Quantum gravity predictions for black hole interior geometry
Emanuele Alesci1,∗ Sina Bahrami1,† and Daniele Pranzetti2‡ 1Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park, Pennsylvania 16802, USA and 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Dated: September 13, 2019) In a previous work we derived an effective Hamiltonian constraint for the Schwarzschild geometry starting from the full loop quantum gravity Hamiltonian constraint and computing its expectation value on coherent states sharply peaked around a spherically symmetric geometry. We now use this effective Hamiltonian to study the interior region of a Schwarzschild black hole, where a homo- geneous foliation is available. Descending from the full theory, our effective Hamiltonian, though still bearing the well known ambiguities of the quantum Hamiltonian operator, preserves all relevant information about the fundamental discreteness of quantum space. This allows us to have a uniform treatment for all quantum gravity holonomy corrections to spatially homogeneous geometries, unlike the minisuperspace loop quantization models in which the effective Hamiltonian is postulated. We show how, for several geometrically and physically well motivated choices of coherent states, the classical black hole singularity is replaced by a homogeneous expanding Universe. The resultant ge- ometries have no significant deviations from the classical Schwarzschild geometry in the pre-bounce sub-Planckian curvature regime, evidencing the fact that large quantum effects are avoided in these models. In all cases, we find no evidence of a white hole horizon formation. However, various aspects of the post-bounce effective geometry depend on the choice of quantum states.
A primary aim for the quantization program of grav- Firstly, we were still confronted with the fundamental itational field is to determine the fate of classical space- issue of whether the microscopic degrees of freedom re- time singularities as predicted by general relativity. It moved by the symmetry reduction at the classical level has been speculated that quantum gravitational effects could have been relevant, affecting the final physical pre- smooth out spacetime singularities, analogously to how dictions of the theory. In the case of LQC techniques ap- an ultraviolet completion of quantum fields tames ultra- plied to the Schwarzschild interior, this was clearly mani- violet divergences. Fortunately, the canonical approach fest in the use of point holonomies for the construction of to loop quantum gravity (LQG) is now sufficiently devel- the reduced kinematical Hilbert space associated with the oped to systematically resolve this key issue from the full 2-spheres that foliate the spacelike surfaces, which com- theory perspective. pletely eliminated all information about the 3-D struc- To provide some historical context, we point out that ture of the graph. Secondly, the reintroduction, by hand, the first generation of quantum gravity predictions were of some ‘elementary plaquette’ at the end of the quanti- made in cosmology and within the minisuperspace quan- zation procedure, in order to fix the quantum parameters tization scheme. There the loop quantization program entering the construction of the states, was an ambigu- was implemented by applying LQG inspired techniques ous prescription, with different choices yielding different to a (model dependent) symmetry reduced phase space physical scenarios, as well as undesirable features. of general relativity parametrized by the Ashtekar con- In order to construct a more robust and reliable model nection and densitized triad [1]. The resultant models, of quantum black hole geometry that addresses the afore- which later became known as loop quantum cosmology mentioned issues, a new framework, the so-called ‘quan- (LQC), unanimously predict that the classical spacetime tum reduced loop gravity’ approach [15–20], has recently singularity is replaced by a quantum bounce [2,3]. They been developed in which one starts from the full LQG offered the first glimpse of how quantum gravity could
arXiv:1904.12412v2 [gr-qc] 12 Sep 2019 theory and then performs the symmetry reduction at the resolve spacetime singularities. quantum level by means of coherent states encoding the However, attempts to generalize the loop quantization information about a semi-classical spherically symmetric program to a black hole geometry were not equally suc- black hole geometry. More precisely, computing the ac- cessful at producing generally accepted predictions [4– tion of the quantum Hamiltonian operator on a partially 14]. The starting point of all these previous attempts was gauge fixed kinematical Hilbert space results in a semi- the observation that the Schwarzschild black hole interior classical, effective Hamiltonian constraint H that