Big Ed on Classical and Quantum Information Theory

I'll have to carve out some time this summer to look at these :-) Perhaps on an airplane...

When I visited IAS earlier in the year, Witten was sorting out Lieb's (nontrivial) proof of strong subadditivity. See also Big Ed.

A Mini-Introduction To Information Theory
https://arxiv.org/abs/1805.11965

This article consists of a very short introduction to classical and quantum information theory. Basic properties of the classical Shannon entropy and the quantum von Neumann entropy are described, along with related concepts such as classical and quantum relative entropy, conditional entropy, and mutual information. A few more detailed topics are considered in the quantum case.

Notes On Some Entanglement Properties Of Quantum Field Theory
https://arxiv.org/abs/1803.04993

These are notes on some entanglement properties of quantum field theory, aiming to make accessible a variety of ideas that are known in the literature. The main goal is to explain how to deal with entanglement when – as in quantum field theory – it is a property of the algebra of observables and not just of the states.

Years ago at Caltech, walking back to Lauritsen after a talk on quantum information, with John Preskill and a famous string theorist not to be named. When I asked the latter what he thought of the talk, he laughed and said Well, after all, it's just linear algebra :-)

Seminars, Colloquia, and Slides I have known

I think I've made this Google drive folder publicly readable. It contains slides for many talks I've given over the years, going back almost to 2000 or so.

Topics include black hole information, monsters in curved space, entanglement entropy, dark energy, insider's guide to startups, the financial crisis of 2008, foundations of quantum mechanics, and more.

(Second slide is from this talk given at the Institute for Quantum Information at Caltech.)

Monsters in AdS

Slides I used for the workshop discussion yesterday: What Part of the Asymptotically AdS Gravitational Phase Space is Dual to a CFT?

In classical general relativity one can construct configurations with fixed ADM mass but arbitrarily large entropy. These objects collapse into black holes but have more entropy than the area of the resulting black hole. Their interpretation in quantum gravity and in the AdS/CFT duality is an open question.

This topic is also nicely discussed by Don Marolf here and here.

It was great to have Bill Unruh in the audience, whom I had never met in person before.

Entropy of black and white holes

Below are some remarks on black (and white) hole entropy which may help clarify the discussion (see also comments) elicited by Lubos Motl's blog post on my recent paper.

Apologies to most of my readers, who are not theoretical physicists! We will return to blogging on other topics soon :-) If you like this kind of discussion, see Lubos' blog -- he is one of the few people on the planet who can write with clarity and frequency on these topics!

In the discussion below I define entropy to be the logarithm of the number of distinct quantum states which are consistent with a particular coarse grained description of an object. For example, suppose there are N quantum states that correspond (when probed by a coarse detector) to a ball of stuff with ADM energy M and radius R. Then the entropy of this object is S = log N. Also note that I assume purely unitary evolution of quantum states.

In my paper I was interested in white holes described by the time reversal of a black hole spacetime. I was specifically interested in white holes in isolation -- i.e., surrounded by vacuum. If the black hole was formed by the collapse of some object (e.g., a star), the corresponding white hole will "explode" at some point, disgorging the original star. Lubos points out that this sequence of events is unlikely on entropic grounds: if one assumes that the white hole has entropy of order of order its area in Planck units, and the matter disgorged has much less entropy than this, the white hole explosion seems to violate the second law of thermodynamics. Do the white holes I consider actually have area entropy?

Note that there is a one to one mapping (via time reversal) of black hole and white hole interior states. The subset of black holes that are formed by the collapse of ordinary objects like stars has much less than area entropy. In fact, their entropy is bounded by a coefficient not much larger than unity times their area to the 3/4 power: A^{3/4}. This can be deduced from an entropy bound on ordinary matter in nearly flat space (see links below). Using the one to one property mentioned above, this gives us an estimate of the entropy of white holes which explode into ordinary matter like a star or dust ball.

