Special Theory of Relativity in Curved Space Time

Space-time measurements, of gedanken experiments of special relativity need modification in curved spaces-times. It is found that in a space-time with metric $g$, the special relativistic factor $\gamma$, has to be replaced by $\gamma_g=1/\sqrt{g_{\mu \nu} V^\mu V^\nu}$, where $V_\mu=(1,v,0,0)$, is the 4-velocity, and $v$ the relative velocity between the two frames Examples are given for Schwarzschild metric, Friedmann-Robertson-Walker metric, and the G\"{o}del metric. Among the novelties are paradoxical tachyonic states, with $\gamma_g$ becoming imaginary, for velocities less than that of light, due to space-time curvature. Relativistic mass becomes a function of space-time curvature, $m=\sqrt{g_{\mu \nu}P^\mu P^\nu}$, where $P_\mu=(E,p)$ is the 4-momentum, signaling a new form of Mach's principle, in which a global object - namely the metric tensor, is effecting interia.