1 Hawking evaporation↩︎

Over 50 years ago, calculated that Black Holes were not black, but rather emitted radiation with a thermal spectrum whose temperature was proportional to the inverse mass of the Black Hole By using the thermodynamic relation: \[\begin{align} TdS=dE, \end{align}\]

What is this entropy? The usual argument is that this is an entropy possessed by the Black Hole, and as the Black Hole evaporated, this entropy is depleted by the radiation emitted by the Black Hole. The model is either the Planck black-box radiation, where the entropy is essentially that of the oscillators in the inside walls of the oscillator, which, the dipole radiation produced by these oscillators, produces the entropy Electromagnetic radiation. Or, equivalently, it can be modelled as the entropy, like that of of a lump of coal, heated up by some incoming pure-state radiation, which, by non-linear interactions of the constituents of the coal, creates entropy then slowly emits that entropy (and energy). But this analogy leads to the question of where that entropy is stored in the Black Hole. Some ideas that the entropy is related of the surface of the Black Hole represented by the horizon, and that each square Planck area somehow stores unit of entropy. the entropy is contained inside the horizon of the Black Hole and tunnels [1] through the horizon inside and outside are causally disconnected regions). quantum gravity causes the horizon to shake producing the radiation. Or, in string theory[2], the horizon is represented by some sort of null d-brane with the strings penetrating the horizon represent the degrees of freedom which are excited as vacuum fluctuation and represent the entropy of the black holes. All have problems. Since the right answer is known (the Hawking entropy), can be answered by one means or the other. An example of such a problem is that, since each field radiates with the above thermal spectrum, the entropy emitted will be proportional to the log of the number of quantum fields in one’s theory, which could rapidly far exceed the Hawking entropy. \(G\), Newton’s constant, could be renormalized in quantum gravity to logarithmic divergence. However, all have the problem that one seems to be getting more out of the theory than one should, .

2 Analogies↩︎

In my career I have come across a number of similar situations. One was the Bohr-Einstein weighing-of-energy debate where Bohr’s answer seemed to demand that General Relativity must be true if quantum mechanics is to be consistent. G. Opat and I[3] showed that one only needed Einstein’s assumption (namely that energy has weight) to show the consistency of quantum mechanics.

Geroch argued that Bekenstein’s identification of entropy with the area of a Black Hole must be wrong. heat engine which extracted energy by lowering a box of entropy toward the black hole Bekenstein by postulating a new law of physics, that there exists an upper bound of the entropy to energy ratio of the contents of box, which would save the Second law. Wald and I[4] showed that Archimedes’ buoyancy principle

should not in general need to introduce new laws of physics to save apparent paradoxes in physics.

Hawking’s assumption, in his calculation, was that gravity was defined by a smooth spacetime, and certainly not a quantum spacetime. It also said nothing about the statistical nature of the entropy. The argument for the entropy arose solely from thermodynamic arguments. The emission of a thermal stream was a direct consequence of linear quantum field theory in a Black Hole spacetime. The emitted entropy should not require any additional assumptions about the nature of quantum gravity or about the Black Hole containing entropy.

3 Amplifier model↩︎

A number of years ago I presented a model of a amplifier[5]. (Some of these ideas were also considered earlier[6]). Let me summarize . Consider two quantum fields, \(\phi\) and \(\psi\), and harmonic oscillator with dynamic variables \(p,~ q\). let me operate in 1+1 dimensional spacetime , with with boundary condition on the fields that their is zero. The fields come in from , interact with the oscillator at \(x=0\) and are immediately reflected back toward Furthermore, \(\mu^2<\epsilon^2\), which keeps the system stable.

Note that the \(\psi\) field has a negative Hamiltonian. This could be realised if there is a state of the field with a maximum value of the energy (e.g., a collection of spin 1/2 particles with all of the particles in their maximum energy state.) The “vacuum" state of the \(\psi\) field is an energy maximum, rather than a minimum. In a physical system, the \(\psi\) system would be non-linear for large deviations from that”vacuum" but I am assuming that the deviations from the “vacuum" are small..

