# Virtual Black Holes and Space–Time Structure

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## Abstract

In the standard formalism of quantum gravity, black holes appear to form statistical distributions of quantum states. Now, however, we can present a theory that yields pure quantum states. It shows how particles entering a black hole can generate firewalls, which however can be removed, replacing them by the ‘footprints’ they produce in the out-going particles. This procedure can preserve the quantum information stored inside and around the black hole. We then focus on a subtle but unavoidable modification of the topology of the Schwarzschild metric: antipodal identification of points on the horizon. If it is true that vacuum fluctuations include virtual black holes, then the structure of space-time is radically different from what is usually thought.

## Keywords

Microstates Black hole unitarity Firewalls Gravitational backreaction Antipodal identification Virtual black holes Vacuole## 1 Introduction

A theory is needed that blends black holes with other, ordinary forms of matter. Among other things, this requires a treatment that explains what happens to the quantum information that appears to be absorbed by black holes [1, 2, 3]. To do this well, we need a description of black holes in terms of pure quantum states [4, 5, 6, 7, 8]—as opposed to thermodynamical objects. Thermodynamical objects are described as density matrix states, and in particular in the black hole case [9, 10], they lack time reversal symmetry.

Here we show how quantum pureness, as well as time reversal symmetry, are to be naturally restored. Neither General Relativity, nor quantum mechanics, need any essential modifications; all that is needed is an analysis that is slightly more accurate than usual. The fact that a modification is needed in the topology of the spacetime structure of the Schwarzschild solution^{1}, may be seen as an almost inevitable consequence of the unitarity requirement; it was simply overlooked in the earlier treatments of black holes, while, curious as this topology may seem to be at first sight, it is just totally natural.

*via*a Bogolyubov transformation [15, 16, 17]. Hence also the vacuum states (defined as the states where all annihilation operators vanish) do not coincide for these different observers. One finds that the vacuum state \(|\Omega \rangle \) for the local observer near the horizon, turns into the

*Hartle-Hawking state*\(|HH\rangle \) for the distant observer:

The particles are assumed to reside in regions \(I\) and \(II\), but of course are expected to travel on to either region \(III\) or region \(IV\). Now, in a black hole just formed by a collapse, the particles originally in region \(II\) are seen in region \(III\) instead, which is thought to describe the inside of the black hole. Since such particles are invisible for observers in region \(I\), thought to be the outside universe, it seemed appropriate to average over their energies \(E\) and quantum numbers \(n\). The probability of an observation of a particle with energy \(E\) and quantum numbers \(n\) in region \(I\) was therefore expected to be a mixture of the states \(|E,n\rangle \) with Boltzmann factors \(e^{-\beta E}\). This is a thermal, mixed, state. As one also sees in statistical systems, the entire state is a pure, but *entangled* state.

For the distant observer, the energies \(E\) are conserved, and hence they stay small. For a local observer, however, the quantities \(E\) are Lorentz boost eigen values; the energies of the particles are not invariant under Lorentz boosts, but they are sent to ± infinity very rapidly, as time proceeds for the distant observer.

Since we define energy with the positive sign both in regions \(I\) and \(II\), while time runs backwards in region \(II\), we see that the total eigenvalues for these Lorentz boosts are always \(E-E=0\), which was to be expected: the vacuum for the local observer is invariant under Lorentz boosts there.

## 2 Hard and Soft Particles

As soon as the local observer considers states other than the Hartle-Hawking state of Eq. (1.1), he will have particles there whose energies rapidly go to infinity as distant time proceeds. It is inevitable, therefore, that particles in the eyes of local observers become *hard* particles. A hard particle is here defined to be a particle whose mass and/or kinetic energy has become larger that \(M_{\mathrm {Planck}}\), so that it acts as a non negligible source of a gravitational field. Indeed, since these energies go to infinity so rapidly, one certainly cannot allow to neglect these gravitational forces.

The hard particles in question will always line up with either the future or the past event horizon (see Fig. 1), and if, after a few moments, we get very many hard particles there, they will form an impenetrable curtain, or *firewall*. These firewalls appear to partly invalidate Hawking’s original argument for the emergence of the Hartle-Hawking state (1.1), as was noted by Almheiri et al. [11]

Particles whose energies, *in a given Lorentz frame*, are small compared to \(M_{\mathrm {Planck}}\), will have weak gravitational fields, and thus will be called *soft* particles. There is no need to neglect their gravitational fields entirely; it suffices to state that, here, a perturbative treatment of the gravitational forces suffices.

During its entire history, a black hole has in-going matter (including the original implosion event out of which it is borne), as well as out-going matter, consisting of Hawking radiation as well as the residues of its final explosion. All of these we wish to represent in terms of *pure quantum states*. As they deviate from the Hartle-Hawking state, we must expect them to form firewalls both on the future and the past event horizon. Note that we keep our discussion as much as possible symmetric under time inversion. Indeed, the quantum theory is expected to be entirely \(CPT\) symmetric.

