Black Holes: Eliminating Information or Illuminating New Physics?
Abstract
:1. Being Simple is Complex
1.1. If You Can Heat It, It Has Micro-Structure!
1.2. Beginner’s Information Loss
2. The Way It All Began
2.1. Semi-Classical Information Loss
- It is not only about horizon: Hiding is not destroyingTill now we have obtained that at the future times an observer will feel to be set up in thermal setting had she started with a vacuum state. However, this really does not amount to saying the black holes do radiate. Masking off a portion of initial vacuum data can be brought about by any horizon, even by the Rindler horizon at the least (or the de Sitter horizon). Therefore, even in Minkowski space time, such horizons can exist which mask off a portion of the vacuum field configuration from a specified family of observers. This does not amount to say that even Minkowski spacetimes radiate. They do not! There is no real flux of radiation in Minkowski spacetime, precisely because it is homogeneous, isotropic and time translational symmetric.What the horizons are capable of, is to give rise to thermal environment for a special set of observers measuring real flux of radiation. The genesis of radiation is brought about by breaking some of the symmetries the spacetime possess, and this is the story of a black hole.
- It is also about geometry non-trivia:The main difference a black hole has from the other horizon settings we commented above is that the black hole not only breaks the homogeneity due to curvature, it also breaks time reversal invariance in order to give rise to time irreversible phenomenon like radiation. In fact the sitting in Equation (4) is understood to be total time, such that particle creation rate per unit time becomes exactly that of a radiating body. Thus, it is the breaking of these symmetries which allow the flux of energy radiated away to be non zero [59,60] when calculated from . Thus, non trivial geometry of the spacetime added with the presence of horizon makes the black hole radiate thermally (predominantly). This is the point exactly where we make true contact with the thermodynamics of the black hole we kept believing in (even when a concrete proof was not available), and this, unfortunately, is also exactly where the problem at hand starts unfolding.
- ... and about finiteness of size:Now that we have credible theoretical understanding that the black holes leak out energy in thermal flux, we can think of the black hole losing mass in this way and evaporating away in time. We did not require anything extra than our belief in the quantum field theory on a regular manifold (however curved it may be). Therefore, unless the black hole becomes so small where the differential structure of spacetime is no longer applicable, we do not see any other reason to discredit or mistrust our calculations (for the case of primordial black holes, see [61,62,63]). This, then immediately leads us to expect that being compact and hence finite sized, the black holes evaporate away practically completely. Then there is a crisis at our hands!
2.2. The Paradox
3. Hiding Information in Correlations
3.1. The Black Hole History
3.2. No Hiding Theorem—No Information in Correlations between Separable Hilbert Spaces
4. Black Hole Complementarity
4.1. Posing The Problem
4.2. A Possible Resolution
- Formation and evaporation of black hole, viewed by a distant observer can be described entirely using unitary quantum mechanics.
- Outside stretched horizon, semi-classical physics holds good.
- To a distant observer, the black hole is a quantum system with discrete energy levels and number of states being exponential of Bekenstein entropy.
