Two years ago or so, the relativistic community - or more precisely the loop quantum gravity sub-community - was excited about the quasinormal modes. I was asked to explain what was the fuss all about and also why I had anything to do with it. It's a rather interesting story - one whose original goals were doomed from the very beginning but one which may still hide some secrets of Nature - namely mysteries of quantum gravity.
What are quasinormal modes?
First of all, let me say what a quasinormal mode is. Start with quantum mechanics - a particle in the potential which confines it classically, but allows it to tunnel out quantum mechanically (like one of these toy models of alpha-decay - an alpha particle confined in the nucleus). In this case, there are "metastable" modes in which the particle sits in the "hole" for quite a long time, but eventually it escapes by quantum tunneling (the rate of the tunneling exponentially depends on the parameters, which is why the lifetime of various alpha-decays are so incredibly diverse in size).
Mathematically, one can find an associated exact formal solution of the Schrodinger equation with a complex value of energy. The constraint is that at infinity, the particle's wave function is "purely outgoing". The exponentially growing (with x) and oscillating component of the solution at infinity is allowed, but the second linearly independent solution of the equation, the exponentially decreasing and oscillating function, is not allowed. Such a condition is kind of "opposite" but equally constraining as the condition for convergence (decrease at infinity) of the normalizable wave functions of the bound states, and it only has discrete solutions that are called the quasistationary states. Much like the bound states, they describe poles in the transmission amplitude.
The equations for small perturbations of some fields (scalar fields, gravitational fields, other fields) propagating on some background geometry are linear equations that are mathematically analogous to the Schrodinger equation. The solutions of these equations with the same boundary conditions as in the case of quasistationary states are called "quasinormal modes", in analogy with the previous case. The term "normal modes" would indicate that the functions are eigenstates, and "quasi-" means that the eigenvalue may be complex. In both cases, the quasistationary states as well as the quasinormal modes, the imaginary part of the frequency eigenvalue describes the exponential decrease of the wavefunction in time. In the case of the particle in quantum mechanics, such a decrease reflects the decaying probability with time that the particle is "still there" - and the increase at "x = infinity" means that for most of the future, the particle is gone at infinity. The interpretation of quasistationary modes as metastable states is only good if the imaginary part of the frequency is pretty small, but surprisingly, the real focus of this text will be modes with a huge value of the imaginary part - those whose direct physical interpretation is much less direct.
The role of quasinormal modes in the context of string theory and especially the AdS/CFT correspondence was studied in many serious papers, starting from pioneering works such as the article of Horowitz and Hubeny. They will not be the focus of this article.
Black hole entropy in loop quantum gravity
In late 1995, string theorists became able to compute the black hole entropy statistically from the microscopic description of the objects in terms of the ingredients found in string theory. Strominger and Vafa designed the first three-charge supersymmetric black hole in five large dimensions (three charges are needed to make the horizon area positive) whose entropy, including the numerical coefficient 1/4, could be calculated both from general relativity as well as from counting of states of D-brane excitations. Their black hole was constructed from D5-branes, D1-branes, and some units of momenta. At weak coupling (small g), the perturbative D-brane calculations are valid and doable and the states may be counted as in statistical mechanics. At strong coupling (large g), the geometry back-reacts, becomes curved, and the same object is described as a black hole; the thermodynamic tools of Bekenstein and Hawking may be applied. A complete agreement is found.
Many other developments followed. Near-extremal, rotating, and other black holes and black strings were also confirmed. Whole classes of non-supersymmetric black holes in different stringy backgrounds matched up to a numerical pre-factor. Previous articles on this blog about black hole thermodynamics include
Recently, the entropy of a 7-parameter black ring - a black hole of nontrivial topology - has been checked. An exact agreement with the Bekenstein-Hawking formula
- S (entropy) = A (area) / 4G (Newton's constant)
was always found (we always use units in which c=hbar=k_{Boltzmann}=1). Back in 1995-96, loop quantum gravity wanted to offer something comparable to the successes of string theory. The goal was to calculate the entropy of all black holes. However, there was a problem:
- The states in loop quantum gravity are classified by spin networks, and in the hyper-optimistic case in which a nearly flat space follows from loop quantum gravity at long distances, the large degeneracy of spin networks carries a volume-extensive entropy, instead of the area-extensive entropy that is needed to explain the black holes. (Well, the Hilbert space is really non-separable, and therefore the entropy is heavily infinite and physically ill-defined, but let's not discuss reality - our focus will be on loop quantum gravity fantasies.)