How do we produce the black holes that account for the vast majority of the A entropy? Not by ordinary astrophysical processes. Instead, we have to start with a small black hole and allow it to slowly accrete energy. (There are other methods; see links below.) However, a black hole which slowly accretes energy will at the same time emit Hawking radiation. The time reversal of this kind of hole is not an isolated white hole, it is a white hole bathed in incoming radiation from the past light cone. This type of (most entropic) black hole IS quite similar to its time reversed white hole counterpart, unlike the isolated holes I studied. See figure below.



I am making a distinction between two types of coarse grained objects: 1. black hole of mass M, but I know nothing more about the object or its history and 2. black hole of mass M, but I know it was formed from, e.g., a star. The two categories of objects have radically different entropy because in case 2 I can restrict the quantum state to a small subset of the states in case 1 (I have more information about the object).


To summarize, the white holes I studied are obtained via time reversal from a tiny subset all possible black holes. However, these are the black holes actually produced by known astrophysical processes (supernovae, galaxy formation, etc.), and their time reversed evolution does lead to an explosion. Whether such objects are of interest is left up to the reader :-)



Useful links:

Talk slides (source of the figure above).

A review article: Monsters, black holes and the statistical mechanics of gravity, http://arxiv.org/abs/0908.1265.

See also: What is the entropy of the universe? http://arxiv.org/abs/0801.1847, also discussed here (source of figure below).

Figure 3 caption: Ordinary matter (star, galactic core, etc.) collapses to form an astrophysical black hole. Under unitary evolution, the number of final Hawking radiation states that are actually accessible from this collapse is $\sim \exp M^{3/2}$, i.e.~precisely the number of ordinary astrophysical precursors (\ref{th1}). It is therefore much smaller than the the number of $\sim \exp M^2$ states a black hole, and its eventual Hawking radiation, could possibly occupy if nothing about its formation process were known.

White holes, entropy and comments for Lubos

Lubos Motl doesn't like my recent paper (arXiv:1007.2934) on white holes. Lubos is a very smart guy, so I take his remarks seriously.

Before getting to his comments, let me say a few things about what I did in the paper.

A. The "white hole" I analyzed is just a classical background which is the time reversal of (part of) a black hole spacetime. The initial data for this spacetime can be obtained from a spacelike slice across the usual black hole spacetime ("after" the horizon has formed), and need not include the singularity.

B. I imposed the condition of isolation (vacuum) outside the white hole at early times. This is equivalent to requiring no radiation in the future of the original black hole spacetime. An unusual boundary condition, but corresponds to the "isolated" white hole I was interested in investigating.

C. I used Hawking's method (i.e., Bogoliubov transformation of in- and out- modes) to study the future behavior of the white hole, or, equivalently, the initial preparation of the black hole required to prevent it from radiating.

It should be clear from the summary that my paper can be read as merely analyzing the black hole spacetime with a nonstandard future boundary condition and then interpreting the results in time reversed language. ("Look ma, no white hole"!) That is, the methods I use have exact time reversal invariance built in. In the paper I note that an isolated white hole and an isolated black hole behave differently. That is not a violation of time reversal invariance applied to an entire spacelike slice because the time reversal of an isolated black hole (which radiates into its future) is not an isolated white hole -- rather, it is a white hole bathed in incoming radiation (from its past) at the Hawking temperature (see figures in the paper). However it does contradict the idea that the time reversed evolution of the hole can be understood independently of its environment (i.e., what is outside its horizon).


Lubos makes a number of remarks in his blog post. I try to summarize them below, together with my comments. He may make other claims as well that I haven't addressed.

1. We know from string theory that black holes and white holes are the same thing. Well, let me first point out that not everyone believes in string theory as the correct theory of quantum gravity (i.e., describing our universe) at 99.9 percent confidence level. Secondly, if a semi-classical calculation like mine suggests differences between the behavior of an isolated white hole and an isolated black hole, isn't it interesting to reconcile that with what AdS/CFT predicts? Although I am not an expert on AdS/CFT I suspect that the time reversal invariance of the CFT boundary state only implies time reversal of the entire bulk state (i.e., on an entire spacelike slice) and not of the black hole alone. If so, there is no contradiction with my results -- see above. Perhaps someone can clarify this for me?