4.3. Problem with These Postulates: Firewall?
4.4. The Role of In-Falling Vacuum
4.5. The Problem of Projection Operator: Can Firewall Exist?
5. The Overreach of Quantum Gravity
5.1. The Fuzzball Paradigm
5.1.1. The No No-Hair!
5.1.2. Violation of Semi-Classical Approximation
5.1.3. Free fall: Fuzzball Complementarity
5.2. A Stable and a Proportionate Remnant?
- One bound on the size of the remnant can be found from appealing to the maximum compressibility of information in a given region. Giddings [112] argued that if there exists an upper bound on entropy density, it should better be that of a Planck scale sized remnant, i.e.,Now, once some matter enters a macroscopic black hole horizon, it inevitably gets drifted towards the central spacelike singularity. Classically all such matter would ultimately reach the central singularity, making it also an infinite entropy density place. However, since we have already committed that the maximum entropy density should always be given by Equation (36), it is reasonable to expect that a quantum theory of gravity, respecting the entropy bound will replace the central singularity by a region of maximum entropy density. The size of the region of such a maximum entropy density corresponding to a black hole of mass M would beThis depicts the region of a core where the entropy of mass M black hole is stored with the maximum capacity. However the fact that the spacetime behind event horizon is not static is in tension with the concept of a core. To remove the spacelike singularity one needs to introduce additional matter fields modifying the “going to be” singular point within the spacelike region. This possibly requires the core to have some non-trivial matter fields, e.g., non-linear electrodynamics may be one viable candidate [113].From Equation (37), it is clear that the size of the core scales with the initial data (aka the mass). Once the black hole starts evaporating, the event horizon starts shrinking. However, since the core is packed with the maximum density allowed, it does not have a capacity to further accommodate any bits carrying the information of evaporation14. Thus, while the horizon shrinks, the core persists. Therefore, there will come a stage when the horizon meets the core and then the Hawking process loses its credibility. What happens next is up for speculation as it caters to regime of Planck scale physics, which we hardly have any clue of (note that the entropy density is at the Planck scale, while the physical size can still be macroscopic). There can be a plethora of various exotic possible outcomes when the horizon comes crashing down to the Planck density core, which has replaced the classical singularity [114]. It can also be thought that the gravity shows some phase transition at this leg of evaporation, and end up in some sort of crystal as a leftover remnant [115]. It is well known that once systems undergo a phase transition into a crystalline phase, quantum correlations become pretty long range. Therefore, it is possible that effects of quantum gravity, in the similar spirit, become important even at large length scales leading to a mass proportionate remnant. Either we need to know the physics inside these remnants or we will have to further wait for the time scale which a remnant of this kind lives for, before jumping back to the revival of (or declaring a convincing triumph over) the problem of information loss.
- In canonical splitting of gravity, we end up with various constraints on the variables of the theory, we use to describe the same. These constraints appear as operator equations at the quantum level, if we keep using these variables in quantum gravity. Using the method of discretization of spacetime lattice [116], one can argue that the solution space, consistent with diffeomorphism constraints, with positive frequency, is obtained through states of the kindUsing this approach of canonical quantization, it can be argued [116] that any collapse of matter in the exterior of the apparent horizon of the collapsing data is accompanied by an out going radiation in the interior of the apparent horizon (and vice versa)16[116]. It is further argued [117,118] that, the absence of firewall must require that only one branch of solutions exists, i.e., a collapsing shell in the exterior accompanied by a thermal radiation in the interior. There should not be any branch that has a collapsing shell in the interior accompanied by a Hawking radiation in the exterior. Therefore, the profile of mass distribution of the shell should be such that its collapse is effectively counterbalanced and gets supported by the interior thermal radiation. Such a mass profile, the size of a resulting “stable structure” and its relation to gravitational mass of the cloud is estimated for different cases [118,119]. Therefore, in this scenario, the collapse halts at the apparent horizon and the black hole never forms in the first place. All what is left is a dust cloud-ball hinging at the apparent horizon, as a stable remnant, being supported by a radiation pressure from the inside necessitated by quantum gravity. This cloud-ball also hides in its inside, a negative mass singularity which gives rise to the radiation flux in the interior.
6. The Quantum Framework: A Re-Look Required(?)
6.1. Can Non-Locality Save the Day?
6.1.1. ER=EPR
6.1.2. Bargaining Micro-Causality: State-Dependent Operators
6.1.3. A Final State for Black Holes
6.2. Bargaining Unitarity: Modifications of Quantum Physics
- Output of a measurementThe standard quantum theory we are familiar with tells us that the time evolution of a closed system is unitary in nature,
- A single copy of a closed systemAnother possible explanation of measurement process can come through the idea of decoherence. If we consider only a subsystem of a large system, the effective dynamics of the subsystem may be non unitary such that the evolution of the full system is still unitary. Also making many copies of subsystem, we can make sense of a statistical interpretation of quantum theory. However, this becomes a problem when we look at a closed system, which has only a single realization. In this case, we have neither system-subsystem division nor many copies of this system to apply statistical description. Therefore, the quantum theory, as it is, fails to give any deterministic prediction. Problem becomes more prominent if such a system evolves on its own to a classical description. Then we are really in a fix. This is a setting where quantum theory is not designed to be applied upon. Our universe makes a concrete example of such a system. Thus, non-unitary modifications to quantum theory derives motivation also from quantum cosmology [157,158,159,160,161,162,163].
