PHYSICAL REVIEW D 100, 124038 (2019)
Ingoing Eddington-Finkelstein metric of an evaporating black hole
† ‡ Shohreh Abdolrahimi,1,2,* Don N. Page,1, and Christos Tzounis1,2, 1Theoretical Physics Institute, University of Alberta, Edmonton, Alberta T6G 2E1, Canada 2Department of Physics and Astronomy, California State Polytechnic University, 3801 West Temple Ave., Pomona, California 91768, USA
(Received 27 July 2017; published 16 December 2019)
We present an approximate time-dependent metric in ingoing Eddington-Finkelstein coordinates for an evaporating nonrotating black hole as a first-order perturbation of the Schwarzschild metric, using the linearized backreaction from a realistic approximation for the stress-energy tensor for the Hawking radiation in the Unruh quantum state.
DOI: 10.1103/PhysRevD.100.124038
I. INTRODUCTION and zero net energy flux, with the outgoing Hawking radiation balanced by incoming radiation from an exter- The physics of black holes is an abundant field in nal heat bath at the Hawking temperature. In [6], a fairly which the convergence of gravitation, quantum theory, good closed-form approximation for the energy density and and thermodynamics takes place. The original derivation stresses of a conformal scalar field in the Hartle-Hawking of Hawking radiation [1] from black holes is based on state everywhere outside a static black hole can be found. semiclassical effective field theory. Normally, quantum In the Unruh state, there is the absence of incoming fields are considered test fields in the curved spacetime radiation at both past null infinity and the past horizon, plus of a classical background geometry. A quantum field regularity of the stress-energy tensor on the future event theory constructed on a curved background spacetime horizon in the frame of a freely falling observer, represent- experiences gravitationally induced vacuum polarization ing a black hole formed from gravitational collapse, with and/or particle creation. These effects induce a nonzero nothing falling into the black hole thereafter. There have expectation value for the stress-energy tensor. The renor- been many calculations of the quantum stress-energy tensor malized expectation value of the complete quantum stress- in the Unruh state in the Schwarzschild spacetime, both energy tensor outside the classical event horizon has been for a massless scalar field and for the electromagnetic field calculated by various authors [2–43], usually using a [8–12,17–23]. A method for computing the stress-energy framework established by Christensen and Fulling [3]. tensor for the quantized massless spin-1=2 field in a general In this framework, the assumptions are that the stress- static spherically symmetric spacetime was presented in energy tensor is time independent, satisfies local stress- [24–26]. The quantum stress-energy tensor has also been energy conservation, and has a trace determined solely by investigated in the Kerr metric [22,28,36,38,42]. the conformal or Weyl anomaly [44] for fields that are One of the important questions that one wants to answer classically conformally invariant (such as the conformally concerns the effect of quantized matter on the geometry of coupled massless scalar field and the electromagnetic field, black holes. Such effects in the Hartle-Hawking state have but not the gravitational field [45]). The quantum state been studied [50] (for similar work, see [51–53]), using the considered is usually either the Hartle-Hawking state [46] approximation found in [6] for the expectation value of the or the Unruh state [47,48]. (For discussions of the various renormalized thermal equilibrium stress-energy tensor of a quantum field theory states outside a black hole, see [49].) free conformal scalar field in a Schwarzschild black hole In the Hartle-Hawking state, one has thermal equilibrium, background as the source in the semiclassical Einstein equation. The backreaction and new equilibrium metric are found perturbatively to first order in ℏ. The new metric is *[email protected] † [email protected] not asymptotically flat unless the system is enclosed by a ‡ [email protected] reflecting wall. The nature of the modified black hole spacetime was explored in subsequent work [54–56]. James Published by the American Physical Society under the terms of Bardeen [57] considered radial null geodesics in a black the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to hole geometry modified by Hawking radiation backreac- the author(s) and the published article’s title, journal citation, tion, showing that the event horizon is stable and shifted and DOI. Funded by SCOAP3. slightly in radius from the vacuum background.
2470-0010=2019=100(12)=124038(19) 124038-1 Published by the American Physical Society ABDOLRAHIMI, PAGE, and TZOUNIS PHYS. REV. D 100, 124038 (2019)
In this paper, we construct the first-order backreaction on Then μðuÞ can be written in terms of v and r as well, since the metric in the Unruh state, using the expectation value of μ3 ≈−3α þ 6α − 12αμ þ 12αμ r the quantum stress-energy tensor in the Unruh state as the v r ln 2μ source in the spherically symmetric Einstein equations. This metric represents the first-order approximation to the 1 ¼ −3αv þ 12αμ − ln z − 1 ; ð5Þ metric of an evaporating black hole. z
II. METRIC ANSATZ where To construct a metric using the expectation value of the 2μ z ≡ : ð6Þ quantum stress-energy tensor in the Unruh state as a source r in the spherically symmetric Einstein equations, we first need to find an appropriate metric ansatz. To do so, we In terms of the zeroth-order solution begin (but do not end) with the outgoing Vaidya metric. 1 The outgoing Vaidya metric describes a spherically sym- μ0ðvÞ ≡ ð−3αvÞ3 ð7Þ metric spacetime with radially outgoing null radiation. Here, we consider a black hole evaporating by Hawking and the function emission. The outgoing Vaidya metric can be written in 4α 1 outgoing Eddington-Finkelstein coordinates as ϵð Þ ≡ − − 1 ð Þ v; z 2 ln z 8 μ0 z 2 2μðuÞ 2 2 2 ds ¼ − 1 − du − 2dudr þ r dΩ : ð1Þ ¼ 2μ ¼ 1 r (which vanishes on the apparent horizon r or z ), the mass μ is given approximately by the solution of the Here u is the retarded or outgoing null time coordinate and cubic equation (5) as Ω2 ¼ θ2 þ 2 θ ϕ2 ℏ ¼ ¼ d d sin d . We use Planck units, c 1 1 1 1 1 1 2 3 1 1 2 3 G ¼ 1. 3 3 μðv; zÞ ≈ μ0 þ − ϵ þ μ0 − − ϵ : When the black hole mass μ is much larger than the 2 4 2 4 reciprocal of the masses of all massive particles, the ð9Þ Hawking emission is almost entirely into massless particles (e.g., photons and gravitons for astrophysical mass black Figure 1 shows a spacetime diagram for the evaporating holes), and the Hawking emission rate is given by black hole and the null coordinates u and v. These null coordinates are chosen so that u ¼ 0 is the event horizon of dμ α μ0 ≡ ≈− ; ð2Þ du μ2 where α is a constant coefficient that has been numerically evaluated to be about α ≈ 3.7474 × 10−5 [58–62] for the emission of massless photons and gravitons. Then, we have
