Shahar Hod's bound on the fine-structure constant

  • #1
mitchell porter
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TL;DR Summary
does the fine structure constant saturate a quantum bound on the properties of charged black holes?
Shahar Hod is an Israeli physicist who has specialized in heuristic reasoning about the properties of quantum black holes, one might say in the tradition of Bekenstein and Hawking. By this I mean that, rather than trying to develop a detailed theory of quantum gravity, he has tried to obtain results through the use of general principles, e.g. the correspondence principle.

His biggest hit (over 800 citations) is a 1998 paper in which he argues that the areas of black holes (in natural units) are quantized in multiples of 4 hbar ln 3. For a while (Dreyer 2002, backstory here) there was excitement that loop quantum gravity could provide a detailed theory of quantum gravity in which this is true. That faded away, but Hod's work did lead to proofs (Motl 2002, Motl and Neitzke 2003) that Hod's factor of ln 3 appears in the frequencies of certain ringing modes of black holes ("quasinormal modes" that can lose energy e.g. via gravitational waves). To this degree, I believe, his 1998 work led to something that is now generally accepted.

Meanwhile, I have only now come across a 2010 work by Hod

"Gravitation, Thermodynamics, and the Fine-Structure Constant"

in which he adds a few extra principles and obtains a bound on the fine-structure constant that is extremely close to the measured value. The only comment on this argument that I can find, is a post by Lubos Motl. Lubos expresses his respect for Hod's work, says that his 2010 argument is similar in spirit to some of the hypotheses coming out of the "swampland" research program in quantum gravity, but that nonetheless he doesn't believe this particular argument.

Lubos offers two reasons for disagreeing. The most cogent is as follows. One of Hod's principles is that the minimum mass of a black hole should be one Planck mass. Lubos agrees that the lightest black holes should be of order the Planck mass, but the coefficient doesn't have to exactly equal 1, it's just going to have that order of magnitude, and that spoils the exactness of Hod's deduction.

His other reason is that he thinks there would be vacua in string theory, in which the fine-structure constant would violate Hod's bound, e.g. he considers alpha ~ 1/150 to be quite conceivable. But he doesn't go into any detail. I guess his reasoning is just "in the real world, the coupling starts at something like 1/24 at the GUT scale and runs down to 1/137 for electromagnetism, and I don't see why it couldn't have gone a bit further".

But the closeness of reality to Hod's bound - and the simplicity of his argument - is striking to me. I think it deserves a second look. And we are supposed to be in a new era of understanding the evaporation of near-extremal charged black holes; we had a thread about it just last month. If I had time, I would definitely be trying to evaluate Hod's argument from the perspective of JT gravity models of Reissner-Nordstrom black holes, which are the kind of black hole discussed in the thread, and also the kind of black hole which features in Hod's argument.

(P.S. For some reason I can't get the math markup right, if I get time I will fix it later.)
 
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  • #2
mitchell porter said:
one might say in the tradition of Bekenstein and Hawking.
As a complete aside, both of these men are now deceased, having died long before the life expectancy they had at birth. Many key mentors of both men survived them.
 
  • #3
I've studied a bit more of Hod's argument. He's looking at RN black holes, so charged, non-rotating black holes which have an inner and outer horizon that coincide when the mass radius equals the charge radius.

If the black hole emits a quantum with specific charge and mass, the black hole charge and mass will change, and so will the RN geometry. In particular we care about the distance between inner and outer horizons. They coincide if the mass radius and charge radius are the same. This actually means that the event horizon goes away, leaving just a naked singularity. This is often considered something to avoid, and so people seek a mechanism to enforce what Penrose called "cosmic censorship", according to which singularities are always hidden behind event horizons.

Hod instead focuses on the Hawking temperature of the black hole, which goes to zero when the horizons coincide. Hod says it would violate the third law of thermodynamics if the temperature can go to absolute zero in finite time, therefore Hawking radiation that would merge the two horizons must be forbidden. So in the end it's just another motivation for cosmic censorship (the classical motivation was that determinism fails at the singularity).