6.2.1. How Much of Non-Unitarity in Quantum Mechanics is Tolerable ?
6.2.2. Continuous Spontaneous Localization Evolution of the Quantum State
7. Black Holes Have (Soft) Hairs
7.1. Supertranslation
7.2. Lorentz Transformation and Superrotation
7.3. Soft Photon Hair on Black Holes: An Illustration
8. A Simple Way to Extract Information—Non-Vacuum Distortions
8.1. Hard (Quantum) Hairs on Black Holes?
8.2. Information of Black Hole Formation: Correlation Function
8.3. Radiation from Black Hole: Information about the Initial State
9. Information Regain?
9.1. Late Time Flash
- Between the initial and final bits radiated, the information about the in-fallen matter is effectively deleted from each individual subsystem (interior or early or late time radiation considered individually).
- Prior to bits are radiated (for any positive ) the information of infalling matter resides in the interior only with fidelity .
- It is in only the final bits (for any positive ), the information is imprinted in the late time radiation considered separately with fidelity .
9.2. Old Enough Black Hole Behaves as a “Mirror”
10. Black Hole Scattering Matrix
11. What Have We Learnt?
- Initial belief was that a black hole is no different than a black body and after a certain time (Page time), i.e., when the Hilbert spaces of the early Hawking radiation and the remaining black hole have similar dimensionality the information starts leaking out of the hole [70]. This view was challenged by the no-hiding theorem of Brustein et al., [71] where they showed that there exists virtually no information in the correlations of a bipartite system. It was subsequently realized that black hole can actually be thought as a tripartite system, and this tripartite system can have correlations storing information. However, then one can ask the question, which of these correlations have the full information of what has fallen in the black hole. It turns out the attempt to answer this challenge leads to the complementarity proposal [209].
- In the complementarity proposal, one assumes that the same copy of information is present both for the static and infalling observer, but since none of them can access the both the copies, there is no violation of the no-cloning theorem. This virtually reduces the no-cloning theorem, for a single causal patch. For the static observer the information is assumed to be stored in a stretched horizon, Planck length away from the hole, while for the infalling observer the information is residing inside the horizon. However the above scenario invites the firewall puzzle. Since the region near the stretched horizon is entangled with both black hole interior and the early Hawking radiation, thus violating the strong sub-additivity of entropy. Whether a freely falling observer has sufficient time before she crosses the horizon to do quantum computation to realize this violation remains a debated subject as of now. One possible resolution being breakdown of equivalence principle, resulting in a non-vacuum structure at the horizon—the firewall [77].
- There have been numerous works, both supporting and opposing the firewall argument. Many have agreed on the violation of strong sub-additivity of entropy, but suggesting some third alternative rather than leaving equivalence principle or quantum theory at stake. There have also been claims that the firewall argument is wrong—possibly it does not describe a classical world. Till now no unique consensus has been reached.
- A very straightforward plausible resolution of the information puzzle is to suggest that the black holes themselves do not exist. This seemingly is the case in the context of certain string theory approach, leading to the fuzzball paradigm [56]. According to this picture, there is no spacetime region inside the event horizon, instead the spacetime caps off through compactified extra dimensions and the resulting curvature acts as a proxy for the gravitating mass. Then evaporation of black holes becomes exactly like burning of coal (nothing separated by causal curtain) and as a result, there is no paradox. As of future, one might want to study the geometry of a fuzzball state, and see if there is any deviation from the Schwarzschild solution. Such a modification have to confront the solar system tests and many more, which the general relativity allowed geometries have done successfully. It will be interesting to see, what the fuzzball paradigm has to offer when the horizons have no tussle with quantum theory. For example, it would be interesting to see what happens for other horizons in general relativity (e.g., Rindler horizon, de Sitter horizon) because they also hide information and have entropy so as to give rise to large number of microscopic configurations. There are also other ideas [112,115,118] which call for a drastic changes to the classical understanding of black holes to fight the information loss they seemingly cause. A geometric study of the fuzzball may provide further insight of such objects.
- Non-local character of quantum correlations can also play a very crucial role towards resolving both the information paradox and the firewall puzzle. Quantum correlations, themselves, are not constrained by causality and spill over from the interior part as well. One may postulate that any two systems, which have quantum correlations (like EPR correlations) are connected by an wormhole like structure in spacetime, famously known as Einstein-Rosen bridge. Thus, the early and late Hawking quanta are connected by ER bridge and hence can exchange information. Also the existence of firewall depends on the static observer, who is playing with the early Hawking radiation. If she decides to decipher the correlations in the early Hawking radiation, the infalling observer will experience firewall, otherwise it will be a smooth transition through the horizon. It would be interesting if one can arrive at an estimate of the probability for existence of a firewall from such an approach. Further, the traversibility of such wormholes must also be properly accounted for.
