Black hole evaporation: information loss but no paradox

Abstract

The process of black hole evaporation resulting from the Hawking effect has generated an intense controversy regarding its potential conflict with quantum mechanics’ unitary evolution. A recent set of works by a collaboration involving one of us, have revised the controversy with the aims of, on one hand, clarifying some conceptual issues surrounding it, and, at the same time, arguing that collapse theories have the potential to offer a satisfactory resolution of the so-called paradox. Here we show an explicit calculation supporting this claim using a simplified model of black hole creation and evaporation, known as the CGHS model, together with a dynamical reduction theory, known as CSL, and some speculative, but seemingly natural ideas about the role of quantum gravity in connection with the would-be singularity. This work represents a specific realization of general ideas first discussed in Okon and Sudarsky (Found Phys 44:114–143, 2014) and a complete and detailed analysis of a model first considered in Modak et al. (Phys Rev D 91(12):124009, 2015).

Appendices

Appendix 1: Non-Hadamard behavior in the $\varvec{out}$ region

In this section we shall follow the standard Hadamard prescription to prove the non-Hadamard behavior of an arbitrary particle state (including the zero particle state, i.e, vacuum) in the out quantization defined in the case of evaporating black hole.

1.1 The two-point function

We use the discrete basis by working with wave packets. In this basis the modes are orthonormalized unlike the continuous counterpart. The field operator has the following form

$$\begin{aligned} \hat{f}^{L/R}(x)= & {} \displaystyle \sum _{n,j}\frac{1}{\epsilon ^{1/2}}\int _{j\epsilon }^{(j+1)\epsilon } d\omega \left( e^{2\pi i \omega n/ \epsilon }~\hat{b}_{\omega }^{L/R} v_{\omega }^{L/R}+ e^{-2\pi i \omega n/ \epsilon }~ \hat{b}^{L\dag /R\dag }_{\omega }v_{\omega }^{L*/R*} \right. \nonumber \\&+ \left. e^{2\pi i \omega n/ \epsilon }~ \hat{\tilde{b}}^{L/R}_{\omega }\tilde{v}_{\omega }^{L/R}+ e^{-2\pi i \omega n/ \epsilon }~\hat{\tilde{b}}^{L\dag /R\dag }_{\omega }\tilde{v}_{\omega }^{L*/R*}\right) \end{aligned}$$

(92)

$$\begin{aligned}= & {} \hat{f}_{1}^{L/R} (x) + \hat{\tilde{f}}_{2}^{L/R} (\tilde{x}). \end{aligned}$$

(93)

In the above expression we again used tilde to denote that these modes are defined interior to the black hole.²⁴ The wave packets defined exterior to black hole, similar to Eq. (30), are given by

$$\begin{aligned} v_{jn}^{L/R}=\epsilon ^{-1/2}\int _{j\epsilon }^{(j+1)\epsilon }d\omega e^{2\pi i\omega n/\epsilon }v_{\omega }^{L/R}, \end{aligned}$$

(94)

where the modes $v_{\omega }^{L/R}$ are given in Eq. (109) and Eq. (111). Below we provide an explicit calculation only for the right-moving modes since the other left-moving sector is fully analogous.

We calculate the two-point function in an arbitrary state, say, $\left| \psi \right\rangle =\left| F \right\rangle ^{int}\otimes \left| F \right\rangle ^{ext}$ of the joint basis. The two-point function takes the following form

$$\begin{aligned} \left\langle \psi \right| \hat{f}^R(x) \hat{f}^R(x') \left| \psi \right\rangle = \left\langle F \right| ^{ext} \hat{f}_{1}^R(x) \hat{f}_{1}^R (x') \left| F \right\rangle ^{ext} + \left\langle F \right| ^{int} \hat{\tilde{f}}_{2}^R (\tilde{x}) \hat{\tilde{f}}_{2}^R (\tilde{x}') \left| F \right\rangle ^{int}\nonumber \\ \end{aligned}$$

(95)

Note that the contributions from the exterior (region II, Fig. 1) and interior (region III, Fig. 1) of the black hole decouples from one another. Let us first focus on the first term defined exterior. Calculations involving the other term defined in the interior to the black hole is exactly similar. Writing this term explicitly yields

$$\begin{aligned}&\left\langle F \right| ^{ext} \hat{f}_{1}^R(x) \hat{f}_{1}^R (x') \left| F \right\rangle ^{ext} = \left\langle F \right| ^{ext} \displaystyle \sum _{n,j} \displaystyle \sum _{n',j'} \frac{1}{\epsilon } \int _{j\epsilon }^{(j+1)\epsilon }d\omega \int _{j'\epsilon }^{(j'+1)\epsilon }d\omega ' \nonumber \\&\quad \times \left( e^{2\pi i (n \omega - n' \omega ')/ \epsilon }~v_{\omega } v_{\omega '}^{*} \hat{b}_{nj}{{\hat{b}}^{\dagger }}_{n'j'} + e^{-2\pi i (n \omega - n' \omega ')/ \epsilon }~v_{\omega '} v_{\omega }^{*} \hat{b}^{\dagger }_{nj}\hat{b}_{n'j'}\right) \left| F \right\rangle ^{ext} \end{aligned}$$