where 1 and 2 represent any two solutions of the equations of motion, and \(\sigma\) represent the collection of \(\phi,~\pi_\phi,~\psi,~\pi_\psi,~q,~p\) for each solution of the equations. (Note that \(\pi_{\psi}=-\partial_t\psi\) because of the negativity of its Hamiltonian/Lagrangian). Modes \(\sigma\) which have positive norms are associated with annihilation operators and the negative norms are associated with creation operators for the fields. This symplectic norm is just the generalization of the Klein-Gordon norm for a single scalar field. For the \(\phi\) field the positive norms will be associated with time dependence of \(e^{-i\omega t}\) with \(\omega>0\), while for the \(\psi\) field positive norm corresponds to \(\omega<0\)

If we take modes such that their incoming parts (from \(x=-\infty\)) are of the form then two sets of normalized solutions for \(\omega>0\) and for \(x\) far to the left, are \[\begin{align} \phi_{0\omega}={1\over\sqrt{2\pi|\omega|}}; \psi_{0\omega}=0; ~~~[{\mathfrak A}]\\ \psi_{0\omega}={1\over\sqrt{2\pi|\omega|}}; \phi_{\omega}=0; ~~~[{\mathfrak B}] \end{align}\] The first, \(\mathfrak A\), have unit positive norm (with the continuum normalization), while the second, \(\mathfrak B\), are unit negative norm solutions. Solutions for the opposite norms will be the complex conjugate of these. However both \({\mathfrak A}\) and \(\mathfrak B\) have the same value of \(\omega>0\).

Installing a mirror with Neumann boundary conditions just to the right of the oscillator at , the solutions for each field are \[\begin{align} \phi_{\omega}(t,x)&=& \lim_{\lambda=0^+}\phi_{0\omega}(e^{-i\omega (t-x)})+e^{i\omega(t+x-2\lambda)}) \nonumber \\ &&+{1\over 2}\epsilon q_\omega (e^{-i\omega(t+x)}+e^{-i\omega(t+x-2\lambda)} \\ \psi_{\omega}(t,x)&=& \lim_{\lambda=0^+}\psi_{0\omega}(e^{-i\omega (t-x)})+e^{i\omega(t+x-2\lambda)}) \nonumber\\ &&-{1\over 2}\mu q_{\omega} e^{-i\omega(t+x)}+e^{-i\omega(t+x-2\lambda)} \end{align}\] where \(\lambda=0^+\) means were are taking the limit as \(\lambda\) approaches 0 from positive values of \(\lambda\).

The \(q\) equation at \(x=0\) is given by \[\begin{align} \left[\omega^2 q_{\omega}- i\omega[(\epsilon(2\phi_{0\omega} +\epsilon q_\omega+\mu(2\psi_{0\omega}-\mu q_{\omega})]e^{-i\omega t)}\right] \end{align}\] from which we get \[\begin{align} q_\omega={(2i(\epsilon\phi_0+\mu\psi_0)\over(i\epsilon^2-i\mu^2+\omega)} \end{align}\] and with \(\phi_{{\rm out}\omega}\) being the portion that goes as \(e^{-\omega x}\) \[\begin{align} \phi_{{\rm out}\omega} &=& \\ &&-e^{-i\omega(t+x)}{(i\epsilon^2+i\mu^2-\omega)\phi_0+2i\epsilon\mu\psi_0\over(-i\epsilon^2+i\mu^2-\omega)}\\ \end{align}\] where “out" designates the part going as \(e^{-i\omega (t+x)}\)

To quantize the this, we choose incoming mode, it is associated with creation operators. Also defining operators \({\boldsymbol{c}},~{\boldsymbol{d}}^\dagger\) for the output fields \(\Phi_{{\rm out}\omega}\) in a similar way \[\begin{align} {\boldsymbol{c}}_\omega=\cosh(r(\omega)){\boldsymbol{a}}_\omega+\sinh(r(\omega)) {\boldsymbol{b}}^\dagger_\omega\\ i{\boldsymbol{d}}^\dagger=\cosh(r(\omega)){\boldsymbol{b}}^\dagger_\omega +\sinh(r(\omega)){\boldsymbol{a}}_\omega \end{align}\] Since \(\cosh(r)>1\), we find that the amplitude output \(\phi\) field is greater than the input \(\phi\) field

However, as a quantum system, we note that the outgoing annihilation operators of \(\phi\) field are a linear combination of the ingoing \(\phi\) annihilation operator and the ingoing \(\psi\) creation operator. This is a two-mode squeezed state.

\[\begin{align} \cosh(r(\omega))&=&\sqrt{\omega^2+(\epsilon^2+\mu^2)^2\over \omega^2+(\epsilon^2-\mu^2)^2}\\ \sinh(r(\omega))&=&\sqrt{4\epsilon^2\mu^2\over \omega^2+(\epsilon^2-\mu^2)^2} \end{align}\]

