Monday, April 08, 2013

Black holes and the Planck length

According to Special Relativity, an object in motion relative to you appears shortened. The faster it is, the shorter it appears. This is effect known as Lorentz-contraction. According to General Relativity, an object that has a sufficiently high mass-density in a small volume collapses to a black hole. Does this mean that if a particle moves fast enough relative to you it turns into a black hole? No, it doesn't. But it's a confusion I've come across frequently. Take Craig Hogan's recent "paper", where he writes:
"[B]elow the Planck length... it is no longer consistent to ignore the quantum character of the matter that causes space-time to curve. Even a single quantum particle of shorter wave-length has more energy than a black hole of the same size, an impossibility in classical relativity..."
The wave-length of a particle depends on the motion you have relative to it. For every particle there is a reference frame in which the wave-length of the particle appears shorter than the Planck length. If it was true what Hogan writes, this would imply that large relative velocities are a problem with classical general relativity. Of course they are not, for the following reasons.

A black hole is characterized by the existence of an event horizon. The event horizon describes the causal connectivity of space-time. It's a global property. Describing an object from the perspective of somebody moving relative to this object is a coordinate transformation. A coordinate transformation changes the way the physics appears, but not the physics itself. It just makes things look different. You cannot create an event horizon by a change of coordinates. Ergo, you cannot create a black hole just by looking at a particle that is moving rapidly relative to you.

There are three points I believe contribute to this confusion:

First, one can take the Schwarzschild metric for a black hole and describe it from the perspective of an observer moving relative to it. This is known as the Aichelburg-Sexl metric. The Aichelburg-Sexl metric is commonly used to handle black hole formation in particle collisions. The argument about the Planck length being a minimal length makes use of black hole formation too. But note that in these cases there isn't one, but at least two particles. These particles have a center-of-mass energy. They create a curvature which depends on the distance between them. They either do or don't form an horizon. These are statements independent on the choice of coordinates. This case should not be confused with just looking at one particle.

Second is forgetting that black holes have no hair. Leaving aside angular momentum, they're spherically symmetric which implies there are preferred frames. Normally one uses a frame in which the black hole is in rest, which then leads to the normal nomenclature with the Schwarzschild radius and so on.  But you better don't apply an argument about concentrating energy inside a volume that you'd have in the static case to the metric in a different coordinate system.

Third is a general confusion about the Planck length being called a "length". That the Planck length has the dimension of a length does not mean that it behaves the same way as a length of some rod. Neither is it generally expected that something funny happens at distance scales close by the Planck length - as we already saw above, this statement doesn't even have an observer-independent meaning.

The Planck length appears in General Relativity as a coupling constant. It couples the curvature to the stress-energy tensor. Most naturally, one expects quantum gravitational effects to become strong, not at distances close by the Planck length, but at curvatures close to one over Planck length squared. (Or higher powers of the curvature close to the appropriately higher powers of the inverse Planck length respectively.) The curvature is an invariant. This statement is therefore observer-independent.

What happens in the two particle collisions is that the curvature becomes large, which is why we expect quantum gravitational effects in this case. It is also the case that in the commonly used coordinate systems these notions agree with each other. Eg, in the normal Schwarzschild coordinates the curvature becomes Planckian if the radius is of Planck length. This also coincides with the mass of the black hole being about the Planck mass. (No coincidence: there is no other scale that could play a role here.) Thus, Planck mass black holes can be expected to be quantum gravitational objects. The semi-classical approximation (that treats gravity classical) breaks down at these masses. This is when Hawkings calculation for the evaporation of black holes runs into trouble.

For completeness, I want to mention that Deformed Special Relativity is a modification of Special Relativity which is based on the assumption that the Planck length (or its inverse respectively) does transform like the the spatial component of a four-vector, contrary to what I said above. In this case one modifies Special Relativity in such a way that the inverse of the Planck length remains invariant. I've never found this assumption to be plausible for reasons I elaborated on here. But be that as it may, it's an hypothesis that leads to consequences and that can then be tested. Note however that this is a modification of Special Relativity and not the normal version.


  1. Donald Moffitt's sci-fi, Crescent in the Sky, A Gathering of Stars - apart from imagining an interstellar human culture where Islam predominates, also uses this "paradox".

    One could ask - can a massive particle moving sufficiently fast relative to the Cosmic Microwave Background develop a horizon? I think the answer is no, but consider the question posed.

  2. "an object in motion relative to you appears shortened." Viewing angle interacts - Terrell rotation,
    Proc. Camb. Phil. Soc., 55 137 (1958)
    Phys. Rev. 116(4) 1041 (1959)

    "not at distances close by the Planck length, but at curvatures close to one over Planck length squared." At per area, sounding like surface maximum information density in black hole thermodynamics.

