The black hole information paradox and the black hole entropy are currently extensively researched. The consensus about the solution of the information paradox is not yet reached, and it is not yet clear what can we learn about quantum gravity from these and the related research. It seems that the apparently irreducible paradoxes force us to give up on at least one well-established principle or another. Since we are talking about a choice between the principle of equivalence from general relativity and some essential principles from quantum theory, both being the most reliable theories we have, it is recommended to proceed with caution and search more conservative solutions. These paradoxes are revisited here, as well as the black hole complementarity and the firewall proposals, with an emphasis on the less obvious assumptions. Some arguments from the literature are reviewed, and new counterarguments are presented. Some less considered less radical possibilities are discussed, and a conservative solution, which is more consistent with both the principle of equivalence from general relativity and the unitarity from quantum theory, is discussed.
By applying general relativity and quantum field theory on curved spacetime, Hawking arrives at the conclusion that the information is lost in the black holes, and this breaks the predictability [
Mainly for general relativists the information loss seemed to be definitive and yet not a big problem [
The dominating proposed solution was, for two decades,
The problems related to the black hole information loss are considered important, being seen as a benchmark for the candidate theories of quantum gravity, which are expected to solve these problems.
The main purpose of this discussion is to identify the main assumptions and see if it is possible to solve the problem in a less radical way. I argue that some of the usually made assumptions are unnecessary, that there are less radical possibilities, and that the black hole information problem is not a decisive test for candidate theories of quantum gravity. New counterarguments to some popular models proposed in relation to the black hole information problem are the following. Black hole complementarity is discussed in Section
To my knowledge, the above-mentioned arguments, presented in more detail in the following, are new, and in the cases when I was aware of other results seeming to point in the same direction, I gave the relevant references. While most part of the article may look like a review of the literature, it is a critical review, aiming to point out some assumptions which, in my opinion, drove us too far from the starting point, which is just the most straightforward and conservative combination of quantum field theory with the curved background of general relativity. The entire structure of arguments converges therefore towards a more conservative picture than that suggested by the more popular proposals. The counterarguments are meant to build up the willingness to consider the less radical proposal that I made, which follows naturally from my work on singularities in standard general relativity ([
Hawking’s derivation of the black hole evaporation [
The derivation, as well as the discussion surrounding black hole information, requires the framework of quantum field theory on curved spacetime [
But in the calculations of the Hawking radiation, the gravitational backreaction is ignored, being very small. To have well behaved solutions, the spacetime slicing is such that the intrinsic and extrinsic curvatures of the spacelike slices are considered small compared to the Plank length; the curvature in a neighborhood of the spacelike surface is also taken to be small. The wavelengths of particles are considered large compared to the Plank length. The energy and momentum densities are assumed small compared to the Plank density. The stress-energy tensor satisfies the positive energy conditions. The solution evolves smoothly into future slices that also satisfy these conditions.
The canonical (anti)commutation relations at distinct points of the slice are imposed. A decomposition into positive and negative frequency solutions is assumed to which the Fock construction is applied to obtain the Hilbert space. The renormalizability of the stress-energy expectation value
The Fock space construction of the Hilbert space can be made in many different ways in curved spacetime, since the decomposition into positive and negative frequency solutions depends on the choice of the slicing of spacetime into spacelike hypersurfaces.
Suppose that a basis of annihilation operators is
The Bogoliubov transformation preserves the canonical (anti)commutation relations and expresses the change of basis of the Fock space, allowing us to move from one construction to another. The Bogoliubov transformations are linear but not unitary. They are symplectic for bosons and orthogonal for fermions though. The number of particles is not preserved, so there is no invariant notion of particles.
This is in fact the reason for both the Unruh effect near a Rindler horizon and the Hawking evaporation near a black hole event horizon. Because of the nonunitarity of the Bogoliubov transformation relating the Fock space representations of two distinct observers, particles can be produced [
While Hawking’s derivation of the black hole evaporation is rigorous and the result is correct, the implication that the information is definitively lost can be challenged. In fact, most of the literature on this problem is trying to find a workaround to restore the lost information and the unitarity. The most popular proposals like black hole complementarity and firewalls do not actually dispute the calculations, but rather they add the requirement that the Hawking radiation should contain the complete information.