Hod has what I think is a formula for the de Broglie wave of a charged quantum of Hawking radiation in a given RN geometry. When the black hole emits a quantum of specific charge and mass, the black hole's own charge and mass change, and so does the geometry. So we're looking at a model of Hawking radiation and its consequences, similar to the "old quantum theory" of the hydrogen atom: stochastic transitions between Bohr orbits.

Hod restricts himself to RN black holes with integer charge - you could say that the corresponding RN geometries are his Bohr orbits - and then his resonance formula for the charged field describes a possible instance of Hawking radiation. We are to assume that Hawking radiation which would remove the event horizon is forbidden, and then we deduce the consequences.

In what follows, the number ##ln(3)/(4\pi)## (derived from a coefficient in his resonance formula) is important. First, its reciprocal provides a lower bound on the charge of the black hole: 12 times the charge of the electron (or proton, for positively charged black holes). Then, his lower bound on the fine structure constant is 1/12 times this number, i.e. ##ln(3)/(48\pi)##. And as he observes, that's less than 0.2% below the actual value - so it seems that, for whatever reason, our world is close to saturating the bound.

I find it remarkable that this paper has provoked no online commentary, except for a Stack Exchange remark by Lubos Motl. If we take it seriously, we'd need to ask why is ##\alpha## so close to the bound? Hod's argument is just that there is a bound, not that the bound is saturated. Probably one could fashion an anthropic argument, but to me it suggests that the fine structure constant is dynamical, with some mechanism placing it at the edge of cosmic censorship violation; reminiscent of the criticality of the Higgs boson mass.
 
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  • #4
Hod has a paper today on magnetic black holes with electroweak hair. The best introduction to this topic might be Maldacena 2020... I mention this simply because it would be good to reproduce the argument for the bound in this context; who knows what else would turn up.

edit: The point is that Hod's bound derives from consideration of electrically charged black holes. But in the real world, electromagnetism is embedded within the electroweak unification, and these magnetically charged black holes can be surrounded by an "electroweak corona" within which electroweak symmetry is restored.

Another interesting aspect of these black holes is that they resemble GUT monopoles. Maldacena says they are a way to obtain magnetic monopoles within standard model + gravity, without a need for grand unification. Hod already argues that considering the electric charge of black holes can tell us something about the electromagnetic coupling; can the electroweak hair of black holes similarly imply e.g. something about the Weinberg angle?
 
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  • #5
mitchell porter said:
Hod has a paper today on magnetic black holes with electroweak hair. The best introduction to this topic might be Maldacena 2020... I mention this simply because it would be good to reproduce the argument for the bound in this context; who knows what else would turn up.

edit: The point is that Hod's bound derives from consideration of electrically charged black holes. But in the real world, electromagnetism is embedded within the electroweak unification, and these magnetically charged black holes can be surrounded by an "electroweak corona" within which electroweak symmetry is restored.

Another interesting aspect of these black holes is that they resemble GUT monopoles. Maldacena says they are a way to obtain magnetic monopoles within standard model + gravity, without a need for grand unification. Hod already argues that considering the electric charge of black holes can tell us something about the electromagnetic coupling; can the electroweak hair of black holes similarly imply e.g. something about the Weinberg angle?
I saw the paper but didn't make the connection. Good catch.
 
  • #6
mitchell porter said:
Hod has a paper today on magnetic black holes with electroweak hair. The best introduction to this topic might be Maldacena 2020... I mention this simply because it would be good to reproduce the argument for the bound in this context; who knows what else would turn up.

edit: The point is that Hod's bound derives from consideration of electrically charged black holes. But in the real world, electromagnetism is embedded within the electroweak unification, and these magnetically charged black holes can be surrounded by an "electroweak corona" within which electroweak symmetry is restored.

Another interesting aspect of these black holes is that they resemble GUT monopoles. Maldacena says they are a way to obtain magnetic monopoles within standard model + gravity, without a need for grand unification. Hod already argues that considering the electric charge of black holes can tell us something about the electromagnetic coupling; can the electroweak hair of black holes similarly imply e.g. something about the Weinberg angle?
I guess those monopoles are magnetic.
So many papers on something that may not exist in nature. :cool:
 
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  • #7
mad mathematician said:
I guess those monopoles are magnetic.
So many papers on something that may not exist in nature. :cool:
The claim is actually that monopole-like magnetic black holes can form in a standard model universe, even though there are no fundamental magnetic monopoles in the standard model. However, the devil is in the details.