now can be described in terms of a contracting anisotropic eff replaces the Hamiltonian constraint of general relativ- Kantowski–Sachs model, allowing one to apply LQC ity. Unlike its predecessors, the newly obtained effective techniques to this highly symmetric geometry. This ap- Hamiltonian constraint for the Schwarzschild geometry proach presented two major drawbacks. [20] descends from the LQG quantum Hamiltonian con- straint. It retains all information about the fundamental degrees of freedom associated with the quantum space ∗ [email protected] † [email protected] that is inherited from the full theory. ‡ [email protected] The effective Hamiltonian in [20] was derived for a gen- 2 eral spacetime foliation, well defined for both the exte- tions and dΩ2 is the unit round 2-sphere line element. rior and the interior Schwarzschild regions. Solving for Due to spatial homogeneity, the only gauge freedom is the effective geometry in both regions cannot be done to rescale proper time by choosing a lapse function N. within the minisuperspace quantization program as a ho- As commonly done, in order to avoid having a diver- mogeneous slicing that covers the entire spacetime is not gent symplectic structure, we require x ∈ [0,L0] where available. One is forced to use an inhomogeneous slic- L0 is some infrared cut-off in the x direction. The co- ing and subsequently work with an infinite dimensional variant phase space of solutions consists of the met- phase space. In particular, the effective scalar constraint ric functions Λ and R and their conjugate momenta equation Heff = 0 is now a highly non-linear, non-local PΛ,PR. The symplectic 2-form for this phase space is ODE. While work is in progress to develop numerical ω = L0(dR∧dPR +dΛ∧dPΛ), whence the Poisson brack- techniques to solve this equation, a simpler task is to ets assume the familiar form {R,PR} = {Λ,PΛ} = 1/L0. restrict attention to the interior region where the stan- To avoid confusion with the previous LQG literature, we dard Schwarzschild foliation is now homogeneous and the emphasize that the L0 factor appearing in the symplec- coherent states are peaked around this homogenous ge- tic structure is not absorbed in the phase space variables. ometry. In this way, the effective Hamiltonian derived in Hence, in all subsequent equations, the phase space vari- [20], as well as the Hamilton’s evolution equations that ables must be regarded as invariants under a rescaling by it generates, assume a more tractable form which can be this cut-off. In particular, as it will become clear below, solved through a combination of analytical and numeri- the effective dynamics is L0-independent and the unde- cal tools. In this letter we report on the results of this sirable features related to this scale dependence that ap- analysis. pear in some of the previous minisuperspace models are Let us emphasize that our restricting to the interior absent in our treatment. minisuperspace for the sake of solving the interior effec- In the case of a Schwarzschild spacetime with a gravi- tive dynamics is not equivalent to the minisuperspace tational mass m, the above phase space variables assume quantization models of the previous literature. In fact, as the following trajectories (in the rest of the letter we de- it will be elucidated in detail, our Heff provides a uniform note classical solutions with a subscript c, and we work treatment of the quantum gravity SU(2) holonomy cor- in c = ~ = 1 units unless otherwise stated) rections to spatially homogeneous geometries. It should τ/2Gm be noted that a number of ambiguities present in the Rc(τ) = 2Gm e , quantization procedure for the Hamiltonian constraint p −τ/2Gm Λc(τ) = e − 1, in LQG [21] inevitably percolate in our construction as 1 h −τ/2Gmi well, implying non-uniqueness for the form of the quan- PRc (τ) = 2 − e , tum corrections that we derive. However, this level of 2G τ/4Gmp τ/2Gm ambiguity is a direct reflection of important open issues PΛc (τ) = −2m e 1 − e , (2) in the full theory, unlike the quantization ambiguity in the polymer approach that is due to a choice to work for with point holonomies. Therefore, our derived Heff repre- R2 sents a far better motivated alternative to the previously Nc = − (3) 2G2mP used Hamiltonian constraint of the minisuperspace loop Λ quantization procedure, which so far has been postulated as the choice for the lapse function. Here the range based on the example of homogeneous and isotropic cos- −∞ < τ < 0 covers the entire interior region of the mology [22]. At the same time, having the full theory Schwarzschild black hole, with τ = 0 corresponding to (graph) structure to begin with, our analysis provides a the horizon and τ = −∞ to the classical singularity. In consistent and intuitive geometrical framework in which the effective theory, the phase space