Relativistic effects are those predicted by Einstein's special theory of relativity, which come into play when objects – such as spacecraft – travel at speeds that are a substantial fraction of the speed of light. These effects include time dilation, mass increase, and length contraction.
Time dilation would, in principle, allow astronauts to travel vast distances well within their own lifetimes. However, relativistic mass increase would make it more and more difficult to continue to accelerate a spacecraft. The factor that determines the amount of mass increase and other relativistic effects is called γ (gamma). For an object moving with speed v relative to an observer considered to be at rest (for example, on Earth), γ is given by:
γ = 1/√(1 - v2/c2)
where c is the speed of light.
The relativistic mass, m, of a body moving at velocity, v, is then
m = γm0 = m0/√(1 - v2/c2)
where m0 is the rest mass. Note that when v = 0, this reduces to the non-relativistic result, m = m0. The impossibility of accelerating an object up to the speed of light is shown by the fact that when v = c, m becomes infinite.
Similarly, the relativistic time dilation is given by:
t = t0 = t0/γ = t0√(1 - v2/c2)