- In order to describe the black hole interior using local bulk operators in the AdS/CFT correspondence, it seems essential for the mapping of CFT operators corresponding to the bulk, to depend on the state of the quantum field in CFT. This also leads to an explicit realization of complementarity, where operators in two causally disconnected regions are related to each other. These state dependent operators lead to violation of micro-causality, i.e., commutator between operators inside and outside the event horizon vanishes for low point correlation but not for high point ones. The above realization of complementarity was performed in the context of empty AdS, it would be interesting to investigate the situation for more general setups with non-maximal symmetric cases. Moreover, the loss of micro-causality at high point correlations should be thoroughly investigated. Whether there is any signature of violation of such micro-locality can transcend down to visible scales is an interesting topic of research.
- One can also, though very reluctantly, think of giving up unitarity, which is the main point of tussle between black hole and the quantum theory. One has to tackle this situation with extreme care, since then one has to confront the very successful unitary quantum theory as well. However it turns out that there exist two features which are not well explained in the context of standard quantum theory—(a) the measurement process, resulting in collapse of wave function itself appears non-unitary and (b) the probabilistic interpretation fails when we have a single copy of a closed system, e.g., the universe as a whole. Till date, the attempts to obtain a consistent non-unitary theory explaining the quantum world have yielded a single non-unitary and non-relativistic generalization, known as the Continuous Spontaneous Localization theory with a stochastic, classical source field driving the quantum system helping in collapse of the wave function. A variation of this model when applied to evaporating black hole results in collapse of the wave function to a number eigenstate, such that the resultant density matrix is thermal. Thus, in this picture one has a thermal radiation and loss of information, but there is nothing paradoxical as fit for a standard quantum system, where measurement leads to collapse of the wave function. Clearly, a thorough understanding of quantum theory, particularly when we move towards larger systems, will help settling out this issue. More so over, if a modification on this account is indeed required, relativistic version of that modified theory must first be sought for, before we can tune such modifications in the black hole setting.
- It has been more or less an agreed upon stance that in unitarily evaporating black hole, most of the information comes very late in time and comes out as a flash. In fact, calculations also show that once the black hole enters the “revealing phase” it literally becomes incapable of holding anything which is then thrown in. Therefore, after a certain time it actually adapts the blackbody character of radiation.
- As advocated recently, in the semi-classical and the quantum domain the no-hair theorems fail and black holes do have additional hairs, storing information about what has fallen into the hole. There can be hairs due to non-vacuum character of the matter fallen in, leading to non-vacuum characters of quantum correlations and also there can be soft hairs, due to infinitely many diffeomorphisms the horizon can afford. The non-vacuum hairs encode the departure of the in-fallen state to be different from the vacuum state. Whenever a matter field carry non-zero stress energy into the black hole, that clearly gets reflected in its late time frequency correlation. With added information of the initial symmetry of the in-data, it is even possible to fully reconstruct the state. A detailed analysis from the point of quantum teleportation and availability of classical channels, particularly at low energies, is definitely required.
- The soft hairs originate due an infinite number of degenerate vacuum states with the same energy but differing in the existence of additional soft photons or gravitons. These are the Goldstone bosons originating from additional symmetries at the black hole horizon, similar to the supertranslations and superrotations for the asymptotically flat spacetimes. The conservation of charges associated with these symmetries following the soft graviton theorem of Weinberg leads to these infinitely inequivalent vacuum states. Understanding of these symmetries and the resulting soft quanta for an arbitrary null surface (of which the black hole horizon is a special case) could be a very insightful exercise for progress.
- One can also invoke more subtle effects to recover the information thrown in the black hole. One such possibility being the Aichelburg-Sexl boost, in which a test particle is dragged along the trajectory of a passing by photon and the displacement suffered depends on the energy of the photon. This shows that as a particle falls into the black hole, it drags the outgoing Hawking quanta along it and hence pass over the information to the quanta, resulting in recovery of information fallen into the hole.