1 μðuÞ ≈ ð−3αuÞ3: ð3Þ
We are setting u ¼ 0 at the final evaporation of the black hole, so that u is negative for the part of the spacetime being considered. Moreover, we are assuming that the black hole mass is infinite at negative infinite u, so going back in retarded time, the mass grows indefinitely, rather than having a black hole that forms at some particular time. However, for a black hole that forms at some initial mass M0, the metric we find should be good for values of u when μðuÞ 124038-2 INGOING EDDINGTON-FINKELSTEIN METRIC OF AN … PHYS. REV. D 100, 124038 (2019) ¼ ¼ 0 −2 the black hole, and u v is the final evaporation event 4αð−3αvÞ 3 ≪ z ≪ 1, we can take z ≈ 2μ0=r, so that in −2 of the hole, when and where the horizon radius goes to any case in which −3αv ≫ 1 and z ≫ 4αð−3αvÞ 3, r ¼ 0. We shall consider the region outside the black hole horizon, where r>2μ,so0 2μ 2μ 2μ ds2 ≈− 1 − e2ψ˜ðv;rÞdv2 þ 2eψ˜ðv;rÞ 2 1 þ 1 − eψ˜ðv;rÞ − 1 dvdr r r r 2μ 2μ 2μ þ 4 1 þ eψ˜ðv;rÞ 1 − 1 þ 1 − eψ˜ðv;rÞ dr2 þ r2dΩ2: ð15Þ r r r For 1 ≪ 2μ0 ≪ r ≪−3v=2, we thus have Einstein equation as a deviation from Schwarzschild to linear order in the stress-energy tensor of the Hawking 2μ ds2 ≈− 1− e2ψ˜ ðv;rÞdv2 þ2eψ˜ ðv;rÞdrdvþr2dΩ2: ð16Þ radiation, which itself can be well approximated by what r it would be for a fixed Schwarzschild metric back- ground. Furthermore, when one goes out from the black 2μ ≪ When 0 r is not true, one is close enough to the hole along the past-directed null direction with black hole that the small deviations of the metric from v ¼ const ≪−1, the energy of the Hawking radiation the Schwarzschild metric are not well approximated by that one crosses increases the mass so that it is no longer the Vaidya metric, since the stress-energy tensor is not 1 μ ð Þ ≡ ð−3α Þ3 ≡ 2μ well approximated by purely outgoing null radiation as it well approximated by 0 v v ,asz =r ϵð Þ ≡ ð4α μ2Þð −1 − would be in the Vaidya metric. However, for 1 ≪ 2μ0, becomes sufficiently small that v; z = 0 z the metric is close to a slowly varying Schwarzschild ln z − 1Þ does not remain small. Then Eq. (14) is no metric, so it is an excellent approximation to solve the longer a good approximation for what ψ˜ðv; rÞ should be. 124038-3 ABDOLRAHIMI, PAGE, and TZOUNIS PHYS. REV. D 100, 124038 (2019) Therefore, we modify the metric (16) with corrections and hðzÞ to allow us to get a better approximation for the going as 1=μ2 with coefficients going mainly as functions metric of an evaporating black hole with the stress-energy of z in order to match the stress-energy tensor of the tensor of massless Hawking radiation, valid everywhere Hawking radiation under the approximation that it is pro- outside the black hole (and also slightly inside) that is not duced by a slowly varying sequence of Schwarzschild to the causal future of the Planckian region where the black metrics. In particular, we write the general spherically hole has shrunk to near the Planck mass. symmetric metric in ingoing Eddington-Finkelstein coor- For z ¼ 1, we shall make the gauge choice of setting dinates as gð1Þ¼hð1Þ¼ln z˜ð1Þ¼0, so that there 2m ≈ 2μ ¼ r and ψ ≈ 0. z ¼ 1 is then the approximate location of the 2mðv; zÞ apparent horizon. For z → 0 (radial infinity backwards ds2 ¼ −e2ψðv;zÞ 1 − dv2 þ 2eψðv;zÞdvdr r along the ingoing radial null curve of fixed v), we have gðzÞ, hðzÞ, and z˜ approaching the constants g0, h0, and z˜0, þ r2dΩ2; ð17Þ respectively, so in this limit of infinitely large r=μ, m=μ → 1 þ μ2 ψ → ð − 4α ˜ Þ μ2 and motivated by the approximate forms above for the h0= 0 and g0 ln z0 = 0. It is only an ψ μ variables μ and ψ˜ in the metric (16), we make the following approximation that and m are functions just of and ansatz for the metric coefficients of the evaporating black z of this form, but for a large and hence very slowly hole: evaporating black hole, it seems to be a very good approximation. 3 1 For r ≪−v ¼ μ0=ð3αÞ, which implies that μ ≈ μ0 ≡ ψ ≈ ½ ð Þ − 4α ˜ ð Þ 1 2 g z ln z ; 18 ð−3α Þ3 μ0 v , Eqs. (2) and (19) above show that at fixed z the mass of the black hole changes as hðzÞ m ≈ μ 1 þ ; ð19Þ dm α μ2 ≈− : ð20Þ 0 dv μ2 where μðv; zÞ is given by Eq. (9), or approximately by From Eqs. (6)–(9) and (17)–(19), we get that the area of Eq. (10) for 2r ≪−3v, z ≡ 2μ=r, and z˜ is a function of the apparent horizon (at z ¼ 1) of the evaporating black μ=μ0 that is approximately z for 2r ≪−3v but whose form hole is for larger r will be evaluated later. ð Þ¼4π½2 ð ¼ 2 Þ 2 ≈ 4π½2μð ¼ 1Þ 2 Once we find the functions gðzÞ, hðzÞ, and z˜ðμ=μ0Þ to A v m v; r m v; z solve the Einstein equation for the small deviation of the 2 2 ¼ 16πμ0 ¼ 16πð−3αvÞ3: ð21Þ metric from a sequence of Schwarzschild metrics with a slowly decreasing mass, the resulting metric (17) should be Note that Bardeen [57] considers a quasistationary a good approximation for the exact nonrotating black hole approximation of the black hole, which is justified as long metric for all v ≪−1 (i.e., for all advanced times before the as the black hole mass is much larger than the Planck mass black hole shrinks down to near the Planck mass) and for m ≡ ðℏc=GÞ1=2, which we are setting equal to unity by ≡ 2μ ≲ 1 p z =r (i.e., for all radii outside and even slightly using Planck units. In this case, Bardeen has the black hole inside the black hole), since the