2 b ( ) δ c c(t) = γloδ c 2 8m e 2T, p c(t) = 4m 2( e 2T + γ2 Lo 2 δ2 c ) 64m 2 e 2T, bo = (1 + γ 2 δ 2 b) 1/2. (40), (39) where The spacetime metric is sin(δ c c) γl oδ c p c = m = 1 2 [ sinh(δ b b) δ b + γ2 δ b sinh(δ b b) ] pb γl o (41) g ab dx a dx b d s 2 = p2 b p clo 2 dx 2 + Ñ 2 τ dτ 2 + p c(dθ 2 + sin 2 θdφ 2 ). (42) The horizon is located at T = 0 and the spatial infinity at T = +. The exterior and interior regions match smoothly across the horizon. 18/21

20 Quantum extension of the Kruskal diagram T J + B J + i 0 IV III i 0 J J + W T B J J + i 0 II I i 0 J W T J 19/21

21 Summary We provide a quantum extension (effective description a la LQC) of an eternal black hole with the following properties. i) Physical quantities are insensitive to the fiducial structures necessary to construct the classical phase space. ii) It admits an infinite number of trapped, anti-trapped and asymptotic regions. iii) Consecutive asymptotic regions of macroscopic black holes have same ADM mass (macroscopic BHs). iv) High-curvature regions are regular (singularity resolution). v) The simplest curvature scalars have mass-independent upper bounds (macroscopic BHs). vi) At low curvatures, quantum effects are small. None of the previously proposed models so far meet all these properties simultaneously. 20/21

22 Limitations and future directions Full quantum dynamics need to be understood: i) The quantum Hamiltonian constraint has a complicated action. Fortunately, we have a proposal worth to be explored. ii) Classical and quantum Hamiltonian framework have been recently discussed in the literature for SU(1, 1) connections in 3+1 gravity. Interesting possibility: Application to our model. iii) Then, one could derive effective equations using the LQC strategy. This is not a collapsing black space-time. But many relevant calculations already use the Kruskal space-time (vacuum states of linear field theories, renormalized stress energy tensor, etc.). Extension to more general spacetimes. 21/21

23 Implementation as functions of Dirac observables The quantum parameters δ b and δ c have been judiciously chosen as Dirac observables. Subtlety: Since H depends on δ b, δ c and they have to be so chosen that along dynamical trajectories determined by H, they are constants of motion. We have found a way to meet this subtle self-consistency requirement: First extend the phase space by adding δ b, δ c as new phase space variables, together with their conjugate momenta. Then restrict δ b, δ c to be desired functions of the original phase space variables via new first class constraints. Finally, identify suitable gauge fixing conditions that eliminate those new constraints (via gauge fixing) so that the reduced phase space is naturally isomomorphic to the original one. To implement this strategy, it is simplest to replace (b, p b ), (c, p c ) with other canonical coordinates (to get the solutions in closed form). 21/21

24 Implementation as functions of Dirac observables Let us consider new coordinates in the phase space Γ o O 1 := 1 [ sin δb b + γ2 δ b 2γ δ b sin δ b b ] pb L o, O 2 := and suitable conjugated momenta P i such that {O i, P j } = δ ij. [ sin δcc ] p c. (43) γl oδ c Let us extend the phase space (Γ o, Ω o) to (Γ, Ω) where δ b and δ c are now phase space variables together with their momenta {δ b, P δb } = 1 and {δ c, P δc } = 1. Let us now introduce the desired conditions on δ b and δ c as constraints (first class with the Hamiltonian). Φ 1 = O 1 F b (δ b ), Φ 2 = O 2 F c(δ c). (44) The total Hamiltonian reads H T = LoÑ G (O 2 O 1 ) + λ 1 Φ 1 + λ 2 Φ 2, (45) We now require gauge fixing conditions such that i) the Hamiltonian vector field is tangential to the constraint - gauge surfaces and ii) the pullback of Ω on those surfaces is symplectomorphic to Ω o. One can see that the gauge fixing conditions that satisfy these requirements are φ 1 = P δb = 0 and φ 2 = P δc = 0. The resulting EOMs are equivalent to the ones we use in our approach. 21/21

25 Determination of δ b and δ c 10 3 p b (T t )δ b δ c δc(δb ) 2π δ b The roots of δ b obtained from solving the area conditions for m = In the large m limit, the central roots are extremely well approximated by previous equations. The leftmost and rightmost roots turn out to have unphysical properties. 21/21