Additional motivation for unitarity comes from the AdS/CFT correspondence [
The favorite scenario among high-energy physicists was, for two decades, the idea of
Assuming that unitarity is to be restored by evaporation alone, the infalling information should be found in the Hawking radiation or should somehow remain above the black hole event horizon, forming the
At first sight, it may seem that the black hole complementarity solves the contradiction by allowing it to exist, as long as no experiment is able to prove it. Alice and Bob’s lightcones intersect, but none of them is included in the other, and they cannot be made so. This means that whatever slicing of spacetime they choose in their reference frames, the Hilbert space constructions they make will be different. So it would be impossible to compare quantum information from the interior of the black hole with the copy of quantum information escaping it. And it is impossible to conceive an observer able to see both copies of information—this would be the so-called
An early objection to the proposal that Alice and Bob can never compare the two copies of quantum information was that the escaping observer Bob can collect the escaping copy of the information and jump into the black hole to collect the infalling copy. This objection was rejected because, in order to collect a single bit of infalling information from the Hawking radiation, Bob should wait until the black hole loses half of its initial mass by evaporation—the time needed for this to happen is called the
The argument based on the Page time works well, but it applies only to black holes of the Schwarzschild type (more precisely this is an Oppenheimer-Snyder black hole [
This objection is relevant, because for the black hole to be of Schwarzschild type, two of the three parameters defining the black hole, the angular momentum and the electric charge, have to vanish, which is very unlikely. The things are even more complicated if we take into account the fact that, during evaporation or any additional particle falling in the black hole, the type of the black hole changes. Usually particles have nonvanishing electric charges and spin, and even if an infalling particle is electrically neutral and has the spin equal to 0, most likely it will not collide with the black hole radially. This continuous change of the type of the black hole may result in changes of type of the singularity, rendering the argument based on the Page time invalid.
In Section
Because of the principle of equivalence, Susskind’s argument should also hold for Rindler horizons in Minkowski spacetime. The equivalence implies that Bob is an accelerated observer, and Alice is an inertial observer, who crosses Bob’s Rindler horizon. Because of the Unruh effect, Bob will perceive the vacuum state as thermal radiation, while for Alice it would be just vacuum. Bob can see Alice being burned at the Rindler horizon by the thermal radiation, but Alice will experience nothing of this sort. But since they are now in the Minkowski spacetime, Bob can stop and go back to check the situation with Alice, and he will find that she did not experience the thermal bath he saw her experiencing. While we can just say that the complementarity should be applied only to black holes, to rule it out for the Rindler horizon and still maintain the idea of stretched horizon only for black holes, this would be at odds with the principle of equivalence which black hole complementarity is supposed to rescue.
The resolution proposed by black hole complementarity appeals to the fact that the Hilbert spaces constructed by Alice and Bob are distinct, which would allow quantum cloning, as long as the two copies belong to distinct Hilbert spaces and there is no observer to see the violation of the no-cloning theorem. This means that the patches of spacetime covered by Alice and Bob are distinct, such that apparently no observer can cover both of them. If there was such an “omniscient” observer, he or she would see the cloning of quantum information and see that the laws of quantum theory are violated.
Yet, there is such an observer, albeit moving backwards in time (see Figure
(a) The Penrose diagram of black hole evaporation, depicting Alice and Bob and their past lightcones. (b) The Penrose diagram of a backwards in time observer Charlie, depicting how he observes Alice and Bob, and the quantum information each of them caries, even if this information is cloned, therefore disclosing a violation of quantum theory.
One can try to rule Charlie out, on the grounds that he violates causality or more precisely the second law of thermodynamics [
After two decades since the proposal of black hole complementarity, this solution was disputed by the
The firewall argument takes place in the same settings as the black hole complementarity proposal, but this time it involves the
One can argue that if the firewall experiment is performed, it creates the firewall, and if it is not performed, Alice sees no firewall, so black hole complementarity is not completely lost. Susskind and Maldacena proposed the
Various proposals to rescue both the principle of equivalence and unitarity were made, for example, based on the entropy of entanglement across the event horizon in [
Having to give up the principle of equivalence or unitarity is a serious dilemma, so it is worth revisiting the arguments to find a way to save both.
In the literature about black hole complementarity and firewalls, by the assumption or requirement of “unitarity,” we should understand “unitarity of the Hawking radiation” or, more precisely, “unitarity of the quantum state exterior to the black hole.” Let us call this
The idea that unitarity should be restored from the Hawking radiation alone, ignoring the interior of the black hole, was reinforced by the holographic principle and the idea of stretched horizon [
In fact, considering both the exterior and the interior of the black hole is behind proposals like remnants and baby universes. But we will see later that there is a less radical option.
