The information paradox

It is difficult to avoid addressing the information loss puzzle (cf e.g. [47]) when discussing the long time evolution of BHs. The main two alternatives are (1) the information is lost, as proposed by Hawking [48], and (2) some mechanism exists that prevents the information loss. Most people with a field (or string) theory background favour the second variant. In the following it will be discussed very briefly which of the proposed solutions fits into the current approach.

The quantum hair solution does not fit into the current formalism since the formation process has been neglected. Also the possibility that the information is emitted together with the Hawking radiation is not applicable here, at least not without specifying the matter content of the theory; neither is the baby universe scenario, at least not in an obvious way. Since a remnant geometry has been predicted there remain two natural solutions: either the information is retained in the remnant or it emerges at the end of the evaporation process. Due to the fact that the proposed remnant geometry is flat it is hard to see where any information could be stored (apart from the scenario with a cosmic string remnant), so the only possibility seems to be that the information is released during the final stages. Indeed, this is what the Carter–Penrose diagram figure3.2 seems to suggest, as well as the discussion in section 3.6. It will be studied in more detail below.

The information paradox manifests itself typically in a pathological feature of Carter–

Penrose diagrams related to BH evaporation (cf e.g. figure 1 of [49] for the classical example; in that reference it is argued in favour of a violation of locality). Thus, it is worthwhile to study the present Carter–Penrose diagram figure 3.2 and to check what, if any, pathological features are encoded there. To this end the consideration of test particles will be useful. Note that in the language of the S-matrix approach to quantum black holes [50] the ‘hard matter’ components have implicitly been taken into account by our assumptions. ‘Hard’ refers to the fact that its backreaction upon geometry cannot be neglected, as opposed to the case for ‘soft’ matter, which can be treated like test particles.

It will be assumed for simplicity that the test particles are massless.

Let us start at some value of advanced timev =v_inatI⁻and follow the light rays until they are reflected at the origin and eventually emitted to a point at I⁺ corresponding to

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a certain value of the retarded timeu=u_out. Ifv_inis sufficiently close to i⁻ thenu_out <0, i.e. the outgoing ray will pass solely through region I, encounter the infalling matter flux and finally pass through II. Similarly, for values of v_in sufficiently close to i⁰ the light ray will pass through regions II and III, encounter the outgoing shock wave and end up in region IV where eventually it will be emitted to I⁺ close to i⁺. Actually, this is true for all values vin > v0. Of course, rays starting closer to v0 will additionally pass the apparent horizon, but so far no pathological features have been found. At v = v₀ there is the problem of the potential singularity encountered at the origin. Recently, imposing a boundary condition at the singularity has been proposed [51] (cf [52] for a critical comment), as a possible path to a solution to information loss; in the present context it is not quite clear whether the singularity exists at all as it lies in the very region we could only speculate about. So let us disregard it and focus on the region at I⁻ that has not been covered so far: v_in< v₀ but still sufficiently far away fromi⁻ that light rays are not scattered to region II.

Such a test particle will trespass on the line u = 0 and be reflected at the origin still in region I. However, it crosses the infalling matter shock wave between u = 0 and u = u_f thus entering region III. It will follow a curved path more or less parallel to the apparent horizon and then pass the outgoing shock wave. In region IV it moves first again towards the origin and after the reflection it finally reaches I⁺. Such test particles are remarkable for three reasons: (1) they are reflected twice at the origin, (2) they cross infalling and outgoing shock waves, (3) for each test particle travelling in this way there exists exactly one ‘partner particle’ emitted atv_in > v₀. The ‘partner particle’ will end up at the same point at I⁺. Thus, tracing back the light rays from I⁺ there exists a region between u=u_f and some larger (but finite) value ofu where the light ray splits into two as soon as the outgoing shock wave is encountered from above: one directly going back to I⁻ and the other one entering region III. In this sense, the outgoing shock wave causes birefringence. The line v = v0 acts like a mirror regarding the two ‘partner particles’: if one of them is close to (far from) v₀, so is the other.