mass will be changing at mass at r ∼ 2μ0, M ≡ μðv; r ¼ 2μ0Þ, being proportional a timescale much longer than the timescale of the mass to ð−vÞ1=3. ≫ 1 itself. For r (radii large in Planck units), the metric The metric (17) can be written in the form should also be good for v ≳ −1 [at advanced times after the black hole shrinks down to near the Planck mass, where 2M˜ the metric has the approximate outgoing Vaidya form of ds2 ¼ −e2Ψ 1 − dv2 þ 2eΨdvdr þ r2dΩ2; ð22Þ r Eq. (1)]. That is, the metric (17) with ψ given by Eq. (18) ð Þ ð Þ ˜ðμ μ Þ and m given by Eq. (19), with g z , h z , and z = 0 to be where Ψ ≡ Ψðv; rÞ¼ψðv; z ¼ 2μ=rÞ and M˜ ≡ M˜ ðv; rÞ¼ found below, should be a good approximation for the mðv; z ¼ 2μ=rÞ. Components of the Einstein tensor for this metric everywhere outside, and even slightly inside, a black metric have the following form: hole evaporating from infinite mass, except in the space- time region near the final evaporation event of the black 2 Gv ¼ − M˜ ; ð23Þ hole and to the causal future of this region (where one v r2 ;r would expect quantum gravity effects to be important). In summary, we started with the outgoing Vaidya metric 2 Gr ¼ M˜ ; ð24Þ as a first approximation for a spherically symmetric black hole v r2 ;v metric evaporating by the emission of massless Hawking 2 radiation, and then we switched to ingoing Eddington- Gv ¼ e−ΨΨ ; ð25Þ Finkelstein coordinates and introduced the functions gðzÞ r r ;r 124038-4 INGOING EDDINGTON-FINKELSTEIN METRIC OF AN … PHYS. REV. D 100, 124038 (2019) pffiffiffiffiffiffiffiffiffiffi ω0ˆ ¼ 1 − ð Þ 2M˜ 2Ψ 2M˜ zdt; 32 Gr ¼ 1 − ;r − ;r ; ð26Þ r r r r2 1 ˜ ω1ˆ ¼ pffiffiffiffiffiffiffiffiffiffi ð Þ ϕ θ 2M 2 dr; 33 G ϕ ¼ G θ ¼ 1 − ðΨ þ Ψ Þ 1 − z r ;r ;rr 1 ˜ 1 þ 1 þ M − 3 ˜ Ψ − ˜ þ −ΨΨ M;r ;r M;rr e ;rv: ˆ ˆ r r r ω2 ¼ rdθ; ω3 ¼ r sin θdϕ; ð34Þ ð27Þ can be written in the following form: Taking v and z to be the independent coordinates (at fixed angles θ and ϕ), we can rewrite the metric (17) for 0 ˜ 1 2r ≪−3v [the region where gðzÞ, hðzÞ, and z˜ have ρ˜ f 00 significant variation; outside this region we can set B ˜ ˜ C μˆ νˆ B f P 00C g ≈ g0, h ≈ h0, and z˜ ≈ z˜0 in the metric (17)]as hψjT jψi¼B C: ð35Þ @ 00p˜ 0 A hðzÞ 000p˜ 2 ≈− 2ψðv;zÞ 1 − 1 þ − 2 ψðv;zÞ 2 ds e z 2 e r;v dv μ0 4μ2 Note that we are using a capital P˜ for the radial pressure and þ 2eψðv;zÞr dzdv þ dΩ2 ;z z2 a lowercase p˜ for the transverse pressure. Dimensional 2 2 2 analysis shows that for massless fields at fixed z, the ¼ −Adv þ 2Bdvdz þ r dΩ ; ð28Þ dependence of the orthonormal components of the stress- energy tensor on the mass μ of the Schwarzschild metric at where fixed z goes as μ−4, so for the slowly evolving metric (28), we shall assume that the stress-energy tensor (35) has 2α 8α2 1 − z z − z ≈− þ ln ð Þ ρðzÞ r;v 2 2 4 ; 29 approximately the following form with functions , zð−3αvÞ3 z ð−3αvÞ3 fðzÞ, PðzÞ, and pðzÞ that are dimensionless even without setting G ¼ 1 (and hence independent of the scale set 2 1 8α 2 − z ln z by μ0): ≈− ð−3α Þ3 − ð Þ r;z 2 v 3 1 : 30 z z ð−3αvÞ3 0 1 ρðzÞ fðzÞ 00 Although this form of the metric does not apply every- B C 1 B fðzÞ PðzÞ 00C where outside a large black hole as the metric (17) does, it hψj μˆ νˆ jψi¼ B C ð Þ T μ4 @ A: 36 applies where the stress-energy tensor of the Hawking 0 00pðzÞ 0 radiation gives significant contributions to the functions 000pðzÞ gðzÞ, hðzÞ, and z˜ in Eqs. (18) and (19). Therefore, this form of the metric will be used to get approximate solutions of the Einstein equations for gðzÞ, hðzÞ, and z˜ with the stress- According to Christensen and Fulling [3], the stress- energy tensor of the Hawking radiation of massless fields in energy tensor in the case of the Schwarzschild spacetime the Unruh quantum state. Then these will be inserted back can be decomposed into four separately conserved quan- into Eqs. (18) and (19) to give the functions in the metric tities. We follow the analysis of Matt Visser [21] and (17), which will be a good approximation for the metric use his form of the stress-energy tensor, which has a everywhere outside a large black hole evaporating by slightly different basis for its decomposition from that of Hawking radiation of massless fields. Christensen and Fulling and is given by The stress-energy tensor for the spherically symmetric jψi hψj μˆ νˆ jψi¼½ μˆ νˆ þ½ μˆ νˆ þ½ μˆ νˆ þ½ μˆ νˆ Unruh quantum state on the spherically symmetric T Ttrace Tpressure Tþ T− ; curved background of the Schwarzschild spacetime with ð Þ 2μ ≡ rz constant, 37 dr2 where, with TðzÞ, HðzÞ, and GðzÞ being a further set of ds2 ¼ −ð1 − zÞdt2 þ þ r2dΩ2; ð31Þ 1 − z functions of z ≡ 2μ=r that are dimensionless without setting the Newtonian gravitational constant to be unity, in the standard static orthonormal frame and with fþ and f− being dimensionless constants, 124038-5 ABDOLRAHIMI, PAGE, and TZOUNIS PHYS. REV. D 100, 124038 (2019) 0 2 1 − ð Þþ z ð Þ 000 seems to exclude any outgoing radiation. Nevertheless, we T z 1−z H z B 2 C z can get outgoing radiation by making f− negative. It is 4 μˆ νˆ B 0 HðzÞ 00C 4 μ ½ ≡ B 1−z C β pð0Þ¼μ p˜ðr ¼ ∞Þ 0 Ttrace @ A; convenient to define to be what 0000would be for massless scalar radiation in the Hartle- 0000Hawking state at large radii (ignoring the backreaction to the Schwarzschild metric), ð38Þ Z 1 1 1 1 ð¯Þ β ≡ ≡ ; ð46Þ ð Þ ≡ T z ¯ ð Þ 213325π2 368 640π2 H z 2 dz; 39 2 z z¯ and set f− ¼ −βf0, where f0 is a positive quantity. In what μ4½T μˆ νˆ 0 pressure0 1 follows, we also define z2 2pðzÞþ GðzÞ 000 2 B 1−z C z B 2 C fðzÞ ≡ βf0 : ð47Þ B 0 z GðzÞ 00C 1 − ≡ B 1−z C; z B 00ð Þ 0 C @ p z A Thus, 000ð Þ p z 0 1 − ð Þ ð Þ 00 ð Þ f z f z 40 B C B fðzÞ −fðzÞ 00C Z μ4½ μˆ νˆ þ μ4½ μˆ νˆ ≡ B C ð Þ 1 2 3 0 Tþ 0 T− @ A: 48 ð Þ ≡ − ð¯Þ ¯ ð Þ 0000 G z ¯3 ¯2 p z dz; 41 z z z 0000 0 1 1100 Therefore, we have 2 B 1100C 4 μˆ νˆ z B C μ0½Tþ ≡ fþ B C; ð42Þ 2 1 − z @ 0000A 4 0ˆ 0ˆ z μ0T ¼ ρðzÞ¼2pðzÞþ ½HðzÞþGðzÞ 0000 1 − z 0 1 − TðzÞ − fðzÞ; ð49Þ 1 −100 2 B −1100C 4 1ˆ 0ˆ 4 0ˆ 1ˆ 4 μˆ νˆ z B C μ0T ¼ μ0T ¼ fðzÞ; ð50Þ μ ½T− ≡ f− B C: ð43Þ 0 1 − z @ 0000A 2 ˆ ˆ z 0000 μ4T1 1 ¼ PðzÞ¼ ½HðzÞþGðzÞ − fðzÞ; ð51Þ 0 1 − z The decompositions (38) and (40) make sense if the μ4 2ˆ 2ˆ ¼ μ4 3ˆ 3ˆ ¼ ð Þ ð Þ integrals