variables are evolved we identify the quantum parameters entering the effective in time τ via the following Poisson brackets: dynamics. This allows us to investigate, in a controlled way, how various choices of quantum coherent states re- 1 ∂Heff [N] R˙ = {R,Heff [N]} = , sult in different physical predictions of the theory. L0 ∂PR 1 ∂Heff [N] Phase space and effective dynamics. We are inter- P˙R = {PR,Heff [N]} = − , ested in an effective description for the interior geom- L0 ∂R etry of a spherically symmetric black hole. Practically, 1 ∂Heff [N] Λ˙ = {Λ,Heff [N]} = , this amounts to considering quantum gravitational ef- L0 ∂PΛ fects for which the metric continues to be spatially ho- ˙ 1 ∂Heff [N] mogeneous. Therefore, let us specialize to a coordinate PΛ = {PΛ,Heff [N]} = − , (4) L0 ∂Λ system in which the metric is where dot denotes differentiation with respect to τ and 2 2 2 2 2 2 2 ds = −N(τ) dτ + Λ(τ) dx + R(τ) dΩ . (1) Heff [N] is the smearing of the effective Hamiltonian con- straint by the lapse function. As gravity is described by Here Λ(τ) and R(τ) are the two dynamical metric func- a constrained Hamiltonian system, we also have Heff = 0 3 at all times. This constraint equation is implied by the The appearance of the Struve function in (5) represents evolution equations, provided that it is satisfied at some the main departure from the minisuperspace quantiza- initial time. tion models in the previous literature. It is a direct reflec- In order to find the solutions to the evolution equa- tion of including ab initio the fundamental discreteness tions, let us introduce Heff and discuss in some detail of quantum geometry on the 2-spheres in the quantum the graph structure that we have adapted to constant τ reduced kinematical Hilbert space. The other significant slices. By specializing the general effective Hamiltonian difference, also originating from the SU(2) holonomies derived in [20] to the spatially homogeneous metric (1) along angular links, is encoded in the term proportional and integrating over the angular coordinates (amounting to γ2 inside the second curly brackets. This term derives to averaging out quantum fluctuations around spherical from the Lorentzian part of the LQG Hamiltonian con- symmetry), we obtain straint and it contains corrections to all orders in , while in its minisuperspace counterpart only the leading term L γG [P R − P Λ] − 0 x R Λ is present 1. Heff = 2 2 R sin 2 (5) 4γ Gx R In order to distinguish the classical regime, in which γGP γGP general relativity is expected to be a valid description × 2 sin Λ + πH Λ R 0 R of gravitational field, from the region where quantum ef- 2 fects are expected to become relevant, we rely on the 2 γGPΛ γGPΛ +xΛ 8γ cos () sin +π sin H0 . Kretschmann scalar which for the classical Schwarzschild 2 R R metric functions given in Eq. (2) becomes
Here H0(x) is the Struve function of zeroth order and γ − 3τ abcd 3 e Gm is the Immirzi parameter that, for the sake of numerical Kc := RabcdR = . (8) calculation presented below, we fix it to be approximately 4(Gm)4 0.274 in consistency with (some) black entropy calcula- Quantum gravity heuristics suggest that any effective tions in LQG [23, 24]. The two quantum parameters in metric encoding quantum gravity corrections should be- (5), and x, are the angular and longitudinal coordi- gin to deviate significantly from the classical one in nate lengths of the so-called plaquettes, cubic cells that the regime where curvature becomes (super-) Planck- are sewn together to make up a graph or a discrete ge- 4 ian, namely as Kc & 1/`p which happens for times ometric structure on constant τ surfaces. The graph is τ τ = (Gm/3) log [3`4/(4G4m4)]. We can thus de- such that each 2D hypersurface in a given leaf of our foli- . ? p fine the parameter ation identified by a pair of coordinates is tessellated by a large number of plaquettes, rendering these coordinate ρ := Rc(τ)/Rc(τ?) (9) lengths very small. More precisely, due to how the par- tially gauged fixed LQG kinematical Hilbert space was and expect quantum effects to become dominant at about constructed in [20] and the peakedness properties of the ρ ∼ 1. coherent states defined on the associated graph structure, The quantum parameters and x defined in (6) are these parameters are related to the semi-classical data as functions of the effective phase space variables (in the old literature, this corresponds to the so-calledµ ¯-scheme [6]). √ It follows that the quantum parameters have non-trivial α p p β 8πγ`p j contributions to the Poisson brackets defining the evo- = , α = 8πγ`p j0; x = , β = √ , R Λ j0 lution equations and lead to a very complicated system (6) of ODEs that is difficult to analytically integrate. We p 3 where `p = ~G/c is the Plank length and j0 and j are thus numerically solve the effective Hamilton’s evolution the