Quantum Hair During Gravitational Collapse

We consider quantum gravitational corrections to the Oppenheimer-Snyder metric describing time-dependent dust ball collapse. The interior metric also describes Friedmann-Lemaitre-Robertson-Walker cosmology and our results are interpreted in that context. The exterior corrections are an example of quantum hair, and are shown to persist throughout the collapse. Our results show the quantum hair survives throughout the horizon formation and that the internal state of the resulting black hole is accessible to outside observers.


I. INTRODUCTION
The unique quantum gravitational effective action program allows model independent calculations in quantum gravity [1][2][3][4][5][6][7][8][9].This approach has been used to study quantum gravitational corrections to a variety of cosmological [10][11][12][13][14][15][16] and astrophysical models [17][18][19][20][21][22][23][24].A study of quantum gravitational corrections to a static dust ball used to model a star [22,25] (see also [26] for earlier work) revealed the existence of quantum hair.It was found that the quantum gravitational potential of a star depends on the composition of the star at second order in the curvature expansion of the effective action.In [25], we suggested that the quantum hair would also apply to a collapsing star model and thus also to a black hole.
The aim of the paper is to extend our previous work on quantum hair.We shall first present a very generic result that is independent on the chosen energy-momentum tensor, proving that the quantum hair must exist for any energy-momentum tensor T µν .This result is fully model independent: it does not depend on the matter model (i.e., T µν ) or on the high energy completion of the effective action.
We then study a specific model for the gravitational collapse of a dust ball namely the Oppenheimer-Snyder model of gravitational collapse [29] and demonstrate that quantum hair is present in this dynamical model and calculable from first principles.The corrections in r −3 and r −5 are identical to those identified in [25] in the static case.Moreover, the quantum hair persists throughout the gravitational collapse of the star.Our work demonstrates that the resulting black hole has quantum hair.These results are also relevant to Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology, as the inside of the collapsing object is described by the FLRW metric.We calculate for the first time the complete leading order quantum gravitational correction to FLRW, and comment on previous works on FLRW quantum cosmology.This paper is organized as follows.In section 2, we present a model independent proof that quantum hair exists for any energy momentum tensor.In section 3, we review the Oppenheimer-Snyder model.In section 4, we compute the leading quantum gravitational corrections to the interior and exterior metric of this model.In the conclusions we discuss some of the implications of our work to black hole information and long wavelength quantum gravity.

II. QUANTUM HAIR AND GENERIC MATTER DISTRIBUTION
In this section, we use the results presented in [28] and argue that there is quantum hair for any matter distribution.Quantum hair can manifest itself as quantum corrections to classical solutions in general relativity describing the exterior metric of an astrophysical object.In the case of black holes, these quantum corrections can carry information about the interior quantum state, whereas the classical no-hair theorem would forbid this.Hence, the existence of quantum hair bears relevance to the black hole information paradox.
The quantum corrections to classical solutions of general relativity are reliably calculable using quantum corrected field equations obtained from the variation of the Vilkovisky-DeWitt unique effective action of quantum gravity as long as curvature invariants remain weak.At second order in curvature, the effective action is given by with a local part where M P = /G N denotes the Planck mass, and a non-local part For simplicity, we set the cosmological constant to zero.In addition, we ignore the boundary term associated with c 4 , as it does not contribute to the field equations.Then, after applying the local and non-local Gauss-Bonnet identities [9], we obtain The quantum gravitational field equations to second order in curvature can be derived from this action, and are given by where G N is Newton's constant, T µν is the energy-momentum tensor, and With weak curvature invariants, perturbation theory can be applied to solve these complicated coupled partial differential equations, as we can obtain a controlled approximation by expanding in curvature.We thus set gµν = g µν +g q µν where g µν is the classical solution and g q µν the quantum solution one is solving Eq. ( 5) for.The log R µ...ν α...β terms correspond to kernels that are integrated over curvature terms which are functions of the energy-momentum tensor (see e.g.Appendix).
A generic astrophysical body has, relative to its surface or horizon, an interior T µν tensor and an exterior one.It is clear that the quantum corrections of the outside metric due to the H NL µν terms must be dependent on T µν and on the higher curvature terms in the effective action.This fact is independent of the specific type of matter distribution, and thus applies to e.g. a static star, a collapsing star or a real astrophysical black hole (i.e.not a static vacuum solution) [35].Hence, the exterior metric will keep a memory of the interior of the matter distribution, which implies the presence of quantum hair for any gravitational body, and, in particular, for realistic black holes.This hair is expressed in terms of deviation from the 1/r Newtonian potential.These deviations are due to quantum gravitational corrections to Newton's law.This is an explicit realisation of the observation that the asymptotic graviton state of an energy eigenstate source is determined at leading order by the energy eigenvalue and that the quantum gravitational fluctuations (i.e., graviton loops) produce corrections to the long range potential whose coefficients depend on the internal state of the source [27].We shall consider an explicit application of this result to the Oppenheimer-Snyder gravitational collapse model and calculate the leading order quantum hair to that classical solution.

