Some Consequences of a Time Dependent Speed of Light

For reasons connected with both cosmology (the flatness and horizon problems) and atomic physics (n-body Dirac equation, etc.), various proposals have been made to modify general or special relativity(SR) to accommodate a cosmologically decreasing light speed [J. Magueijo, Rep. Prog. Phys. \textbf{66}, 2025 (2003)]. Two such theories, projective SR [S.N. Manida, gr-qc/9905046; S. S. Stepanov, physics/9909009 and Phys. Rev. D, \textbf{62}, 023507 (2000)] and symmetric SR [F.T. Smith, Ann. Fond. L. de Broglie, \textbf{30}, 179 (2005)] adapt special relativity to in different ways to an expanding, hyperbolically curved position space and predict time-dependences of $c$ within reach of measurement but differing by a factor of two. Both theories bring in a new constant $\lambda ^{-1}=\sigma =c^2H_0 ^{-1}$. As Magueijo points, out the role of $c$ in physics and cosmology is so profound that many deep changes must follow if is not absolutely invariant in space and time. In particular, symmetric SR brings a new light to the Dirac large-number relationship between the constants of gravitation and atomic physics.