(96)

while the other terms do not contribute. The next step would be to use the commutator relation between the creation and annihilation operators and orthonormality condition $\left\langle F_{nj} \right| {F_{n'j'}}\rangle ^{ext} = \delta _{nn'}\delta _{jj'}$. This simplifies Eq. (96) to the following form

$$\begin{aligned} \left\langle F \right| ^{ext} \hat{f}_{1}^R (x) \hat{f}_{1}^R (x') \left| F \right\rangle ^{ext} = \displaystyle \sum _{n,j} \frac{(2F_{nj} + 1)}{\epsilon }~I_{1}(\omega ) I_{1}^{*}(\omega ') \end{aligned}$$

(97)

where,

$$\begin{aligned} I_{1} (\omega )= & {} \int _{j\epsilon }^{(j+1)\epsilon }\frac{d\omega }{\sqrt{4\pi \omega }} e^{\frac{i\omega }{\Lambda }\log (-\Lambda x)+ 2\pi i n \omega /\epsilon } \end{aligned}$$

(98)

$$\begin{aligned} I_{1}^{*} (\omega ')= & {} \int _{j\epsilon }^{(j+1)\epsilon }\frac{d\omega '}{\sqrt{4\pi \omega '}} e^{\frac{-i\omega '}{\Lambda }\log (-\Lambda x)- 2\pi i n \omega '/\epsilon }. \end{aligned}$$

(99)

In the expression Eq. (97), the integration over the term that includes $F_{nj}$ gives

$$\begin{aligned}&\displaystyle \sum _{n,j} \frac{F_{nj}}{\epsilon \left( \frac{1}{\Lambda }\log (-\Lambda x)+ \frac{2\pi n}{\epsilon }\right) ^{1/2}\left( \frac{1}{\Lambda }\log (-\Lambda x')+ \frac{2\pi n}{\epsilon }\right) ^{1/2}}\nonumber \\&\qquad \times \left( \gamma \left[ \frac{1}{2},(j+1)\epsilon \right] - \gamma \left[ \frac{1}{2},j\epsilon \right] \right) ^2 \end{aligned}$$

(100)

Note that since $\left| F \right\rangle ^{ext}$ is a well defined state in the out Fock space it may have arbitrary number of particles but in no case it can be infinity. Therefore not all $F_{nj}$-s are nonzero when n and j are changing their values. On the other hand the term without $F_{nj}$ in Eq. (97) can be simplified in the following manner. We first use the identity

$$\begin{aligned} \displaystyle \sum _{n} e^{2\pi i n (\omega -\omega ')/ \epsilon } = \epsilon \delta (\omega - \omega '). \end{aligned}$$

(101)

As a result the integrations of the two integrals over $\omega$ and $\omega '$ transforms into a single integral, and then, by appropriately using the definition of the function $\gamma$ to turn the the sum over j into an integral, we obtain the final expression for the two-point function

$$\begin{aligned}&\left\langle F \right| ^{ext} \hat{f}_{1}^R (x) \hat{f}_{1}^R (x') \left| F \right\rangle ^{ext} \nonumber \\&\quad = - \frac{1}{4\pi } \log |\log (x/x')| + \displaystyle \sum _{n,j} \frac{F_{nj}}{2\pi \epsilon h(x) h(x')} \left( \gamma \left[ \frac{1}{2},(j+1)\epsilon \right] -\gamma \left[ \frac{1}{2},j\epsilon \right] \right) ^2,\nonumber \\&\quad h(x) = \left( \frac{1}{\Lambda }\log (-\Lambda x)+ \frac{2\pi n}{\epsilon }\right) ^{1/2}. \end{aligned}$$

(102)

In order to check the Hadamard behavior, as explained in Sect. 3.3, we need to construct the symmetrized two-point function

$$\begin{aligned} G^{(1)} (x, x') = \frac{1}{2}(\left\langle F \right| ^{ext} \hat{f}_{1}(x) \hat{f}_{1} (x') \left| F \right\rangle ^{ext} + \left\langle F \right| ^{ext} \hat{f}_{1}(x') \hat{f}_{1} (x) \left| F \right\rangle ^{ext}) \end{aligned}$$

(103)

and subtract the “Hadamard ansatz”. If the quantum state is Hadamard then we expect no singular behavior after the above said subtraction.