III. OPPENHEIMER-SNYDER MODEL: CLASSICAL SOLUTION
We will now consider the Oppenheimer-Snyder model of gravitational collapse [29].In the exterior region, the metric is defined by the line element [30] with where M is the total ADM mass of the ball, and R is the areal radius with R ∈ [R s (t), ∞).
The energy-momentum tensor vanishes: T µν = 0.In the interior region, the metric is defined by the line element with r ∈ [0, r s ], where the scale factor is given by with τ < τ s and τ s the time at which the ball collapses to a singularity.This time can be calculated and is given by The scale factor corresponds to a Hubble scale Furthermore, the energy-momentum tensor is that of a perfect fluid: The dust ball model assumes p = w ρ with w = 0.

IV. QUANTUM CORRECTIONS
Our goal here is to describe the gravitational collapse of a star, and to show that we can compute, in a controlled approximation, the quantum gravity corrections to the metric during the formation of a black hole.Note that the initial formation of an astrophysical black hole (i.e.non-quantum black hole) does not require large curvatures anywhere in the dust ball.Specifically, this means that we can use the flat space kernel function (see Appendix) to compute quantum corrections arising from the effective action.The only region in the dust ball collapse spacetime with large curvature is the future black hole singularity, but the quantum corrections to the external metric from this region are small as the integrals over the singularity at r = 0 are well behaved [33,34].
Thus, we will work again with the Vilkovisky-DeWitt unique effective action of quantum gravity at second order in curvature.However, to calculate the quantum correction to the interior metric, it is easiest to use the Weyl basis, in which case one has with ĉ1 = c1 + 1 3 c2 , ĉ2 = 1 2 c2 , α = α + 1 3 β and β = 1 2 β.

A. Interior corrections
In the interior we have a FLRW metric.For this metric the Weyl tensor vanishes, and it is thus convenient to work in the Weyl basis.Quantum corrections to this metric have been studied before in the literature.Corrections due to the R 2 term have for example been studied in [11,31,32], while non-local corrections due to the R log( )R terms have been studied in [8,[10][11][12][13].However, none of the previous studies considered the local and non-local corrections together.We will explain that this is crucial for obtaining a consistent result.By considering the local corrections to the action, one obtains the modified Friedmann equation [11] with t P the Planck time.We solve this perturbatively, i.e. we set H = H c + H L , where ) and H c solves the classical Friedman equation.It is thus given by ( 13) and Solving for H L then yields To obtain the non-local corrections, one must evaluate the log R term in the field equations, which is discussed in the Appendix.This leads to the modified Friedmann equation [11] given by Solving this perturbatively yields Gathering the local and non-local corrections, we find This equation is renormalization group invariant because we have considered the local and non-local corrections together.This had not been done previously in the literature, implying that previous studies of FLRW cosmology within this framework are flawed.We can now solve for a(τ ).We find where τ ∈ [0, τ s ).This is the quantum correction to the interior metric.Note that it depends on both the non-local and the local Wilson coefficients.The latter are not calculable within the effective theory approach that we have used, as it depends on the ultra-violet completion of the effective action.
Because we have considered the full effective action to second order in curvature, our result differs from previous studies of quantum cosmology within this framework, such as that in [10] where only the non-local contributions were considered.The phenomenology clearly needs to be considered again and the question of a big bounce should be investigated anew.We now turn our attention to the exterior solution and its quantum gravitational corrections.

B. Exterior corrections
We discuss the kernel in spherically symmetric coordinates in the Appendix.The calculation follows the methodology of that presented in [22] with the notable complication that we now have a space-time dependent problem.In the Appendix we show that ln( )R can be approximated by eq. ( 41) where Ṙs (t r ) = dR s (t r ) dt r (23) and that any non-zero energy momentum tensor will produce quantum hair in the form of quantum corrections to the classical spacetime resulting from the Einstein equations.
While the general proof is model independent, we have illustrated this result with a direct calculation in the case of the Oppenheimer-Snyder collapse model.We have shown by explicit calculation that the quantum corrections to the Oppenheimer-Snyder classical solution are sensitive to the density of the matter distribution.The outside gravitational field contains information about the collapse process that is stored in the quantum hair.In principle, a distant observer could measure the deviation from the Newton potential.This work is a further demonstration that all classical solutions in general relativity, including black holes, are hairy in quantum gravity.