aa r X i v : . [ g r- q c ] J a n Gravitational collapse in quantum gravity
Viqar Husain ∗ Department of Mathematics and Statistics,University of New Brunswick, Fredericton, NB E3B 5A3, Canada (Dated: June 2007)We give a review of recent work aimed at understanding the dynamics of gravitational collapse inquantum gravity. Its goal is to provide a non-perturbative computational framework for understand-ing the emergence of the semi-classical approximation and Hawking radiation. The model studiedis the gravity-scalar field theory in spherical symmetry. A quantization of this theory is given inwhich operators corresponding to null expansions and curvature are well defined. Together with theHamiltonian, these operators allow one to follow the evolution of an initial matter-geometry stateto a trapped configuration and beyond, in a singularity free and unitary setting.
I. INTRODUCTION
One of the outstanding problems in theoretical physics is the incomplete understanding at the quantum level of theformation, and subsequent evolution of black holes in a quantum theory of gravity. Although a subject of study forover three decades, it is fair to say that, in spite of partial results in string theory and loop quantum gravity, thereis no widely accepted answer to many of the puzzles of black hole physics. This is largely because there has been nostudy of quantum dynamical collapse in these approaches. Rather, progress has focused mainly on explanations of themicroscopic origin of the entropy of static black holes from state counting. A four-dimensional spacetime picture ofblack hole formation from matter collapse, and its subsequent evolution is not available in any approach to quantumgravity at the present time.This paper summarizes an attempt to address this problem in the context of Hawking’s original derivation of blackhole radiation: spherically symmetric gravity minimally coupled to a massless scalar field. This is a non-linear 2dfield theory describing the coupled system of the metric and scalar field degrees of freedom. Gravitational collapsein the classical theory in this model has been carefully studied numerically [1, 2], but its full quantization has neverbeen addressed.Hawking’s semi-classical calculation [3] uses the eikonal approximation for the wave equation in a mildly dynamicalbackground, where the dynamics centers on the surface of a star undergoing collapse. The essential content of it isthe extraction of the phase of the ingoing mode from an outgoing solution of the scalar wave equation as a classicallycollapsing star crosses its Schwarzschild radius. According to this calculation, emitted particles appear to originatenear the event horizon. This means that an emitted particle observed by a geodesic observer is transplankian atcreation origin due to the gravitational redshift (which is infinite at the horizon). Its back reaction is therefore notnegligible, bringing into question the entire approximation.It is likely that a complete understanding of quantum evolution in this system will resolve all the outstandingproblems of black hole physics in the setting in which they originally arose. The following sections contain a summaryof the work described in refs. [4–7].
II. CLASSICAL THEORY
The phase space of the model is defined by prescribing a form of the gravitational phase space variables q ab and˜ π ab , together with falloff conditions in r for these variables, and for the lapse and shift functions N and N a , such thatthe ADM 3+1 action for general relativity minimally coupled to a massless scalar field S = 18 πG Z d xdt h ˜ π ab ˙ q ab + ˜ P φ ˙ φ − N H − N a C a i (1) ∗ Electronic address: [email protected] is well defined. The constraints arising from varying the lapse and shift are H = 1 √ q (cid:18) ˜ π ab ˜ π ab −
