Black Hole Evaporation Rates without Spacetime

Samuel L. Braunstein

Author: Samuel L. Braunstein

Affiliation: Professor of Quantum Computation, University of York, UK


Why black holes are so important to physics

In the black hole information paradox, Hawking pointed out an apparent contradiction between quantum mechanics and general relativity so fundamental that some thought any resolution may lead to new physics. For example, it has been recently suggested that gravity, inertia and even spacetime itself may be emergent properties of a theory relying on the thermodynamic properties across black hole event horizons [1]. All these paradoxes and prospects for new physics ultimately rely on thought experiments to piece together more detailed calculations, each of which themselves only give a part of the full picture. Our work "Black hole evaporation rates without spacetime" adds another calculation [2] which may help focus future work.

The paradox, a simple view

In its simplest form, we may state the paradox as follows: In classical general relativity, the event horizon of a black hole represents a point of no return - as a perfect semi-permeable membrane. Anything can pass the event horizon without even noticing it, yet nothing can escape, even light. Hawking partly changed this view by using quantum theory to prove that black holes radiate their mass as ideal thermal radiation. Therefore, if matter collapsed to form a black hole which itself then radiated away entirely as formless radiation then the original information content of the collapsing matter would have vanished. Now, information preservation is fundamental to unitary evolution, so its failure in black hole evaporation would signal a manifest failure of quantum theory itself. This "paradox" encapsulates a profound clash between quantum mechanics and general relativity.

To help provide intuition about his result Hawking presented a heuristic picture of black hole evaporation in terms of pair creation outside a black hole's event horizon. The usual description of this process involves one of the pair carrying negative energy as it falls into the black hole past its event horizon. The second of the pair carries sufficient energy to allow it to escape to infinity appearing as Hawking radiation. Overall there is energy conservation and the black hole losses mass by absorbing negative energy. This heuristic mechanism actually strengthens the "classical causal" structure of the black hole's event horizon as being a perfect semi-permeable (one-way) membrane. The paradox seems unassailable.

Scratching the surface of the paradox

This description of Hawking radiation as pair creation is seemingly ubiquitous (virtually any web page providing an explanation of Hawking radiation will invoke pair creation).

Nonetheless, there are good reasons to believe this heuristic description may be wrong [3]. Put simply, every created pair will be quantum mechanically entangled. If the members of each pair are then distributed to either side of the event horizon the so-called rank of entanglement across the horizon will increase for each and every quanta of Hawking radiation produced. Thus, one would conclude that just as the black hole mass were decreasing by Hawking radiation, its internal (Hilbert space) dimensionality would actually be increasing.

For black holes to be able to eventually vanish, the original Hawking picture of a perfectly semi-permeable membrane must fail at the quantum level. In other words, this "entanglement overload" implies a breakdown of the classical causal structure of a black hole. Whereas previously entanglement overload had been viewed as an absolute barrier to resolving the paradox [3], we argue [2,4] that the above statements already point to the likely solution.

Evaporation as tunneling

The most straightforward way to evade entanglement overload is for the Hilbert space within the black hole to "leak away". Quantum mechanically we would call such a mechanism tunneling. Indeed, for over a decade now, such tunneling, out and across the event horizon, has proved a useful way of computing black hole evaporation rates [5].

Spacetime free conjecture

In our paper [2] we suggest that the evaporation across event horizons operates by Hilbert space subsystems from the black hole interior moving to the exterior. This may be thought of as some unitary process which samples the interior Hilbert space; picks out some subsystem and ejects it as Hawking radiation. Our manuscript primarily investigates the consequences of this conjecture applied specifically to event horizons of black holes.

At this point a perceptive reader might ask how and to what extent our paper sheds light on the physics of black hole evaporation. First, the consensus appears to be that the physics of event horizons (cosmological, black hole, or those due to acceleration) is universal. In fact, it is precisely because of this generality that one should not expect this Hilbert space description of evaporation at event horizons to bear the signatures of the detailed physics of black holes. In fact, as explained in the next section we go on to impose the details of that physics onto this evaporative process. Second, sampling the Hilbert space at or near the event horizon may or may not represent fair sampling from the entire black hole interior. This issue is also discussed below (and in more detail in the paper [2]).