Is there some way how an area-extensive result for the entropy may be recovered from such a volume-extensive starting point? Sure, there is. Simply assume that the entropy only comes from the horizon. Ignore the entropy of the black hole interior as well as the rest of the Universe. There is another problem:
- There is nothing special about the horizon, and it therefore contributes no extra entropy.
This problem is also easy to solve. Simply add new degrees of freedom at the horizon. Loop quantum gravity wants to describe the four-dimensional gravitational field as an SU(2) gauge field. The horizon is a 3-dimensional submanifold of the spacetime, so why don't we put some three-dimensional theory with a gauge field - namely Chern-Simons theory - on the horizon? This is a pretty natural thing to do.
Indeed, when we do it, the entropy of the horizon - calculated from the Chern-Simons theory - will be proportional to the area. That's a fascinating result that can be summarized as follows:
- The black hole entropy in loop quantum gravity may be shown to be proportional to the horizon area assuming that we subtract all contributions that are not proportional to the area, and assuming that we attach new degrees of freedom to the area.
Another thing we would like to check is the numerical coefficient. Recall that we want to obtain the Bekenstein-Hawking formula
- S = A / 4G
In the Chern-Simons theory, however, the entropy will be dominated by the intersections of the horizon with the spin network, especially those the links that carry "j=1/2" (these have maximal entropy per area). At least, it was believed in the middle 1990s that the higher spin intersections could be neglected, and we will clarify this point later in the text.
The object with spin "j=1/2" has two possible microstates (polarizations), namely "m=+1/2" and "m=-1/2". Correspondingly, the entropy from such an object (a quantum of the area) is "ln(2)". The result is that the coefficient calculated in loop quantum gravity was proportional to "ln(2)", instead of "1/4" (and some powers of pi, and so forth). Obviously, it does not work. How do you repair the overall coefficient? Well, you just insert an extra parameter "gamma" to various formulae, and call it the Barbero-Immirzi parameter. This parameter, if chosen properly, fixes the incorrect ratio between the right entropy and the calculated entropy.
Now, the value of "gamma" for which the Hamiltonian constraint simplifies is "gamma=+i" or "gamma=-i". For all other choices of "gamma", the Hamiltonian constraint of general relativity, translated to the "new variables", gives you such a non-linear, UV-singular mess that the virtues of the "new variables" evaporate completely. This was explained e.g. in a meaningful paper on loop quantum gravity. But of course, as a loop quantum gravity theorist, you don't care because the Hamiltonian constraint is a hopeless mess anyway and because of the principle that says that
- loop quantum gravity is a sequence of research steps in which the previous not-too-promising speculations are ruled out and identified as nothing more than a motivation for other, even less likely, less intriguing, and more hopeless speculations.
When you assume that "gamma" is something like "ln(2)/sqrt(3).pi" or so, you get the right proportionality factor to agree with the Bekenstein-Hawking formula. Note that there is not a single consistency check or a nice surprise in either of these calculations. The entropy came out proportional to the area only because we suppressed all contributions that are not proportional to the area and added a new artificial term, absent in the original theory, that is proportional to the area. And the pre-factor is only right because we rescaled it by the multiplicative discrepancy to make it right. This is not what Richard Feynman would call science, for example.
When a leader of loop quantum gravity, a very nice person, was visiting Harvard in late 2002, I told her or him that the proposed entropy calculation can only be interesting and non-trivial if the coefficient "gamma" is explained by other means. She or he said that it was a great idea to try to calculate this coefficient differently. She or he had never thought about it before, I was told. ;-)
It was Friday. In the morning, she or he explained me that the three-dimensional Chern-Simons theory was equivalent to M-theory. That was a very surprising conjecture, so I was asking why should this be the case. The answer was something like this:
- I don't believe that M-theory is complicated. M-theory must be simple. The Chern-Simons theory is also simple, and therefore they're equivalent.
Then we went to have lunch in the Society of Fellows. She or he asked me whether I was interested in politics. She or he informed me that she or he had also done scientific research of political science, and the main question beyond her or his research was Why is it exactly so that the U.S. society was such a complete disaster?
That was a very nice discussion - you may imagine what I think about those who try to paint this sort of opinions as "deep scientific research"- so I asked: Why don't we return to physics? So she or he told me that she or he considered the dualities in string theory and the whole second superstring revolution to be uninteresting, and too much ado about nothing. Very nice. So I suggested, for the sake of peace, that we would return to our discussions about politics. ;-) At lunch, she or he asked the mathematicians in the Society whether they agree with her or him that the axioms about integers become inconsistent when sufficiently large numbers are looked at.
