Pathological Monsters! Cried the Terrified Mathematician

by Domenic 2007-11-14T03:48:00

What's this now? Doing my weekly browsing through seminar calendars in search of something interesting, I come across an event titled "Curved Space, Monsters and Black Hole Entropy" in the Caltech High Energy Physics Seminar series. Check this out:

We use curved space to construct objects of extremely high entropy -- potentially higher than that of a black hole of equal mass. Due to their pathological properties, we refer to these objects as monsters. However they seem to be legitimate physical configurations and should be part of the Hilbert space of gravity. 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.

A Calculation Excursion

Now, the way I'd always heard it, black holes were objects with maximum entropy for their given volume, not mass, so I was a little confused here. I started working out the equations, which led to me installing a TeX(-like) equation renderer on this blog, so now both you and I can use <math /> tags to have equations in posts and comments! Oh, we lead such an exciting life.

Right then, let's start working through things. This is basic algebra, but what the variables represent might be unfamiliar to you, so I suggest poking around Wikipedia if that's the case. Also, we're going to use Planck units, since they make life easy. And finally, note that the geometric quantities here—viz. radius, area, and volume—refer to the event horizon of the black hole, and not the singularity itself (for which all of these are zero, at least until quantum gravity shows up to save the day).

For black holes, the entropy is given by S = \frac14 A, and (for non-rotating ones) the radius is r = 2 m. A = 4 \pi r^2 = 4 \pi (2m)^2 = 16 \pi m^2, so S_{\text{BH}} = \frac14 (16 \pi m^2) = 4 \pi m^2. We also get V_{\text{BH}} = \frac43 \pi r^3 = \frac 43 \pi (2m)^3 = \frac{32}{3} \pi m^3, and hence \rho_{\text{BH}} = \frac{m}{V_{\text{BH}}} = \frac{3}{32 \pi m^2}. This latter "density" is necessarily the mean density of matter within the black hole's event horizon, for (as mentioned above) all of this mass is actually concentrated at the singularity.

So the way that I've heard it, S_{\text{BH}}/V_{\text{BH}} = \frac{4 \pi m^2}{\frac{32}{3} \pi m^3} = \frac{3}{8 m} is maximal. That is, given any other object of mass m and volume V, that object has S/V < \frac{3}{8m}.

The article is claiming that there is some "monster" object such that S_{\text{M}}/m is higher than the same value would be for a black hole, which is S_{\text{BH}}/m = 4 \pi m. Our question is, under what conditions does this give S_{\text{M}}/V > S_{\text{BH}}/V? Well,

S_{\text{M}}/m > S_{\text{BH}}/m \Longleftrightarrow \rho_{\text{M}} S_{\text{M}}/m > \rho_{\text{M}} S_{\text{BH}}/m \Longleftrightarrow S_{\text{M}}/V > \rho_{\text{M}} 4 \pi m.

This will imply S_{\text{M}}/V > S_{\text{BH}}/V = \frac{3}{8 m} iff \rho_{\text{M}} 4 \pi m \geq \frac{3}{8 m}. Simplifying, this means \rho_{\text{M}} \geq \frac{3}{32 \pi m^2}. But this latter expression is exactly equal to \rho_{\text{BH}}!

So! These "monsters" only violate our usual maximum entropy principle if they have a greater density than black holes do. Can such a configuration exist? We'll have to read on to find out...

The Story

All right, now that we're done having fun with our new equation renderer, let's try to figure out what's really going on. Well, it appears that the arXiv has a paper of almost the same title, with one of the coauthors being the person who will be giving the upcoming talk. And if we check the trackbacks, we find that this coauthor has his own blog, where he has a post on the paper! So now we have several sources of summarization: the Caltech talk summary; the arXiv abstract; and the blog post. But then again, the paper's only five pages—four, if you don't count references—so you might just want to go read it. I'll do that now.

Well huh, that was interesting. The abstract really doesn't tell the whole story here. Here are some highlights, keeping in mind their definition the crucial quantity \epsilon(r) = 1 - 2 M(r)/r, where M(r) is the "energy within radius r."

As demonstrated, curved space configurations can have greater entropy than their flat space counterparts of the same mass or size. This is because of their small \epsilon(r): the configurations have proper surface area A \sim M^2, but have internal proper volume much larger than A^{3/2}. Equivalently, they have very large proper mass M_p relative to mass M. It is easy to see that the ratio M/M_p can be made as small as desired if \epsilon(r) approaches zero for large r. The large negative gravitational binding energy allows us to pack substantially more proper mass into the region than suggested by a flat space analysis.

And the arbitrarily large entropy we were promised:

Without a constraint on how close \epsilon(r) can get to zero, S can be made arbitrarily large. Invoking quantum effects, one might require that a Planck length uncertainty in the proper radial distance not cause horizon formation [...]. This implies \epsilon(r) > r^{-2} [...]. This is still potentially problematic for the area entropy of black holes. A limit of S < A would require that \epsilon(r) > r^{-1}. This would be the consequence of the previous logic if one assumed a Planck length uncertainty in the radial coordinate r rather than the proper radial distance r \epsilon(r)^{-1/2} (or equivalently an uncertainty in proper radial distance which grows as \epsilon(r)^{-1/2}). This seems unphysical, but nevertheless cannot be excluded as a consequence of quantum gravity.