Imposing black hole physics

We rely on a few key pieces of physics about black holes: the no-hair theorem and the existence of Penrose processes. We are interested in a quantum mechanical representation of a black hole. At first sight this may seem preposterous in the absence of a theory of quantum gravity. Here, we propose a new approach that steers clear of gravitational considerations. In particular, we derive a quantum mechanical description of a black hole by ascribing various properties to it based on the properties of classical black holes. (This presumes that any quantum mechanical representation of a black hole has a direct correspondence to its classical counterpart.) In particular, like classical black holes our quantum black hole should be described by the classical no-hair properties of mass, charge and angular momentum. Furthermore, these quantum mechanical black holes should transform amongst each other just as their classical counterparts do when absorbing or scattering particles, i.e., when they undergo so-called Penrose processes. By imposing conditions consistent with these classical properties of a black hole we obtain a Hilbert space description of quantum tunneling across the event horizons of completely generic black holes. Crucially, this description of black hole evaporation does not involve the detailed curved spacetime geometry of a black hole. In fact, it does not require spacetime at all. Finally, in order to proceed to the next step of computing the actual dynamics of evaporation, we need to invoke one more property of a black hole: that of its enormous dimensionality.

Tunneling probabilities

The Hilbert space dimensionalities needed to describe a black hole are vast (at least 10¹⁰⁷⁷ for a stellar-mass black hole). For such dimensionalities, random matrix theory tells us that the statistical behavior of tunneling (as a sampling of Hilbert space subsystems) is excellently approximated by treating tunneling as a completely random process. This immediately imposes a number of symmetries onto our description of black hole evaporation. We can now completely determine the tunneling probabilities as a function of the classical no-hair quantities [2]. These tunneling probabilities are nothing but the black hole evaporation rates. In fact, these are precisely the quantities that are computed using standard field theoretic methods (that all rely on the curved black hole geometry). Thus, the calculation of tunneling probabilities provides a way of validating our approach and making our results predictive.

The proof of the pudding: validation and predictions

Our results reproduce Hawking's thermal spectrum (in the appropriate limit), and reproduce his relation between the temperature of black hole radiation and the black hole's thermodynamic entropy.

When Hawking's semi-classical analysis was extended by field theorists to include backreaction from the outgoing radiation on the geometry of the black hole a modified non-thermal spectrum was found [5]. The incorporation of backreaction comes naturally in our quantum description of black hole evaporation (in the form of conservation laws). Indeed, our results show that black holes that satisfy these conservation laws are not ideal but "real black bodies" that exhibit a non-thermal spectrum and preserve thermodynamic entropy.

These results support our conjecture for a spacetime free description of evaporation across black hole horizons.

Our analysis not only reproduces these famous results [5] but extends them to all possible black hole and evaporated particle types in any (even extended) gravity theories. Unlike field theoretic approaches we do not need to rely on one-dimensional WKB methods which are limited to the analysis of evaporation along radial trajectories and produce results only to lowest orders in ℏ.

Finally, our work quite generally predicts a direct functional relation exists between the irreducible mass associated with a Penrose process and a black hole's thermodynamic entropy. This in turn implies a breakdown in Hawking's area theorem in extended gravity theories.


And the paradox itself

The ability to focus on events horizons is key to the progress we have made in deriving a quantum mechanical description of evaporation. By contrast, the physics deep inside the black hole is more elusive. If unitarity holds globally then our spacetime free conjecture can be used to describe the entire time-course of evaporation of a black hole and to learn how the information is retrieved (see e.g., [6]). Specifically, in a unitarily evaporating black hole, there should exist some thermalization process, such that after what has been dubbed the black hole's global thermalization (or scrambling) time, information that was encoded deep within the black hole can reach or approach its surface where it may be selected for evaporation as radiation. Alternatively, if the interior of the black hole is not unitary, some or all of this deeply encoded information may never reappear within the Hawking radiation. Unfortunately, any analysis relying primarily on physics at or across the horizon cannot shed any light on the question of unitarity (which lies at the heart of the black hole information paradox).