12 ˜ π (cid:19) − √ q R ( q )+4 πG (cid:18) √ q ˜ P φ + √ qq ab ∂ a φ∂ b φ (cid:19) = 0 , (2) C a = D c ˜ π ca − ˜ P φ ∂ a φ = 0 , (3)where ˜ π = ˜ π ab q ab and R is the Ricci scalar of q ab . The falloff conditions imposed on the phase space variables aremotivated by the Schwarzschild solution in Painleve-Gullstand (PG) coordinates, which itself is to be a solution in theprescribed class of spacetimes. These conditions give the following falloff for the gravitational phase space variables(for ǫ > q ab = e ab + f ab ( θ, φ ) r / ǫ + O ( r − ) , π ab = g ab ( θ, φ ) r / + h ab ( θ, φ ) r / ǫ + O ( r − ) , (4)where f ab , g ab , h ab are symmetric tensors, π ab = ˜ π ab / √ q , and q = det q ab .In this general setting we use the parametrization q ab = Λ( r, t ) n a n b + R ( r, t ) r ( e ab − n a n b ) (5)˜ π ab = P Λ ( r, t )2Λ( r, t ) n a n b + r P R ( r, t )4 R ( r, t ) ( e ab − n a n b ) , (6)for the 3-metric and conjugate momentum for a reduction to spherical symmetry, where e ab is the flat 3-metric and n a = x a /r . Substituting these into the 3+1 ADM action for general relativity shows that the pairs ( R, P R ) and(Λ , P Λ ) are canonically conjugate variables. We note for example the Poisson bracket n R f , e iλP R ( r ) o ≡ (cid:26)Z ∞ Rf dr, e iλP R ( r ) (cid:27) = i Gλf ( r ) e iλP R ( r ) , (7)which is the bracket represented in the quantum theory (described below).The falloff conditions induced on these variables from (4), together with those on the lapse and shift functions,ensure that the reduced action S R = 12 G Z dtdr (cid:16) P R ˙ R + P Λ ˙Λ + P φ ˙ φ − constraint terms (cid:17) + Z ∞ dt ( N r Λ P Λ ) (8)is well defined. This completes the definition of the classical theory.At this stage we perform a time gauge fixing using the condition Λ = 1 motivated by PG coordinates. This issecond class with the Hamiltonian constraint, which therefore must be imposed strongly and solved for the conjugatemomentum P Λ . This gauge fixing eliminates the dynamical pair (Λ , P Λ ), fixes the lapse as a function of the shift, andleads to a system describing the dynamics of the variables ( R, P R ) and ( φ, P φ ) [5]. The reduced radial diffeomorphismgenerator C red ≡ P ′ Λ ( R, φ, P R , P φ ) + P φ φ ′ + P R R ′ = 0 (9)remains as the only first class constraint. It also gives the gauge fixed Einstein evolution equations via Poissonbrackets, for example ˙ φ = { φ, R dr N r C red } . III. QUANTUM GRAVITY
The quantization route we follow is unconventional in that field momenta are not represented as self-adjoint op-erators; rather only exponentials of momenta are realized on the Hilbert space. This is similar to what happensin a lattice quantization, except that, as we see below, every quantum state represents a lattice sampling of fieldexcitations, with all lattices allowed. This quantization allows definitions of bounded inverse configurations operatorssuch as 1 /x , which for quantum gravity leads to the mechanism for curvature singularity resolution described below.A quantum field is characterized by its excitations at a given set of points in space. The important difference fromstandard quantum field theory is that in the representation we use, such states are normalizable. A basis state is | e i P k a k P R ( x k ) , e iL P l b l P φ ( y l ) i ≡ | a . . . a N ; b . . . b N i , (10)where the factors of L in the exponents reflect the length dimensions of the respective field variables, and a k , b l arereal numbers which represent the excitations of the scalar quantum fields R and φ at the radial locations { x k } and { y l } . The inner product on this basis is h a . . . a N ; b , . . . b N | a ′ . . . a ′ N ; b ′ . . . b ′ N i = δ a ,a ′ . . . δ b N ,b ′ N , if the states contain the same number of sampled points, and is zero otherwise.The action of the basic operators are given byˆ R f | a . . . a N ; b . . . b N i = L X k a k f ( x k ) | a . . . a N ; b . . . b N i , (11) \ e iλ j P R ( x j ) | a . . . a N ; b . . . b N i = | a . . . , a j − λ j , . . . a N ; b . . . b N i , (12)where a j is 0 if the point x j is not part of the original basis state. In this case the action creates a new excitation atthe point x j with value − λ j . These definitions give the commutator h ˆ R f , \ e iλP R ( x ) i = − λf ( x ) L \ e iλP R ( x ) . (13)Comparing this with (7), and using the Poisson bracket commutator correspondence { , } ↔ i ~ [ , ] gives L = √ l P ,where l P is the Planck length. There are similar operator definitions for the canonical pair ( φ, P φ ). A. Singularity resolution