On top of that, the later the test particle is emitted at I⁻ in region I (i.e. the closer v_in is tov₀) theearlier it arrives at I⁺ in region IV (i.e. the closer it is to u=u_f). Thus, causality is violated (or rather: inverted), but this violation is not noticed atI⁺ until the BH has evaporated completely!

This is the pathological feature we were looking for (or hoping to avoid): there is something very strange about I⁺: while the regions u < 0 and close to i⁺ are unproblematic, there is no way to send information to the region u_f > u > 0 by means of test particles—this coincides with the region dominated by Hawking radiation caused by ‘hard matter’. Moreover, there exists a region u_e > u > u_f with some finite u_e where information may arrive in two ways: either directly from I⁻ with only one reflection or with two reflections and inverted causality, as described in the previous paragraphs. It is worthwhile to study these features in more detail, but such a study would lead beyond the scope of this paper.

4. Relaxing the conditions and conclusions

Relaxations within 2D dilaton gravity. There are three generic possibilities regarding the first assumption: first, one can assume the complete absence of deformations with the

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usual ensuing problems; second, one can allow for consistent deformations only as in the present approach (with the important caveat that only deformations of the spherically reduced theory have been considered); third, one can allow for arbitrary deformations which enlarges the ‘phase space’ of possible models but at the same time reduces the predictability. It is sound to start with the second possibility until some fundamental theory dictates the precise form of the allowed deformations.

Relaxing the second assumption is possible in principle—e.g. a BH immersed in (A)dS or Rindler space–time can be considered along these lines. However, the fact that asymptotics is not changed essentially during the evaporation process should be maintained, because otherwise the notion of a long time asymptotic observer will be challenged. For the third assumption similar considerations apply: one can allow e.g. for two Killing horizons and thus is able to discuss the Reissner–Nordstr¨om case. Such a generalization is not as harmless as it might seem at first glance: once two or more horizons are allowed extremal ones may emerge during the evaporation process. Since they do not produce Hawking radiation extremal geometries are additional candidates for possible end-points of the evolution. Because stringy methods are capable of describing (nearly) extremal BHs very well (cf e.g. [53]) one could imagine that the non-extremal evolution is described as discussed in the current work and from the extremal point on string theory takes over. However, there is classical and semi-classical evidence that nearly extremal BHs do not evolve towards the extremal limit, which can be taken as a justification of our neglect of extremal horizons (cf section3.3).

Relaxing the crucial fourth assumption does not seem plausible from a physical point of view—either the asymptotic flux would have to be negative at some point (‘anti-evaporation’) or it would have to grow without bound.

Obviously it is desirable to generalize from thea–b family to a more general class (or the most general class) of possible models. Some steps in that direction are presented in appendix B. As long as no extremal horizon is present, evidence points again towards the previously discussed remnant geometry as the end-state of evaporation.

Beyond 2D dilaton gravity. The restriction to PSMs⁹, even though it allows for phenomenologically relevant models like the SS BH or generalizations thereof (with charge and/or cosmological constant), could be overcome by leaving the comfortable realm of 2D. Due to the rigidity result [17] mentioned in footnote 2 a natural framework would be a scalar–tensor theory `a la Jordan–Brans–Dicke [55] or its more recent incarnation as quintessence [56]. A brief discussion of scalar–tensor theories in the framework of 2D models can be found in [57]. It should be possible to apply the methods of this work to the dimensionally reduced models treated in that reference.

To be a bit more concrete: an action of aD-dimensional scalar tensor theory,

MD

d^Dx√

−g[XR−U(X)(∇X)²+ 2V(X)], (4.1)

contains two arbitrary functions U, V of the dilaton X which usually are adjusted either by hand or by some guiding principle from a more fundamental theory. Tinkering with the assumptions, it should be possible to generalize the present work from D = 2 to

9 A generalization to graded PSMs in order to describe dilaton supergravity models should be straightforward;

cf [54].