Writing \(\vert 0 \rangle_{in}\) in terms of the number states for the out and the initial state is the vacuum state \({\boldsymbol{a}}_{\omega in}\vert 0 \rangle={\boldsymbol{b}}_{\omega in}\vert 0 \rangle=0\) then the outgoing vacuum state is \({\boldsymbol{c}}_{\omega out}\vert 0 \rangle_{out}={\boldsymbol{d}}_{\omega out}\vert 0 \rangle=0\) is

If we look only at the outgoing \(\phi_0\) field, thus tracing out over the \(\psi_0\) field. we get the density matrix for the \(\phi\) operators \[\begin{align} \rho_{\phi_{out}}={1\over\cosh(r)^2}\sum_n e^{n ln(\tanh(r)^2}\vert n \rangle_c\langle n \vert_c \end{align}\] where For every mode this is exactly a thermal density matrix with temperature of (i.e., the emission is not thermal across the modes.) The entropy of each mode is then \[\begin{align} S_\omega=- \int \sum_n \tanh(r)^{2n}n \ln\left(tanh(r)^2\right) \end{align}\] Since \(|\tanh(r)|<1\), the sum is negative giving a positive entropy emitted by the amplifier per unit time.

This incoming mode is linearly separated into positive and negative norm states ( and thus the description by a non-trivial Bogoliubov transformation). It is this model, not that of of a hot lump of coal, which best describes the entropy emission from a Black Hole as I shall argue in the next section. As we shall see in the next section, the Black Hole operates very similarly except that the temperature is not a function of the frequency. Each mode is thermal, and the temperature of each mode is the same.

4 Black Hole Entropy Emission↩︎

In the following I am going to look at a Black Hole. To simplify the analysis, I will make user of a 1+1 dimensional toy model which we recently published[7].

Consider a 1+1 dimensional Schwarzschild Black Hole metric. \[\begin{align} \label{toymetric} ds^2&=& {x-2M\over x}dt^2-{x\over x-2M}dx^2\nonumber \\ &\approx& 16M^2\left[({x\over2M}-1)^2d({t\over 4M})^2-(d{{x\over 2M}-1}))^2\right]; \nonumber \\ &&\hskip 4cm (x~ {\rm near}~0) \nonumber \\ &\approx& 16M^2\left[d({t\over 4M})^2-(d{x/4M})^2\right] ; ~~{\rm near}~\infty \end{align}\tag{1}\]

This continuous (in \(r\)) approximate metric maintains the key structure of the 2D Schwarzschild metric near the horizon and at infinity. has flat Minkowski spacetime in the vicinity of the horizon, and a different flat spacetime near infinity. It also maintains the Hawking radiation at infinity. \[\begin{align} ds^2&=&(4M)^2(\rho)^2 d\tau^2-d\rho^2 ;~~0<\rho<{ 1}\\ ds^2&=& (4M)^2 (d\tau^2-d\rho^2);~~\rho>{1} \end{align}\] where Also we have \(\tau=t/4M\). This metric has components which are continuous everywhere except at \(\rho=0\), the horizon.

This metric looks like the Schwarzschild metric with a horizon at \(\rho=0\), but near the horizon, (\(\rho<1\)) it is just a form of the Rindler metric, a coordinate transformation of a flat Minkowski spacetime. Outside \(\rho=1\), it is again just flat spacetime in the usual Minkowski coordinates, but a different flat spacetime than near the horizon. a metric which, as we shall see, has Hawking thermal radiation just as does the Schwarzschild metric. Since the metric is conformal flat, massless fields have the same field equations as they would have in flat spacetime. The metric is a glueing together of two flat spacetimes. \(\rho=1\) is a curve of constant acceleration and is a straight timelike geodesic for \(\rho= 1\) as seen in the metric for \(\rho>1.\) Reference[7] gives an embedding of this spacetime into a 3-D globally Minkowski flat spacetime.