Exterior unitarity, or the proposal that the full information and purity are restored from Hawking radiation alone, simply removes the interior of the black hole from the reference frame of an escaping observer, consequently from his Hilbert space. This type of unitarity imposes a boundary condition to the quantum fields, which is simply the fact that there is no relevant information inside the black hole. So it is natural that, at the boundary of the support of the quantum fields, which is the black hole event horizon, quantum fields behave as if there is a firewall. This is what the various estimates revealing the existence of a highly energetic firewall or horizon singularity confirm. Note that since the boundary condition which aims to rescue the purity of the Hawking radiation is a condition about the final state, sometimes its consequences give the impression of a conspiracy, as sometimes Bousso and Hayden put it [
While I have no reason to doubt the validity of the firewall argument [
The initial Hilbert spaces of Alice and Bob are not necessarily distinct. Even if they and their Fock constructions are distinct, each state from one of the spaces may correspond to a state from the other. The reason is that a basis of annihilation operators in Alice’s frame, say
Thus, one may hope that the Hilbert spaces of Alice and Bob may be identified, even though through a very scrambled vector space isomorphism, so that black hole complementarity saves the day. However, exterior unitarity imposes that the evolved quantum fields from the Hilbert spaces have different supporting regions in spacetime. While before the creation of the black hole they may have the same support in the spacelike slice, they evolve differently because of the exterior unitarity condition. Bob’s system evolves so that his quantum fields are constrained to the exterior of the black hole, while Alice’s quantum fields include the interior too. Bob’s Hilbert space is different, because when the condition of exterior unitarity was imposed, it excluded the interior of the black hole. So even if the initial underlying vector space is the same for both the Hilbert space constructed by Alice and that constructed by Bob, their coordinate systems diverged in time, so the way they slice spacetime became different. While normally Alice’s vacuum is perceived by Bob as loaded with particles in a thermal state, this time in Bob’s frame Alice’s vacuum energy becomes singular at the horizon. This makes the firewall paradox a problem for black hole complementarity. A cleaner argument based on purity rather than monogamy is made by Bousso [
An interesting issue is that Bob can infer that if the modes he detects passed very close to the event horizon, they were redshifted. So, evolving the modes backwards in time, it must be that the particle passes close to the horizon at a very high frequency, maybe even higher than the Plank frequency. Does this mean that Alice should feel dramatically this radiation? There is the possibility that, for Alice, Bob’s high frequency modes are hidden in her vacuum state. This is also confirmed by acoustic black holes [
It seems that the strength of the firewall proposal comes from rendering black hole complementarity unable to solve the firewall paradox. They are two competing proposals, both aiming to solve the same problem. While one can logically think that proposals that take into account the interior of black holes to restore unitarity are good candidates as well and that they may have the advantage of rescuing the principle of equivalence, sometimes they are dismissed as not addressing the “real” black hole information paradox. I will say more about this in Section
The purposes of this section are to prepare for Section
The entropy bound of a black hole is proportional to the area of the event horizon [
The black hole entropy bound (
It is tempting to think that the true entropy of quantum fields in spacetime should also include the areas of the event horizons. In fact, there are computational indications that the black hole evaporation leaks the right entropy to compensate the decrease of the area of the black hole event horizon.
But there is a big difference between the entropy of quantum fields and the areas of horizons. First, entropy is associated with the state of the matter (including radiation, of course). If we look at the phase space, we see that the entropy is a property of the state alone, so it is irrelevant if the system evolves in one direction of time or the opposite; the entropy corresponding to the state at a time
On the other hand, the very notion of event horizon in general relativity depends on the direction of time. By looking again at Figure
This is consistent with the usual understanding of entropy as hidden information; indeed, the true information about the microstates is not accessible (only the macrostate), and this is what entropy stands for. But it is striking, nevertheless, to see that black holes do the same, yet in a completely time-asymmetric manner. This is because the horizon entropy is just a bound for the entropy beyond the horizon; the true entropy is a property of the state.
