- The Box-Problem in Deformed Special Relativity, arXiv:0912.0090v1 [gr-qc]
Deformed Special Relativity
An energy-dependent speed of light without the introduction of a preferred frame is a feature of what's become known as "Deformed Special Relativity" (DSR). If the energy-dependence is first oder in the photon's energy over the Planck energy, then it would become observable in the travel-time of highly energetic photons from distant gamma ray bursts. This prediction has received a considerable amount of attention. We previously discussed it here, here, here, here and most recently here. The reason that a tiny Planck scale effect grows to observable magnitude is that it adds up over the long distance the photons travel.
The amazing thing about this prediction is that there is actually no (agreed upon) DSR model in position space. DSR, pioneered by Giovanni Amelino-Camelia and later Kowalski-Glikman, Lee Smolin and João Magueijo, is a theory that is formulated in momentum space. It is motivated by the desire to modify the usual Lorentz-transformations such that the Planck energy is an invariant of the transformation, and it would appear the same for all observers. This works very well in momentum space. One obtains in these models a modified dispersion relation from which there follows the energy-dependence of the speed of light.
It is indeed interesting that it is possible to do that in an observer-independent way. The new transformations assure that the dispersion relation in its new form is the same for all observers. The one important equation to take from this is that also the energy-dependence of the speed of light is invariant when changing reference frames. Since the energy changes this means it's the functional relation that all observers agree on: if I change from one inertial frame to the other and the energy of a photon changes (by use of the new transformation) from E to E', then the speed of the photon changes to c'(E') which is the same as c(E'). That's what it means for the speed to be observer-independent. You cannot achieve this invariance of different speeds with ordinary Lorentz-transformations but the "deformed" ones do the trick.
But that's all in momentum space. So how was the prediction made for the photons propagating from the gamma ray burst towards Earth, arguably in position space? Well, one just used the c(E) that one obtains from the momentum space treatment. Now there's several technical arguments one can raise for why this is not a good idea (see eg this paper and references therein), but it later occurred to me there's a simpler way to see why this doesn't work. For this, let's just assume that one can indeed have c'(E') = c(E), and I'll tell you what a mess you get.
The Box Problem
Here's a very simple thought experiment. It is an extreme case but exemplifies the problem that is also present in less extreme situations. Consider you have a photon with Planck energy and a speed of light that decreases with increasing energy. Since the function c(E) should be monotonic and we don't want to introduce another scale, it's plausible that it drops to zero. So a photon with Planck energy, which is now the maximal possible energy, has zero speed, it doesn't move. I take that photon and put it inside a box. The box represents a classical, macroscopic system, one for which we know there's no funny things going on. It also takes into account that there's a finite precision to our measurements (the size of the box).
Now I change into a different reference frame, one in which the box moves relative to me. The photon's energy is by construction invariant under the transformation. If its speed is also invariant, it means the photon is in rest also in the other frame. This means however it can't remain inside the box. This is sketched in the space-time diagram below, the grey shaded area is the box, the red dashed line is the funny photon:
This peculiar transformation behavior was clear to Giovanni Amelino-Camelia already since the early days of DSR. In a 2002 paper he wrote:
"It is unclear which type of spacetime picture is required by the DSR framework. We should be prepared for a significant “revolution” in the description of spacetime ... In typical DSR theories ... one observer could see two particles ... moving at the same speed and following the same trajectory (for [this observer the particles] are “near” at all times), but the same two particles would have different velocities according to a second observer so they could be “near” only for a limited amount of time."
So far, so good. Then he writes
"For the particles we are able to study/observe, whose energies are much smaller than the Planck scale, and for the type of (relatively small) boosts we are able to investigate experimentally, this effect can be safely neglected."
And this unfortunately is not true for the following reason.
In the above figure you might not be too worried about the lines diverging. After all, it's hard to get two worldlines to be and stay perfectly parallel anyway, and it could take a very long time for them to diverge. But let's bring a third particle into the game. You can think of it as an electron that interacts with the photon inside the box which could be a detector. For simplicity think of that particle as being low energetic, such that additional DSR effects are irrelevant (that's not a crucial point). Then look at the same system from a different restframe (the green line is the electron):
What happens is that what was one space-time event (the black circle) in one restframe splits up into three different events. This generically happens if you have three lines. It is a very special case if they meet in one point. You make a small change one line (a different transformation behavior than the other lines), they'll fail to meet in one point and instead pairwise meet in three points. What this means is that not only is there no clear definition of "rest" or what it means for two particles to be "near." It's far worse: the notion of an event itself is ill-defined; it becomes an observer-dependent notion. Note that with the finite size of the box I've acknowledge that there's some fundamentally finite precision with which we can decide what's an event. But the DSR non-locality is well above that limit. The non-locality is in fact macroscopically large.
And that's bad. That's really bad because it totally messes up particle physics in regimes where we've tested it to very high accuracy. Whether two particles did or didn't interact inside a detector better not be observer-dependent, because that interaction could have real-world consequence. You can't talk it away. In my paper the particle interaction in the box triggers a bomb. It blows up the lab in one frame and not the other.
The later sections of my paper study a more realistic setting of the same problem with actually achievable particle energies and relative velocities. It turns out that, if DSR had a first-order modification in the speed of light that was indeed observable in the measurements from gamma-ray-bursts, then the mismatch in the location of the events would be of the order of a km. Not exactly what I'd call safely negligible. The irony is that what makes this mismatch so large is what makes the time-delay for the gamma ray burst's photons observable in the first place: the long distance traveled.
The setup of the experiment in my paper might seem rather complicated, but that's because I've been very careful to chose a situation in which there's no loophole. So far nobody has found one. It follows from my considerations that DSR with an energy-dependent speed of light that has modifications to first order in the energy over Planck mass are ruled out.
It should be emphasized that the here outlined problem does not occur in theories with a modified dispersion relation that actually break Lorentz-invariance and introduce a preferred frame. But then there are tight constraints on Lorentz-invariance violations already. There are also versions of DSR where the speed of light is not energy-dependent. These form a subclass of the general case and also do not suffer from the here discussed problem. This subclass is what I've been working on for some while. Needless to say I still think it's the only case that makes sense :-)
I'm curious to hear what arguments will be raised in the discussion this afternoon.