2. Hawking clearly said the same in his 1976 paper. That was my impression on first reading, but since all of his arguments center on the case of a black hole in equilibrium with a bath at equal temperature, it is unclear (at least to me) how this can be generalized to an isolated white hole. That was one motivation for my investigation.

3. Entropic arguments imply that white holes (as obtained via (A) above) are extremely unlikely: specifically, a highly entropic white hole should not explode into lower entropy ejecta. I understand the argument but don't place as much confidence in it as Lubos does. The uncertainty is not about the 2nd Law but about the interpretation of black or white hole entropy.

Lubos does not want me to consider classical spacetimes generated by the initial data obtained in (A) above. Even if one accepts that such spacetimes are highly improbable, that does not mean that they shouldn't be studied. (For example, if you are a many worlder there are some branches on which exploding white holes are observed!) Apparently it is in bad taste to think about (exploding) white holes, but perhaps Lubos should tell this to, e.g., Frolov and Novikov.

Now, a little analysis of the cognitive dissonance (conflicting priors ;-) between Lubos and me. When I say "white hole" I mean the time reversal of some classical black hole spacetime. I consider this (time reversed) spacetime of theoretical interest, even if it results from strange initial conditions. I use what I know (general relativity + quantum fields in slightly curved space) to probe the more mysterious issues (black hole entropy, quantum gravity, ... ). This follows Wheeler's approach of "radical conservatism" -- take the physics you trust with high confidence, and extrapolate to extreme conditions until something interesting happens! Lubos is a true believer in string theory, so to him a black hole is this stringy thing about which we already know almost everything, including that its entropy is due to countable internal microstates, it is dual to some YM configuration through holography, etc. This will likely elicit a shriek of anger from Lubos (or he will just call me dumb), but I consider all of those claims plausible but perhaps not true in our universe: string theory may turn out not to describe Nature.


Finally, there is also some discussion of my paper here and here, but it seems that both authors are slightly confused about the results (perhaps this is my fault for not being clear :-). For example, the requirement that white holes "explode" is not a consequence of my analysis, but just follows from time reversal of the black hole formation event (see, e.g., Frolov and Novikov or figures in the paper). I am only studying the quantum effects (i.e., equivalent of Hawking radiation), which are a correction to the classical evolution.

Further discussion in a follow up post about white/black hole entropy.

Classical and Quantum Gravity 2009

The most read articles of 2009 in the journal Classical and Quantum Gravity -- see under JULY :-)

JANUARY
What is a particle?
D Colosi and C Rovelli
http://www.iop.org/EJ/abstract/0264-9381/26/2/025002

FEBRUARY
Arithmetical chaos and quantum cosmology
L A Forte
http://www.iop.org/EJ/abstract/0264-9381/26/4/045001

MARCH
Local Hawking temperature for dynamical black holes
S A Hayward, R Di Criscienzo, M Nadalini, L Vanzo and S Zerbini
http://www.iop.org/EJ/abstract/0264-9381/26/6/062001

APRIL
The double pulsar system: a unique laboratory for gravity
M Kramer and N Wex
http://www.iop.org/EJ/abstract/0264-9381/26/7/073001

MAY
LISA Pathfinder: the experiment and the route to LISA
M Armano et al.
http://www.iop.org/EJ/abstract/0264-9381/26/9/094001

JUNE
Status of NINJA: the Numerical INJection Analysis project
L Cadonati et al.
http://www.iop.org/EJ/abstract/0264-9381/26/11/114008

JULY
What is the entropy of the universe?
P H Frampton, S D H Hsu, T W Kephart and D Reeb
http://www.iop.org/EJ/abstract/0264-9381/26/14/145005

AUGUST
Testing gravitational-wave searches with numerical relativity waveforms: results from the first Numerical INJection Analysis (NINJA) project
B Aylott et al.
http://www.iop.org/EJ/abstract/0264-9381/26/16/165008

SEPTEMBER
Polarized spots in anisotropic open universes
R Sung and P Coles
http://www.iop.org/EJ/abstract/0264-9381/26/17/172001

OCTOBER
Present status of the Penrose inequality
M Mars
http://www.iop.org/EJ/abstract/0264-9381/26/19/193001

NOVEMBER
The information paradox: a pedagogical introduction
S D Mathur
http://www.iop.org/EJ/abstract/0264-9381/26/22/224001

DECEMBER
Casimir energy and gravitomagnetism
F Sorge
http://www.iop.org/EJ/abstract/0264-9381/26/23/235002

What is the entropy of the universe?