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arbitrary D, in particular to D= 4. If again the phenomenon of ‘dilaton evaporation’ to a constant dilaton vacuum occurs, the final state will be a solution of ordinary vacuum Einstein theory with a cosmological constant Λ, the value of which depends on the function V(X) of the attractor solution: Λ ∝ V(XCDV)/XCDV, where XCDV is a solution of the equation 2V(XCDV) =XCDVV(XCDV). The existence of such a solution poses non-trivial constraints on the dilaton potential V.

What are the experimental consequences? Since for each SS BH (independent of its original mass, as long as it was larger than M_Planck/2) a cosmic string with a total mass of maximal M_Planck/2 remains, this puts an upper bound on the total amount of primordial BHs. Actually, this bound is well known [58] because several models predict a relic with a mass of order of M_Planck [59]. Apart from their total mass, cosmic strings contribute to density fluctuations which should be visible in the observed cosmic microwave background anisotropies (cf e.g. [60] for an analysis of the BOOMERanG data). Finally, the thunderbolt with a total energy of maximal M_Planck/2 should lead to observable consequences, possibly in the form of γ-ray bursts. The sum of the energy of the thunderbolt and the mass of the cosmic string remnant is exactlyMPlanck/2 provided that energy conservation holds. However, none of these consequences provides a ‘smoking gun signature’. Moreover, as mentioned before, it is not unlikely that eventually non-spherical modes will become relevant during the evaporation process, which further limits the phenomenological relevance of the present work.

Brief summary. Let us finally repeat the lines of reasoning underpinning the approach presented in this paper: it is generally expected that during the final stages of BH evaporation some drastic changes will occur, since the semi-classical theory extrapolated to the very end predicts its own failure. In order to discuss the issue of long time BH evaporation in more detail either quantum gravity must be applied or some trick must be used. It has been proposed to follow the latter route by assuming an isothermal observer at i⁰, i.e. the asymptotic Hawking flux (and hence the Hawking temperature) as measured by this observer does not change during the whole evaporation process. It can be replaced by a much weaker condition, namely boundedness and positivity of the asymptotic Hawking flux. As a consequence—together with ‘natural’ assumptions being made regarding the asymptotics, the causal structure, the nature of allowed quantum deformations of gravitational symmetries and some simplifying technical assumption further restricting the allowed deformations—an evolution of the SS BH towards a remnant geometry (2.9) with B = 0 has been predicted. We speculated about the final flicker, argued in favour of a first-order phase transition to a constant dilaton vacuum, discussed briefly implications for the information paradox and concluded with possible generalizations of our postulates.

If nothing else, the present approach shows that quantum induced deformations of the gauge symmetries of gravity can play a pivotal role in the understanding of the long time behaviour of black holes.

Acknowledgment

This work was supported by projects P-14650-TPH and J-2330-N08 of the Austrian Science Foundation (FWF). I would like to thank my long time collaborators on 2D

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gravity, W Kummer and D Vassilevich, for numerous stimulating discussions in the past, present and future and for a careful reading of the manuscript. Moreover, I am grateful to H Balasin for his interest, valuable comments and his important role as advocatus diaboli. I render special thanks to T Strobl for bringing reference [20] to my attention and to F Brandt for e-mail correspondence on consistent deformations. I have profited from conversations and e-mail exchanges with D J Schwarz on cosmological topics. I thank A Barrau, A Bonanno, D Easson, B Harms and S Nojiri for pointing out some references and/or for correspondence. Several comments by the anonymous referee led to a considerable improvement of the presentation. Finally, I am grateful to V Frolov for discussions on cosmic strings and SS BHs and to O Zaslavskii for a useful pedagogical suggestion during the workshop ‘Gravity in two dimensions’ at the International Erwin–

Schr¨odinger Institute.

Appendix A: deformations of dilaton gravity

This appendix is meant to be self-contained. Its purpose is to review concisely some features of ordinary and deformed dilaton gravity which are (at least implicitly) relevant for the rest of this paper.