1.2 The Hadamard Ansatz

In two dimensions it seems to be a non-agreement about precise form of the “Hadamard ansatz” [88] and [89]. Usually the differences in opinion comes through the finite terms that are present in the Hadamard ansatz. Nevertheless, for our purpose we are interested to extract the singular behavior due to coincidence limit. We restrict to the following singular behavior of the Hadamard ansatz

$$\begin{aligned} H (x, x') = -\frac{1}{4\pi } \log \sigma , \end{aligned}$$

(104)

where $\sigma$ is half of the geodesic distance square between two close points x and $x'$ in the normal neighborhood.

Now let us proceed to calculate the geodesic distance between x and $x'$. We will do it in null coordinates. The calculations of this section are based on section 2 of chapter II of [90]. The geodesic distance is given by

$$\begin{aligned} \sigma =\frac{1}{2}u_{1}^{2}g_{\mu '\nu '}U^{\mu '}U^{\nu '}, \end{aligned}$$

(105)

where $u_{1}$ is the affine parameter of the geodesic at $x'$, $U^{\mu '}=\frac{dx}{du}$ is the tangent vector to the geodesic and $g_{\mu '\nu '}$ is the metric of space-time. Now we use that

$$\begin{aligned} u_{1}U^{\mu '}=\xi ^{\mu }+\frac{1}{2}\Gamma ^{\mu '}_{\alpha '\beta '}\xi ^{\alpha }\xi ^{\beta }, \end{aligned}$$

(106)

where $\xi ^{\mu }=x^{\mu }-x^{\mu '}$ and $\Gamma ^{\mu '}_{\alpha '\beta '}$ are the usual Christoffel symbols. By using Eq. (106) in Eq. (105) we obtain

$$\begin{aligned} \sigma =\frac{1}{2}g_{\mu '\nu '}\left( \xi ^{\mu }+\frac{1}{2} \Gamma ^{\mu '}_{\alpha '\beta '}\xi ^{\alpha }\xi ^{\beta }\right) \left( \xi ^{\nu }+\frac{1}{2}\Gamma ^{\nu '}_{\gamma '\eta '}\xi ^{\gamma }\xi ^{\eta }\right) . \end{aligned}$$

(107)

Up to cubic order in $\xi ^{\mu }$ we obtain

$$\begin{aligned} \sigma =\frac{1}{2}\left( g_{\mu '\nu '}\xi ^{\mu }\xi ^{\nu }+\frac{1}{2}g_{\mu '\nu '}\Gamma ^{\mu '}_{\alpha '\beta '}\xi ^{\alpha }\xi ^{\beta }\xi ^{\nu }+\frac{1}{2}g_{\mu '\nu '}\Gamma ^{\nu '}_{\gamma '\eta '}\xi ^{\mu }\xi ^{\gamma }\xi ^{\eta }+ ...\right) . \end{aligned}$$

(108)

Now we find that the only Christoffel symbols different from zero are²⁵

$$\begin{aligned} \Gamma ^{x}_{xx}=x^{+}\Omega \Lambda ^{2},\quad \Gamma ^{x^{+}}_{x^{+}x^{+}}=x\Omega \Lambda ^{2}, \end{aligned}$$

(109)

where $\Omega =\frac{1}{\frac{M}{\Lambda }-xx^{+}\Lambda ^{2}}$ and $g_{xx^{+}}=g_{x^{+}x}=-\frac{1}{2}\Omega$. Then the geodesic distance is given by

$$\begin{aligned} \sigma= & {} \frac{1}{2}\left( 2g_{x'x'^{+}}\xi ^{x}\xi ^{x^{+}}+\frac{1}{2}g_{x'x'^{+}}\Gamma ^{x}_{xx}\xi ^{x}\xi ^{x}\xi ^{x^{+}}+\frac{1}{2}g_{x'x'^{+}}\Gamma ^{x'^{+}}_{x'^{+}x'^{+}}\xi ^{x^{+}}\xi ^{x^{+}}\xi ^{x}\right. \nonumber \\&+ \left. \frac{1}{2}g_{x'x'^{+}}\Gamma ^{x'^{+}}_{x'^{+}x'^{+}}\xi ^{x}\xi ^{x^{+}}\xi ^{x^{+}}+\frac{1}{2}g_{x'^{+}x'}\Gamma ^{x'}_{x'x'}\xi ^{x^{+}}\xi ^{x}\xi ^{x}\right) . \end{aligned}$$