Define

All of

The metric \(\rho<1\) and \(\rho>1\) can be written as \[\begin{align} ds^2 &=&16M^2(-dUdV)\\ &=& -16M^2 e^{v-u}dudv; ~~\rho<1 \\ &=&-16M^2 e^{v-u}d\tilde{v} d\tilde{u}{\rm~~\rho>1} \end{align}\] with equations of motion \[\begin{align} \partial_U\partial_V\Phi=\partial_u\partial_v\Phi=\partial_{\tilde{u}}\partial_{\tilde{v}}\Phi=0 \end{align}\] and with mode solutions \[\begin{align} \phi_L(U)={e^{-i\Omega U}\over\sqrt{2\pi|\Omega|}};~~~\phi_R(V)={e^{-i\Omega V}\over\sqrt{2\pi\Omega}}\\ \phi(u)=\alpha {e^{-i\omega u}\over \sqrt{2\pi\omega }}\Theta(-U) +\beta {e^{-i\omega' \tilde{u}}\over\sqrt{2\pi|\omega'|}}\Theta(-U) \end{align}\] and similarly for \(v\) and \(\tilde{v}\).

The usual Minkowski quantization chooses \(\Omega>0\) for the positive norm modes proportional to analytic in the upper half \(U\) plane only if \(\tilde{\omega}=-\omega\) and \({\beta\over \alpha}= e^{-2\pi\omega}\) for all \(\omega\), positive or negative. In the quantization using the \(\phi(u)\) modes, this means that the positive norm \(U\) modes must be two-mode squeezed states when expressed in Rindler coodinates, just as the \(\phi\) modes are for the amplifier. Since, by experiment and prejudice, the Minkowski vacuum has zero energy, this means that the this two-mode squeeze state near the horizon will have zero energy, and zero energy flux for \(0<\rho<1\). However for \(\rho>1\), the outgoing mode of frequency \(\omega\) would be in the vacuum state, only if one assumed that the \(\omega\) modes for the \(u\) dependent would have the annihilation operators associated with the \(\omega>0\) modes. After those outgoing (\(U\) dependent) modes flows through \(\rho=1\), the energy-momentum tensor will be non-zero, and one will have a flux of energy out from the Black Hole which seems to originate at the curvature delta function, just as was argued in [7].

Why would the delta function curvature change the vacuum state inside \(\rho=1\) to a thermal flux for \(\rho>1\)?

As we know[8], the state of the field as seen by an observer depends on the motion of the observer. The energy as seen by an observer[9] is determined by geodesic point splitting, in which it is the geodesics which determine the regularization of the field. Outside the horizon, it is the \(u\) and \(v\) vacua which have no particles with respect to the geodesics. On the inside, it is the U and V vacua which have no particles. But at the curvature delta function, the modes change from being the ones associated with the vacuum state for U and V to u and v, from a vaccum state to a thermal state.

This model behaves just as does the amplifier model, with the curvature playing the role of coupling between the positive and negative modes of the field in the amplifier. Here the entaglement is between the outgoing (\(U<0\) depedent) modes to the right of the horizon, and the outgoing modes inside the horizon (\(U<0\)), where we need \(\tilde{o}mega=-\omega\).

What of the entropy? Taking the entropy to be the heat flow divided by the temperature, and the heat flow in both the amplifier and Black Hole being just the energy, this means that the entropy flux will just be the regularized energy flow divided by the temperature. Thus the net flow of the entropy out of the Black Hole will just be the integral of the energy flow divided by the temperature. The total entropy is not the entropy that resides in the black hole (with all of the problems associated with where that entropy is supposed to hide). Rather it is the total entropy created because of the two-mode squeezed-state behaviour of the field in the presence of the horizon This has a number of implications. The search for where the entropy is stored in the Black Hole is not a useful endeavour. The Page curve[10], for example, is based on the wrong analogy. However, the explanation for how the Geroch heat engine works– as being due to the acceleration temperature by an accelerated observer– [11] still works in the same way as it does in the Schwarzschild metric. The accelerated box sees the Minkowski vacuum as a thermal state with the required buoyancy force.

Throughout the years as I have given lectures on analog Black Holes, one question almost always arises– can these analogues say anything about the "information paradox’, and in particular about the entropy of Black Holes? My answer was always "No", because the temperature in the fluid analog had nothing to do with the energy. While true, the analogue holes would still produce entropy by the same mechanism as Black Holes do– splitting the outputs of two-mode squeezed states, with one going out to infinity and the other flowing into the horizon of the analog horizon. The total entropy emitted will not be related to the energy emitted, because the temperature has nothing to do with the energy that has flowed out of the dumb hole, but it will still produce entropy by the same mechanism as black-holes do. By measuring the entropy or the two-mode squeezed nature of the created particles by the dumb hole, one can test the same mechanism as occurs in a Black Hole.

