QUANTUM EFFECTS IN BLACK HOLE INTERIORS

arXiv:gr-qc/9210013v1 23 Oct 1992Alberta-Thy-36-92QUANTUM EFFECTS IN BLACK HOLE INTERIORSWarren G. Anderson, Patrick R. Brady, Werner Israel,and Sharon M. MorsinkCanadian Institute for Advanced Research Cosmology Program,Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2J1The Weyl curvature inside a black hole formed in a generic collapse grows, clas-sically without bound, near to the inner horizon, due to partial absorption andblueshifting of the radiative tail of the collapse. Using a spherical model, we exam-ine how this growth is modified by quantum effects of conformally coupled masslessfields.PACS numbers: 9760L, 0460Typeset Using REVTEX1

Classical models of generic black hole interiors [1]- [8] have made progress in unravellingthe nature of the internal geometry up to the onset of singular behavior at the inner (Cauchy)horizon. At this lightlike hypersurface, which corresponds to infinite external advanced time,the “Coulomb component” |Ψ2| of the Weyl curvature diverges exponentially with advancedtime.

(For spherical symmetry, |Ψ2| ≈m/r3 in terms of the Schwarzschild local massfunction m.)Here we report on a first attempt to gauge the influence of quantum effects on thisscenario; in particular, to examine whether vacuum polarization and pair creation will actso as to damp the classical rise of curvature and possibly limit it to sub-Planck values.The classical divergence of |Ψ2| is caused by the partial absorption, and infinite blueshift-ing [9] at the Cauchy horizon (CH), of gravitational radiation from the radiative tail [10] ofthe collapse, in concert with radiative outflow from the star as it shrinks within the hole. (The outflow has the merely catalytic role of focussing the generators and initiating thecontraction of the CH.

)Ori [3] has devised a simple model, involving a spherical charged black hole, which ap-pears to capture the essence of the physics. The influx of gravitational waves is modelledby a radial stream of lightlike particles.

(This “graviton”, “effective stress-energy” descrip-tion [11] is justified by the large blueshift.) The outflow is treated schematically as a thin,transparent lightlike shell Σ within and parallel to the event horizon (Fig.

1).The metric in each of the domains V−and V+ separated by Σ then has the chargedVaidya form for pure inflow:(ds2)± = dv±(2 dr −f± dv±) + r2 dΩ2 ,f± = 1 −2m±r+ e2r2 . (0.1)The advanced time parameters v+ and v−are unequal.

They are related by noting that theequations of Σ with respect to the two abutting coordinate systems aref+ dv+ = f−dv−= 2 dralong Σ . (0.2)2

Continuity of the influx across Σ requires thatf −2+dm+dv+= f −2−dm−dv−. (0.3)From (0.2) and (0.3),dm+f+= dm−f−along Σ ,(0.4)which clearly shows the divergence of the interior mass function m+ as one approaches theCH (f−→0, v−→∞).The ansatzm−(v−) = m0 −(const.) × v−(p−1)−(0.5)reproduces the correct, power-law decay dm−/dv−∼v−p−of the externally observed gravi-tational wave flux (p=12 for quadrupole waves [10]).Integration of the foregoing equations is now straightforward, and yields [3,8] to leadingorder (with m+ written simply as m from here on),m(v+) = m0| ln (−v+/m0)|−p(−v+/m0)−1(v+/m0 →0−) ,(0.6)−v+ = (const.) × exp(−κ0v−)(v−→∞) ,(0.7)Here, κ0 = (m20 −e2)1/2/r20 is the surface gravity (and r0 the radius) of the initial, staticsegment of the inner horizon in V−, and we have set a dimensionless numerical coefficient in(0.6) (depending on the luminosity and initial deformation of the collapsing star) equal tounity.A Vaidya geometry with metric of the form (0.1) has the Ricci curvatureRαβ ≈2 ˙m(v+)r−2(∂αv+)(∂βv+)(0.8)where the contribution of the electrostatic field has been ignored, and−Ψ2 = 12Cθφθφ = [m(v+) −e2/r]r−3(0.9)3

is the sole non-vanishing Newman-Penrose component of the Weyl curvature.To study effects of quantum corrections we aim to estimate the expectation value ⟨Tαβ⟩(in the Unruh state) of the stress-energy for conformally coupled massless fields on thisVaidya background when the mass function has classically the diverging form (0.6).In general, finding ⟨Tαβ⟩is a problem of notorious difficulty [12]. It becomes tractable inthe present instance because of a number of special circumstances: (i) We are only interestedin the asymptotic form of ⟨Tαβ⟩near the CH.

(ii) The singularity is relatively mild (Ψ2 isa diverging but integrable function of v) [3,4]. (iii) The lightlike character of (0.8) meansthat ordinarily dominant terms nonlinear in Rαβ actually vanish.

(iv) The special form(0.6) of the mass function means that, of two terms containing the same total number ofm-factors and v-derivatives, the term with the smaller number of m-factors is dominant;e.g. ¨m >> m ˙m/r2.