(110)

This simplifies to

$$\begin{aligned} \sigma =g_{x'x'^{+}}\xi ^{x}\xi ^{x^{+}}\left( 1+\frac{1}{2}\Gamma ^{x'}_{x'x'} \xi ^{x}+\frac{1}{2}\Gamma ^{x'^{+}}_{x'^{+}x'^{+}}\xi ^{x^{+}}\right) . \end{aligned}$$

(111)

Using Eq. (109) and the expression for the metric in Eq. (111) we finally obtain

$$\begin{aligned} \sigma =-\frac{1}{2}\Omega \Delta x\Delta x^{+}\left( 1-\frac{\Omega \Lambda ^{2}}{2}(x^{+}\Delta x+x\Delta x^{+})\right) , \end{aligned}$$

(112)

where $\Delta x=x'-x$ and $\Delta x^{+}=x'^{+}-x^{+}$. This geodesic distance is clearly an approximation, but for our purposes is seems to be enough.

1.3 The non-Hadamard behavior

In order to check the Hadamard property we substitute Eq. (112) in Eq. (104) and subtract the result from Eq. (102). We obtain the following expression

$$\begin{aligned} F(x, x')= & {} - \frac{1}{8\pi } \log (xx') - \log \left[ -\frac{1}{2}\Omega \Delta x^{+}\left( 1-\frac{\Omega \Lambda ^{2}}{2}(x^{+}\Delta x+x\Delta x^{+})\right) \right] \nonumber \\&+ \displaystyle \sum _{n,j} \frac{F_{nj}}{2\pi \epsilon h(x) h(x')} \left( \gamma \left[ \frac{1}{2},(j+1)\epsilon \right] - \gamma \left[ \frac{1}{2},j\epsilon \right] \right) ^2. \end{aligned}$$

(113)

Note that with the above expression for the renormalized two-point function we have a well defined coincidence limit with x and $x'$. However while approaching the horizon $x= x^- + \Delta =0$, there appears another divergence of the form $\log x$ which is not of the Hadamard form. Therefore clearly the state $|F\rangle ^{ext}$ is not Hadamard. Similarly one can prove that the particle state $|F\rangle ^{int}$ is also non-Hadamard. This obviously implies that the states $|F\rangle ^{int}\otimes |F\rangle ^{ext}$ of the joint Hilbert space is non-Hadamard.

Appendix 2: Proof of well defined foliation

The intersecting curves Eq. (65) and Eq. (66) are very important since they determine the foliation of the space-time. One might be concerned with the possibility that the foliating surfaces may cross each other at some region of the space-time. If this happens then one fails to associate a well defined evolution of the quantum states. Here we give a proof that such crossings among foliating surfaces does not take place.

The most general situation where this crossing can happen is shown on the left in Fig. 6. From the figure we can see that such crossings can take place both in regions II and III. Let us first consider the situation in region III. A zoomed version highlighting this aspect is shown on the right of Fig. 6. In the figure $T_{r_1} (r_1, X)$ and $T_{r_2}(r_2, X)$ are two $r=const.$ curves and the points A and B are on curves $T_{r_1}$ and $T_{r_2}$ respectively. The slope of the line joining A and B is given by

$$\begin{aligned} m_{BA} = \frac{T_{A} - T_{C}}{X_C - X _B}. \end{aligned}$$

(114)

This can be easily expressed in terms of the slopes of the line joining the points E with A and E with C such that

$$\begin{aligned} m_{BA} = m_{EA} - m_{EC}. \end{aligned}$$

(115)

Now making use of the mean-value theorem and $\Delta X = X_C - X _B \rightarrow 0$ we can associate the slope $m_{BA}$ with the slope of the guiding curve (generated by all intersecting points) and $m_{EA}$ with the curve $T_{r_1}$. Then Eq. (115) clearly tells us that the slope of the guiding curve should be less than the slope of $r=const.$ curve. But this is clearly in contradiction with our choice Eq. (65). Actually, it is trivial to verify that the slope of Eq. (65) is always greater than the slope of Eq. (64) provided $r<r_h$. With this we rule out the crossing behavior showing in Fig. 6. Similarly, for region-II, in order to have a crossing it is necessary that the slope of the guiding curve is less that the $t= const.$ curve. However, by comparing the slopes of Eq. (66) and Eq. (63) one can conclude that this situation does not appear given the condition $X>T$ in region II.