quantum numbers associated respectively with the equations starting with classical initial data very near the 2/3 longitudinal and the angular links of the coherent states. black hole event horizon, namely when ρ ∼ (Gm/`p) . It is important to emphasize that the above quantum pa- We specialize to the following choice of lapse function rameters are local quantities that are invariant under a γR rescaling of global quantities, such as the infrared cut-off N = − h i , (10) γGPΛ π γGPΛ L0. Therefore, the dynamics derived from the effective Gm sin R + 2 H0 R Hamiltonian (5) with the expressions (6) for the quan- tum parameters does not contain any information about L0, rendering our key physical predictions independent 1 of this scale as well. At the same time, this is the only contribution from the Lorentzian term that survives in the homogeneous interior fo- Let us point out that, not surprisingly, in the classical liation. In all cases considered in this letter, it can be verified limit where ~ → 0, and x vanish, the discreteness of that this contribution does not play a dominant role in determin- quantum geometry disappears, and the effective Hamil- ing the asymptotic behavior for the effective metrics. Therefore, tonian in Eq. (5) reduces to its classical value we expect the ambiguities surrounding the quantization prescrip- tion for the Lorentzian piece of the Hamiltonian constraint in the H GP P GΛP 2 Λ full theory not to have a significant impact on the physical pre- c − R Λ Λ − = + 2 . (7) dictions of the effective theory for the interior geometry. L0 R 2R 2G 4
6× 10 12 12 10 mp and j = j0 = 10. The qualitative behavior of 5× 10 12 the two metric functions remain the same for different 4× 10 12 values of the mass and spin numbers as long as m is
12 3× 10 R reasonably above the Planck mass (a stellar mass black 38 2× 10 12 Rc hole has m ≈ 10 mp). The plots in FIG.1 show how
1× 10 12 the effective metric agrees with the classical one in the low curvature region which spans all the way till τ ∼ τ , 0 ? -6× 10 13 -4× 10 13 -2× 10 13 0 where the Planck regime is reached and quantum geome- τ 600 000 try corrections become dominant, as expected. In partic- ular, R decreases with τ till it reaches a minimum value 500 000 Rmin = R(τb). It then bounces and grows exponentially 400 000 as τ → −∞. Therefore, the area of 2-spheres foliating 300 000 Λ constant τ surfaces never shrinks to zero, i.e. the singu- Λc 200 000 larity is effectively resolved. The surface τ = τb marks 100 000 the transition for the 2-spheres from being trapped in
0 the past (τb < τ < 0) to being anti-trapped in the future -5× 10 13 -4× 10 13 -3× 10 13 -2× 10 13 -1× 10 13 0 (−∞ < τ < τ ). In order to determine if a white hole τ b horizon forms in the future of τb, we need to analyze the FIG. 1. Plots of the effective metric functions R, Λ (blue lines) behavior of the second metric function Λ. in comparison with the classical metric function Rc, Λc (red As can be seen in FIG.1, numerical integration sug- 12 lines) for m = 10 mp in the first regime α = β for the choice gests that Λ approaches a constant non-zero value for 13 of j = j0 = 10. The bounce occurs at τb = −3.9 × 10 (with τ 0. An analysis of the Hamilton’s dynamical equa- 13 τ? = −3.7 × 10 ). The quantities are reported in the unit tions confirms this indication. In fact, it can be shown 2 mp = `p = 1. that as τ → −∞ one has PR/RΛ and 2PΛ/R approach- ing −2π/γα, which in turn implies that R/R˙ → −1/2m and Λ˙ → 0. We obtain the following estimates for the as it simplifies the analysis we perform in the asymptotic metric functions in this limit: regime. This lapse reduces to the classical Nc given above √ in the limit → 0. In presence of a non-zero and for j`p ~ p N(τ) → constant ∼ , m much larger than the Planck mass mp = ~c/G, the Gm 1 black hole event horizon is still nearly at τ = 0. τ can be Gm 3 extended all the way to −∞ unless the coordinate system Λ(τ) → constant ∼ √ , j` e g p breaks down at a finite value of τ due to, . , reaching a 1 Killing horizon. 2 4!3 j `p − τ The only remaining freedom in specifying the coherent R(τ) → e 2Gm . (11) Gm states is in the choice of spin numbers j0 and j. Different choices of α and β can appear depending on what one Asymptotically, as can be seen from the above equation, assumes for the quantum numbers j0 and j. It turns the interior geometry becomes a product of a 2D pseudo- out that the qualitative behavior of the solutions to the Euclidean space and a round 2-sphere whose proper area effective dynamics depends only the value of the ratio is blowing up exponentially. Note that as ~ → 0 the clas- η ≡ α/β and not on the specific value of the two spin sical singularity reappears. It is clear from Eq. (11) that numbers 2. We now present results for both when η = 1 the interior metric is not asymptotically flat. In