Also in the case of 4-D Black Holes, ( or in 2D for massive fields) the modes that come out of the horizon do not simply flow completely out to infinity as do the massless modes in 2-D. Instead the angular momentum barrier and the gravitational barrier cause the low frequency modes ( as seen some distance away from the horizon) instead get reflected back to the Black Hole. But the reflected modes carry energy back to the Black Hole. In addition the incoming vacuum state also gets reflected by the same barriers. This decreases the negative energy which they carry toward the Black Hole. Thus the mass of the Black Hole will shrink more slowly, because the mass decreases more slowly.

In conclusion, it seems that Black Holes themselves have no entropy. They simply continuously create entropy It is like a diner, which does not have a supply of pre-cooked eggs which they feed to the customers until they are all gone. It is rather like having a short order cook, who creates the cooked eggs on demand.

The analogies to amplifiers are that amplifiers, just like Black Holes emit a thermal state in each component of the field, if the input was the vacuum state. That thermal state comes about because of structure of a two-mode squeezed state, with one of the states (the \(\phi\) modes, and the outgoing modes in the Black Hole case). They differ in that the two parts of the mode, the entaglement between the \(\phi\) and \(\psi\) fields, could in principle be measured. In the case of the black hole, such a measurement is impossible because because the analogue of the negative energy \(\psi\) field is hidden behind the horizon. They also differ in that the amplifier model has a temperature which depends strongly on the frequency of the mode, while the the black-hole, the temperature is constant for all the outgoing modes, xince \(T=1\over 8\pi M\).

It is important that in both cases, just as in Hawking’s original derivation was for a linear field. There is no "mixing" of modes which would require non-linear processes to produce the entropy, as happens within a lump of coal.

The above depends critically on the idea that General Relativity is good approximation to the true theory of gravity near the horizon. If, for example, string theory is a better approximation to the reality of gravity, then the string theory arguments about the relation between the entropy of Black Holes and, for example, D-Branes explanations, may be closer to reality. In that model, perhaps Black Holes do have entropy.

5 Acknowledgements↩︎

I thank the IQSE and Hagler Institute at the Texas A&M University for support while goading me to think more deeply about the Black Hole evaporation and the relationship to quantum optics. I especially thank the M Aspelmeyer who invited me to spend some time at the IQOQI and the University of Vienna where this paper was mostly written. I thank the UBC administration who came through with support when NSERC cancelled my funding. Finally, I thank my wife, Patricia Unruh, whose support and help has been crucial during my trips and during my writing of this paper.

6 END NOTES↩︎

6.1 Entropy↩︎

The density matrix is \[\begin{align} \rho\propto\int \sum_{n_\omega}e^{n_\omega\omega} \vert n_\omega,\tilde{\omega} \rangle\langle n_\omega,8\pi M \tilde{\omega} \vert d\tilde{\omega} \end{align}\] where \(\omega=8\pi M \tilde{\omega}\) and \(\tilde{\omega}\) is the energy of mode. The sum over the probabilities \(\sum_{n_\omega} e^{n_\omega\omega} ={1\over {1-e^{-\omega}}}\) is precisely the square of the factor in front of the mode \[\begin{align} J_(t,x)_{\tilde{\omega}}&=& {i\over 4\pi|\tilde{\omega}|}\left(\phi(t-x)^*_{\tilde{\omega}}\partial_x\phi(t-x)\phi(t-x)_{\tilde{\omega}}\nonumber\right. \nonumber\\ &~&~~~~~\left. -\phi(t-x)_{\tilde{\omega}}\partial_x\phi(t-x)\phi(t-x)_{\tilde{\omega}}\right) \end{align}\] the flux of the Klein Gordon norm for the mode. Thus the flux of energy is just \[\begin{align} <{\rm J}_E>=\int\tilde{\omega} {\rm J_{\tilde{\omega}}}(t,x) d\tilde{\omega} \end{align}\] and the Entropy flux is \[\begin{align} <{\rm J}_S(t,x)>= \int 8\pi M \tilde{\omega} {\rm J_{\tilde{\omega}}}(t,x) d\tilde{\omega} =8\pi M <{\rm J}_E> \end{align}\] which, with \(T= {1\over 8\pi M}\), the Hawking temperature, is just \[\begin{align} <{\rm J}_S(t,x)>= <{\rm J}_E>/T \end{align}\]

6.2 Minkowski Rindler modes↩︎

References↩︎