The bigger picture

At this stage we might take a step back and ask the obvious question: Does quantum information theory really bear any connection with the subtle physics associated with black holes and their spacetime geometry? After all we do not yet have a proper theory of quantum gravity. However, whatever form such a theory may take, it should still be possible to argue, either due to the Hamiltonian constraint of describing an initially compact object with finite mass, or by appealing to holographic bounds, that the dynamics of a black hole must be effectively limited to a finite-dimensional Hilbert space. Moreover, one can identify the most likely microscopic mechanism of black hole evaporation as tunneling. Formally, these imply that evaporation should look very much like our sampling of Hilbert space subsystems from the black hole interior for ejection as radiation [2,4,6]. Although finite, the dimensionalities of the Hilbert space are immense and from standard results in random unitary matrix theory and global conservation laws we obtain a number of invariances. These invariances completely determine the tunneling probabilities without needing to know the detailed dynamics (i.e., the underlying Hamiltonian). This result puts forth the Hilbert space description of black hole evaporation as a powerful tool. Put even more strongly, one might interpret the analysis presented as a quantum gravity calculation without any detailed knowledge of a theory of quantum gravity except the presumption of unitarity [2].

Hints of an emergent gravity

Verlinde recently suggested that gravity, inertia, and even spacetime itself may be emergent properties of an underlying thermodynamic theory [1]. This vision was motivated in part by Jacobson's 1995 surprise result that the Einstein equations of gravity follow from the thermodynamic properties of event horizons [7]. For Verlinde's suggestion not to collapse into some kind of circular reasoning we would expect the physics across event horizons upon which his work relies to be derivable in a spacetime free manner. It is exactly this that we have demonstrated is possible in our manuscript [2]. Our work, however, provides a subtle twist: Rather than emergence from a purely thermodynamic source, we should instead seek that source in quantum information.

In summary, this work [2,4]:

• shows that the classical picture of black hole event horizons as perfectly semi-permeable almost certainly fails quantum mechanically
• provides a microscopic spacetime-free mechanism for Hawking radiation
• reproduces known results about black hole evaporation rates
• authenticates random matrix theory for the study of black hole evaporation
• predicts the detailed black hole spectrum beyond WKB
• predicts that black hole area must be replaced by some other property in any generalized area theorem for extended gravities
• provides a quantum gravity calculation based on the presumption of unitarity, and
• provides support for suggestions that gravity, inertia and even spacetime itself could come from spacetime-free physics across event horizons

References
[1] E. Verlinde, "On the origin of gravity and the laws of Newton", JHEP 04 (2011) 029. Abstract.
[2] S.L. Braunstein and M.K. Patra, "Black Hole Evaporation Rates without Spacetime", Phys. Rev. Lett. 107, 071302 (2011). Abstract. Article (pdf).
[3] H. Nikolic, "Black holes radiate but do not evaporate", Int. J. Mod. Phys. D 14, 2257 (2005). Abstract; S.D. Mathur, "The information paradox: a pedagogical introduction", Class. Quantum Grav. 26, 224001 (2009). Abstract.
[4] Supplementary Material to [2] at http://link.aps.org/supplemental/10.1103/PhysRevLett.107.071302.
[5] M.K. Parikh and F. Wilczek, "Hawking Radiation As Tunneling", Phys. Rev. Lett. 85, 5042 (2000). Abstract.
[6] S.L. Braunstein, S. Pirandola and K. Życzkowski, "Entangled black holes as ciphers of hidden information", arXiv:0907.1190.
[7] T. Jacobson, "Thermodynamics of Spacetime: The Einstein Equation of State", Phys. Rev. Lett. 75, 1260 (1995). Abstract.