Black hole evaporation rates without spacetime
Samuel L. Braunstein
University of York, United Kingdom
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
. 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  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 . 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 , 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
Spacetime free conjecture
In our paper  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
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.
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 .
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
. 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 
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 h-bar.
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., ).
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
Hints of an emergent gravity
Verlinde recently suggested that gravity, inertia, and even spacetime
itself may be emergent properties of an underlying thermodynamic theory
. 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 .
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
. 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
- 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
- provides support for suggestions that gravity, inertia and
even spacetime itself could come from spacetime-free physics across
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