(The ratio of the two sides is “merely” a logarithmic factor; howeverthis factor does become infinite as v →0−, and for v = −1010 Planck times in a solar massblack hole, it has already grown to 1023 ! )This conjunction of circumstances permits us to treat the geometry as a linear pertur-bation of flat space.The terms linear in derivatives of m(v), which we retain, actuallydominate the neglected nonlinear terms.Barvinsky and Vilkovisky [13] have developed the first stages of a systematic generalformalism applicable in such circumstances, i.e., terms linear (and, more generally of alge-braically lower order) in curvature and derivatives dominate over higher order terms havingthe same dimension.

In the work reported here, we shall apply a simpler, first-order pre-scription due to Horowitz [14].First, we address and remove a potential source of difficulty. The Weyl curvature ap-proaches Planck levels, |Ψ2| →t−2p , as the advanced time coordinate v+ →vp < 0, where|vp| = b−2ǫtp[ln (b/ǫ2)]−p ,b ≡r0/m0 ,ǫ ≡tp/r0 ,(0.10)and tp = 10−43s is the Planck time.

One may therefore hope that, for |v+| >> |vp|, effectsof quantum gravity will remain small, and the spacetime geometry effectively classical. The4

strongly divergent expression (0.8) raises some doubt on this score, suggesting that Riccicurvature may dominate Weyl curvature and surpass Planck levels at times much earlierthan vp. This statement is, of course, not coordinate independent: an observer with fourvelocity uα measures a Ricci curvature Rαβuαuβ which can be arbitrarily small if he fallsinward at nearly the speed of light, whereas the Weyl curvature scalar is boost-invariant.Nonetheless, it might be cause for worry that the growth of Ricci curvature in some physically“reasonable” frame (however defined) will make our semiclassical analysis meaningless.It is easy to allay such doubts for all practical purposes.

The conformal transformationds2 = (r/r0)2ds2∗(0.11)generates a new Ricci tensor R∗αβ free of the strongly divergent ˙m terms (see (0.15) below),while merely multiplying Ψ2 by a factor (r/r0)2 of order unity. We shall obtain ⟨Tαβ⟩in theconformal metric, then use Page’s formula [15] to transform back to the physical metric.It is helpful to rescale the null coordinate v+ so that it more nearly represents Planckscales.

We setv ≡ǫv+ ,u ≡2ǫr20/r ,θ ≡ǫ−1θ ,x + i y ≡tpǫ−1ei φ sin ǫθ ,(0.12)so that r20dΩ2 ≈dx2 + dy2, i.e. the sphere r = r0 is nearly flat on Planck scales.

Further,since m(v+) is much larger than |e| and r near the CH, we may use the approximationf ≈−2m/r.The conformal metric now takes the Kerr-Schild (flat plus lightlike) formds2∗= −dudv + dx2 + dy2 + 2L(u, v)dv2 ,(0.13)L = 18ku3m(v+) ,k = (tpr30)−1 ,(0.14)which is manifestly almost flat for |Ψ2| << t−2p . The Ricci curvature for (0.13) isR∗αβ = 4Luu(2L lαlβ −l(αnβ))(0.15)where lα = −∂αv, nα = −∂αu and the u-subscripts denote partial derivatives.5

A general argument due to Horowitz [14] shows that, to linear order in a nearly flatspacetime, the in-in vacuum expectation value ⟨Tµν⟩of the stress-energy for a conformallycoupled massless field is given in terms of the retarded integral over the past light-cone ofthe point x in the flat background;t−2p ⟨Tµν⟩ren = aZH(x −x′)Aµν(x′)d4x′ + αAµν(x) + βBµν(x) . (0.16)Here, a is a positive numerical coefficient whose value is known for different spins [14]; αand β are arbitrary numbers; Aµν and Bµν are the (linearized) variational derivatives δ/δgµνof the actions associated with C2αβγδ and R2 respectively – explicitly,Aµν = −2✷Gµν −23Gαα , µν + 23ηµν✷GααBµν = 2ηµν✷Gαα −2Gαα , µν .

(0.17)The past light-cone distribution H is given byH(x −x′) = δ′(σ)θ(t −t′) ,σ ≡12(xµ −x′µ)2 . (0.18)Explicitly, for any function f,ZH(x −x′)f(x′)d4x′ =Z4πdΩZ 0−∞dU" ∂f∂U ln−Uλ+ 12∂f∂V#V =0,(0.19)where U, V , dΩare spherical lightlike coordinates centered on x, so that the past lightconeof x is V = 0.