Three weeks later or so, I received an e-mail from her or him that stated that the problem of the value of "gamma" had been solved. The solution was written in a provoking paper by Olaf Dreyer which was equivalent to the following argument:
- If we change the gauge group from SU(2) to SO(3), then the minimal allowed "j" in the spin network will be "j=1" which carries three polarizations, i.e. the entropy dominated by the minimal spin links is going to change to "ln(3)". Because "S=A/4G", the quantum of area must be "4.ln(3).G". When the four-dimensional Schwarzschild black hole reduces its area by "4.ln(3).G", its mass decreases by "ln(3).T_{Hawking}" by the well-known simple relation between the mass and the area. This is not a random value of mass (or frequency): numerically, it was found to be the asymptotic real part of the highly-damped quasinormal frequencies of a neutral black hole. That's therefore an independent calculation of the parameter.
Of course I did not believe that the black hole evaporates a delta-function-like spectrum whose frequencies were multiples of "ln(3).T_{Hawking}" which is as non-thermal as you can get. Such a massive violation of thermality would force us to alter Hawking's calculation drastically, and if we did so, the resulting entropy would be different anyway and the original motivation (a numerical agreement) for altering Hawking's calculation would be absent. But the agreement between these two numbers was definitely an interesting numerical coincidence - it was not obvious why the same number "ln(3)/8.pi" should have been found by two very different ways, and this coincidence could have meant something.
The value of "ln(3)" appearing in the quasinormal modes was only known numerically and Shahar Hod noticed in a heuristic paper from 1998 that the value was probably "ln(3)". Matthew Schwartz and Alex Maloney taught me how to use the cycles in Mathematica and they reconstructed a simple program for the recursive calculation of the continued fractions, originally done by Leaver and Nollert. But the real goal was not just to play with the old numerical simulations, but to get an analytical answer which took a few weeks. The first working method to prove that "ln(3)" appears in the quasinormal modes was based on the analysis of the behavior of the continued fractions and their critical behavior - plus patching different partial solutions together. The second proof which uses analytical continuation and the monodromy method was completed by Andy Neitzke and by me after having tortured ourselves for three weeks with the details of the proof, properties of Bessel's functions, and the weird Stokes' phenomenon. The functional methods of the latter paper were more flexible and allowed us to calculate also the Reissner-Nordstrom black hole and the higher-dimensional Schwarzschild black holes.
What happened afterwards?
People have computed various types of the asymptotic quasinormal modes for various black holes in various gravitational theories; in the asymptotic case, they were using the monodromy method. The number "ln(3)" was shown to occur for some excitations of different spin but not others; it was shown that in various modified theories of gravity, this number changes with the coupling, and so forth. The rotating black hole was analyzed in several papers, for example in a paper by
One side of the conjectured duality "quasinormal modes - black hole thermodynamics" collapsed because more reliable computations falsified all the simple conjectures.
On the other hand, the old calculation of the black hole entropy in loop quantum gravity turned out to be flawed, too. The original assumption from the middle 1990s that the spin "j" higher than "1/2" could be neglected was shown erroneous by the Polish physicists
whose improved calculation does not neglect the higher spins and replaces "ln(2)" by a completely transcendental number. Even these new calculations from July 2004 are argued to be incorrect in the papers from November 2004 by
who propose yet another transcendental result. It's not quite clear to me whether the newest papers are really the most reliable ones. But one thing is clear: the hard-science calculations can't give the answers that would be encouraging for loop quantum gravity in any rational sense. As far as science goes, loop quantum gravity seems to be pretty safely falsified as a candidate for a consistent quantum theory of gravity including black holes.
However, there remain many interesting topics and observations connected with these calculations of quasinormal modes:
- the "monodromy method" calculation depends on an analytical continuation near the black hole singularity even though the result is a feature of the quasinormal modes that propagate between the horizon and infinity. Does it mean that the exterior of the black hole knows about the interior? Is the interior of the black hole, in some sense, an analytical continuation of the exterior? Can it solve the black hole information puzzles?
- the asymptotic spacing of the imaginary part of the quasinormal frequencies seem to be "2.pi.T_{Hawking}" every time they behave periodically in the highly-damped limit. This shows that the modes "know" about the temperature of the black hole - the corresponding Green's functions resemble the thermal Green's functions in a certain limit. Is there a complete understanding of the behavior of the imaginary parts?
