Abstract
We find finite-boost transformations DSR theories in first order of the Planck length lp, by solving differential equations for the modified generators. We obtain corresponding dispersion relations for these transformations, which help us classify the DSR theories via four types. The final type of our classification has the same special relativistic dispersion relation but the transformations are not Lorentz. In DSR theories, the velocity of photons is generally different from the ordinary speed c and possess time delay, however in this new DSR light has the same special relativistic speed with no delay. A special case demonstrates that any search for quantum gravity effects in observations which gives a special relativistic dispersion relation is consistent with DSR.
Highlights
DSR theories have been proposed for quantum gravity (QG) modifications to Einstein’s special relativity [1–5]
From Eq (14), we find differential equations which govern the evolution of energy and momentums in momentum space, and we solve the differential equations for the p0 and p1 components together and for the p2 and p3 components together
A second example of the transformations which have the special relativistic dispersion relation in the first order of the Planck length is found by taking β4 = 1 instead while keeping β0 = B and β2 = 0 as before
Summary
DSR theories have been proposed for quantum gravity (QG) modifications to Einstein’s special relativity [1–5]. We continue this study by finding the finite-boost DSR transformations to leading order of the Planck length from the solutions of the differential equations, and by looking at the observational consequences of these transformations. Maguijio and Smolin obtained a different realization of DSR theories by non-linear action of the Lorentz group on the energy-momentum space [4] They gave a procedure for finding corresponding transformations for any given modified dispersion relation [5]. We show that the Amelino-Camelia (AC) [1], and the Maguejo-Smolin (MS) [4] DSR theories in first order of the Planck length lp, are special cases of our DSR finite-boost transformations. Finite-boost transformation DSR theories and classifications of first order DSR theories (by use of a dispersion relation) provide more possibilities for investigating quantum gravity effects in observations. The freedom comes from the use of adjustable parameters in the finite-boost transformations
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