To this end (only in this appendix) the first-order version of (2.1) [19, 61]

L⁽¹⁾ =

M2

[Xa(D∧e)^a+Xd∧ω+V(XaX^a, X)], (A.1) will be employed where X is the dilaton field, e^a is the zweibein 1-form, is the volume 2-form. The 1-formωrepresents the spin connectionω^a_b =ε^a_bω withε_ab being the totally antisymmetric Levi-Civita symbol. The action (A.1) depends on two auxiliary fieldsX^a. It is a special case of a PSM [19, 62]:

L^PSM =

M2

[dXⁱ∧A_i+¹₂P^ijA_j ∧A_i] (A.2) with the Poisson tensor (f and C are arbitrary functions of Xⁱ)

P^ij =ε^ijkfC,k :=ε^ijkf ∂C

∂X^k (A.3)

which fulfils the (generalized) Jacobi identity

PⁱⁿP_,n^jk+ cycl.(i, j, k) = 0, (A.4)

and a three-dimensional target space, the coordinates of which are X, X^a. In light cone coordinates (η₊₋ = 1 = η₋₊, η₊₊ = 0 = η₋₋) the first (‘torsion’) term of (A.1) is given by

X_a(D∧e)^a=η_abX^b(D∧e)^a =X⁺(d−ω)∧e⁻+X⁻(d+ω)∧e⁺. (A.5) The function V is an arbitrary potential depending solely on Lorentz invariant combinations of the target space coordinates, namely X and X⁺X⁻. To match the first-order with the second-order formulation one has to chooseV =U(X)X⁺X⁻+V(X) with the same functions U, V as in (2.1). To match the PSM with the first-order version one has to choose f = 1/I(X), C =I(X)X⁺X⁻+w(X) with

I(X) := exp X

U(y) dy, w(X) :=

X

I(y)V(y) dy. (A.6)

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Variation of the fields yields the equations of motion

dXⁱ+P^ijA_j = 0, (A.7)

d∧A_i− ¹₂P_,i^jkA_k∧A_j = 0. (A.8)

Under the symmetry variation (note that ε = ε(x^µ, Xⁱ), where x^µ are the world sheet coordinates andXⁱ the target space coordinates)

δ_εXⁱ =P^ijε_j, δ_εA_j =−dε_j −P_,j^knε_nA_k, (A.9) the action (A.2) transforms into a total divergence

δεL=

∂M2

dXⁱεi. (A.10)

The commutator of two symmetry variations closes only on shell in general:

[δ_ε1, δ_ε2]Xⁱ =δ_ε3Xⁱ, (A.11)

[δ_ε1, δ_ε2]A_i =δ_ε3A_i+ (dX^j+P^jkA_k)P_,ji^nmε_1mε_2n, (A.12) with

ε3i =P,ijk

ε1kε2j +P^jk[εk,(ε,j)i]12, (A.13) and [f(ε), g(ε)]₁₂ :=f(ε₁)g(ε₂)−f(ε₂)g(ε₁).

The introduction of the Schouten–Nijenhuis bracket [63]

{Xⁱ, X^j}=P^ij, {Xⁱ,•}=P^ij∂_j(•), (A.14) allows one to re-express e.g. the conservation law as {Xⁱ,C} = 0, ∀i. According to Izawa [20], consistent deformations in the sense of Barnich and Henneaux are those which leave the bracket structure (A.14) essentially intact, i.e. the Poisson tensor may change its functional dependence on the fields Xⁱ, but its dimension (and the fact that one stays within the realm of PSMs) remains fixed.

Since Izawa’s result is of some importance for the present approach his proof will be sketched briefly: the starting point is 2D BF theory (the conventions of [20] have been translated accordingly):

S₀ =

M2

dXⁱ∧A_i, (A.15)

which is just a very trivial special case of a PSM (A.2) with vanishing Poisson tensor.

Then, the minimal solution ¯S to the classical master equation and the generator of BRST symmetry s are constructed. A deformation L₁ of the Lagrangian must obey [5]

sL1+da1 = 0. (A.16)

This, together with its descent equations sa₁+da₀ = 0, sa₀ = 0, allows one to construct the most general consistent L₁. It is given by equation (14) of [20]. The deformed action

S= ¯S+

M²

L₁ =S₀+ ¹₂

M²

P^ijA_j∧A_i+ antifield terms, (A.17) satisfies the classical master equation and reduces to (A.2) for vanishing antifields.