The four laws of black hole mechanics are the following [
The analogy between the laws of black hole mechanics and thermodynamics is quite impressive [
These laws of black hole mechanics are obtained in purely classical general relativity but were interpreted as laws of black hole thermodynamics [
Interestingly, despite their analogy with the laws of thermodynamics, the laws of black hole mechanics hold in purely classical general relativity. While we expect general relativity to be at least a limit theory of a more complete, quantized one, it is a standalone and perfectly selfconsistent theory. This suggests that it is possible that the laws of black hole mechanics already have thermodynamic interpretation in the geometry of spacetime. And this turns out to be true, since black hole entropy can be shown to be the Noether charge of the diffeomorphism symmetry [
This is not to say that the interpretations of the laws of black hole mechanics in terms of thermodynamics of quantum fields do not hold, because there are strong indications that they do. My point is rather that there are thermodynamics of the spacetime geometry, which are tied somehow with the thermodynamics of quantum matter and radiation. This connection is probably made via Einstein’s equation or whatever equation whose classical limit is Einstein’s equation.
Classically, black holes are considered to be completely described by their mass, angular momentum, and electric charge. This idea is based on the
In classical general relativity, the black holes radiate gravitational waves and are expected to converge to a no-hair solution very fast. If this is true, it happens asymptotically, and the gravitational waves carry the missing information about the initial shape of the black hole horizon, because classical general relativity is deterministic on regular globally hyperbolic regions of spacetime.
Moreover, it is not known what happens when quantum theory is applied. If the gravitational waves are quantized (resulting in gravitons), it is plausible to consider the possibility that quantum effects prevent such a radiation, like in the case of the electron in the atom. Therefore, it is not clear that the information about the infalling matter is completely lost in the black hole, even in the absence of Hawking evaporation. So we should expect at most that black holes converge asymptotically to the simple static solutions, but if they would reach them in finite time, there would be no time reversibility in GR.
Nevertheless, this alone is unable to provide a solution to the information loss paradox, especially since spacetime curvature does not contain the complete information about matter fields. But we see that we have to be careful when we use the no-hair conjecture as an assumption in other proofs.
While black hole mechanics suggest that the entropy of a black hole is limited by the Bekenstein bound (
Because of the
It is often suggested that there are some horizon microstates, either floating above the horizon but not falling because of a
Other counting proposals are based on counting string excited microstates [
Another interesting possible origin of entropy comes from
But, following the arguments in Section
Since in Section
The interest in the black hole information paradox and black hole entropy is not only due to the necessity of restoring unitarity. This research is also motivated by testing various competing candidate theories of quantum gravity. Quantum gravity seems to be far from our experimental possibilities, because it is believed to become relevant at very small scales. On the other hand, black hole information loss and black hole entropy pose interesting problems, and the competing proposals of quantum gravity are racing to solve them. The motivation is that it is considered that black hole entropy and information loss can be explained by one of these quantum gravity approaches.
On the other hand, it is essential to remember how black hole evaporation and black hole entropy were derived. The mathematical proofs are done within the framework of quantum field theory on curved spacetime, which is considered a good effective limit of the true but yet to be discovered theory of quantum gravity. The calculations are made near the horizon; they do not involve extreme conditions like singularities or planckian scales, where quantum gravity is expected to take the lead. The main assumptions are quantum field theory on curved spacetime the Einstein equation, with the stress-energy tensor replaced by the stress-energy expectation value
For example, when we calculate the Bekenstein entropy bound, we do this by throwing matter in a black hole and see how much the event horizon area increases.
These conditions are expected to hold in the effective limit of any theory of quantum gravity.
But since both the black hole entropy and the Hawking evaporation are obtained from the two conditions mentioned above, this means that any theory in which these conditions are true, at least in the low energy limit, is also able to imply both the black hole entropy and the Hawking evaporation. In other words, if a theory of quantum gravity becomes in some limit the familiar quantum field theory and also describes Einstein’s gravity, it should also reproduce the black hole entropy and the Hawking evaporation.