Fig. 6

Possible crossing between hyper-surfaces due to a bad choice of foliation. We show this situation does not take place in our construction. For details see the text

Appendix 3: Useful integrals to define $\zeta$

Here we provide the explicit results of various integrals defined in Sect. 6. These integrals measure the distance of a point in a particular foliating hypersurface. This distance is the parameter $\zeta$ Eq. (67) used together with $\tau$ for locating a point in CGHS space-time.

$$\begin{aligned} \zeta _{A}= & {} \left( \frac{A}{M/\Lambda - \Lambda ^2 A}\right) ^{1/2} \ln \left| \frac{c-e}{f}\right| \\ A= & {} \frac{e^{2\Lambda r (c,e)}}{\Lambda ^2},\\ \zeta _{B}= & {} \frac{1}{\Lambda } \log \left( -2 c \Lambda \right. \\&+\left. 2 \sqrt{\frac{-2 c \Delta \Lambda ^3-2 c \Lambda ^3 x+M+\Lambda ^3 x^2+\Delta \Lambda ^3 x}{\Lambda }}+\Delta \Lambda +2 \Lambda x\right) _{c-e-\Delta }^{c-d-\Delta } \\ \zeta _{C}= & {} \frac{1}{\Lambda }\log \left( \Delta \Lambda +2 \Lambda x \right. \\&+ \left. 2 \sqrt{\frac{-a M+a \Lambda ^3 x^2+a \Delta \Lambda ^3 x+b M+b \Lambda ^3 x^2+b \Delta \Lambda ^3 x}{\Lambda (a+b)}}\right) _{c-d-\Delta }^{a-b}. \end{aligned}$$

Appendix 4: The SSC formalism and the sudden collapse of the quantum state

When one considers a description of a situation requiring a quantum treatment of matter fields and at the same time requires a classical picture of space-time, allowing the consideration of questions such as those that usually as arise in the context of the BHIP; one is in the realm of semi-classical gravity. As discussed in this can not considered as a truly fundamental characterization of the physical situation, but only as an effective description. Nevertheless it is convenient among other reasons for the sake of conceptual clarity to have a well defined framework, where various issues can be studied in a self consistent manner. This can be thought as analogous to, say, the Navier Stokes equation in hydrodynamics, which even though can not be considered as a fundamental description of a fluid, and which is known not to be valid under various kind of circumstances, leads to an internally self consistent description, that can be the subject of rigorous mathematical analysis. A scheme which is hoped to offer a similar characterization has been developed for semi-classical gravity, and it is defied as follows.

Definition

The set $\left\{ g_{\mu \nu }(x),\hat{\varphi }(x),\hat{\pi }(x),{\mathscr {H}},\vert \xi \rangle \ in {\mathscr {H}}\right\}$ represents a Semiclassical Self-Consistent Configuration SSC if and only if $\hat{\varphi }(x)$, $\hat{\pi }(x)$ and ${\mathscr {H}}$ correspond to a quantum field theory constructed over a space-time with metric $g_{\mu \nu }(x)$ (as described in, say [69]), and the state $\vert \xi \rangle$ in ${\mathscr {H}}$ is such that

$$\begin{aligned} G_{\mu \nu }[g(x)]=8\pi G\langle \xi \vert \hat{T}_{\mu \nu }[g(x),\hat{\varphi }(x),\hat{\pi }(x)]\vert \xi \rangle \end{aligned}$$

(116)

for all the points x in the space-time manifold.

The point however is that such rigid definition does not allow for the incorporation of something like a collapse of the quantum state of the matter fields. However we can extend the scheme by incorporating a transition from one SSC to a another associated to a sudden change in the state of the system, as an analog of the matching used in the treatment of thin shells in general relativity developed in [93]. That is, just as in the later case, one matches two space-times across a time-like boundary representing the thin matter shell, in the situation at hand, one matches two space-times across a space-like hypesurface taken to represent the excitation of the fundamental underlying DOF of quantum gravity that must occur in association with what we call the collapse of the quantum state of the matter fields.

The basic idea is then that a collapse is associated not just with the sudden transition from one state $\vert \xi _1\rangle$ in ${\mathscr {H}}$ to another $\vert \xi _2\rangle$ in ${\mathscr {H}}$ but with a the transition of a complete SSC to another $SSC_1 \rightarrow SSC_2$, which not only involves a jump in the state but also the jump in the space-time metric and even in the Hilbert space. The analysis performed in [86] indicates that the matching of space-times might be done while requiring continuity of the induced metric across the collapse hypersurfae, but allowing for a discountinuity of the extrinsic curvature.

The above scheme is clearly designed to deal with a single spontaneous collapse of the quantum state, but it seems in principle extendable to deal with proposals involving continuous reduction processes such as those considered in CSL.

It is clear however that further research on the details of these formalism as applied to theories like CSL is required.