    Do something densely energetic like pedal to teh firewall RHIC or LHC with Pb-208 (zero nuclear spin); muons or electrons in a long opposed linear collider. A Planckian singularity cannot ingest faster than it decays (water swirling down a drain).

    Coin funnels are hyperbolic, gravitating space is elliptic. The only way to know is to look.

  3. This comment has been removed by the author.

  4. You could however compare the *Compton wavelength* of a massive particle to its Schwarzschild diameter:

    \hbar/Mc and 4GM/c^2

    The mass of a particle for which these are equal is twice the usual Planck mass. A massive particle may be used as a probe of lengths comparable to (or smaller than) its Compton wavelength, but in this case that cannot be done.

    Perhaps I am being generous, but the quote you provided was sufficiently vague that it could be talking about this possibility.

  5. Amitabha beat me to it. The minute I saw it I thought that the guy has Compton wavelength in mind and its relation to Schwarzschild radius.

    Otherwise it doesn't make much sense...

  6. Giotis, Amitabha,

    Yes, you can. That's the estimate I did here, one of the easiest ways to arrive at the Planck scale. I don't know what he had in mind, but whatever it was the way he expressed it is nonsense. Best,


  7. Hi Arun,

    Since the CMB is dilute but physically present, there's a tiny but nonzero probability that your massive particle, if sufficiently fast, will hit a CMB photon and form a black hole. So then it has a horizon. But I'm not sure that's what you had in mind. Best,


  8. Who is this Hogan guy? The statement you quoted would probably be rejected from the sci.physics.research newsgroup as "too speculative" (a euphemism for "not even wrong"). Sounds like typical crackpot stuff.

  9. OK, I thought I remembered him from a previous post here. Let's see: "Craig Hogan is Director of the Fermilab Center for Particle Astrophysics, where he is also a member of the scientific staff and the Theoretical Astrophysics Group. He is also a professor at the University of Chicago, where he is on the faculty of the Department of Astronomy and Astrophysics, the Enrico Fermi Institute, and the Kavli Institute for Cosmological Physics.
    Hogan earned his doctorate at the University of Cambridge in 1980. After postdocs at Chicago, Caltech and Cambridge, he joined the faculty of Steward Observatory at the University of Arizona in 1985. In 1990 he moved to the Physics and Astronomy Departments at the University of Washington in Seattle, where in time he served as department Chair, divisional Dean, and Vice Provost for Research. He moved to Chicago in 2008.

    Hogan’s theoretical work has encompassed many areas of astrophysical cosmology: the origin of the elements, cosmic phase transitions and defects, magnetic fields, background radiation, cosmic reionization, gravitational lensing, cosmic structure and dark matter, global cosmological parameters, and gravitational waves. He was a co-founder of the Large Synoptic Survey Telescope Corporation, and is a US member of the LISA International Science team, which is planning a gravitational wave observatory in space. His research has been recognized by prizes including an Alexander von Humboldt Research Award, and the Gruber Cosmology Prize, awarded to the High-z Supernova Search Team for the co-discovery of cosmic Dark Energy."

    OK, more credentials than both of us together. Impressive. But a particle moving so fast that it has a short de Broglie wavelength doesn't become a black hole. What has happened to this guy? Another case of Josephson decline?

  10. I've just opened the paper. The quoted text refers to figure 1 where it is clear (well not so clear) that he is talking about the Compton wavelength and its relation to Schwarzschild radius.

    Of course he should state this clearer in his text too. So my overall impression is that this is a misunderstanding.

  11. Thanks, Bee!

    Just as with colliding with a CMB photon, can in principle a particle collide with a graviton and produce a black hole?

  12. Reading this article, I came to realise that I have actually only done some calculations in GR given a Schwarzschild-like metric. Come to think of it, I have never seen anything like the dynamical process of the gravitation collapse of a bunch of matter to form a black hole. Does anyone know any proper references?

  13. Hogan is closer with his holographic model to AWT, than most of other physicists - I mean phenomenologically - from logical perspective the holography is
    crippled model anyway. Of course Hogan's theory is hyperdimensional, so it MUST violate general relativity - not to say about special relativity. In AWT the holographic noise not only exists, but it's equivalent with CMBR noise. In AWT the fast moving charged bodies exhibit drag to vacuum fluctuations at highly luminal speed and they're losing energy with radiation of scalar waves or even their solitons (neutrinos) during acceleration. So-called Dirac fermions, like the electrons inside of superconductors are already moving fast, so they're susceptible to this interaction. This is the model, in which I'm explaining [url=]gravitational beams[/url] of Podkletnov. This view is consistent with [url=]R. Wayne's theory[/url] too and theories of [url=]superconductive mirrors[/url] for gravitational waves. So we have quite wide both theoretical, both experimental background for Hogan's ideas already.

