Tuesday, July 04, 2017

Cosmology and entropy

After a hiatus of approximately seven years, I've just written a new paper, 'Cosmology and entropy: in search of further clarity'. Here's an extract:

It is a widespread belief amongst modern physicists that black holes, or their horizons, possess temperature and entropy. The putative black-hole temperature is inversely proportional to the surface area of the horizon, while the entropy is proportional to the surface area. In natural units, the entropy of a non-rotating black hole is (Penrose 2016, p271):
S_{bh} = \frac{1}{4} A = 4 \pi M^2 \,,
$$ where $A$ is the area and $M$ is the mass.

The concept that a black hole could be the bearer of entropy is often justified by claiming that the black-hole entropy compensates for the 'loss of information', or the 'lost degrees of freedom', associated with matter and radiation falling into the black hole, never to be seen again. Bekenstein's original argument went as follows:

"Suppose that a body carrying entropy $S$ goes down a black hole...The $S$ is the uncertainty in one's knowledge of the internal configuration of the body...once the body has fallen in...the information about the internal configuration of the body becomes truly inaccessible. We thus expect the black hole entropy, as the measure of the inaccessible information, to increase by an
amount $S$
," (Bekenstein 1973).

Presumably, the idea is that one loses both the actual entropy and the maximum possible entropy associated with these extinguished dimensions of phase-space. However, as Dougherty and Callender (2016) point out, Bekenstein-type arguments express an epistemic and operationalistic interpretation of entropy. They rightly complain that "The system itself doesn't vanish; indeed, it had better not because its mass is needed to drive area increase...there is no reason to believe that a body slipping past an event horizon would lose its entropy...no compensation is necessary...we could observe the entropy of steam engines and the like that fall behind event horizons. Just jump in with them!"

We can make the objection more precise in general relativistic terms. For example, take the Oppenheimer-Snyder spacetime for a star collapsing to a black hole, or the Schwarzschild spacetime for a black hole itself. In each case, the spacetime is globally hyperbolic, hence it can be foliated by a one-parameter family of spacelike Cauchy hypersurfaces $\Sigma_t$, and the entire spacetime is diffeomorphic to $\mathbb{R} \times \Sigma$.

Each Cauchy surface is a complete and boundaryless 3-dimensional Riemannian manifold. There is no sense in which any Cauchy surface intersects the singularity. Each Cauchy surface which contains a region inside the event horizon also contains a region outside the horizon. Moreover, every inextendible causal curve in a globally hyperbolic spacetime $\mathbb{R} \times \Sigma$ intersects each Cauchy surface $\Sigma_t$ once and only once. Particles follow causal curves, hence because each particle will intersect each Cauchy surface exactly once, assuming that none of those particles form bound systems, it follows that no degrees of freedom are lost. The future may well be finite inside the event horizon, but that doesn't entail that any degrees of freedom are lost from the universe.

The entropy of one part of the universe can decrease, just as the entropy of a volume of water decreases when it transfers heat to some ice cubes immersed within it. Similarly, if a material system possessing entropy falls into a black hole, whilst the region of the universe exterior to the black hole loses entropy, the total entropy does not decrease from one spacelike Cauchy hypersurface to the next. To echo Dougherty and Callender, there is no reason for the event horizon of a black hole to possess entropy; there is simply no loss to compensate for.

Penrose, however, argues that "the enormous entropy that black holes possess is to be expected from...the remarkable fact that the structure of a stationary black hole needs only a very few parameters [mass, charge and angular momentum] to characterize its state. Since there must be a vast volume of phase space corresponding to any particular set of values of these parameters, Boltzmann's formula suggests a very large entropy," (2010, p179).

This appeal to the 'no-hair' theorem of black holes is based upon a sleight of hand: it is the space-time geometry of the stationary, asymptotically flat, vacuum solutions which are classified by just three parameters. Such vacuum solutions are useful idealisations for studying the behaviour of test particles in a black hole spacetime, but they do not represent the history of actual black-holes.