fact, and when η ∈ (1 − ζ, 1 + ζ) for ζ 1, as one would the Ricci scalar approaches the non-vanishing asymp- 2 2 2 2 expect from the regularity condition on the geometry in totic value 3/(2G m N ) ∼ 1/(j0`p). Some of the other ab the large number of plaquettes limit. curvature invariants, such as the Ricci squared RabR and the Kretschmann scalar remain non-vanishing but The case η = 1. The effective metric functions R(τ) abcd bounded everywhere, while the Weyl squared CabcdC and Λ(τ) are plotted in FIG.1 for the choice of m = vanishes asymptotically. Interestingly enough, the upper bounds as well as the asymptotic values of the aforemen- tioned curvature scalars are all independent of the black 2 Let us remark that the coherent states we used to derive the hole mass m and only carry information about the quan- effective Hamiltonian (5) are peaked already for spin numbers tum structure. It follows without much difficulty from of order 10 (see, e.g., [25]). In the cases considered here, the the asymptotic relations (11) that the interior metric is bounce happens at an instant of time in vicinity of τ?, where 2/3 2 geodesically complete. The null energy condition is vio- the area of the minimal 2-sphere is of order A ∼ (Gm/`p) `p. 12 Therefore, already for black holes of mass m ∼ 10 mp, the spin lated for τ . τ∗. numbers j ∼ j0 ∼ 10 correspond to the minimal 2-sphere being tessallted by N ∼ 107 plaquettes. This is well within our working The case η ∈ (1 − ζ, 1 + ζ). The qualitative behavior assumption of a large number of degrees of freedom even in the of the metric function R in this case is similar to the pre- high curvature regime. vious one, with a bounce happening around the critical 5 time τ? followed by a subsequent exponential growth. As for Λ, it crucially depends on whether η < 1 or η > 1. In fact, performing an asymptotic analysis of the Hamil- 1.0 ton’s equations indicates that N approaches a constant 0.8 in the τ → −∞ limit while the other two metric functions assume the following time dependence: 0.6 ↵ δ F(η)τ/2Gm cos (κ/η)τ/2Gm 0.4 ∝ ∝
2 2 2 2 3 where we used Eq. (6). Therefore, as in the regulariza- Here L() = 2γ 2 + πH−1(π) /π H0 (π) + O( ) , tion scheme of [14], the two quantum parameters acquire which is numerically found to be greater than zero for a dependence on the gravitational mass as a power of 1, but significantly suppressed compared to unity. −1/3. Numerical investigations of Eq. (4) show that all This asymptotic behavior of Λ is confirmed by the nu- commonly used spacetime curvature invariants, such as merical solution, that shows how Λ reaches a maximum the Kretschmann scalar, remain bounded for all times value around the bouncing time, then decreases until a −∞ < τ < 0, solidifying the resolution of black hole minimum value greater than zero is reached, and then singularity in this model. Moreover, we numerically con- grows very slowly as τ → −∞. Therefore, Λ never goes firm a feature found in [13] that suggests that the upper to zero after the initial instant of time corresponding to bounds for these curvature invariants are mass indepen- the black hole horizon. dent as long as m mp. However, the similarities end here. In particular, by omitting the corrections coming from the 2-sphere quan- Let us conclude by remarking that, as in the previous tum geometry due to using point holonomies for the an- case, the interior metric is geodesically complete but not gular directions, the effective dynamics of [13] predicts asymptotically flat. However, now the components of the appearance of a white hole horizon at a finite instant the Riemann tensor computed in an orthonormal frame of time τ. As emphasized in the beginning of this letter, vanish asymptotically. Moreover, the weak, strong and the effective dynamics derived in [20] accounts for pre- dominant energy conditions are violated only around the cisely these ignored degrees of freedom that are present bounce τ ∼ τb ∼ τ∗, while they are satisfied as τ gets in the full theory. This modification has drastic implica- more negative. tions for predictions of the theory. In fact, our quantum- corrected interior geometry prevents the formation of a white hole horizon at any time. More precisely, the fol- lowing asymptotic estimates that are valid in the limit Acknowledgments. We are grateful to A. Ashtekar τ → −∞ can be obtained: and J. Olmedo for helpful discussions. E. Alesci and S. Bahrami are supported in part by the NSF grant PHY- R x − τ N(τ) ∼ ∼ e 2Gm , 1505411, the Eberly research funds of Penn State, and Gm Gm the Urania Stott fund of Pittsburgh Foundation. Re- Gm − τL() Λ(τ)∼ e 2Gm , search at the Perimeter Institute is supported in part by x the Government of Canada through NSERC and by the − τ R(τ)∼xe 2Gm . (18) Province of Ontario through MRI.
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