GðzÞ and HðzÞ converge. Imposing mild integra- 0T 0T p z : 52 bility constraints on TðzÞ and pðzÞ at the apparent horizon that is very near z ¼ 1, which are satisfied for the Unruh Since the stress-energy tensor is given in the orthonormal state where Tð1Þ and pð1Þ are actually finite, we have frame, we rewrite the Einstein equation in the orthonormal frame, i.e., 1 HðzÞ¼ Tð1Þð1 − zÞþO½ð1 − zÞ2 ; ð44Þ 2 Gμˆ νˆ ¼ 8πTμˆ νˆ : ð53Þ GðzÞ¼−pð1Þð1 − zÞþO½ð1 − zÞ2 : ð45Þ To find the approximate time-dependent metric for an evaporating black hole as a first-order perturbation of This is enough to imply that the two tensors (38) and (40) the Schwarzschild metric, using the linearized backreaction are individually regular at both the past and future horizon. from the stress-energy tensor (49)–(52) of the Hawking In Kruskal null coordinates, ½Tþ is singular on the future radiation in the Unruh quantum state in the Schwarzschild horizon Hþ and regular on the past horizon H−. On the spacetime, we solve Eq. (53) up to relative corrections of − 2 other hand, ½T− is singular on the past horizon H and the order of 1=μ in the Planck units that we are using. To þ regular on the future horizon H . The two tensors (42) and bring Gμν to the orthonormal frame, note that we have (43) correspond to outgoing and ingoing null fluxes, ˆ ˆ ˆ ˆ respectively. The constants fþ and f− determine the overall ds2 ¼ −ðω0Þ2 þðω1Þ2 þðω2Þ2 þðω3Þ2 flux. The Unruh state must be regular on the future horizon, ¼ −Adv2 þ 2Bdvdz þ r2dΩ2; ð Þ so we need fþ ¼ 0. However, such a condition naïvely 54 124038-6 INGOING EDDINGTON-FINKELSTEIN METRIC OF AN … PHYS. REV. D 100, 124038 (2019) where From the one-one component of the Einstein equation (53), pffiffiffiffi we get ˆ B ω0 ¼ Adv − pffiffiffiffi dz; ð55Þ A z2 PðzÞ¼ ˆ B 32πð1 − Þ ω1 ¼ pffiffiffiffi dz; ð56Þ z A 2 2 2 × ½2αð1 − 8z þ 6z Þ − 2zð1 − zÞ g;z þ z ð1 − zÞh;z ˆ ˆ ω2 ¼ rdθ; ω3 ¼ r sin θdϕ: ð57Þ 1 1 ¼ ð Þð1 − 8 þ 6 2Þ − 3ð1 − Þ þ 4 f z z z 16π z z g;z 32π z h;z: We have that the Einstein tensor is ð66Þ 0ˆ 0ˆ 0ˆ 1ˆ 1ˆ 0ˆ 1ˆ 1ˆ G ¼ G0ˆ 0ˆ ω ω þ G0ˆ 1ˆ ω ω þ G1ˆ 0ˆ ω ω þ G1ˆ 1ˆ ω ω 2ˆ 2ˆ 3ˆ 3ˆ From the two-two component, we get þ G2ˆ 2ˆ ω ω þ G3ˆ 3ˆ ω ω ¼ 2 þ 2 þ 2 þ θ2 þ ϕ2 3 Gvvdv Gvzdvdz Gzzdz Gθθd Gϕϕd : ð Þ¼ z ½16α þð2 − 5 Þ − 2 p z 64π z g;z zh;z ð58Þ 2 þ 2zð1 − zÞg;zz − z h;zz Therefore, we obtain ¼ 4fðzÞzð1 − zÞ ¼ Gvv ð Þ 3 G0ˆ 0ˆ ; 59 þ z ½ð2 − 5 Þ − 2 þ 2 ð1 − Þ − 2 A 64π z g;z zh;z z z g;zz z h;zz : Gvz Gvv Gˆ ˆ ¼ Gˆ ˆ ¼ þ ; ð60Þ ð67Þ 0 1 1 0 B A ¼ AGzz þ 2 Gvz þ Gvv ð Þ Now let us suppose that in the stress-energy tensor G1ˆ 1ˆ 2 ; 61 B B A components (49)–(52), the functions pðzÞ and TðzÞ are Gθθ explicitly given. We then solve for the metric functions hðzÞ Gˆ ˆ ¼ ; ð62Þ 2 2 r2 and gðzÞ in terms of pðzÞ and TðzÞ. Moreover, we are ð1Þ¼ ð1Þ¼0 Gϕϕ making the gauge choice of setting g h . From G3ˆ 3ˆ ¼ : ð63Þ Eq. (64),wehavefðzÞ. r2 sin2 θ From G0ˆ 0ˆ ¼ 8πT0ˆ 0ˆ and G1ˆ 1ˆ ¼ 8πT1ˆ 1ˆ , we get Deriving the Einstein tensor components Gvv, Gvz, Gzz, and Z 1 2αð1 − 4¯ þ 2¯2Þ ð¯Þþρð¯Þ Gθθ – z z P z z for the metric (28) and using Eqs. (59) (63), we obtain gðzÞ¼− − þ 16π dz¯ the Einstein tensor components in the orthonormal frame. z¯ð1 − z¯Þ2 z¯3ð1 − z¯Þ Zz 1 ð¯Þþ ð¯Þ ð¯Þ − 2 ð¯Þ ¼ − 32π H z G z − 16π T z p z III. SOLUTION FOR THE METRIC z¯ð1 − z¯Þ2 z¯3ð1 − z¯Þ z COEFFICIENTS 4α − dz;¯ ð68Þ We now solve the Einstein equation (53) to first order in z¯ the perturbation of the metric from the Schwarzschild Z 1 2αð1 − 2¯z2Þ 32πρðz¯Þ metric, using the stress-energy tensor whose components hðzÞ¼ − dz¯ are proportional to 1=μ4. From the zero-one component of z¯2ð1 − z¯Þ z¯4 Zz the Einstein equation, we get 1 ð¯Þþ ð¯Þ ð¯Þ − 2 ð¯Þ ¼ −32π H z G z þ 32π T z p z 2 ¯2ð1 − ¯Þ ¯4 αz z z z z fðzÞ¼ : ð64Þ 16πð1 − zÞ ð1 þ z¯Þ þ 4α dz:¯ ð69Þ z¯2 Comparing this to Eq. (47), we have f0 ¼ α=ð16πβÞ. Therefore, the Hawking radiation luminosity of the black Note that the functions HðzÞ and GðzÞ are given in 2 2 hole is not only L ¼ −dm=dv ≈ α=μ but also 16πβf0=μ . terms of TðzÞ and pðzÞ by Eqs. (39) and (41). For any From the zero-zero component, we find conformally invariant quantum field, the trace of the stress tensor is known exactly and is given by the conformal z2 ρðzÞ¼ ½2αð1 − 2z2Þ − z2ð1 − zÞh anomaly. In the Schwarzschild spacetime, the dimension- 32πð1 − Þ ;z 4 α z less trace is TðzÞ¼μ Tα, where z4h ¼ ð Þð1 − 2 2Þ − ;z ð Þ 6 6 f z z 32π : 65 TðzÞ¼γz ¼ βξz ; ð70Þ 124038-7 ABDOLRAHIMI, PAGE, and TZOUNIS PHYS. REV. D 100, 124038 (2019) with β ≡ 1=ð213325π2Þ. Here ξ is a dimensionless gravitons, which are not conformally invariant [45]. coefficient of the trace. For spins 0, 1=2,1,3=2, and 2 Equation (39) then gives (with particles identical to antiparticles, so that for γ βξ ð Þ¼ ð1 − 5Þ¼ ð1 − 5Þ ð Þ each momentum, there is a single one-particle state for H z 10 z 10 z : 71 spin 0 and two for higher spins, one for each of the two helicities), ξ is, respectively, 96, 168, −1248, −5592, and Therefore, we can derive gðzÞ and hðzÞ using these special 20352 [49], but this does not give the full trace for forms for TðzÞ and HðzÞ, Z 1 ð¯Þ ð¯Þ 8 16 ð Þ¼32π G z þ p z ¯ þ πγð1 − Þð29 þ 17 þ 8 2Þþ 4α − πγ ð Þ g z 2 3 dz z z z ln z; 72 z z¯ð1 − z¯Þ z¯ ð1 − z¯Þ 15 5 Z 1 1 ¯2 8 ð1 − Þ 16 4α ð Þ¼32π 2 ð¯Þþ z ð¯Þ ¯ þ πγ z ð2 − 3 − 5 2 − 6 3Þþ 4α − πγ − ð1 − Þ ð Þ h z 4 p z G z dz z z z ln z z : 73 z z¯ 1 − z¯ 5 z 5 z A five-term polynomial is believed to be a good approximation for the function pðzÞ. There is evidence [63,64] that pðzÞ may start off at order z3 (but see below), and the anomalous trace introduces a term of z6. Also, for spin-1 particles, there exists evidence [63] for a z7 term. Therefore, we consider pðzÞ to be a polynomial of the form 3 4 5 6 7 pðzÞ¼βðk3z þ k4z þ k5z þ k6z þ k7z Þ: ð74Þ 3 Figure 2 gives graphs of pðzÞ=z from numerical values of the ki coefficients given later, below Eq. (85). Using Eq. (74) for pðzÞ in Eq. (41) gives ð Þ¼βð1 − Þ k3 ð1 − 3 Þ − 2 − k5 ð1 þ þ 2 þ 9 3Þ G z z 2 z k4z 12 z z z − k6 ð1 þ þ 2 þ 3 þ 6 4Þ − k7 ð1 þ þ 2 þ 3 þ 4 þ 5 5Þ ð Þ 10 z z z z 10 z z z z z : 75 Finally, we can write fðzÞ, ρðzÞ and PðzÞ in the following form: z2 fðzÞ¼βf0 ; ð76Þ 1 − z ξ ρð Þ¼− ð Þþβ 2 ð1 þ þ 2 þ 3 − 9 4Þþk3 ð1 þ Þþ 2 − k5 ð1 þ þ 2 − 15 3Þ z f z z 10 z z z z 2 z k4z 12 z z z − k6 ð1 þ þ 2 þ 3 − 14 4Þ − k7 ð1 þ þ 2 þ 3 þ 4 − 15 5Þ ð Þ 10 z z z z 10 z z z z z ; 77 3 FIG. 2. (a) Behavior of the function pðzÞ=z for spin 0, black dotted line for the case of nonzero k3, k4, k5, and k6 given by [63] and red 3 solid line for the case of nonzero k3, k4, k5, and k6 given by [64]. (b) Behavior of the function pðzÞ=z for spin 1, blue dashed line for the case of nonzero k3, k4, k5, k6, and k7 given by [63] and black dash-dotted line for the case of nonzero k3, k4, k5, and k6 given by [64]. 