  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.

General Physics