To address the singularity avoidance issue, we first extend the manifold on which the fields R etc. live to includethe point r = 0 , which in the gauge fixed theory is the classical singularity. We then ask what classical phase spaceobservables capture curvature information. For homogeneous cosmological models, a natural choice is the inverse scalefactor a ( t ). For the present case, a guide is provided by the gauge fixed theory without matter where it is evident thatit is the extrinsic curvature that diverges at r = 0, which is the Schwarzschild singularity. This suggests, in analogywith the inverse scale factor, that we consider the field variable 1 /R as a measure of curvature. A more natural choicewould be a scalar constructed from the phase space variables by contraction of tensors. A simple possibility is˜ π = 12 (cid:18) P Λ R + P R Λ R (cid:19) . (14)The small r behaviour of the phase space variables ensures that any divergence in ˜ π is due to the 1 /R factor. Wetherefore focus on this. A first observation is that the configuration variables R ( r, t ) and φ ( r, t ) defined at a single pointdo not have well defined operator realizations. Therefore we are forced to consider phase space functions integratedover (at least a part of) space. A functional such as R f = Z ∞ drf R (15)for a test function f provides a measure of sphere size in our parametrization of the metric. We are interested inthe reciprocal of this for a measure of curvature. Since R ∼ r asymptotically, the functions f must have the falloff f ( r ) ∼ r − − ǫ for R f to be well defined. Using this, it is straightforward to see that 1 /R f diverges classically for smallspheres: we can choose f > f ∼ r <<
1, which for large r falls asymptotically to zero. Then R f ∼ r and 1 /R f diverges classically for small spheres.A question for the quantum theory is whether 1 /R f can be represented densely on a Hilbert space as 1 / ˆ R f . This ispossible only if the chosen representation is such that ˆ R f does not have a zero eigenvalue. If it does, we must represent1 /R f as an operator more indirectly, using another classically equivalent function. Examples of such functions areprovided by Poisson bracket identities such as1 | R f | = (cid:18) iGf ( r ) e − iP R ( r ) (cid:26)q | R f | , e iP R ( r ) (cid:27)(cid:19) , (16)where the functions f do not have zeroes. The representation for the quantum theory described above is such thatthe operator corresponding to R f has a zero eigenvalue. Therefore we represent 1 /R f using the r.h.s. of (16). Thecentral question for singularity resolution is whether the corresponding operator is densely defined and bounded. Thisturns out to be the case.Using the expressions for the basic field operators, we can construct an operator corresponding to a classicalsingularity indicator: d | R f | ≡ l P f ( x j ) \ e − iP R ( x j ) " \ q | R f | , \ e iP R ( x j ) . (17)The result is that basis states are eigenvectors of this operator, and all eigenvalues are bounded. This is illustratedwith the state | S a i ≡ | e ia P R ( r =0) i , (18)which represents an excitation a of the quantum field ˆ R f at the point of the classical singularity:ˆ R f | S a i = (2 l P ) f (0) a | S a i , (19) d | R f | | S a i = 2 l P f (0) (cid:16) | a | / − | a − | / (cid:17) | S a i which is clearly bounded. This shows that the singularity is resolved at the quantum level. In particular if there isno excitation of R f at the classical singularity, ie. a = 0, the upper bound on the eigenvalue of the inverse operatoris 2 /l P . B. Quantum black holes
The event horizon of a static or stationary black hole is a global spacetime concept. It does not provide a usefullocal determination of whether one is inside a black hole. The fundamental idea for defining a black hole locally isthat of a trapped surface, first introduced by Penrose. One considers a closed spacelike 2-surface in a spacetime,and computes the expansions θ + and θ − of outgoing and ingoing null geodesics emanating orthogonally from thesurface. If θ + > θ − <
0, the surface is considered normal. On the other hand if θ + ≤ θ − <
