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 $X$. ...