P H Frampton, S D H Hsu, T W Kephart and D Reeb

http://www.iop.org/EJ/abstract/0264-9381/26/14/145005

Abstract. Standard calculations suggest that the entropy of our universe is dominated by black holes, whose entropy is of order their area in Planck units, although they comprise only a tiny fraction of its total energy. Statistical entropy is the logarithm of the number of microstates consistent with the observed macroscopic properties of a system, hence a measure of uncertainty about its precise state. Therefore, assuming unitarity in black hole evaporation, the standard results suggest that the largest uncertainty in the future quantum state of the universe is due to the Hawking radiation from evaporating black holes. However, the entropy of the matter precursors to astrophysical black holes is enormously less than that given by area entropy. If unitarity relates the future radiation states to the black hole precursor states, then the standard results are highly misleading, at least for an observer that can differentiate the individual states of the Hawking radiation.

Supermassive black holes and the entropy of the universe

New Scientist has an article about a recent paper by two Australian researchers (http://www.arxiv.org/abs/0909.3983), which contains detailed estimates of the entropy of various components of the universe (black holes, neutrinos, photons, etc.). This paper is related to some work I did with with Frampton, Kephart and Reeb: What is the entropy of the universe?. (Discussion on Cosmic Variance.)

Our work did not focus on the numerical values of various contributions to the entropy (we made some simple estimates), but rather what the physical meaning is of this entropy -- in particular, that of black holes; see excerpt below.

New Scientist: Mammoth black holes push universe to its doom

30 September 2009 by Rachel Courtland

THE mammoth black holes at the centre of most galaxies may be pushing the universe closer to its final fade-out. And it is all down to the raging disorder within those dark powerhouses.

Disorder is measured by a quantity called entropy, something which has been on the rise ever since the big bang. Chas Egan and Charles Lineweaver of the Australian National University in Canberra used the latest astrophysical data to calculate the total entropy of everything in the universe, from gas to gravitons. It turns out that supermassive black holes are by far the biggest contributors to the universe's entropy. Entropy reflects the number of possible arrangements of matter and energy in an object. The number of different configurations of matter a black hole could contain is staggering because its internal state is completely mysterious.

Egan and Lineweaver found that everything within the observable universe contains about 10^104 units of entropy (joules per Kelvin), a factor of 10 to 1000 times higher than previous estimates that did not include some of the biggest known black holes (www.arxiv.org/abs/0909.3983, submitted to The Astrophysical Journal).

If entropy were ever to reach a maximum level, that would mean the heat death of the universe. In this scenario no energy can flow, because everything is the same temperature and so life and other processes become impossible. "Our results suggest we're a little further along that road than previously thought," Egan says.

But although black holes do boost the universe's total entropy, it is not clear whether they will hasten its heat death. Supermassive black holes don't contribute much to the flows of heat that even out temperature throughout the universe, says physicist Stephen Hsu at the University of Oregon in Eugene.

It's true that these black holes will slowly evaporate by releasing Hawking radiation, particles created near the boundary of the black hole. And this radiation could move the universe towards heat death.

Black holes may evaporate via Hawking radiation, tipping the universe towards its heat death. However, it will take some 10^102 years for a supermassive black hole to evaporate. "The entropy inside those black holes is effectively locked up in there forever," Hsu says. So we may have reached a state approaching heat death long before, as stars burn out and their matter decays.

The large result obtained by the Egan and Lineweaver for the entropy of the universe is primarily due to supermassive black holes. How do we interpret that entropy? Here is an excerpt from our paper (excuse the latex).