124038-8 INGOING EDDINGTON-FINKELSTEIN METRIC OF AN … PHYS. REV. D 100, 124038 (2019) ξ ð Þ¼− ð Þþβ 2 ð1 þ þ 2 þ 3 þ 4Þþk3 ð1 − 3 Þ − 2 − k5 ð1 þ þ 2 þ 9 3Þ P z f z z 10 z z z z 2 z k4z 12 z z z − k6 ð1 þ þ 2 þ 3 þ 6 4Þ − k7 ð1 þ þ 2 þ 3 þ 4 þ 5 5Þ ð Þ 10 z z z z 10 z z z z z : 78 We want at asymptotic infinity the stress-energy tensor to be that of an outgoing flux of positive radiation, requiring ρðzÞ → fðzÞ asymptotically as z → 0 [3]. Picking up the dominant terms ½Oðz2Þ in (76) and (77), we see that ξ ¼ þ k3 − k5 − k6 − k7 ð Þ f0 20 4 24 20 20 : 79 In the case that k3 ¼ 0, this constraint is the same as Eq. (29) in [21]. If now we apply the constraint (79), we can replace one of the constants ξ, k3, k4, k5, k6,ork7. For example, we can write 5 5 ξ ¼ 20 − 5 þ þ þ ¼ 2113252πα − 5 þ þ þ ð Þ f0 k3 6 k5 k6 k7 k3 6 k5 k6 k7; 80 5 5 ¼ −20 þ ξ þ 5 − − ¼ −2113252πα þ ξ þ 5 − − ð Þ k6 f0 k3 6 k5 k7 k3 6 k5 k7; 81 5 5 ¼ −20 þ ξ þ 5 − − ¼ −2113252πα þ ξ þ 5 − − ð Þ k7 f0 k3 6 k5 k6 k3 6 k5 k6: 82 Then, the equations for ρðzÞ and PðzÞ can be written as follows: 2 βz ξ 4 5 k3 2 2 ρðzÞ¼ ð1 − 20z þ 18z Þþ ð1 − 2z Þþk4z ð1 − zÞ 1 − z 20 4 − k5 ð1 − 32 3 þ 30 4Þ − k6 ð1 − 30 4 þ 28 5Þ − k7 ð1 − 32 5 þ 30 6Þ 24 z z 20 z z 20 z z 2 f0 4 5 k3 2 2 2 k5 3 k6 4 k7 4 ¼ βz ð1 − 20z þ 18z Þ − z ð1 þ z − 9z Þþk4z þ z ð16 − 9zÞþ z − z ð2 − 3zÞ 1 − z 2 12 2 2 2 f0 4 5 ξ 4 k3 2 2 2 k5 3 3k7 4 ¼ βz ð1 − 30z þ 28z Þþ z − z ð1 þ z − 14z Þþk4z þ z ð8 − 7zÞ − z ð1 − zÞ 1 − z 2 2 6 2 f0 ξ k3 ¼ βz2 ð1 − 32z5 þ 30z6Þ − z4ð2 − 3zÞ − z2ð1 þ z þ z2 − 15z3Þ 1 − z 2 2 3 þ 2 þ k5 3ð1 þ Þð16 − 15 Þþ k6 4ð1 − Þ ð Þ k4z 12 z z z 2 z z ; 83 2 βz ξ 5 k3 2 2 PðzÞ¼ ð1 − 2z Þþ ð1 − 8z þ 6z Þ − k4z ð1 − zÞ 1 − z 20 4 − k5 ð1 þ 16 3 − 18 4Þ − k6 ð1 þ 10 4 − 12 5Þ − k7 ð1 þ 8 5 − 10 6Þ 24 z z 20 z z 20 z z 2 f0 5 k3 2 3 2 k5 3 k6 4 k7 5 ¼ βz ð1 − 2z Þ − zð4 þ z þ z þ z Þ − k4z − z ð8 − zÞ − z − z 1 − z 2 12 2 2 2 f0 4 5 ξ 4 k3 2 3 2 k5 3 k7 4 ¼ βz ð1 þ 10z − 12z Þ − z − zð4 þ z þ z þ 6z Þ − k4z − z ð4 − 3zÞþ z ð1 − zÞ 1 − z 2 2 6 2 2 f0 5 6 ξ 5 k3 2 3 4 2 k5 3 2 k6 4 ¼ βz ð1 þ 8z − 10z Þ − z − zð4 þ z þ z þ z þ 5z Þ − k4z − z ð8 − z − 5z Þ − z ð1 − zÞ : ð84Þ 1 − z 2 2 12 2 124038-9 ABDOLRAHIMI, PAGE, and TZOUNIS PHYS. REV. D 100, 124038 (2019) Expressions (83) and (84) are each presented in four the energy-momentum tensor show that for spin-0 particles forms, first in terms of ξ and the ki coefficients, then an expansion of (74) with terms up to k6 are appropriate. in terms of f0 and the ki coefficients and then by However, for spin-1 particles, an expansion of (74) with substituting either k6 or k7 from the constraint. This is terms up to k7 are used in [63]. Now, solving the Einstein done because current fits to the numerical computations of equation (53), we obtain 1 2 2 3 gðzÞ¼ ð1 − zÞ½2ðξ − k6Þð29 þ 17z þ 8z Þ − 5k5ð7 þ 3zÞ − k7ð73 þ 49z þ 31z þ 15z Þ 2113352π 1 ¼ ð1 − Þ½6ð4 − Þð29 þ 17 þ 8 2Þþ8 ð1 þ þ 2Þ − 9 ð1 þ Þð1 þ 2Þ 211345π z f0 k3 z z k5 z z k7 z z 1 2 2 3 ¼ ð1 − zÞ½−18ðξ − k6Þð1 þ zÞð1 þ z Þþ6ð4f0 − k3Þð73 þ 49z þ 31z þ 15z Þ 212345π 2 3 þ k5ð31 þ 31z þ 31z þ 15z Þ ; 1 2 2 hðzÞ¼ ð1 − zÞ½−6ξð3 þ 5z þ 6z Þþ120k4 þ 5k5ð13 þ 15zÞþ2k6ð19 þ 25z þ 28z Þ 2113352π 2 3 þ k7ð23 þ 35z þ 41z þ 45z Þ 1 ¼ ð1 − Þ½−6ð4 − Þð3 þ 5 þ 6 2Þþ24 þ 2 ð5 þ 5 − 3 2Þþ4 ð1 þ þ 2Þ 211335π z f0 k3 z z k4 k5 z z k6 z z 2 3 þ k7ð1 þ z þ z þ 9z Þ 1 2 2 ¼ ð1 − zÞ½12ξð1 þ z þ z Þ − 6ð4f0 − k3Þð19 þ 25z þ 28z Þ 211345π 2 2 þ 72k4 þ 4k5ð5 þ 5z − 7z Þ − 9k7ð1 − zÞð1 þ 2z þ 3z Þ 1 ¼ ð1 − Þ½6ξð1 þ þ 2 þ 9 3Þ − 6ð4 − Þð23 þ 35 þ 41 2 þ 45 3Þ 212345π z z z z f0 k3 z z z 2 3 2 þ 144k4 þ k5ð55 þ 55z − 41z − 45z Þþ18k6ð1 − zÞð1 þ 2z þ 3z Þ : ð85Þ We remind the reader that ξ is the value of the trace us in 2016 [65] and published in 2017 [63]: k3 ¼ 0.264, anomaly coefficient from Eq. (70). The constants k3, k4, k5, k4 ¼ 25.438, k5 ¼ −57.460, k6 ¼ 37.503, and f0 ¼ 5.385. k6, and k7 appear in Eq. (74) for the transverse pressure. In 2018, Bardeen [64] presented revised fits of k3 ¼ The expressions for gðzÞ and hðzÞ are each presented in 0.2524, k4 ¼ 25.5439, k5 ¼ −57.6663, k6 ¼ 37.6172, four forms, first in terms of ξ and the ki coefficients, then in and f0 ¼ 5.385 for conformally invariant scalars terms of f0 and the ki coefficients, and then by substituting (spin 0). either k6 or k7 from the constraint (except that the second For the electromagnetic field (massless spin 1) in the and third forms for gðzÞ coincide, so that form is written Unruh state, Bardeen [63] used the results of Jensen and only once). The constant f0 is related to the Hawking Ottewill [16] and Jensen et al. [17] to give k3 ¼ 114.62, ¼ − ≈ α μ2 ¼ 16πβ μ2 luminosity by L dm=dv = f0= , with k4 ¼ −1186.24, k5 ¼ 1393.96, k6 ¼ −2537.42 and k7 ¼ β ≡ 1 ð213325π2Þ = . 