Nevertheless, some candidate theories to quantum gravity do not actually work in a dynamically curved spacetime, being, for example, defined on flat or AdS spacetime, yet they still are able to reproduce a microstructure of black hole entropy. This should not be very surprising, given that, even in nonrelativistic quantum mechanics, quantum systems bounded in a compact region of space have discrete spectrum. So it may be very well possible that these results are due to the fact that even in nonrelativistic quantum mechanics entropy bounds hold [
The entropy bound (
Sometimes it is said that the true black hole information paradox is the one following from Don Page’s article [ This is now a real problem. Evaporation causes the black hole to shrink and thus to reduce its surface area. So
I think there are some assumptions hidden in this argument. We compare the von Neumann entropy of the black hole calculated during evaporation with the black hole entropy calculated by Bekenstein and Hawking by throwing particles in the black hole. While the proportionality of the black hole entropy with the area of the event horizon has been confirmed by various calculations for numerous cases, the two types of processes are different, so it is natural that they lead to different states of the black hole and hence to different values for the entropy. This is not a paradox; it is just an evidence that the entropy contained in the black hole depends on the way it is created, despite the bound given by the horizon. So it seems more natural not to consider that the entropy of the matter inside the black hole reached the maximum bound at the beginning but rather that it reaches its maximum at the Page time, due to the entanglement entropy with the Hawking radiation. Alternatively, we may still want to consider the possibility of having more entropy in the black hole than the Bekenstein bound allows. In fact, Rovelli made another argument pointing in the same direction that the Bekenstein-Bound is violated, by counting the number of states that can be distinguished by local observers (as opposed to external observers) using local algebras of observables [
We have seen in the previous sections that some important approaches to the black hole information paradox and the related topics assume that the interior of the black hole is irrelevant or does not exist, and the event horizon plays the important role. I also presented arguments that if it is to recover unitarity without losing the principle of equivalence, then the interior of the black hole should be considered as well, and the event horizon should not be endowed with special properties. More precisely, given that the original culprit of the information loss is its supposed disappearance at singularities, then singularities should be closely investigated. The least radical approach is usually considered the avoidance of singularity, by modifying gravity (
But singularities are accompanied by divergences in the very quantities involved in the Einstein equation, in particular the curvature and the stress-energy tensor. So even if it is possible to reformulate the Einstein equation in terms of variables that do not diverge, remaining instead finite at the singularity, the question remains whether the physical fields diverge or break down. In other words, what are in fact the true, fundamental physical fields, the diverging variables, or those that remain finite? This question will be addressed soon.
An earlier mention of the possibility of changing the variables in the Einstein equation was made by Ashtekar, for example, in [
In [
However, it turns out that, on the space obtained by factoring out the subspace of isotropic vectors, an inverse can be defined in a canonical and invariant way and that there is a simple condition that leads to a finite Riemann tensor, which is defined smoothly over the entire space, including at singularities. This allows the contraction of a certain class of tensors and the definition of all quantities of interest to describe the singularities without running into infinities and is equivalent to the usual, nondegenerate semi-Riemannian geometry outside the singularities [
An essential difficulty related to singularities is given by the fact that, despite the Riemann tensor being smooth and finite at such singularities, the Ricci tensor
Another difficulty this approach had to solve was that it applies to a class of degenerate metrics, but the black holes are nastier, since the metric has components that blow up at the singularities. For example, the metric tensor of the Schwarzschild black hole solution, expressed in the Schwarzschild coordinates, is
For the horizon
This is not to say that physics depend on the coordinates. It is similar to the case of switching from polar to Cartesian coordinates in plane or like the Eddington-Finkelstein coordinates. In all these cases, the transformation is singular at the singularity, so it is not a diffeomorphism. The atlas, the differential structure, is changed, and in the new atlas, with its new differential structure, the diffeomorphisms preserve, of course, the semiregularity of the metric. And just like in the case of the polar or spherical coordinates and the Eddington-Finkelstein coordinates, it is assumed that the atlas in which the singularity is regularized is the real one, and the problems were an artifact of the Schwarzschild coordinates, which themselves were in fact singular.
Similar transformations were found for the other types of black holes (Reissner-Nordström, Kerr, and Kerr-Newman) and for the electrically charged ones the electromagnetic field also no longer blows up [
Returning to the Schwarzschild black hole in the new coordinates (
An analytic extension of the black hole solution beyond the singularity.
The resulting spacetime does not have Cauchy horizons, being hyperbolic, which allows the partial differential equations describing the fields on spacetime to be well posed and continued through the singularity. Of course, there is still the problem that the differential operators in the field equations of the matter and gauge fields going through the singularity should be replaced with the new ones. Such formulations are introduced in [
It is an open problem whether the backreaction will make the spacetime to curve automatically so that these conditions are satisfied for all possible initial conditions of the field. This should be researched in the future, including for quantum fields. It is to be expected that the problem is difficult, and what is given here is not the general solution but rather a toy model. Anyway, no one should expect very soon an exact treatment of real case situations, so the whole discussion here is in principle to establish whether this conservative approach is plausible enough.