Along with the Reissner-Nordstorm solution for an electrically charged, non-rotating black hole, there are "dyonic" solutions in which the black hole has both electric and magnetic charge. The math is the same, you just reinterpret the charge as a dyonic combination of electric and magnetic, rather than purely electric.

However, the question is whether a magnetically charged black hole can form, if physics doesn't have any fundamental magnetic monopoles, only electric charges and magnetic fields. Here I should mention the two main types of monopole in theoretical physics, the Dirac monopole which is a point object, and the 't Hooft-Polyakov monopole which is smooth.

If the only way to get a standard model magnetic black hole was via a Dirac monopole configuration, then it might actually be unphysical, in the sense that you could write it down mathematically, but it couldn't actually form. In my search for a way to form magnetic black holes, the only method I have found so far is mentioned in

"Nontopological Magnetic Monopoles and New Magnetically Charged Black Holes" (K. Lee & E. Weinberg 1994)

which says on page 10 that in a theory without fundamental monopoles, magnetic black holes would need to be formed in pairs by a kind of quantum tunneling during the decay of larger black holes.

If a magnetic black hole did manage to form in such a universe, it would be far longer-lived than the electric black hole, because the electric black hole can lose charge by emitting electrically charged particles. But if there are no magnetically charged particles, the magnetic black hole can't lose its magnetic charge except by emitting smaller magnetic black holes, or perhaps by some quantum tunneling process that is not clear to me.

Another interesting aspect of this, is that magnetic monopoles - and dyons, the dyonic perspective of objects carrying electric and/or magnetic charges is really the most general one - are commonplace in grand unified theories. I found it very intriguing that even in a theory without grand unification, but with gravity, dyon-like objects might nonetheless exist. It would suggest a deep principle of quantum field theory at work. But it's still not completely clear to me whether dyonic black holes really can form even in the standard model, or whether they are just formal classical possibilities (i.e. you just suppose that the black hole singularity has magnetic charge as well as mass, electric charge, and angular momentum).

Since we are talking about electroweak, another thing that deserves to be mentioned is the so-called "electroweak monopole" discovered by Maison and Cho. I believe this was initially a singular field configuration similar to the Dirac monopole, but there are claims that it has a smooth form with a finite mass that might be detectable around a few TeV. In the present context, just as magnetically or dyonically charged black holes appear to be natural monopoles/dyons (especially if we think about micro black holes with magnetic or dyonic charge), these magnetic black holes with an electroweak corona seem to be "giant Maison-Cho electroweak monopoles".

Incidentally, the electroweak symmetry restoration is predicted to occur in sufficiently strong magnetic fields. There is transition from the magnetic fields in the usual standard model vacuum with a nonzero Higgs vev, to hypermagnetic fields in a vacuum where electroweak symmetry has been restored - hypermagnetic fields being the magnetic fields of the hypercharge gauge field.

One final note: Maldacena seems to have initially studied magnetic black holes in the context of trying to make a wormhole out of entangled black holes. In the thread on evaporation of extremal electrically charged black holes, we discussed how astrophysically unnatural they are. But if such extremal holes are part of the process of making magnetic black holes, which can then be entangled to create wormholes, that would provide an incentive for a galactic civilization to technologically create the artificial conditions under which they might be produced. Although there is a downside, the time to travel through these traversable Maldacena wormholes is at least as long as it takes to travel at light speed the normal way. So no time travel or causality violation!
 
  • #8
mad mathematician said:
So many papers on something that may not exist in nature. :cool:

Yeah, like neutrinos, or Higgs boson, or gravitational waves... Oh, wait...
 
  • #9
weirdoguy said:
Yeah, like neutrinos, or Higgs boson, or gravitational waves... Oh, wait...
You want to tell me that every physical phenomena ever suggested to exist in the literature must exist in nature?!
:oldbiggrin: I understand you username now...
 
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