652.20, with f0 ¼ 2.435. In 2018, Bardeen [64] obtained For the case of a conformal scalar field (spin 0) in the access provided privately by Visser from the calculations of Unruh state, Matt Visser [21] has performed a least-squares [17], and this led to an improved fit with k3 ¼ 81.80, fit to the transverse pressure data of Jensen et al. [17] and of k4 ¼ −770.42, k5 ¼ 65.38, and k6 ¼ −942.18, with f0 ¼ Anderson et al. [18], giving the constants k4 ¼ 26.5652, 2.4346 (using data from [59]), with a nonzero k7 no longer k5 ¼ −59.0214, k6 ¼ 38.2068 (to six digits, though not seen necessary. For our graphs, we use the Bardeen [63,64] claimed accurate beyond 1%), and f0 ¼ 5.349 (close to Elster’s value [8] of 5.385 that is probably more accurate). data as probably the most careful synthesis of previous 2 From the summary in [21] of the data of [17,18] and of calculations. Figures 3–5 graph gðzÞ, hðzÞ, and μ ψ [given private communications to Visser from some of those by Eq. (18)] for the numerical values of the ki coefficients. authors, Bardeen discovered that an improved fit to that Now that we have calculated the hðzÞ and gðzÞ functions data and to the scalar luminosity results [8,12,22] could be of the metric (28), we can calculate the Kretschmann scalar αβγδ made by including a nonzero k3, which he kindly provided K ≡ RαβγδR , which has the following form: 124038-10 INGOING EDDINGTON-FINKELSTEIN METRIC OF AN … PHYS. REV. D 100, 124038 (2019) FIG. 3. Behavior of the function gðzÞ for spin 0, black dotted ð Þ line for the case of nonzero k3, k4, k5, and k6 given by [63], and FIG. 4. Behavior of the function h z for spin 0, black dotted red solid line for the case of nonzero k3, k4, k5, and k6 given by line for the case of nonzero k3, k4, k5, and k6 given by [63] and [64]. For spin 1, blue dashed line for the case of nonzero k3, k4, red solid line for the case of nonzero k3, k4, k5, and k6 given by k5, k6, and k7 given by [63] and black dash-dotted line for the [64]. For spin 1, blue dashed line for the case of nonzero k3, k4, case of nonzero k3, k4, k5, and k6 given by [64]. k5, k6, and k7 given by [63] and black dash-dotted line for the case of nonzero k3, k4, k5, and k6 given by [64]. K ¼ 12 2 þ 4 ð2 a − iÞþð a − iÞ2 þ 2 a b F F Ga Gi Ga Gi GbGa 2 0 1 2 0 2 1 2 0 1 0 1 2 0 1 2 ¼ 12F þ 8FðG0 þ G1 − G2Þþ3ðG0Þ þ 3ðG1Þ þ 2G0G1 þ 4G1G0 − 4G2ðG0 þ G1 − G2Þ 2 r v θ r 2 v 2 r v r v θ r v θ ¼ 12F þ 8FðGr þ Gv − GθÞþ3ðGrÞ þ 3ðGvÞ þ 2GrGv þ 4GvGr − 4GθðGr þ Gv − GθÞ 48 2 128π ≈ m − m ½ ð Þ − ð Þþρð Þ ð Þ 6 3 4 p z P z z : 86 r r μ0 Here, F ¼ 2m=r3, and the indices a and b correspond to 0 and 1 while i and j correspond to 2 and 3. The first term in K is equal to the Kretschmann scalar for the Schwarzschild solution. In Fig. 6, we have plotted 210π C ≡ r6K − 48m2 ≈ ½PðzÞ − ρðzÞ − pðzÞ z3 210π ¼ ½TðzÞ − 3pðzÞ : ð87Þ z3 We further define 1 1 ð Þ¼μ8 ab − c ab − d ð Þ S z 0 T 2 T cg Tab 2 T dgab ; 88 where lowercase Latin letters a, b, etc., run only over 0 and 1, or t and r. In terms of ρ þ þ 2 2 P f 2 3 FIG. 5. Behavior of the function μ ψ for spin 0, black dotted JðzÞ¼ ¼ 4f0ð1 þ 2z þ 3z þ 4z Þ βz2ð1 − zÞ line for the case of nonzero k3, k4, k5, and k6 given by [63] and red solid line for the case of nonzero k3, k4, k5, and k6 given by 2 3 2 3 4 − k3ð2z þ 3z þ 4z Þþ k5z − k7z ; ð89Þ [64]. For spin 1, blue dashed line for the case of nonzero k3, k4, 3 k5, k6, and k7 given by [63] and black dash-dotted line for the this gives case of nonzero k3, k4, k5, and k6 given by [64]. 124038-11 ABDOLRAHIMI, PAGE, and TZOUNIS PHYS. REV. D 100, 124038 (2019) 6 αβγδ 2 10 −3 FIG. 6. (a) Behavior of the function C ≡ r RαβγδR − 48m ≈ 2 πz ½PðzÞ − ρðzÞ − pðzÞ for spin 0, black dotted line for the case of nonzero k3, k4, k5, and k6 given by [63] and red solid line for the case of nonzero k3, k4, k5, and k6 given by [64]. (b) Behavior of the function C for spin 1, blue dashed line for the case of nonzero k3, k4, k5, k6, and k7 given by [63] and black dash-dotted line for the case of nonzero k3, k4, k5, and k6 given by [64]. 1 1 ð Þ¼ ðρ þ Þ2 − 2 2 ¼ β2 4 ð Þ½ð1 − Þ2 ð Þ − 4 S z 2 P f 2 z J z z J z f0 1 ¼ β2 5½12 ð1 þ 2 þ 3 2 þ 4 3Þ − 3 ð2 þ 3 2 þ 4 3Þþ2 3 − 3 4 18 z f0 z z z k3 z z z k5z k7z 3 4 3 4 2 3 4 3 4 5 × ½−12f0ð5z − 4z Þ − 3k3ð2 − z − 5z þ 4z Þþ2k5ðz − 2z þ z Þ − 3k7ðz − 2z þ z Þ 4 5 4 ¼ β2 −4 5 þð−6 þ 2 2Þ 6 þ −8 þ þ 2 2 7 þ −40 2 − 2 þ 2 − 8 f0k3z f0k3 k3 z f0k3 3 f0k5 k3 z f0 f0k7 2 k3 3 k3k5 z 2 7 2 þ −48 2 þ 40 − 7 2 þ þ 2 9 þ −56 2 þ 28 − 2 − þ 2 10 f0 f0k3 k3 3 k3k5 k3k7 z f0 f0k3 2 k3 k3k7 9 k5 z 40 10 4 2 þ −64 2 þ 32 − − 4 2 þ − 2 − 11 f0 f0k3 3 f0k5 k3 3 k3k5 9 k5 3 k5k7 z 32 8 2 4 1 þ 128 2 − 64 þ þ 20 þ 8 2 − − 5 þ 2 þ þ 2 12 f0 f0k3 3 f0k5 f0k7 k3 3 k3k5 k3k7 9 k5 3 k5k7 2 k7 z 2 1 þ −16 þ 4 − − 2 13 þ 2 14 ð Þ f0k7 k3k7 3 k5k7 k7 z 2 k7z : 90 Near the horizon, for z → 1, we have 4 28 ð Þ ∼ β2 −80 þ 18 − þ 2 þ 480 − 112 þ − 16 ð1 − Þ ð Þ S z f0 f0 k3 3 k5 k7 f0 k3 3 k5 k7 z : 91 Figure 7 shows the behavior of JðzÞ, and Fig. 8 graphs stress could be calculated numerically. Therefore, although SðzÞ=z5. Note also that very far from the black hole, z ≪ 1, this term indeed improves the fit for the finite range of r where the leading contributions (lowest powers of z) are SðzÞ ∼ the numerical calculations were made, possible small sys- 2 5 2 2 6 2 −4β f0k3z þ β ð−6f0k3 þ 2k3Þz þ β ð−8k3f0 þð4=3Þ× tematic errors in these calculations might have been respon- 2 7 3 f0k5 þ 2k3Þz . Since for both spin 0 and spin 1, Bardeen’s sible for the fit by a nonzero k3z term, rather than indicating 3 fits give k3 > 0, if there is indeed such a positive 1=r term an actual asymptotic term of this form [66].Marolf[67] and in the transverse stress, this would imply that SðzÞ is Horowitz [68] have expressed scepticism about the existence 3 negative for sufficiently small z (large r=ð2μ0Þ ≡ 1=z). of a 1=r term in the transverse stress, and now Ori [69] has 3 However, the k3z term in Bardeen’s fits to the transverse given us a detailed argument against such a term. Therefore, if 4 stress does not dominate over the k4z term for either the indeed asymptotically k3 ¼ 0, the leading term for SðzÞ 2 7 conformal massless spin 0 or the massless spin 1 until one would be ð4=3Þβ f0k5z , which is negative for conformal goes to values of r much larger than the range where the massless spin 0 but positive for massless spin 1. 124038-12 INGOING EDDINGTON-FINKELSTEIN METRIC OF AN … PHYS. REV. D 100, 124038 (2019) FIG. 7. (a) Behavior of the function JðzÞ for spin 0, black dotted line for the case of nonzero k3, k4, k5, and k6 given by [63] and red solid line for the case of nonzero k3, k4, k5, and k6 given by [64]. (b) Behavior of the function JðzÞ for spin 1, blue dashed line for the case of nonzero k3, k4, k5, k6, and k7 given by [63] and black dash-dotted line for the case of nonzero k3, k4, k5, and k6 given by [64]. 