Note the entropy used in this paper describes the uncertainty in the precise quantum state of a system. If the system is macroscopic the full quantum state is only accessible to a kind of ``super-observer'' who is unaffected by decoherence \cite{decoherence}. Individual observers within the system who have limited experimental capabilities can only detect particular decoherent outcomes. These outcomes arise, e.g., from an effective density matrix that results from tracing over degrees of freedom which are out of the experimenter's control (i.e., which form the ``environment''). In \cite{BID} the experimental capabilities necessary to distinguish decoherent branches of the wavefunction, or, equivalently, the precise quantum state of Hawking radiation from a black hole, are discussed. It is shown that a super-observer would either need (at minimum) the capability of making very precise measurements of accuracy $\exp(- M^2 )$ (see also the proposal of Maldacena \cite{eternal} for a specific measurement to determine whether black hole evaporation is unitary), or alternatively the capability of engineering very precise non-local operators, which measure a large fraction of the Hawking radiation at once, including correlations (i.e., as opposed to ordinary particle detectors, which only measure Fock state occupation numbers and are not sensitive to phase information).

An observer who lacks the capabilities described in the previous paragraph would be unable to distinguish the states in the $S = M^{3/2}$ subspace in Fig.~\ref{figure1} from those in the larger $S = M^2$ subspace, assuming the unitary evaporation resembles, in gross terms, Hawking evaporation, with the information hidden in correlations among the emitted quanta. In that case, the future uncertainty for ordinary (non-super) observers might be better characterized by the larger $S = M^2$ entropy. Putting it another way, an ordinary (non-super) observer is forced (due to experimental limitations) into a coarse grained description of the radiation; they cannot distinguish between most of the radiation states, and for them the $S = M^2$ entropy is appropriate. For a super-observer, however, due to unitary evolution, the uncertainty in the quantum state does not increase. For them, black holes do not have greater entropy than the precursor states from which they formed.

For the super-observers described above, the large black hole entropies in Table I do not reflect the actual uncertainties in the (current and future) state of the universe and are in that sense misleading. A black hole of mass $M$ whose formation history is typical for our universe (e.g., it originated from gravitational collapse of a star or galactic core) satisfies the bound S [less than] M^{3/2} \cite{MI}. Thus, re-evaluating the numbers in Table I, the total entropy of all black holes in our universe is not bigger than the total matter entropy: the dominant uncertainty in the precise state of the universe, at least as far as arises from known physics, is, in fact, due to CMB photons or neutrinos.

Figure 3 caption: Ordinary matter (star, galactic core, etc.) collapses to form an astrophysical black hole. Under unitary evolution, the number of final Hawking radiation states that are actually accessible from this collapse is $\sim \exp M^{3/2}$, i.e.~precisely the number of ordinary astrophysical precursors (\ref{th1}). It is therefore much smaller than the the number of $\sim \exp M^2$ states a black hole, and its eventual Hawking radiation, could possibly occupy if nothing about its formation process were known.

Final mysterious comment, maximally compressed for the cognoscenti: assuming unitarity, black holes do not push us closer to heat death (equilibrium) in the multiverse, but can contribute (albeit very slowly) to the (coarse grained) heat death experienced by a non-super observer (i.e., an observer subject to decoherence). See here for more on equilibrium in the multiverse.

Monsters in Modern Physics Letters

My student David Reeb and I were asked to write a short review of our recent work on monsters for the journal Modern Physics Letters. The review is now on arxiv. You can find slides from a recent talk on this subject here (given at Fermilab).

We included some new material in the second part of the paper. In the last few years there has been significant progress in the foundations of statistical mechanics, in which thermodynamic properties are seen to emerge as a consequence of entanglement and the high dimensionality of Hilbert space. Even the Second Law can be deduced in a probabilistic sense from underlying dynamics that is fundamentally time-symmetric. We discuss the possibility that a similar approach can be applied in gravity to deduce, e.g., the Generalized Second Law of Thermodynamics, which governs black hole entropy as well as that of ordinary matter.

Monsters, black holes and the statistical mechanics of gravity

Authors: Stephen D. H. Hsu, David Reeb

http://arxiv.org/abs/0908.1265

Abstract: We review the construction of monsters in classical general relativity. Monsters have finite ADM mass and surface area, but potentially unbounded entropy. From the curved space perspective they are objects with large proper volume that can be glued on to an asymptotically flat space. At no point is the curvature or energy density required to be large in Planck units, and quantum gravitational effects are, in the conventional effective field theory framework, small everywhere. Since they can have more entropy than a black hole of equal mass, monsters are problematic for certain interpretations of black hole entropy and the AdS/CFT duality.

