Astronomy Without A Telescope – Doubly Special Relativity


General relativity, Einstein’s theory of gravity, gives us a useful basis for mathematically modeling the large scale universe – while quantum theory gives us a useful basis for modeling sub-atomic particle physics and the likely small-scale, high-energy-density physics of the early universe – nanoseconds after the Big Bang – which general relativity just models as a singularity and has nothing else to say on the matter.

Quantum gravity theories may have more to say. By extending general relativity into a quantized structure for space-time, maybe we can bridge the gap between small and large scale physics. For example, there’s doubly special relativity.

With conventional special relativity, two different inertial frames of reference may measure the speed of the same object differently. So, if you are on a train and throw a tennis ball forward, you might measure it moving at 10 kilometers an hour. But someone else standing on the train station platform watching your train pass by at 60 kilometers an hour, measures the speed of the ball at 60 + 10 – i.e. 70 kilometers an hour. Give or take a few nanometers per second, you are both correct.

However, as Einstein pointed out, do the same experiment where you shine a torch beam, rather than throw a ball, forward on the train – both you on the train and the person on the platform measure the torch beam’s speed as the speed of light – without that additional 60 kilometers an hour – and you are both correct.

It works out that for the person on the platform, the components of speed (distance and time) are changed on the train so that distances are contracted and time dilated (i.e. slower clocks). And by the math of Lorenz transformations, these effects become more obvious the faster than train goes. It also turns out that the mass of objects on the train increase as well – although, before anyone asks, the train can’t turn into a black hole even at 99.9999(etc) per cent of the speed of light.

Now, doubly special relativity, proposes that not only is the speed of light always the same regardless of your frame of reference, but Planck units of mass and energy are also always the same. This means that relativistic effects (like mass appearing to increase on the train) do not occur at the Planck (i.e. very small) scale – although at larger scales, doubly special relativity should deliver results indistinguishable from conventional special relativity.

The Planck spacecraft - an observatory exploring the universe and named after the founder of quantum theory. Coincidence? Credit: ESA.

Doubly special relativity might also be generalized towards a theory of quantum gravity – which, when extended up from the Planck scale, should deliver results indistinguishable from general relativity.

It turns out that at the Planck scale e = m, even though at macro scales e=mc2. And at the Planck scale, a Planck mass is 2.17645 × 10-8 kg – supposedly the mass of a flea’s egg – and has a Schwarzschild radius of a Planck length – meaning that if you compressed this mass into such a tiny volume, it would become a very small black hole containing one Planck unit of energy.

To put it another way, at the Planck scale, gravity becomes a significant force in quantum physics. Although really, all we are saying that is that there is one Planck unit of gravitational force between two Planck masses when separated by a Planck length – and by the way, a Planck length is the distance that light moves within one unit of Planck time!

And since one Planck unit of energy (1.22×1019 GeV) is considered the maximal energy of particles – it’s tempting to consider that this represents conditions expected in the Planck epoch, being the very first stage of the Big Bang.

It all sounds terribly exciting, but this line of thinking has been criticized as being just a trick to make the math work better, by removing important information about the physical systems under consideration. You also risk undermining fundamental principles of conventional relativity since, as the paper below outlines, a Planck length can be considered an invariable constant independent of an observer’s frame of reference while the speed of light does become variable at very high energy densities.

Nonetheless, since even the Large Hadron Collider is not expected to deliver direct evidence about what may or may not happen at the Planck scale – for now, making the math work better does seem to be the best way forward.

Further reading: Zhang et al. Photon Gas Thermodynamics in Doubly Special Relativity.

23 Replies to “Astronomy Without A Telescope – Doubly Special Relativity”

  1. I haven’t read the paper, so I just judge from this article here.

    However, I am at least very cautious with such claims. For examples: Electrons are much lighter than the Planck mass, with a mass of m_e=9.1*10^-31 kg. That’s 23 orders of magnitude lighter, but the electron obeys special relativity, otherwise all of our particle colliders wouldn’t have worked.

    So, why should a Planck mass (which is not terribly light) obey such a strange behaviour, while things much lighter do not?

    1. Hi Doc,

      The ‘strange behaviour’ arises when you compress the mass into a volume with a Planck length radius (allegedly). Otherwise, there’s nothing remarkable about the mass of a flea’s egg.

  2. The best way forward for what, understanding cosmology and the early universe? That should be along the ongoing lines AFAIK. Investigating the standard cosmology. Inflation and other phenomena may inform of high energy events/small spacetime scales in such cases as well.

    Btw, I believe it is a mistake to claim that GR doesn’t inform of the early universe. Standard cosmology is a GR theory par excellence, and it does so.

    Further I would argue that in its simplest form this cosmology argues against a singularity and for semiclassical worldline continuity, as that is the simplest interpretation of the inflation that it incorporates. (Which is strengthened by string theory making the same predictions as a more general theory of GR + particles.)

    The “GR + singularity” picture has been left behind by history. I don’t think it is illuminating to start from other physics than the current accepted, it is richer than that, and it seems likely that it implies the old ideas were as wrong as they could possible be. :-~

    By extending general relativity into a quantized structure for space-time, maybe we can bridge the gap between small and large scale physics.