The arbitrary length scale λ may be considered to reflect the arbitrariness ofα in (0.16), and could be adjusted so as to absorb α.The in-vacuum expression (0.16) needs to be supplemented by a local, conserved tensorrepresenting initial conditions appropriate to the Unruh state for an evaporating black hole.Inside the hole, this is just the lightlike influx of negative energy that accompanies thethermal outflux to infinity [12]. However, this remains negligible up to the moment v+ = vpwhen the classical curvature becomes Planckian if the black hole is larger than 100kg [4],and it will therefore be ignored.According to (0.17), (0.15) and (0.14), the functions to be inserted in (0.16) are6

Auu = Buu = 0 ,Avv = 13Bvv = 4kud2m/dv2 ,(0.20)Auv = 13Buv = −12Axx = −12Ayy = 112Bxx = 112Byy = 4kdm/dv ,with m given by (0.6).It is now straightforward to express the spherical lightlike coordinates U, V , dΩof (0.19)in terms of the plane coordinates of (0.13) and then evaluate the integrals approximately byLaplace’s method.From (0.20) it is immediate that ⟨Tuu⟩= 0. Transforming back to the physical metric(0.11) does not change this result significantly.

Page’s formula [15] yields for the quantumoutfluxDT αβE(∂αv+)(∂βv+) ∼t2pm/r5 ,(0.21)which is of order t2pΨ2 times the classical outflux from the collapsing star. The smallness ofthis result has the important consequence that the classical picture of the CH contractingvery slowly (on Planck scales) under irradiation by the star is not affected by the quantumcorrections up to the time when curvatures become Planckian (cf [7]).The dominant contribution to ⟨Tαβ⟩comes from the logarithmic term in (0.19) withf = Avv:DTv+v+E≈a∗t2pr−3 ¨m(v+) ln (−v+/λ)(0.22)where a∗is a positive number, approximately a∗= 72982π331/2 a.

(The result quoted is forthe physical metric. The only effect of the conformal transformation on the leading term ofDTv+v+Eis to rescale the arbitrary length scale λ.

)There are now two essentially different possibilities. If λ >> |vp| (vp < 0 is the coordinatetime (0.10) at which the classical curvature approaches Planck levels), the logarithm in(0.22) is negative as v+ approaches vp, and (0.22) predicts damping of the classical growthof curvature due to quantum effects.

If, on the other hand, λ <∼|vp|, quantum effects wouldfurther destabilize the classical plunge toward a curvature singularity.7

Unfortunately, this ambiguity cannot be resolved within the present theoretical frame-work, which provides no information about λ [16]. The quantum theory of massless fieldspropagating on a fixed classical background has no inherent length-scale.Something further can be said if one is willing to entertain an arguable hypothesis aboutthe origin of this incompleteness of the semi-classical theory [16].Quantum effects of the gravitational field itself have not yet been included.

A successful(renormalizable or finite) quantum theory of the gravitational field would be expected tohave the effect, at moderate curvatures, of modifying the Einstein-Hilbert Lagrangian byterms quadratic in curvature,16πLG = t−2p R + α1C2αβγδ + β1R2 ,(0.23)where α1 and β1 are constants of order unity. (General arguments due to Fradkin andVilkovisky [17] suggest that α1 is negative.) The added terms induced in the effective fieldequations,Gµν + t2p(α1Aµν + β1Bµν) = 8πnT classµν+ ⟨Tµν⟩o(0.24)are of the same form as the terms left arbitrary in (0.16).

This suggests that the incomplete-ness of the semi-classical theory is related to the neglect of quantum gravitational effects.It is arguable that a quantum theory of massless fields which includes gravity wouldbecome a complete theory, with the net coefficients of Aµν and Bµν determined [16].Suppose this is true. Adjust λ in (0.19) so that α = 0 in (0.16).

Then α1 in (0.24)is expected to be of order unity, and the term α1Aµν may be interpreted as representingeffects associated with gravitational vacuum polarization. Now, it is reasonable to expectthat, once the quantum gravitational degrees of freedom are activated (i.e.

when v+ ≈vp),gravitons will have effects not too disimilar from photons and other massless fields, i.e. that⟨Tµν⟩∼−α1t2pAµνfor v+ ≈vp .

(0.25)Comparison of (0.20) and (0.22) now shows that the logarithmic factor is of order unity, i.e.that λ ≈|vp|. (The circumstance that this is much shorter than a Planck time tp has no8

physical significance, because the scale of the null coordinate v+ (which λ normalizes) hasno intrinsic local meaning. )If this conclusion is correct, (0.22) should be interpreted as an intensification (ratherthan a damping) of the classical influences tending to produce a curvature singularity atthe CH, at least for the Ori spherical model considered here.

It would indicate stronglythat the ultimate, quantum stage of evolution of the hole is inaccessible to semi-classicalconsiderations. A detailed account of this work is in preparation.It is a pleasure to thank R. Balbinot, C. Barrab`es, A. Barvinsky, R. Camporesi, S.Droz, D. Page and E. Poisson for stimulating discussions.This work was supported bythe Canadian Institute for Advanced Research and the Natural Sciences and EngineeringResearch Council of Canada.

One of us (SMM) wishes to acknowledge financial supportfrom the Alberta Heritage fund, an Avadh Bhatia Fellowship and the Killam Foundation.9

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FIGURESFIG. 1.Ori model.

Infalling radiation passes through a transparent, “outgoing” lightlike shellΣ inside a charged spherical hole. EH is the event horizon, and CH the Cauchy horizon.12

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