And the real shocker:

To obtain entropy scaling faster than A^{3/4}, we must consider configurations in which \epsilon(r) is close to zero in regions containing significant entropy and energy density. We now show that such configurations have the following pathological properties.

  1. They inevitably evolve into black holes, even in the absence of any outside perturbation.
  2. Even their time-reversed evolution leads to black hole formation.

They are therefore neither ordinary black holes nor ordinary matter configurations. We refer to them as monsters.

Finally, from the comments in the linked blog post, Steve adds:

Note these configurations have so much entropy (number of possible states) that they can't be accommodated in any kind of holographic dual description.

I'll say! So if these things exist—which apparently could be possible as a result of quantum tunneling from an ordinary matter configuration—then it looks like the holographic principle is doomed. At least, as far as I can tell...

Tying Things Together

How does this fit with our above derivation? Well, the authors do not mention an entropy/volume ratio at all, which at this point makes me think that all the cool physicists must be thinking about entropy/mass ratios instead and only us amateurs who are constantly referring to Wikipedia end up thinking about volume. But really, this is pretty immaterial: the main points don't relate to the entropy/volume ratio, but instead to the entropy/area ratio—because this is what's really of interest when we're trying to overthrow the holographic principle, for instance. Nevertheless, I'll try to extract some information on these monsters' densities, to tie things back to where we started.

They give two specific examples of such monster configurations, both spherically symmetric. The first, a "blob of matter," has S scaling with A, but interestingly enough has a density less than \rho_{\text{BH}} = \frac{3}{32 \pi m^2} = \frac{3}{32 \pi (r/2)^2} = \frac{3}{8 \pi r^2}. We can see this from the fact that at the very densest region, the uniformly-dense core (outside of which the density tapers off as (r_0/r)^2), the density \rho_0 is chosen to relate to r_0 via \rho_0 = \frac{1}{8 \pi r_0^2}. So this object has entropy greater than any ordinary matter (which is bounded by A^{3/4}, as discussed early in the paper), and indeed entropy on the order of a black hole, but has density less than a black hole, and so by our opening reasoning does not violate the (supposed) maximality of black-hole S/V. Interesting!

The second configuration consists of a thin shell of mass, with inner radius R and outer radius R + d. The authors give two instances of this: one which obeys the ordinary-matter bound S < A^{3/4}, and one which can take on arbitrarily high entropy, independent of the area, in the region r \in (R_1, R + d) for some R_1 \in (R, R + d). And this time, it appears to happen via arbitrarily high density. Actually, the details on this—which are too technical to really do here; ask me in the comments if you insist—are pretty interesting. They seem to imply that, by choosing differing values of d and R_1 for these shells, you would be dealing with a density that was either higher or lower than black hole density depending on the radius R of your object. For example, take d = 2, R_1 = R + d/2: then if R < 1.5 (approximately), the overall density of the shell is greater than the density of a black hole of the same radius, whereas if R > 1.5 then it is less.

This becomes really fascinating once you realize that we're still working in Planck units. So, that 1.5? That's 1.5 Planck lengths. A popular conjecture (i.e., something that people heuristically argue will follow from quantum gravity) is that all length must be quantized in integer multiples of the Planck length, and that the "minimum length" of anything is one Planck length. So what we have here, for this denser-than-a-black-hole shell, is R = 1, R_1 = 2, R + d = 3. That is, an empty sphere of radius 1, surrounded by a width-1 shell of bounded entropy, surrounded by another width-1 shell of arbitrarily high entropy. This is more or less a concrete picture of what such a monster configuration would look like, if I'm not mistaken.

Now What?

That was pretty fun. I'm going to have to write another blog post about the process of what I just did—find a thing, play with it, find a paper on the thing, work through it, come up with interesting stuff—because it's pretty much representative of the sort of free-to-dabble "research" that undergraduates get to indulge in. (Other groups in academia might do this too, but I would imagine after a certain point you start becoming locked in to your specialty?)

But seriously. Where do I go from here? Well, I'll certainly attend that seminar next Thursday—I'll have to make sure to wake up before 16:00, but that's usually doable. I'll play around with these equations some more, and see if I can get any more conceptual insight into both "monsters" and black holes in general. (I encourage you to do the same!) I'll follow some references in that article: the sentence "see [18] for a discussion of highly entropic objects and their effect on black hole thermodynamics" points to a particularly intriguing-looking paper.

And, well, I'm almost embarrassed to admit this, but despite having just discoursed extensively on black holes, I still don't really know general relativity. So I'd better continue moving through my (excellent) textbook on the subject. I'm about a third of the way through... two more chapters until we start talking about curved spacetime. I think I'd better get on that.

P.S.: Kudos to anyone who got the (probably very obscure) reference that the title was making.

P.P.S.: there's a video too!! Be sure to listen to the last ten seconds. Most excellent.

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Categories: Contains Math, General Relativity, Thermodynamics

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