However, I would like to propose here a different, more general argument, which avoids the difficulties given by the necessity that the field equations should satisfy at the singularities special conditions like the sufficient conditions found in [
First consider Fermat’s principle in optics. A ray of light in geometric optics is straight, but if it passes from one medium to another having a different refraction index, the ray changes its direction and appears to be broken. It is still continuous, but the velocity vector is discontinuous, and it appears that the acceleration blows up at the surface separating the two media. But Fermat’s principle still allows us to know exactly what happens with the light ray in geometric optics.
On a similar vein, I think that, in the absence of a proof that the fields satisfy the exact conditions [
The least action principle involves the integration of the Lagrangian densities of the fields. While the conditions the fields have to satisfy at the singularity in order to behave well are quite restrictive, the Lagrangian formulation is much more general. The reason is that integration can be done over fields with singularities, also on distributions, and the result can still be finite.
Consider first classical, point-like particles falling in the black hole, crossing the singularity, and exiting through the white hole which appears after the singularity disappears. The history of such a test particle is a geodesic, and to understand the behavior of geodesics, we need to understand first the causal structure. In Figure
(a) The causal structure of the Schwarzschild black hole in coordinates
If the test particle is massless, its path is a null geodesic. In [
If the test particle is massive, its history is a timelike geodesic. In this case, a difficulty arises, because in the new coordinates the lightcones are squashed. This allows for distinct geodesics to intersect the singularity at the same point and to have the same spacetime tangent direction. In the Schwarzschild case, this does not happen for timelike geodesics, but in the Reissner-Nordström case [
But the least action principle allows this to be solved regardless of the specific local solution of the problem at the singularity. The timelike geodesics are tangent only at the singularity, which is a zero-measure subset of spacetime. So we can apply the least action principle to obtain the history of a massive particle and obtain a unique solution. The least action principle can be applied for classical test particles because a particle falling in the black hole reaches the singularity in finite proper time, and similarly a finite proper time is needed for it to get out. Moreover, the path integral quantization will consider anyway all possible paths, so even if there would be an indeterminacy at the classical level, it will be removed by integrating them all.
For classical fields, the same holds as for point-like classical particles; only the paths are much more difficult to visualize. The least action principle is applied in the configuration space even for point-like particles, and the same holds for fields, the only difference being the dimension of the configuration space and the Lagrangian. The points from the singularity form again a zero-measure subset compared to the full configuration space, so finding the least action path is similar to the case of point-like particles. The Lagrangian density is finite at least at the points of the configuration space outside the singularities, which means almost everywhere. But the volume element vanishes at singularities, which improves the situation. So its integral can very well be finite, even if the Lagrangian density would be divergent at the singularities. It may be the case that the fields have singular Lagrangian density at the singularity and that when we integrate them it is not excluded that even the integral may diverge, but in this case the least action principle will force us anyway to choose the paths that have a finite action density at the singularities, and such paths exist, for example, those satisfying the conditions found in [
So far we have seen that the principle of least action allows determining the history of classical, point-like particles or fields, from the initial and final conditions, even if they cross the singularity. This is done so far on fixed background, so no backreaction via Einstein’s equation is considered, only particles or fields. But the Lagrangian approach extends easily to include the backreaction; we simply add the Hilbert-Einstein Lagrangian to that of the fields or point-like particles. So now we vary not only the path of point-like particles or fields in the configuration space but also the geometry of spacetime in order to find the least action history. This additional variation gives even more freedom to choose the least action path, so even if on fixed background the initial condition of a particular field will not evolve to become, at the singularity, a field satisfying the conditions from [
Now let us consider quantum fields. When moving to quantum fields on curved background, since the proper time of all classical test particles is finite, we can apply the path integral formulation of quantum field theory [
Of course the background geometry should also depend on the quantum fields. Can we account for this, in the absence of a theory of quantum gravity? We know that at least the framework of path integrals works on curved classical spacetime (see,
The proposal I described in this section is still at the beginning, compared to the difficulty of the remaining open problems to be addressed. First, there is obviously no experimental confirmation, and it is hard to imagine that the close future can provide one. The plausibility rests mainly upon making as few new assumptions as possible, in addition to those coming from general relativity and quantum theory, theories well established and confirmed, but not in the regimes where both become relevant. For some simple examples, there are mathematical results, but a truly general proof, with fully developed mathematical steps and no gaps, does not exist yet. And, considering the difficulty of the problem, it is hard to believe that it is easy to have very soon a completely satisfying proof in this or other approaches. Nevertheless, I think that promising avenues of research are opened by this proposal.
Everything is included; no additional data is needed; it is a hep-th manuscript.
The author declares that there are no conflicts of interest.