    As I believe I have commented here before, it is all very well to quantize gravity as an effective field theory but there is an unwarranted jump to the idea that therefore spacetime should be quantized. Indeed, the way the quantized theory breaks down at precisely such scales implies instead that this is a problem!

    Further it seems from supernova photon timing events that special relativity is valid on all scales, as predicted by it and its instantiation as string theory.

    For example, there’s doubly special relativity.

    Oh, that was some time since I last heard of that! IIRC the problem is that you can’t scale it up to aggregates of particles, there is no natural superposition principle that makes it valid for matter.

    Besides, AFAIU the same supernova photon timings as above also rejected the relevant scales of energy dependent light speeds that was predicted. Interesting that it is to note, thanks for positing this btw, frankly I’m also amazed people still dabbles in this; observations, such as they are, doesn’t encourage it.

    1. You know what, maybe (just maybe) it is because it is one of the modern observations that I find most interesting, our first Planck scale observations (!), I suggest them as a good UT article topic someday.

      They are buried in an arxiv article somewhere, but I think written up by bona fide astronomers as a one off observation that shouldn’t go to waste.

      Also, I believe I have seen someone in the theoretical physics know how (Maldacena, perhaps?) discuss somewhere how they don’t constrain theory. I don’t get that, since I believe the observation marginally within ~ 1 Planck time difference was more than 3 sigma which should test theory – if one accepts the observation. The one observation within ~ 0.01 Planck time delta was less certain, unfortunately.

      So, an interesting set of observations, an arguable reception and a unique result. Ripe for blogging.

    2. We might have some quantum underpinning to gravitation, but we might in fact not really have quantum gravity. It is possible that gravitation is an emergent phenomenon from a quantum field theoretic substratum, where the continuity of spacetime might be similar to the large scale observation of superconductivity or superfluidity. The AdS/CFT is a matter of classical geometry and its relationship to a quantum field theory. So the AdS_4/QFT suggests a continuity of spacetime which has a correspondence with the quark-gluon plasma, which has a Bjorken hydrodynamic scaling. The fluid dynamics of QCD, currently apparent in some LHC and RHIC heavy ion physics, might hint at this sort of connection.

      So we might not really have a quantum gravity as such. or if there are quantum spacetime effects it might be more in the way of quantum corrections to fluctuations with some underlying quantum field. Currently there are models which give quantum gravity up to 7 loop corrections, or 8 orders of quantization. Of course the tree level of quantum gravity is formally the same as classical gravity. I think it is then quite possible that gravitation is a large scale effect from bosonization of D0-branes (fermionic partons) which form a sort of condenstate, similar to superconductivity.


      1. Thanks for the considered response! I’m not sure my musings are worthy.

        The AdS/CFT is a matter of classical geometry and its relationship to a quantum field theory.

        Good point, but again I get the feeling that this singles out QFT (because we know it). I believe the holographic principle is larger, incorporating string theory which is only a QFT “when it needs to be” AFAIU.

        I think it is then quite possible that gravitation is a large scale effect

        I agree, FWIW.

    3. Agree standard cosmology is a GR theory par excellence. However, I’m not sure GR has much to say about the Planck epoch (which is also standard cosmology).

      I think DSR has internal inconsistencies, as I pointed out – and am not promoting it as a strong candidate for filling the gap. I am not immediately sure what is a strong candidate.

      1. Thanks for the considered response! I’m not sure my musings are worthy.

        I’m not sure GR has much to say about the Planck epoch (which is also standard cosmology).

        Agreed, but I merely wanted to point out that it says more than “nothing” if the theoreticians can describe inflation as generic and our universe as a local outcome, use semiclassical world lines a lá Linde to describe some of its physics, et cetera,

        It may be pitiful, but it is all we got so far, and it is better than before. And it looks to be testable (testing inflation itself; testing different inflation mechanisms; et cetera), which in itself is way better than a singularity hidden behind a Planck epoch. So I’m (not so quietly) happy with this.

        Btw, I enjoy these weekend postings (especially since I don’t own a telescope, nor have the inclination to do so), so I want to balance my fun in trampling all over the weaker terrain (um, “swampland”? :-D) to others not appreciating my footprints all over. I’m not sure I am too successful. :-/

  3. Uh, oh this. There is of course a problem with this. In particular the Fermi measurements of the simultaneous arrival of widely different frequencies of light were a negative result on these ideas. Joao Magueijo and Lee Smolin proposed this in the context of loop quantum gravity (LQG). The idea is that anything Lorentz contracted to 10^{-33} com can’t be contracted further, and that there is an actual transformation rule for this. The problem is that this assigns degrees of freedom to every region of spacetime that gives a huge entropy to spacetime. This is frankly a bit similar to the ether ideas of the pre-20 th century physics.

