Justin Pombrio

Faster Than Light

To travel back in time is to exceed the speed of light, and to exceed the speed of light is to travel back in time.

Let us work within the framework of the theory of special relativity, and call the speed of light c.

To say that an object exceeds the speed of light is to say that it is present at two events which are space-like separated. So if the distance between the events is d and the time elapsed between the events is t, then d > ct.

Now suppose that an observer is moving with a velocity v relative to the observer who declared the object to have exceeded the speed of light, such that

c2t/d < v < c

The existence of such a velocity follows from the fact that d > ct. Using the Lorentz transformation, we can see that according to this second observer the elapsed time between the events will be,

γt - γvd/c2 = γd/c2(c2t/d - v) < 0

Thus there is a frame of reference moving subluminally with respect to the first and according to which the object travelled back in time.