Abstract
Over the years, the so-called black hole information loss paradox has generated an amazingly diverse set of (often radical) proposals. However, 40 years after the introduction of Hawking’s radiation, there continues to be a debate regarding whether the effect does, in fact, lead to an actual problem. In this paper we try to clarify some aspect of the discussion by describing two possible perspectives regarding the landscape of the information loss issue. Moreover, we advance a fairly conservative point of view regarding the relation between evaporating black holes and the rest of physics, which leads us to advocate a generalized breakdown of unitarity. We conclude by exploring some implications of our proposal in relation with conservation laws.
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Notes
There is, though, one issue that is raised with some frequency: the trans-Planckian character of the modes involved. In our view, however, the mere formulation of such a concern is misguided simply because the is nothing intrinsically trans-Planckian about any mode in the absence of an ad hoc specification of a preferential local inertial frame. In other words, any mode is trans-Planckian when characterized in a suitable local Lorentz frame. And, of course, the assignment of a privileged status to one such frame comes in directly contradiction local Lorentz invariance, a feature that underlies both general relativity and quantum field theory.
According to Maudlin, this change in topology evades Geroch’s theorem because, due to the strange geometry at the Evaporation Event, the spacetime in question is not a manifold, and thus the global manifold structure presupposed in that theorem is not longer there (see [16] for details).
In fact, to be precise, one should consider \(B-L\), baryon number minus lepton number, which is a quantity that is conserved both at the perturbative and non-perturbative levels in the Standard Model of particle physics. The conservation of B itself is violated through highly suppressed “sphaleron” mediated processes. In the rest of the discussion we refer for simplicity to B although, strictly speaking, we should be talking about \(B-L\).
In [16] it is argued that the firewall argument is not sound because it relies on the false assumption that \(\Sigma _2\) is a Cauchy hypersurface. It very well might be the case that in [1] such an assumption is mistakenly held, but the firewall argument itself does not require it, it only requires the assumption that all the information is encoded in the outgoing radiation.
A full clarification of all these issues would requires a detailed technical analysis, something which goes beyond the scope of the present paper.
Since different convex sums of pure states can correspond to the same mixed state, convex sums are better suited to encode epistemic uncertainty.
The Eigenvector/Eigenvalue rule states that a physical system possesses the value \(\lambda \) for a property represented by the operator O if and only if the quantum state assigned to the system is an eigenstate of O with eigenvalue \(\lambda \).
Regarding proper mixtures, two ensembles of electrons, one with half of them spin-up and half of them spin-down along z and the other with half spin-up and half spin-down, this time along y, are assigned the same density matrix. As for improper mixtures, a system of two electrons with state either \( \frac{1}{\sqrt{2}} \left( |+ \rangle ^{(1)}_{z} |- \rangle ^{(2)}_{z} + |-\rangle ^{(1)}_{z} |+ \rangle ^{(2)}_{z} \right) \) or \(\frac{1}{\sqrt{2}} \left( |+ \rangle ^{(1)}_{z} |+\rangle ^{(2)}_{z} + |-\rangle ^{(1)}_{z} |- \rangle ^{(2)}_{z} \right) \) leads to the same reduced density matrix for, say, the second electron. It is interesting to note that all of the examples considered above lead to exactly the same density matrix; of course, this does not mean that the physical situations described are all equal (see [21] for a thorough discussion of this issue).
To avoid confusion, it is worth pointing out that the proper way to represent ignorance mathematically is a probability measure over pure states and not a mixed state.
Explicit proposal of this kind, including figures characterizing the expected modifications of the classical Penrose diagrams with similar characteristics as Fig. 3, have been advocated in works such as [2, 9, 11, 12]. We think, however, that such figures are to be considered as heuristic at best. That is because quantum gravity is expected to heavily modify the nature of the fundamental degrees of freedom in the regions where a full quantum gravity description is required and thus spacetime concepts, such as the metric, would simply cease to make sense (in the same way that hydrodynamic notions, such as fluid velocity, cease to make sense when the level of description is that of a quantum theory of atoms and molecules or, at an even deeper level, quantum field theory).
Of course, unless one is dealing with a superfluid.
The analogy between black holes and hurricanes could seem a bit strained. Unlike for a hurricane, the criteria for being a black hole are sharply defined, as is the process of black hole evaporation. In it important to remember, though, that such criteria are sharply defined only if general relativity is taken as a non-emergent theory, which is exactly the assumption we are questioning.
In a recent private conversation, R. M. Wald pointed out that the fact that W. G. Unruh and him addressed the issue raised in [3], does not mean that they accept the premises underlying such a work.
Let us clarify further our point of view in this regard. Imagine for a moment that that, at the fundamental level, the spacetime description is something like what is provided by loop quantum gravity. If so, at the fundamental level there is no metric and thus no notion of asymptotic flatness nor of black holes. It is clear, however, that if the theory is viable, it would have to be able to account for the emergence of a spacetime metric, and thus of black holes (i.e., that under certain circumstances, some features of the state of holonomies and fluxes would have to be identified as describing a black hole). Now, imagine that, in order to describe a physical process, one computes the amplitude using some kind of path integration (which in the case of loop quantum gravity would involve spin foams). The standard usage of path integrals involves summing over all possible intermediate paths connecting the initial and final conditions. Thus, any such complete sum over paths can be expected to contain some of the holonomies and fluxes that have been identified as corresponding to a black hole. In other words, we expect virtual black holes as part of any physical process simply because at the basic level there is no distinction between virtual and real processes.
That is, in the absence of gravitation, the configuration of gauge and scalar fields known as the sphaleron is characterized by a certain value of energy. However, such a value of energy is intimately tied with the symmetries of Minkowski spacetime. Within a black hole, and particularly an evaporating one, the notion of energy is not even well-defied. Therefore, the configuration of the filed corresponding to the sphaleron would, quite likely, be rather different from its flat spacetime counterpart. As a result, under such conditions it is unclear exactly how the energetic suppression of the sphaleron mediated processes would manifest itself.
For quantitative reasons, the theory should also lead to enhanced violation of T and CP symmetries, known occur in the standard model of particle physics only in a highly suppressed manner.
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Acknowledgements
We thank T. Maudlin and R. M. Wald for very useful discussions. We acknowledge partial financial support from DGAPA-UNAM Project IG100316. DS was further supported by CONACyT project 101712.
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Okon, E., Sudarsky, D. Losing Stuff Down a Black Hole. Found Phys 48, 411–428 (2018). https://doi.org/10.1007/s10701-018-0154-3
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DOI: https://doi.org/10.1007/s10701-018-0154-3