  14. /*one expects quantum gravitational effects to become strong, not at distances close by the Planck length, but at curvatures close to one over Planck length squared*/

    The human observer scale is exactly in the middle of dimensional scales/curvatures/energy densities of quantum mechanics and general relativity, which means, the realm of quantum gravity is just the human observer scale (all these phenomena violating Lorentz symmetry like the refraction and polarization of light and all forces violating the inverse square law from Cassimir force to force of love...). This is fundamental shift in paradigm. The Planck scale is rather the scale, when all observable signs of relativity disappear, rather than the quantum gravity effects will become apparent.

  15. The thinking of formal physicists like the Bee is similar to attitude of people, who are believing, that the space-time is curved and path of light is straight even at the moment, when they're already wildly ENCIRCLING the black hole together with light around event photon sphere. Of course, they've truth from their perspective, but this perspective is not experimentally testable from outside. To prove the space-time is curved near black hole we should put the clock there and compare their time with clock at distance. From outside we can see only exactly the opposite effect: the space-time is flat and path of light is curved, which is essentially the quantum mechanical perspective - not relativistic one. Unfortunately for vanilla relativists, this is the only perspective, which we can measure and observe experimentally. The contemporary theorists are like explorers lost in the depth of rainforest at the riverside of Amazon, who are still believing, they're walking around free coast of Brazil. Because Amazon is wide river and visibility is low there, such a mistake is quite easy to do.

  16. BTW This [url=]Felber's study[/url] is closely related to the Hogan's proposal too. Just compare the outcome of all these seemingly unrelated models: for object moving faster than 70% c Felber predict push speed, which exerts to massive bodies, Wayne predicts drag for such an object and Hogan predicts the emanation of "holographic noise". In AWT the noise is just manifestation of scalar waves and Podkletnov observed the radiation of scalar wave beam. Of course, we shouldn't forget the theory of Unruh radiation and GZK cuttoff in this context.

  17. I believe that black holes are characterized by a curvature singularity where the weyl curvature diverges to infinity. In an extreme case of the Kerr metric, describing rotating black holes, admits a naked singularity with no event horizon. I believe it is when the angular momentum contribution in larger than the mass. They are not thought to physically exist due to collapse issues and thus cosmic censorship, but the math still describes the spacetime as a black hole.

  18. Hi Giotis,

    My comment really wasn't so much about Hogan's sentence in particular, but about the role of the Planck length in this kind of arguments more generally. It just came to my mind when I read Hogan's paper. Here's another example, from arxiv:0905.4803:

    "[W]ell-separated particles in one frame will in some other frame be closer together and form a black hole. The reconciliation must be achieved by a new uni fied theory that includes new, unfamiliar transformation properties."

    Again one is lead to believe that the existence of an event horizon depends on the relative velocity of the observer. Best,


  19. Robert:

    I deleted your comment. There is really no need to hand out insults for no reason whatsoever. Best,


  20. Could this have been more simply stated by observing that any coordinate transformation will also transform the event horizon. Thus if a mass extends beyond its Schwarzchild radius in the frame at rest with the centre of mass, then the mass will also extend beyond the Schwarzchild radius under any Lorentz transformation.

  21. Could this have been more simply stated by observing that any coordinate transformation will also transform the event horizon. Thus if a mass extends beyond its Schwarzchild radius in the frame at rest with the centre of mass, then the mass will also extend beyond the Schwarzchild radius under any Lorentz transformation.

    I apologize if this is a re-comment, I think blogspot ate my last comment

  22. Bee, I think that your take on this is essentially correct. However, I want to point out a situation where it looks problematical (altho I have seen good-enough-looking justifications.) That is the apparent "event horizon" for an accelerating observer, O. (In Born coordinates, it's where the required acceleration for "rigidity" becomes infinite.) Well you get trouble if you try to incorporate Hawking radiation into that, because the EH is only "real" for O. Someone sitting where that imaginary plane is, can't find a proper physical reason for particle pairs to "split" around them. That would be presumably an objective event so it doesn't seem to "make sense." However I have heard, with relation to the Unruh effect, that O still experiences the correct "Hawking radiation" anyway per that (and I don't recall the details of how it works.)


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