Wednesday, September 28, 2011

Why FTL violates causality

Suppose we have two observers, one in relative motion with respect to the other. We can represent this in fig 1 with one of the observer at rest -- frame of reference in black -- while the second one in motion -- frame of reference in blue. The line cutting in the middle represents the speed of light, c = 1. In particular, from the point of view of a stationary observer, an observer moving at constant velocity has a coordinate frame whose space and time axes are “tilted” towards the light cone (blue).

Two events a and b are silmultaneous in the moving frame -- they both lie on the x' axis, that is, t' = 0. But for the stationary observer, these are read off at ta ≠ tb. Therefore these two events are not simultaneous. Welcome to the world of Einstein's theory of Special Relativity.

But now suppose that we have invented a device that can send signal at speed greater than c, and to make it easy for illustration purpose, we take that speed to be infinite, that is, sending a signal at P arrives instantaneously as Q. This is indicated by the red line in fig 2.

In fig 3, events P and Q are simultaneous in the stationary frame ( X-T frame), while events Q and R are simultaneous in the moving frame ( X' - T' frame). So one could send a signal to himself, which would arrive in his past!

Or to put in another way, I would receive signals from my future self. This would violate causality.

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