The Quantum of Area?

John C. Baez

Published in Nature 421 (Feb. 13 2003), 702-703.

Among the many curious features of quantum mechanics, one of the first to be discovered was that atoms have a discrete spectrum of energy levels. Could the same be true for black holes? To know for sure, we would need to understand how quantum mechanics and general relativity fit together. But we can already guess the answer, just as Bohr guessed a lot about the spectrum of hydrogen before quantum mechanics was fully worked out. And late last year, much to the shock of the experts, two completely different ways of guessing gave the same answer, for reasons that are still mysterious [1,2].

It all started in 1975, when Hawking [3] predicted that black holes aren't quite black. Taking quantum mechanics into account, they should glow slightly, as if they had a nonzero temperature. Nobody has ever seen this "Hawking radiation" - it's far too faint - but his calculation has been checked in many ways, and seems unassailable. The question is, what to make of it?

If black holes have a temperature, we can study them using thermodynamics. In particular, we can compute their entropy! Confirming Bekenstein's hope [4], the entropy of a black hole turns out to be proportional to its surface area. Even better, the constant of proportionality can be worked out exactly.

Now, the entropy of a system says how many different states it can be in while looking the same on a large scale. For a box of gas, these different "microstates" are easy to explain: they are all the ways the atoms can be wiggling about. But what are the microstates of a black hole?

This question has tortured physicists for decades now, and dozens of answers have been suggested. Since the entropy of a black hole is proportional to its surface area, one might guess that the microstates describe the geometry of its surface. This is not a physical object in the usual sense: it's an imaginary boundary called the "event horizon", which says how close you can get before inevitably being sucked in. Still, in many ways it acts like a flexible membrane [5]. But to count its microstates and get a sensible answer, we must describe it using quantum mechanics.

Some progress along these lines comes from "loop quantum gravity" [6]. In this theory, the fabric of space is like a weave of tiny threads, and area comes in discrete units: each thread poking through a surface gives it a little bit of area [7]. In particular, the surface area of a black hole comes from all the threads puncturing it. The event horizon is flat except at these punctures, where it can flex. The microstates of the black hole describe the different ways the event horizon can flex in or out.

In 1997, Ashtekar and collaborators tried to calculate the entropy of a nonrotating black hole using this theory [8]. But in the process, they had to face up to a curious feature of loop quantum gravity. At the time, this theory was unable to predict the "quantum of area": the smallest amount of area carried by one of the threads from which space is woven. Without knowing this, they could check that the entropy of a black hole is proportional to its area, but they couldn't compute the constant of proportionality.

But when life gives you lemons, sometimes you can make lemonade. After all, Hawking had already worked out this constant of proportionality by other means. So, Ashtekar et al could reverse their calculation and use Hawking's work to determine the quantum of area!

Here things get downright spooky. One year later, Hod [9] calculated the quantum of area a completely different way. Instead of loop quantum gravity, he used a simple argument borrowed from Bohr's work on the hydrogen atom. He assumed that like an atom, a black hole has a discrete spectrum of energy levels. Using computer calculations of the vibrational frequencies of black holes [10], he guessed the spacing between these energy levels. But for a nonrotating black hole, its energy determines its surface area. Using this, Hod was able to work out the quantum of area. His result was 4.39444 times the Planck area (itself a puny unit of area, roughly 10-70 square meters). Finally, in a daring bit of numerology, he observed that this number was very close to 4 times the natural logarithm of 3.

It would be nice to report that this result matched that of Ashtekar et al. Unfortunately, it did not. But last November, Dreyer [1] made a remarkable discovery. If one modifies the loop quantum gravity calculation slightly - in a way that is quite reasonable, but too technical to describe here - it gives a quantum of area equal to exactly 4 ln(3) times the Planck area!

At this point the suspense became almost unbearable: Hod's observation relied on numerical calculations, so the very next digit of his number might fail to match that of 4 ln(3). Luckily, soon after Dreyer's work, Motl [2] showed that the match is exact.

While exciting, these developments raise even more questions than they answer. Nobody knows why the two calculations agree, nor how to extend them to rotating black holes. Could it all be just a coincidence, or have we discovered the quantum of area? Only time will tell - or perhaps space.

[1] Olaf Dreyer, Quasinormal modes, the area spectrum, and black hole entropy, to appear in Phys. Rev. Lett.

[2] Lubos Motl, An analytical computation of asymptotic Schwarzschild quasinormal frequencies.

[3] Stephen Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975), 199-220.

[4] Jacob Bekenstein, Black holes and entropy, Phys. Rev. D7 (1973), 2333-2346.

[5] Kip Thorne, Richard Price, and Douglas Macdonald, eds. Black Holes: the Membrane Paradigm, New Haven, Yale U. Press, 1986.

[6] Carlo Rovelli, Loop quantum gravity, Living Reviews in Relativity 1, 1 (1998).

[7] Carlo Rovelli and Lee Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys. B442 (1995), 593-622. Erratum: Nucl. Phys. B456 (1995), 734.

[8] Abhay Ashtekar, John Baez, Alejandro Corichi and Kirill Krasnov, Quantum geometry and black hole entropy, Phys. Rev. Lett. 80 (1998), 904-907.

[9] Shahar Hod, Bohr's correspondence principle and the area spectrum of quantum black holes, Phys. Rev. Lett. 81 (1998), 4293-4296.

[10] Hans-Peter Nollert, Quasinormal modes of Schwarzschild black holes: the determination of quasinormal frequencies with very large imaginary parts, Phys. Rev. D47 (1993), 5253-5258. © 2003 John Baez