The spacetime of an actual black-hole contains the mass-energy which collapses to form the black hole, and any mass-energy which falls into the black-hole thereafter, including swirling accretion disks of matter and so forth. Hence, actual black holes are represented by variations of the Oppenheimer-Snyder spacetime, not the Schwarzschild space-time. As Dafermos and Rodnianski comment, "It is traditional in general relativity to 'think' Oppenheimer-Snyder but `write' maximally-extended Schwarzschild," (2013, p18).

Whilst the exterior region of a collapse solution is isometric to an exterior region of the vacuum solution, the difference in the interior solution makes all the difference in the world. Spacetimes which represent collapse to a black-hole are not classified by just three parameters; on the contrary, they are classified by a large number of parameters, characterising the specifics of the collapsing matter, including its entropy. The entropy of such black-hole spacetimes is possessed, not by the geometry of the black-hole horizon, but by the infalling mass-energy, just as it should be, (see Figure 1).

Conformal diagram of a black hole, including a pair of Cauchy
surfaces, $\Sigma_1$ and $\Sigma_2$. The shaded region represents the infalling
matter; the thin diagonal line represents the event horizon; and the jagged
line represents the singularity. Cauchy surface $\Sigma_2$ possesses a region
inside the event horizon, and a region outside the event horizon. The shaded
region possesses entropy; the horizon doesn't. (From Maudlin 2017)

Dougherty and Callender also draw attention to a number of conceptual contradictions associated with the notion that black-hole horizons possess entropy and temperature. For example:
  1. The increase in the area of a black-hole horizon, and therefore its purported entropy, is proportional to the mass-energy of the material which falls into the black-hole. Hence, if a massive object with a small entropy falls into the hole, it produces a large increase in black-hole entropy, whilst if a small object with a large entropy falls in, it produces a small increase in black-hole entropy.
  2. Entropy is an 'extensive' thermodynamic property, meaning that it is proportional to the volume of a system. In contrast, black-hole entropy is proportional to the area of the black-hole.
  3. Temperature is an 'intensive' thermodynamic property, meaning it is independent to the size of an object, yet if the size of a black-hole is increased, its temperature decreases.
  4. There is no 'equilibrium with' relationship in black-hole thermodynamics. Individual black-holes can be in equilibrium in the sense that the spacetime is stationary, but one black-hole cannot be in equilibrium with another.
  5. If two black-holes of the same area, and therefore with the same purported temperature, coalesce, then the area of the merged black-hole is greater than each of its progenitors, hence the purported entropy increases. In contrast, thermodynamics dictates that the coalescence of two entities at the same temperature is an isentropic process.
Even if it is accepted that black holes, or their horizons, possess entropy, a belief in black hole entropy is typically twinned with a belief in the eventual evaporation of black holes. For example, Penrose (2010,  p191) asserts that black holes will evaporate by Hawking radiation after the cosmic background radiation cools to a lower temperatures than the temperature of the holes. In this scenario, all the entropy in the universe eventually becomes radiation entropy. Hence, once again, it seems that the clumping of matter is nothing more than an intermediate state. If black holes can evaporate, then black holes are clearly not the ultimate means by which entropy is maximised.

An alternative scenario suggests that large black holes will not evaporate because there is a fundamental lower limit to the temperature of the cosmological radiation field, and this temperature is greater than the possible temperature of large black holes. The belief in such a lower limit is based upon the fact that a universe with a positive cosmological constant $\Lambda > 0$, such as ours currently appears to be, possesses a spacelike future conformal boundary, and the past light cone of each point on this future boundary defines an event horizon. It is then suggested that this event horizon possesses a temperature and an entropy, just as much as the event horizon of a black hole.

However, the reasons for believing that a cosmological event horizon possesses temperature and entropy are much weaker than those for believing a black hole possesses thermodynamic properties. The cosmological event horizon is entirely observer dependent, unlike the case of a black hole event horizon. Moreover, the region rendered unobservable by an event horizon is the region to the future of the event horizon, and in the case of the cosmological event horizon this is the region to the exterior of the past light cone. (In contrast, the
region to the future of the event horizon of a black-hole is the interior of the black hole). 

Penrose (2016, p278-279) points out that the region to the exterior will be of infinite volume if the universe is spatially non-compact, hence its entropy will also be infinite. It therefore makes no sense to interpret the (finite) entropy of a cosmological event horizon as the entropy/information of all the matter and radiation `lost' beyond that horizon.