- (
*i*) -
They would invalidate Hawking’s original derivation of the Hartle-Hawking state, Eq. (1.1), since it assumes that the local observer sees a vacuum, not a firewall. The past firewall might be overcome, since it merely represents the imploding matter, which could be taken into account, but the firewall along the future event horizon, representing the vary late Hawking particles, selected to be in a quantum state different from the late \(HH\) state by the late detections, deviates too much from the vacuum state that was assumed.

- (
*ii*) -
The firewalls represent a strictly infinite number of quantum states, adding particles from Lorentz boosts to the distant past and/or distant future; these are much more states, it seems, than the ones needed to accommodate for the expected Hawking entropy. Indeed, the firewalls would represent a black hole information problem that must be addressed.

- (
*iii*) -
It will be hard to treat the firewalls in a \(CPT\) invariant formalism.

*remove the firewall*. We do not see this yet, but we note two things:

- (
*a*) -
It is perhaps reasonable to suspect that the entire set of pure quantum states of a black hole may be represented by allowing only soft particles in its environment, and

- (
*b*) -
in the real physical world, we never encounter hard particles at all. The most energetic cosmic rays observed ever are still significantly less energetic than the Planck energy.

*justified*by our findings below. So let us start from here.

## 3 The Gravitational Back Reaction

## 4 Particles and Footprints: The Firewall Transformation

Since the gravitational forces acting between soft particles are weak, one may apply standard quantum field theory and perturbative gravity to follow the behaviour of fields and particles throughout this Penrose diagram. As long as our time intervals \(\tau =t/4M\) are of order 1, we can still follow the evolution quite precisely. However, the particles are effectively Lorentz boosted, so we cannot follow the evolution much longer. Sooner or later, some of the particles will cross the borderline between soft and hard. As soon as we have a hard particle, we have to calculate its effect on the other (soft) particles by applying Eq. (3.1), adapted to the fact that we are not seaming flat space-times together, but parts of Schwarzschild space-time.

*change*\(\delta p^-\) in the in-going momentum \(p^-(\Omega )\) and the

*change*\(\delta u^-\) it brings about in the positions \(u^-(\Omega ')\) of the out-going Hawking particles. From here, it is only a small step to postulate that our system started with both \(u^-(\Omega ')\) and \(p^-(\Omega )\) being zero. We then get:

In what follows, we consider the positions \(u^-(\Omega ')\) of the particles going out (the out-particles) as being the *footprints* of the particles going in (the in-particles). Note that, as soon as \(p^-(\Omega )\) exceeds the Planck energy, the positions \(u^-(\Omega ')\) will become large; hence their momenta become small: *The footprints left by the hard particles are themselves soft particles*, and *vice versa*. Thus, what we really have to do is disentangle the hard components of the in- and out-particles from the soft components. This way, we end up with a space-time that contains soft particles only. We have *cis-Planckian* and *trans-Planckian* duality!

Note that the “footprints” were identified as actually being the out-particles. If we would have kept the in-particles as well as their footprints, the out-particles, we would have made a mistake by counting every particle twice. Thus, in-particles that became too hard while entering the horizon, are simply being removed and replaced by the soft ones. This way we verified a posteriori that our initial assumption is verified: all hard particles can be removed, a procedure one could characterise as the *firewall transformation*.

This is a Fourier transformation on the wave functions.

A particle may be replaced by its footprint: particles entering through the future event horizon, leave their footprints on the past event horizon. With the Green function\(f\),the momentum of the in-particle is transformed to the position operators of the out particles.

## 5 Expansion in Spherical Harmonics

Most importantly, the equations decouple entirely; at every value for \(\ell \) and \(m\), we have separate equations for just two operators \(u^\pm \) and two operators \(p^\pm \).

Our next problem is: how exactly should we physically interpret the existence of *two* regions, \(I\) and \(II\) ?

## 6 Regions \(I\) and \(II\)

*I*and

*II*are indicated in Fig. 4. If \(u^+>0\), the particle is in region \(I\), if \(u^+<0\), it is in region \(II\).

It is now very important to realise that, if the in-particle were entirely in region \(I\), so that \(\psi (u^+)=0\) when \(u^+<0\) , then the Fourier variable \(p^-\) must be non-vanishing both when \(p^->0\) and \(p^-<0\). Therefore, the footprint of a particle in region \(I\) necessarily lives both in region \(I\) and in region \(II\). *We cannot keep regions* \(I\) *and* \(II\) *separate;* the wave functions in \(I\) and \(II\) are necessarily connected.

Our equations are to be interpreted as a boundary condition at the origin, where in-particles are replaced by out-particles, their footprints. Now the Fourier transformation is unitary; it preserves the norm of the states, but only if we combine the wave functions in regions \(I\) and \(II\). Therefore, region \(II\) will be absolutely essential for obtaining a unitary evolution law for the black hole.

If region \(II\) were to represent a *different* black hole, then the unitary evolution would directly connect these two black holes. When one black hole would be considered separated from the other, unitarity would fail [14].