    There is another funny thing, which is there is no need for concern over something appearing to Lorentz contract smaller than the Planck length. If one observes something moving at relativistic velocities it actually would not be seen to contract. There would be a Terrell rotation:

    The Lorentz contracting of an object is observed with an object which approaches the event horizon of a black hole. Here the curvature of spacetime acts upon the optical light from the object and the contraction effects are apparent. The limit on the contraction is something called the stretched horizon.


    1. After seeing this youtube clip,. Now I definitely want to fly near light-speed at least once in my life 🙂

    2. Good points, and I see you refer to Fermi as well.

      As for entropy, I had assumed that one could hide the additional degrees of freedom from macroscopic consideration if one wanted. The matter problem looked more as a “there’s no way you can hide this mess, mate” problem to me, but I guess I can see your concern.

      And quite frankly, the problems pop up all over the place, as they are wont to do if it is unrelated to actual physics. Choosing just one is as easy as to choose which particular piece of garbage you want to swallow from the recycling container. 😀

      On the Terrell rotation vs DSR, can you believe I have never (IIRC) connected those two! Thank you for making this clear!

      (Also, now it removes the “GR can be quantized, so spacetime must be too” intuit even further in my mind. The separation between these two concepts is huge!)

    3. Awesome YT btw. After seeing this youtube clip, I definitely want to fly near light-speed like: never on my life. 😀 (Just kidding, it looks like fun. To try once!)

  4. The Fermi results clearly indicate there is no dispersive physics with photons. With Loop Quantum Gravity (LQG) spacetime is sliced up into discrete bits with connection terms that have quantum states associated with them. This means a couple of things: light exhibits a dispersion due to the vacuum and the vacuum of the universe has an enormous amount of quantum information.

    LQG predicts that the Lorentz symmetry of spacetime is violated as one approaches this Planck scale. This means that a photon traveling through space is perturbed by little quantum fluctuations which give a dispersion. So a photon of ordinary wavelength will over a vast distance exhibit dispersion. This dispersion is wavelength dependent. However, Fermi probe found that photons of widely different wavelengths from a galaxy billions of light years distance had not dispersion. So there is experimental evidence against this. Double relativity is a form of breakdown of Lorentz symmetry on a small scale. This would have turned up in these measurements.

    The other problem is that LQG predicts an enormous amount of quantum data in the vacuum. For this reason LQG is not able to recover a classical limit, for any classical limit predicts enough quantum information and entropy to be a black hole — even if the spacetime is a flat spacetime. String theory and holography removes this problem through the identification of quantum information in spacetme by quantum modes horizons of one dimension lower. This also extends to the equivalency between the isometries of the anti de Sitter spacetime and its boundary as the conformal symmetries which are a quantum field.

    It is a funny state of affairs, for LQG seems perfectly natural in many ways. It is closets to the general relativistic ideology of Einstein and spacetime physics. Yet in a funny way it fails quite spectacularly. Understanding this is important, and it really is similar to what happened with classical physics at the end of the 19th century. People assumed that EM radiation propagated in an ether that filled space, as a sort of gas. If so there would have been an enormous amount of “data” in every region of space. However, the solution turned out to be a symmetry principle that required no idea of there being some medium with “data.”


  5. Light can be warped into different layers in which matter can pass around with near perpetual friction powering the course. Writing out the formula would simply give you the answer but I’ll lead you in the right way with a trick to start.

    Use the CERN magnets to balance multiple projectiles that just graze each other but continue to do so with additional projectiles being added almost non-stop. Eventually the power released by these collisions will create a “warp zone” or “wave Flux” inside the CERN unit itself. The computer needed to calculate the exact position of each unit collision & monitor these small waves is not available yet to mankind. Give it about another 5 years, military use first.

  6. Gulp!

    “…since one Planck unit of energy (1.22×10^19 GeV)”

    This is also 1.22×10^28 eV or 12,200,000,000,000,000,000,000,000,000 eV.
    In energy terms this is 19546563426 1.95×10^9 Joules or just on 0.5 Megawatt-hours!

    All this packed into one particle is truly mind-blowing!

    1. It is a large quantum of energy. This is a quantum of a black hole, where the Schwarzschild radius equals the de Broglie wavelength of the mass equivalent of a black hole. The spectrum of elementary particles has a much smaller mass. This involves some physics of BPS black holes, or black holes with some gauge charge or quantum number. In particular this is worked with black holes that are BPS supersymmetric.


  7. The voice on the YT commentary sounds like the computer generated one on my Mac!

  8. Einstein saw gravity as a distortion of spacetime and not a separate force.

    I humbly suggest that the attractive force we experience is due to the natural imbalance of the EM density between one mass and another. There only has to be a difference of one part in10^42 for there to be the attraction we experience.

    1. I must most emphatically tell you that this is wrong. What you say makes absolutely no sense at all. If people want to understand physics they need to study it. It is amazing and annoying to keep getting this endless stream of nonsense from people who have these strange ideas. If you know physics it is not hard to “clock” this stuff as rubbish.


  9. This all sounds very intriguing; but is there any science behind all these far-out statements involving Planck length, energy, etc.? In addition to being intriguing, it unfortunately also sounds like mumbo-jumbo.


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