0, thesurface is called trapped. This provides a criterion for subdividing a spacetime into trapped and normal regions. Theouter boundary of a trapped region may be considered as the (dynamical) boundary of black hole, also known asthe ”apparent horizon” in numerical relativity. It is a function computed in classical numerical evolutions to test forblack hole formation. Similarly, a setting for studying quantum collapse requires an operator realisation of the nullexpansion ”observable,” and a criterion to see if a given quantum state describes a ”quantum black hole.”The classical expansions in spherical symmetry are the phase space functions [6] θ ± = − (cid:0) R ΛΛ ′ ± P Λ + 4Λ RR ′ (cid:1) . (20)Given phase space functions on a spatial hypersurface Σ, the marginal trapping horizon(s) are located by finding thesolution coordinates r = r i ( i = 1 · · · n ) of the conditions θ + = 0 and θ − <
0, (since in general there may be morethan one solution). The corresponding radii R i = R ( r i ) are then computed. The size of the horizon on the slice Σ isthe largest value in the set { R i } .Since only translation operators are available in our quantization, we define P Λ indirectly byˆ P λ Λ = l P iλ (cid:16) ˆ U λ − ˆ U † λ (cid:17) (21)where 0 < λ ≪ U λ denotes exp( iλP Λ /L ). This is motivated by thecorresponding classical expression, where the limit λ → P Λ . λ is perhaps bestunderstood as a ratio of two scales, λ = l p /l , where l is a system size. As for a lattice quantisation, it is evidentthat momentum in this quantisation can be given approximate meaning only for λ ≪ λ is also the minimum valueby which an excitation can be changed.Definitions for the operators corresponding to R ′ and Λ ′ are obtained by implementing the idea of finite differencing.We use narrow Gaussian smearing functions with variance proportional to the Planck scale, peaked at coordinate points r k + ǫl P , where 0 < ǫ ≪ f ǫ ( r, r k ) = 1 √ π exp (cid:20) − ( r − r k − ǫl P ) l P (cid:21) (22)Denoting R f ǫ by R ǫ for this class of test functions we defineˆ R ′ ( r k ) := 1 l P ǫ (cid:16) ˆ R ǫ − ˆ R (cid:17) . (23)Putting all these pieces together, we can construct the desired operatorsˆ θ ± ( r k ) = − ǫl P ˆ R ˆΛ (cid:16) ˆΛ ǫ − ˆΛ (cid:17) ∓ l P iλ (cid:16) ˆ U λ − ˆ U † λ (cid:17) ˆΛ − ǫl P ˆΛ ˆ R (cid:16) ˆ R ǫ − ˆ R (cid:17) , (24)which have a well defined action on the basis states.In analogy with the classical case, we propose that a state | Ψ i represents a quantum black hole if h Ψ | ˆ θ + ( r k ) | Ψ i = 0 , and h Ψ | ˆ θ − ( r k ) | Ψ i < . (25)for some r k . The corresponding horizon size is given by R H = h Ψ | ˆ R ( r k ) | Ψ i .This definition is utilised as follows: Given a state with field excitations at a set of coordinate points { r i } , onewould plot the expectation values in Eqn. (25) as functions of h Ψ | ˆ R ( r k ) | Ψ i , and locate the zeroes, if any, of theresulting graph. The resulting ”quantum horizon” location is invariant under radial diffeomorphisms because theseact on states to shift the coordinate locations of field excitations, but leave the expectation values unchanged – thegraph is a physical observable.It is straightforward to construct explicit examples of states satisfying these quantum trapping conditions. Someexamples appear in [6]. The quantum horizons so determined are not sharp since h θ i 6 = 0. IV. SUMMARY AND OUTLOOK
The results so far from this approach to understanding black hole formation in quantum gravity are threefold: (i)A quantization procedure which allows explicit calculations to be done, (ii) a test for black holes in a full quantumgravity setting which makes no use classical boundary conditions at event horizons, and (iii) singularity free andunitary evolution equations using the Hamiltonian defined in [7].The main computational challenge is to use the formalism to explicitly compute the evolution of a given matter-geometry state until it satisfies the quantum black hole criteria, and then to continue to the evolution to see if andhow Hawking radiation might arise. This work is in progress. [1] M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993).[2] C. Gundlach, Phys. Rept.
339 (2003).[3] S. W. Hawking, Commun. Math. Phys. (1975) 199.[4] V. Husain, O. Winkler, Phys. Rev. D71 104001 (2005).[5] V. Husain, O. Winkler, Class. Quantum Grav. , L127 (2005).[6] V. Husain, O. Winkler, Class. Quantum Grav.22