## 7 The Basic, Explicit, Calculation

As explained above, we have a simple, unitary evolution law; however, it only works if all in- and out-particles can be identified exclusively by their momentum distributions, \(p_{\ell m}^\mp \). This will be assumed to be the case, for the time being. Thanks to the spherical harmonics expansion, all calculations can be done explicitly. From here on, we omit the subscripts \(\ell ,\,m\), since the different \(\ell \) and \(m\) values do not mix.^{2}

The bounce guarantees that soft particles never become hard, both in the far future and in the far past. Thus, we obtain the complete set of (pure) quantum states of the black hole.

As for the range of allowed \(\ell \) values, there are still some things to be sorted out. In practice, it seems, that the total number of \((\ell ,\,m)\) partial waves that is to be included tends to coincide with the total number of Hawking particles emitted during the black hole lifetime.

An other remark is that, after leaving their footprints in the set of out-particles, an in-particle may be seen to continue its ways in regions \(III\) and/or \(IV\). One might be worried that this would violate the no-quantum-cloning principle. Our best answer to that is that regions \(III\) and \(IV\), in all respects, appear to represent ‘time beyond \(\pm \,\)infinity’. Thus, these particles do not over count the quantum states, but merely extend the time line to beyond infinity, without causing any harm to any of the known physical principles.

## 8 The Antipodal Identification

In the above calculations, we have not yet explained the physical interpretation of the quantum states in region \(II\). Up to this point, we treated region \(II\) as a universe that is exactly as real as the one described in region \(I\). Since we projected all pure quantum states of the black hole as soft excitations of the vacuum in the metric described by the Penrose diagram of the eternal black hole, we cannot afford to discard region \(II\). Indeed, the matrix \(F\) in Eq. (7.11) is unitary only if we keep the components that map states from region \(I\) into those of region \(II\) and *vice versa* (the off-diagonal components of the matrix \(F\)).

The option that region \(II\) would describe a different black hole in some other universe, or at least far from the original black hole, is ruled out [14]. The only option we have is to postulate that region \(II\) refers to the same black hole as region \(I\). This, however, would lead to cusp singularities if in-particles would generate out-particles at the same spot of the horizon. A local observer would spot this as a singularity, which is against the philosophy of Einstein’s theory of General Relativity.

Figure 5 illustrates the effects of this antipodal identification. Space-time is divided in half, the two sides are identified. Thus, if we follow a trajectory in space-time (not necessarily a geodesic), we can travel from region \(I\) to region \(II\), which leads us to the opposite side of the same black hole. The effect of this is that, while the original space-time had every point \((r,\,t,\,\Omega )\) mapped into two spacetime points in the regular coordinate system, \((\pm \,x,\,\pm \,y,\,\Omega )\), this is now again reduced to one space-time point.

Most importantly, there should be no points that have other asymptotic regions than the regular ones; presumably one can also demand that all true space-time singularities are to be screened off by some cosmic censorship condition. This is certainly the case in our description of the Schwarzschild metric.

All space-time metrics describing objects such as black holes, must have a single Minkowski space-time in their asymptotic region, such that all points of space-time can be connected to this asymptotic space-time by time-like geodesics.

While entering a black hole, a particle might continue its way in region \(III\) or \(IV\), but its footprint lives on outside the horizon. There is no moment in time that the particle, or its footprint, are at two places at the same time, which is forbidden by the no-cloning condition of quantum mechanics. The particles spend a brief time in regions \(III\) or \(IV\) as enjoying their “after life”, beyond time \(= \pm \,\)infinity.

It is also important to note that, with the antipodal identification, particles emerging at opposite sides of the black hole, will be strongly entangled [14] which also implies a strong deviation from purely thermal behaviour [26].

A remarkable consequence of the antipodal identification is the fact that the variables \(u^\pm \) and \(p^\pm \) all switch signs when followed from region \(I\) to region \(II\), just as what happens in the spherical harmonics with odd \(\ell \). Therefore, in our spherical harmonic expansion, only odd values of \(\ell \) are allowed. As usual, \(m\) can have any integer values between \(-\,\ell \) and \(\ell \).

Also, as we have seen, *time* switches its sign when passing from region \(I\) to region \(II\). In fact, the entire topology of space-time can be described by excising a 3-sphere out of Minkowski space-time and gluing the 4-dimensional antipodes together, see Fig. 6.

## 9 A Time–Like Möbius Strip

## Footnotes

- 1.
In more general black hole configurations, the Reissner Nordström, Kerr, and Kerr Newmann solutions, the topology will have to be adapted in exactly the same way.

- 2.
Some researchers point out that, due to non-linear effects, they do expect mixing; however, such effects would be very small, in particular for sufficiently small \(\ell \) values. Only when \(\ell \) approaches its limiting value, close to \(M\) in Planck units, one might expect difficulties due to transverse gravitational forces, but even here, we expect these to be small and manageable.

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