Comparison of relativity theories with observer-independent scales of both velocity and length/mass

We consider the two most studied proposals of relativity theories with observer-independent scales of both velocity and length/mass: the one discussed by Amelino-Camelia as illustrative example for the original proposal (gr-qc/0012051) of theories with two relativistic invariants, and an alternative more recently proposed by Magueijo and Smolin (hep-th/0112090). We show that these two relativistic theories are much more closely connected than it would appear on the basis of a naive analysis of their original formulations. In particular, in spite of adopting a rather different formal description of the deformed boost generators, they end up assigning the same dependence of momentum on rapidity, which can be described as the core feature of these relativistic theories. We show that this observation can be used to clarify the concepts of particle mass, particle velocity, and energy-momentum-conservation rules in these theories with two relativistic invariants.Comment: 21 pages, LaTex. v2: Andrea Procaccini (contributing some results from hia Laurea thesis) is added to the list of authors and the paper provides further elements of comparison between DSR1 and DSR2, including the observation that both lead to the same formula for the dependence of momentum on rapidit

New varying speed of light theories

We review recent work on the possibility of a varying speed of light (VSL). We start by discussing the physical meaning of a varying $c$, dispelling the myth that the constancy of $c$ is a matter of logical consistency. We then summarize the main VSL mechanisms proposed so far: hard breaking of Lorentz invariance; bimetric theories (where the speeds of gravity and light are not the same); locally Lorentz invariant VSL theories; theories exhibiting a color dependent speed of light; varying $c$ induced by extra dimensions (e.g. in the brane-world scenario); and field theories where VSL results from vacuum polarization or CPT violation. We show how VSL scenarios may solve the cosmological problems usually tackled by inflation, and also how they may produce a scale-invariant spectrum of Gaussian fluctuations, capable of explaining the WMAP data. We then review the connection between VSL and theories of quantum gravity, showing how ``doubly special'' relativity has emerged as a VSL effective model of quantum space-time, with observational implications for ultra high energy cosmic rays and gamma ray bursts. Some recent work on the physics of ``black'' holes and other compact objects in VSL theories is also described, highlighting phenomena associated with spatial (as opposed to temporal) variations in $c$. Finally we describe the observational status of the theory. The evidence is currently slim -- redshift dependence in the atomic fine structure, anomalies with ultra high energy cosmic rays, and (to a much lesser extent) the acceleration of the universe and the WMAP data. The constraints (e.g. those arising from nucleosynthesis or geological bounds) are tight, but not insurmountable. We conclude with the observational predictions of the theory, and the prospects for its refutation or vindication.Comment: Final versio