This theory was proposed by Giovanni Amelino-Camelia in the mid to late 90's.

What is it?

Special Relativity is derived from normal (or Galilean) relativity by introducing an observer independent scale, the speed of light c. Einstein didn't pull this out of his hat but rather was motivated by wishing to unify Maxwell's equations of Electricity and Magnetism. The measurment of a length depends upon the frame of reference of the observer. Amelino-Camelia was motivated to introduce a second observer independent scale into relativity, namely the Planck Length. The motivation is that this is a length derived from fundamental constants and therefore should not change, or at least one could conceed that having this length not changing could be desirable. The name of the theory comes from the two invariants used in deriving the theory. In the paper by Amelino-Camelia that I read he takes great pains to point out that his motivation is to see if such a theory can be constructed consistently. He points out that the construction leads to a family of theories and that they should live or die based on their predictions. What is interesting is that such theories can be consistently created.

Consequences

The consequences are radical. It leads to a speed of light that is dependent upon the energy of the photon. This seems at odds with a theory which puts a constant speed of light in to begin with. The regime in which the variation occurs is almost beyond observational tests. A consequence of a varying speed of light VSL include removing the need for inflation to explain the homogineity of the cosmic microwave background CMB. In the early universe photons at higher energies travelled with higher speeds giving a much larger horizon at that time. Another consequence is that space becomes discritised. This ties in with the work of Lee Smolin on spin networks and quantum gravity, but the details are somewhat beyond me.

In Favour

Keeping the Planck length observer independent is clearly a nice idea. BrianShader asks me to elaborate on this. Since the Planck length is derived from constants having it change from one frame to another means that that we must interpret one of these constants as changing. Furthermore the Planck length is very important for Quantum mecahics. Opening a path towards a working quantum theory of gravity is also an idea worthy of investigation.

Against

VSL is contentious. Most of the predictions lie just outside of the parameter space that we can test. This is traditionally the area occupied by theories which are later discarded and places this theory firmly in the 'we should be justifiably skeptical until' bracket.

What does the evidence say

As I mentioned above the theories predictions are almost untestible. We can look at time of arrival data of very highly energised photons from gamma ray bursts for example. An energy dependence in their speed will give a measurable effect if the photon energy is high enough. Measuring these photons. In a paper titled 'Limits to Quantum Gravity Effects on Energy Dependence of the Speed of Light from Observations of TeV Flares in Active Galaxies' a group of gamma ray astronomers detected no such dependence, however their work (Physical Review Letters, Volume 83, Issue 11, September 13, 1999, pp.2108-2111) was based on 4 photons and so only placed limits on VSL theories.

DSR and VSL provide an alternative to inflation, the new results from the Wilkinson Microwave Anisotropy Probe(WMAP) give direct evidence for inflation. Last week I asked Dave Spergel(that's a gratuitous name drop but hey!), one of the principal investigators on WMAP, what his opinion of DSR was. He was mildly skeptical and pointed out that inflation has to date passed all observational tests.