5 FIG. 8. (a) Behavior of the function SðzÞ=z for spin 0, black dotted line for the case of nonzero k3, k4, k5, and k6 given by [63] and red solid line for the case of nonzero k3, k4, k5, and k6 given by [64]. For spin 0, SðzÞ < 0 for 0 Negative SðzÞ means that the stress-energy tensor is of enough of sky [more than 63.4%] that the flux will be inward Hawking-Ellis Type IV [70], so there is no timelike or in the observer’s frame). Therefore, if indeed there were a 3 null eigenvector. This implies that no matter how fast an nonzero k3z term asymptotically for the transverse stress of observer moves away from the black hole where SðzÞ < 0, the positive sign as given by Bardeen’s fits [63,64],thenfor the Hawking radiation energy flux in the observer’s frame both the massless conformal spin 0 and spin 1, the stress- will always be outward (unlike solar radiation, e.g., where energy tensor would be asymptotically Type IV,with no frame at an average location of the earth at one astronomical unit in which the flux is zero, but if k3 ¼ 0, then the stress-energy from the sun, and using the value of the solar radius from tensor will be asymptotically Type IV for the conformal spin 0 [71], an observer with an outward velocity greater than with k5 < 0 but the ordinary Type I (with a timelike 0.999 993 7453(12) of the speed of light [a gamma fac- eigenvector, for which an observer whose 4-velocity is the tor greater than γ ¼283.737ð26Þ≈31=4 ðastronomical unitÞ= timelike eigenvector of the stress-energy tensor would see ðsolar radiusÞ] relative to the sun, will see the sun cover zero flux) asymptotically for massless spin 1 with k5 > 0. 124038-13 ABDOLRAHIMI, PAGE, and TZOUNIS PHYS. REV. D 100, 124038 (2019) Numerically, we find that the stress-energy tensor for Eq. (2.28) to be the same as twice our S, is given by the massless conformal scalar is Type IVeverywhere outside a semianalytical model in their Eq. (7.41) (which seems to the horizon for both of Bardeen’sfits[63,64] and also for the have many incorrect coefficients), and apparently these earlier fit by Visser [21] with k3 ¼ 0. However, for massless mistakes and/or round-off errors lead to the nonzero spin 1, it is only Type IV for 0 1 T 2ð Þ¼ρ2 þ 2 þ 2 2 − 2 2 ¼ þ 2 2 þ ð2 − Þ2 ð Þ z P p f S p 2 p T : 92 4 T 2ð Þ¼β2 −4 5 þð−6 þ 6 2Þ 6 þ −8 þ þ 2 2 þ 8 7 z f0k3z f0k3 k3 z f0k3 3 f0k5 k3 k3k4 z 5 20 þ −40 2 − 2 þ 2 þ þ 4 2 8 f0 f0k7 2 k3 3 k3k5 k4 z 2 2 9 þð−48f þ 3k − k3k5 þ 6k3k6 þ 8k4k5Þz 0 3 7 5 38 þ −56 2 þ 28 − 40 − 2 þ 10 þ 7 − þ 6 − 2 þ 2 10 f0 f0k3 f0k4 2 k3 k3k4 k3k7 3 k4k5 k4k6 k4k7 9 k5 z 160 40 19 8 þ −64 2 þ 32 − − 4 2 þ þ 8 − 2 þ 6 − 11 f0 f0k3 3 f0k5 k3 3 k3k5 k4k7 9 k5 k5k6 3 k5k7 z 82 þ 328 2 − 164 þ − 20 þ 40 f0 f0k3 3 f0k5 f0k6 f0k7 41 41 41 5 61 5 þ 2 − þ 5 − 10 þ 2 − þ þ 2 − þ 2 12 2 k3 6 k3k5 k3k6 k3k7 72 k5 6 k5k6 6 k5k7 2 k6 k6k7 k7 z 7 9 þ −56 þ 14 − þ 6 − 3 2 13 þ 2 14 ð Þ f0k7 k3k7 3 k5k7 k6k7 k7 z 2 k7z : 93 Near the horizon, for z → 1, we have 74 53 73 T 2ð Þ ∼ β2 120 2 − 122 − 40 − − 20 − 18 þ 2 þ 18 þ þ 11 z f0 f0k3 f0k4 3 f0k5 f0k6 f0k7 2 k3 k3k4 6 k3k5 k3k6 19 193 31 31 5 5 þ 11 þ 4 2 þ þ 6 þ 6 þ 2 þ þ þ 2 þ 5 þ 2 k3k7 k4 3 k4k5 k4k6 k4k7 72 k5 6 k5k6 6 k5k7 2 k6 k6k7 2 k7 748 − β2 1920 2 − 1448 − 400 − − 240 − 264 þ 264 2 f0 f0k3 f0k4 3 f0k5 f0k6 f0k7 k3 166 þ 156 þ 109 þ 114 þ 132 þ 32 2 þ þ 60 þ 68 k3k4 k3k5 k3k6 k3k7 k4 3 k4k5 k4k6 k4k7 155 187 þ 2 þ 56 þ þ 30 2 þ 66 þ 36 2 ð1 − Þ ð Þ 6 k5 k5k6 3 k5k7 k6 k6k7 k7 z : 94 Figure 9 shows the behavior of T 2ðzÞ=z5. 124038-14 INGOING EDDINGTON-FINKELSTEIN METRIC OF AN … PHYS. REV. D 100, 124038 (2019) 2 5 FIG. 9. (a) Behavior of the function T ðzÞ=z for spin 0, black dotted line for the case of nonzero k3, k4, k5, and k6 given by [63] and red solid line for the case of nonzero k3, k4, k5, and k6 given by [64].ForpðzÞ given by Eq. (74) for spin 0 with nonzero k3, k4, k5, k6, and 2 2 k7 given by [63], T ðzÞ > 0 for z>0.88587, but T ðzÞ < 0 for 0 Numerically, we have that for the conformal massless one to calculate an accurate metric (i.e., one whose small scalar field, T 2ðzÞ < 0 (violating the TOSEC defined by departure at each time from the Schwarzschild metric is [35]) for z<0.88587 and T 2ðzÞ > 0 (the more common accurately known) for such an isolated astrophysical black situation, obeying the TOSEC) for 0.88587 124038-15 ABDOLRAHIMI, PAGE, and TZOUNIS PHYS. REV. D 100, 124038 (2019) 4π rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψ ¼ r ð 0ˆ 0ˆ þ 0ˆ 1ˆ þ 1ˆ 0ˆ þ 1ˆ 1ˆ Þ ð Þ 4α x2 þ x þ 1 pffiffiffi 1 x − 1 ;r T T T T : 96 ˜ ≈ 3 −1 pffiffiffi 1 − 2m=r z 2 exp tan : μ0ðx − 1Þ 3 3 x þ 1 ð Þ For the polynomial approximations for the stress-energy 100 tensor given above, one gets that the radial partial derivative 6α ≪ μ3 ≡ −3α ≈ 1 þ 2α μ3 of ψ at fixed ingoing null coordinate v, ψ , is approx- For r 0 v, so that x r= 0 is very near ;r ˜ ≈ imately 1=ðμ2rÞ multiplied by a polynomial in z ¼ 2μ=r. 1, this gives z z, but unlike z, which goes to zero as r goes ˜ Note that we have made the gauge choice of setting to infinity, z decreases only to the very small positive ψðv; z ¼ 1Þ¼0 at the apparent horizon (z ≡ 2μ=r ¼ 1), v-dependent constant so that the value of ψðv; zÞ outside the apparent horizon pffiffiffi 1 ψ 4α 3π (i.e., for z< ) is obtained by integrating ;rdr from the z˜0ðvÞ ≡ z˜ðv; z ¼ 0Þ ≈ pffiffiffi exp 2 6 apparent horizon at r ¼ 2μ to the greater value r ¼ 2μ=z 3μ0 pffiffiffi along a null geodesic with constant v. Because the 4α1=3 3π integrand is approximately 1=ðμ2rÞ multiplied by a poly- ¼ exp : ð101Þ 37=6ð−vÞ2=3 6 nomial in z, we can integrate 1=ðμ2rÞ multiplied by each power of z and then combine the results with the appro- As a result, the function ψ in the ingoing Eddington- priate coefficients from the polynomial in z. When we do Finkelstein metric (17), which we have set to zero [along this for the positive powers of z, μ will stay very near ð Þ ð Þ ¼ 1 1 with g z and h z ] at the apparent horizon at z , does μ0 ≡ ð−3αvÞ3 ≫ 1 for nearly all of the dominant part not diverge as r is taken to infinity and z ≡ 2μ=r is taken to of the integral, so that those integrals will combine to give zero, but rather it goes to the finite (and small) limit of 2 2 gðzÞ=μ0. However, if we assume that μ ≈ μ0 when 1=ðμ rÞ ð − 4α ˜ Þ is multiplied by the constant term in the polynomial in z, ψ ð Þ ≡ ψð ¼ 0Þ ≈ g0 ln z0 ð Þ ð−4α μ 2Þ 0 v v; z 2 : 102 this term integrates to = 0 ln z, which