In the second part of the paper we review recent developments in the foundations of statistical mechanics which make use of properties of high-dimensional (Hilbert) spaces. These results primarily depend on kinematics -- essentially, the geometry of Hilbert space -- and are relatively insensitive to dynamics. We discuss how this approach might be adopted as a basis for the statistical mechanics of gravity. Interestingly, monsters and other highly entropic configurations play an important role.

Excerpt from the paper:

Can the quantum mechanical derivation of statistical mechanics given above be applied to gravity? For example, can we deduce the Second Law of Thermodynamics on semiclassical spacetimes (i.e., including, for example, large black holes)?

This might seem overly ambitious since we currently lack a theory of quantum gravity. However, the results described above are primarily a consequence of the high-dimensional character of Hilbert spaces. If the state space of quantum gravity continues to be described by something like a Hilbert space, then its dimensionality will almost certainly be large, even for systems of modest size. Further, it seems a less formidable task to characterize some aspects of the state space of quantum gravity than to fully understand its dynamics. Indeed, for our purposes here we only consider semiclassical spacetimes.

Early attempts at quantization, culminating in the Wheeler-DeWitt equation, were based on the classical Hamiltonian formulation of general relativity\cite{WDW1,WDW2}. These led to a configuration space (``superspace'') of 3-geometries, modulo diffeomorphisms, and to the wavefunction, $\Psi [ h_{ab}, \phi ]$, of the universe as a functional over 3-metrics $h_{ab}$ and matter fields $\phi$. This description of the state space seems quite plausible, at least in a coarse grained sense, even if the fundamental objects of the underlying theory are something else (strings, loops, etc.). Let us assume that some form of short-distance regulator is in place (or, alternatively, that the dynamics itself generates such a regulator in the form of a minimum spacetime interval), so that we can neglect ultraviolet divergences.

Now consider the set of asymptotically flat, non-compact 3-geometries. Impose conditions on the asymptotic behavior so that the total ADM mass of the system is $M$, and further assume that all the energy density is confined to a region of surface area $A$. This results in a restricted state space ${\cal H}_R$. If the concentration of measure results apply to ${\cal H}_R$, then the observed properties of any small subsystem $X$ are likely to be the same as if the universe were in the equiprobable, maximally mixed state $\rho_* = \mathbbm{1}_R / d_R$. In the flat space case this leads to the usual canonical (Boltzmann) distribution in $X$. ...

Backreaction, black holes and monsters

A must read post on black hole entropy by physicist (Perimeter Institute) and blogger Sabine Hossenfelder! ;-)

Now that she mentions it, "Entropy" is a cool name for a kid (superhero?), maybe even cooler than "Max Talmud" :-)

These days, everybody is talking about entropy. In fact, there is so much talk about entropy I am waiting for a Hollywood starlet to name her daughter after it. To help that case, today a contribution about the entropy of black holes.

To begin with let us recall what entropy is. It's a measure for the number of micro-states compatible with a given macro-state. The macro-state could for example be given by one billion particles with a total energy E in a bag of size V. You then have plenty of possibilities to place the particles in the bag and to assign a velocity to them. Each of these possibilities is a micro-state. The entropy then is the logarithm of that number. Don't worry if you don't know what a logarithm is, it's not so relevant for the following. The one thing you should know about the total entropy of a system is that it can't decrease in time. That's the second law of thermodynamics.

It is generally believed that black holes carry entropy. The need for that isn't hard to understand: if you throw something into a black hole, its entropy shouldn't just vanish since this would violate the second law. So an entropy must be assigned to the black hole. More precisely, the entropy is proportional to the surface area of the black holes, since this can be shown to be a quantity which only increases if black holes join, and this is also in agreement with the entropy one derives for a black hole from Hawking radiation. So, black holes have an entropy. But what does that mean? What are the microstates of the black hole? Or where are they? And why doesn't the entropy depend on what was thrown into the black hole?