Actually, it even satisfies the quantum master equation as there is no contribution from renormalization [64]. No further deformations occur because P^ij fulfils the (generalized)

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Jacobi identity (A.4), i.e. the first-order deformation of Abelian BF theory, which is nothing but a PSM, provides already the most general consistent deformation. Thus, also the most general consistent deformation of a PSM is just another PSM (with the same number of target space coordinates).

It is important to notice that dilaton gravity, if formulated as a PSM, requires additional structure, namely a correspondence between the gauge fields and the line element. The suggestive notationAi = (ω, e⁻, e⁺) together withg = 2η+−e⁺e⁻apparently fixes this structure in the undeformed case, but there is no a priori relation between the three gauge fields and the line element if deformed PSMs are considered. This subtle and important issue is addressed in detail in [11].

Appendix B: the U–V family

For technical reasons the main part of the paper concentrated on the a–b family. In this appendix the results will be generalized to the U–V family, i.e. to models with a Lagrangian (2.1) with arbitrary potentialsU(X) andV(X) which fulfil the MGS condition and which have at most one horizon.

All local and global classical solutions of such models have been discussed extensively in a series of papers by Kl¨osch and Strobl [65]. For a first orientation on classical, semi-classical and quantum dilaton gravity in 2D the review [22] may be consulted.

In the following the notation and results of [37] will be used; for the sake of self-containment we recall two definitions: the ‘integrating factor’ I(X) and the conformally invariant combination w(X) as defined in (A.6). The MGS property reads w ∝ 1/I.

Provided that the equation I = c has exactly one solution for positive values of c and no solution for c ≤ 0 the existence of at most one horizon is guaranteed. The constant c scales like 1/M_BH. The Hawking temperature reads T_H ∝ U(X_h)M_BH where X_h is the solution of I(X_h) = 1/M_BH. It is convenient to introduce the inverse function of the integrating factor, I^∗ ◦I = id, because X_h = I^∗(1/M_BH). With these preliminaries the Hawking temperature for MGS models of the U–V family with one horizon reads

T_H∝ M_BH² (I^∗)|1/MBH

. (B.1)

Now we would like to address the issue of boundedness of T_H. Suppose thatI^∗ is a given function of X (from this one can deduce all other quantities). First, it will be assumed that I^∗ has a Laurent series expansion of the form

I^∗ = N

n=−∞

a_nXⁿ, N ∈Z. (B.2)

If N = 0 then the leading term drops out after differentiation, so in this case the value for N relevant in all subsequent formulae is given by the next to leading order term. For small values of the BH mass the Hawking temperature is dominated by the highest power N in (B.2) (and hence the discussion is essentially reduced to the a–b family),

T_H(M_BH→0)≈a(M_BH)^N+1, a ∈R⁺. (B.3)

Thus, forN ≥ −1 only (but N = 0, because this term drops out after differentiation) the temperature remains bounded for all times. Relevant examples/counter-examples are the

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CGHS model with N = −1 and spherically reduced gravity from D = 4 with N = −2, respectively. The attractor solution does not fit into this scheme because I = constant does not allow for an inverse (but this model is rather trivial anyway).

If there is a non-perturbative prefactor in front of the sum in (B.2) which is regular at M_BH = 0, then again the highest term dominates and the same bound N ≥ −1 is recovered. If there is a different (but at M_BH = 0 regular) non-perturbative prefactor for each term, then still the same conclusion holds. Thus, the bound N ≥ −1 is rather general. If the prefactor is singular a different bound may be obtained (e.g. N > −1 for

If there is a non-perturbative prefactor in front of the sum in (B.2) which is regular at M_BH = 0, then again the highest term dominates and the same bound N ≥ −1 is recovered. If there is a different (but at M_BH = 0 regular) non-perturbative prefactor for each term, then still the same conclusion holds. Thus, the bound N ≥ −1 is rather general. If the prefactor is singular a different bound may be obtained (e.g. N > −1 for

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