diverges as r is μ0 taken to infinity and hence z is taken to zero. This divergent logarithmic integral is of course dominated by very large r, In Figs. (3–5), we have plotted the functions gðzÞ, hðzÞ, 2 where it is not valid to retain the approximation μ ≈ μ0, and μ ψ for spin-0 and spin-1 particles, assuming that μ0 is which is actually only valid for 2r ≪−3v. For larger r, μ sufficiently large that the last plot does not extend into the 2 rises with r, so that the integral of dr=ðμ rÞ remains very large r ≳ −v regime where μ significantly exceeds μ0. bounded by a finite quantity no matter how large the upper Therefore, knowing the value of the trace anomaly limit of r is taken. What we actually get is the convergent coefficients ξ for the appropriate conformal massless integral fields, i.e., scalar and/or electromagnetic, and the constants k3, k4, k5, k6, and k7, we have an approximate time- Z r 4α 0 dependent metric for an evaporating black hole as a first- ð−4α μ 2Þ ˜ ¼ dr ð Þ = 0 ln z 2 0 : 97 order perturbation of the Schwarzschild metric given by 2μ μ r 0 Eqs. (17)–(19) in (v; r) coordinates with z ≡ 2μ=r from Eq. (6), μðv; zÞ from Eqs. (7)–(9),orμðv; rÞ from Eq. (11) This integral will give z˜ ≈ z ¼ 2μ=r for 2r ≪−3v. The (approximately for r ≪ μ3=α), the functions gðzÞ and hðzÞ deviations will only become significant for larger r, for given by Eq. (85), and the function z˜ given by Eq. (100) which the 1=z term on the right-hand side of Eq. (5) with x therein given by Eq. (98). We also remind the reader becomes large in comparison with each of the other two that we have chosen the gauge such that at the apparent terms. Then we can let horizon, where z ¼ 1 in our approximate metric, ϵð1Þ¼ gð1Þ¼hð1Þ¼ln z˜ð1Þ¼0, and that this metric applies 1 1 μ 6α 3 2 3 everywhere outside the black hole before it gets so small ≡ ≈ 1 þ r ¼ 1 þ r ð Þ x 3 98 that quantum gravity effects become important. μ0 μ0 −v IV. COMPARISON OF METRICS become the new independent variable, which has a lower 2 1 2 ¼ 2μ ¼ð1 þ 12α μ Þ3 ≈ 1 þ 4α μ The metric (17) with our expressions for the functions limit at r 0 of xh = 0 = 0, 1 1 ≪ ð−3α Þ3 ≲ which then gives contained therein should be valid for v r. (Indeed, it should apply also for somewhat smaller r, a bit Z inside the event horizon that is very near z ¼ 1, but for z x 3dy − ln z˜ ¼ ; ð99Þ significantly larger than 1, the approximations used for the y3 − 1 xh stress-energy tensor will not be valid.) Next, we want to convert it back to outgoing Eddington-Finkelstein coor- and hence dinates to see how our metric behaves in the coordinate 124038-16 INGOING EDDINGTON-FINKELSTEIN METRIC OF AN … PHYS. REV. D 100, 124038 (2019) system (u; r) in the restricted region μ=r ¼ð−3αuÞ1=3= 2ð−3αuÞ1=3 g ≈−1 þ ; ð111Þ r ≪ 1, where the outgoing Vaidya metric (1) is approx- uu r imately valid. Consider the metric (17) with functions h and g known from Eq. (85), using the relation gur ≈−1; ð112Þ r grr ≈ 0: ð113Þ v ≈ u þ 2r − 4ð−3αuÞ1=3 þ 4ð−3αuÞ1=3 ln : 2ð−3αuÞ1=3 Therefore, for μ=r ¼ð−3αuÞ1=3=r ≪ 1, we get the out- ð Þ 103 going Vaidya metric, We get 2ð−3αuÞ1=3 ds2 ≈ −1 þ du2 − 2dudr þ r2dΩ2: ð114Þ r dv ¼ Ψ1du þ Ψ2dr; ð104Þ 1 This form of the metric applies for 1 ≪ ð−3αvÞ3 ≪ r,so where it does not apply near the black hole horizon, where the more general Eq. (17) does apply, but it is applicable for 4α ½ r ln 1=3 8α arbitrarily large r. It also applies for positive advanced time Ψ ≈ 1 − 2ð−3αuÞ þ ð Þ 1 2 3 2 3 ; 105 v, after the black hole has evaporated, so long as one avoids ð−3αuÞ = ð−3αuÞ = the Planckian region near the final evaporation and its causal future, where quantum gravity effects are expected. 4ð−3αuÞ1=3 Ψ2 ≈ 2 þ : ð106Þ r V. SUMMARY The metric (17) with functions h and g given by Eq. (85) In this paper, we have constructed an approximate time- becomes dependent metric for an evaporating black hole as a first- order perturbation of the Schwarzschild metric, using the 2m linearized backreaction from the stress-energy tensor of ds2 ¼ −e2ψ 1 − Ψ2du2 r 1 the Hawking radiation in the Unruh quantum state in the unperturbed spacetime. We used a metric ansatz in ingo- 2ψ 2m ψ − 2 e 1 − Ψ1Ψ2 − e Ψ1 dudr ing Eddington-Finkelstein coordinates (v; r). Our ansatz is r such that at infinity we get the Vaidya metric in the out- ψ 2ψ 2m 2 2 2 2 going Eddington-Finkelstein coordinates (u; r). We have þ 2e Ψ2 − e 1 − Ψ dr þ r dΩ r 2 solved the corresponding Einstein equations for the metric functions to first order in the stress-energy tensor of the ¼ 2 þ 2 þ 2 þ 2 Ω2 ð Þ guudu gurdudr grrdr r d : 107 unperturbed Schwarzschild metric. Therefore, our metric should be a very good approximation everywhere near to For −3αu ≫ 1,wehave and everywhere outside the event horizon when the mass is large in Planck units. 2ð−3αuÞ1=3 1 g ¼ −1 þ þ ½32α þ 2h þ 4g uu r ð−3αuÞ1=3r ACKNOWLEDGMENTS 1 þ O ; ð108Þ We acknowledge helpful information about the stress- ð−3αuÞ2=3 energy tensor by email from James Bardeen. This work was supported in part by the Natural Sciences and Engineering 8ð−3α Þ2=3 1 Research Council of Canada. During the latter stages of ¼ −1 þ u þ O ð Þ gur 2 2 ; 109 the paper, D. N. P. appreciated the hospitality of the r r Perimeter Institute and of Matthew Kleban at the Center for Cosmology and Particle Physics of New York 16ð−3α Þ2=3 −96α ð−3α Þ4=3 ¼ u þ u þ O u ð Þ University. Research at Perimeter Institute is supported grr 2 3 4 : 110 r r r by the Government of Canada through the Department of Innovation, Science and Economic Development and by Equations (107)–(110) give us the corrections to the out- the Province of Ontario through the Ministry of Research, going Vaidya metric (1) for an evaporating black hole in Innovation and Science. Calculations by D. N. P. during outgoing Eddington-Finkelstein coordinates. When we one stage of the paper were made at the Cook’s Branch consider only terms of the order of unity and of first order Nature Conservancy under the generous hospitality of the in the small quantity μ=r ¼ð−3αuÞ1=3=r, we get Mitchell family and of the George P. and Cynthia W. 124038-17 ABDOLRAHIMI, PAGE, and TZOUNIS PHYS. REV. 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