While virtually nobody in his right mind doubts black hole have an entropy, the interpretation of that entropy is less clear. There are two camps: On the one side those who believe the black hole entropy counts indeed the number of micro-states inside the black hole. I guess you will find most string theorists on this side, since this point of view is supported by their approach. On the other side are those who believe the black hole entropy counts the number of states that can interact with the surrounding. And since the defining feature of black holes is that the interior is causally disconnected from the exterior, these are thus the states that are assigned to the horizon itself. These both interpretations of the black hole entropy are known as the volume- and surface-interpretations respectively. You find a discussion of these both points of view in Ted Jacobson's paper "On the nature of black hole entropy" [gr-qc/9908031] and in the trialogue "Black hole entropy: inside or out?" [hep-th/0501103].

A recent contribution to this issue comes from Steve Hsu and David Reeb in their paper

Black hole entropy, curved space and monsters
Phys. Lett. B 658:244-248 (2008)
arXiv:0706.3239v2

...read the rest...

See related post here, with nice pictures.

What is the entropy of the universe?

New paper! You can find some additional discussion on Cosmic Variance.

What is the entropy of the universe?

Paul Frampton, Stephen D.H. Hsu, Thomas W. Kephart, David Reeb


Abstract: Standard calculations suggest that the entropy of the universe is dominated by black holes, although they comprise only a tiny fraction of its total energy. We give a physical interpretation of this result. Statistical entropy is the logarithm of the number of microstates consistent with the observed macroscopic properties of a system, hence a measure of uncertainty about its precise state. The largest uncertainty in the present and future state of the universe is due to the (unknown) internal microstates of its black holes. We also discuss the qualitative gap between the entropies of black holes and ordinary matter.

Curved space and monsters

New paper!

http://arxiv.org/abs/0706.3239

A simple question: how many different black holes can there be with mass M? Conventional wisdom: of order exp(A), where A is the surface area of the hole and scales as M^2.

Using curved space, we construct objects of ADM mass M with far more than exp(A) microstates. These objects have pathological properties, but, as far as we can tell, can be produced via quantum tunneling from ordinary (non-pathological) initial data. Our results suggest that the relation between black hole entropy and the number of microstates of the hole is more subtle than perhaps previously appreciated.

Update! Rafael Sorkin was kind enough to inform us of his earlier related work with Wald and Zhang. We've added the following end-note to the paper.

Note added: After this work was completed we were informed of related results obtained by Sorkin, Wald and Zhang [25]. Those authors investigated monster-like objects as well as local extrema of the entropy S subject to an energy constraint, which correspond to static configurations and obey $A^{3/4}$ scaling. For example, in the case of a photon gas the local extrema satisfy the Tolman--Oppenheimer--Volkoff equation of hydrostatic equilibrium. In considering monster configurations, Sorkin et al. show that requiring a configuration to be no closer than a thermal wavelength $\lambda \sim \rho^{-1/4}$ from its Schwarzschild radius imposes the bound $S < A$. While this may be a reasonable criterion that must be satisfied for the assembly of an initial configuration, it does not seem to apply to states reached by quantum tunneling. From a global perspective configurations with $S > A^{3/4}$ are already black holes in the sense that the future of parts of the object does not include future null infinity.




Black hole entropy, curved space and monsters

Stephen D.H. Hsu, David Reeb

(Submitted on 21 Jun 2007)

We investigate the microscopic origin of black hole entropy, in particular the gap between the maximum entropy of ordinary matter and that of black holes. Using curved space, we construct configurations with entropy greater than their area in Planck units. These configurations have pathological properties and we refer to them as monsters. When monsters are excluded we recover the entropy bound on ordinary matter $S < A^{3/4}$. This bound implies that essentially all of the microstates of a semiclassical black hole are associated with the growth of a slightly smaller black hole which absorbs some additional energy. Our results suggest that the area entropy of black holes is the logarithm of the number of distinct ways in which one can form the black hole from ordinary matter and smaller black holes, but only after the exclusion of monster states.