# Special Relativity/Faster than light signals, causality and Special Relativity

It is popularly imagined that Special Relativity forbids travel faster than the speed of light or the propagation of signals faster than the speed of light. However, the actual theory does not contain this assumption. The original theory, framed by Einstein in 1905, states that the speed of light in free space is constant in all inertial frames of reference so how did people in general come to believe that this implies a speed limit? The idea of a speed limit comes from two predictions of the theory, that inertia increases towards infinite as velocity approaches light speed and that causality, the succession of cause and effect, is violated if we could signal at speeds above the speed of light.
The inertial constraint does not apply to particles without a rest mass, such as the photon, or to particles that might oscillate between massless and massive forms. The possibility that causality would be violated if signals could travel faster than the speed of light is a more interesting problem however. The relationship between Faster than Light signal speeds and causality will be considered and it will be shown that if a Faster than Light signal were ever discovered then either Special Relativity or Causality will be false.

## Special Relativity, faster than light signals and causality

Prior to the twentieth century physicists believed that sending signals from one observer to another was straightforward. They believed that light was always transmitted from a transmitter to a receiver where the transmitter was earlier than the receiver (the red line in the diagram below) and that signals might be transmitted instantaneously (the blue and purple lines in the diagram). In this pre-relativistic scenario the two observers in the diagram, Bill and Bertha, have total freedom to transmit signals at any rate up to an infinite velocity. Prior to Special Relativity there was no theory for how signals might be sent backwards in time and such a possibility was discounted.
Special Relativity leads to different predictions about the behaviour of signals. In the section of this book on simultaneity it was shown that Special Relativity predicts that clocks on two relatively moving observers will go progressively out of phase with distance along a common x-axis. This effect is shown in the diagram below.
The x' axis on the diagram is all those points that Jim considers to be NOW, events that exist at the present moment. Jim's NOW differs from Bill's NOW. The two observers read the same time at the origin of the graph but clocks differ with distance from the origin. This means that if Jim were able to send a message instantaneously from one place to another, so that it was transmitted and received at the same time, Bill would see the signal to be spanning two different times. The signal would appear to go backwards or forwards in time. This is shown in the diagram below in which Bertha sees the signal begin at one time (point B) and end at another time (point A).
Notice that if Bertha asks Jane to send an instantaneous signal to Jim then Bill, who is right next to Jim when the signal arrives, will register this signal as arriving at an earlier time than the time it was started by Bertha. If it is possible to signal instantaneously then it is possible to transmit messages backwards through time! This transfer of information back through time would apply to any signal that could be sent at a speed faster than the speed of light.
If signals could be sent at faster than light speed then Bertha and Bill could work together to become rich. Bertha could ask Jane and Jim to signal the result of a race or the price of a stock back to Bill then Bill could send this result back to Bertha at a time before the race. Bertha could then place a bet to win a fortune.. This sequence of events is shown in the diagram below, Bertha sees the result of the race at point B, signals this back to Bill at point A, using Jane and Jim to send the signal, then Bill signals back to Bertha how she should bet before the race begins.
This sequence of events might be held to be impossible in physics because it violates the principle of Causality. Causes precede effects according to the principle of Causality but if faster than light signals are possible then effects could precede causes. Notice that the violation of causality would actually be quite limited and would only apply to "space-like" separated events, these are events that are so recent that it would require a signal travelling faster than the speed of light to observe them now. Even if Bill and Jim were as far away as the moon they would only be able to peek a little over a second into Bertha's future if instantaneous signals were possible (the moon is less than 2 light seconds away).
It is well known that weakly interacting, free particles which cannot be observed because they are "space-like" separated from observation are described by the probabilistic predictions of Quantum Theory so it would not be altogether shocking, though certainly surprising, if causality were violated in these circumstances.
Further Reading
Liberati, S., Sonego, S. and Visser, M. (2002) Faster-than-c signals, special relativity, and causality. Annals Phys. 298 (2002) 167-185. http://arxiv.org/abs/gr-qc/0107091
Garrison, J.C., Mitchell, M.W., Chiao, R.Y., Bolda, E.L. (1998) Superluminal Signals: Causal Loop Paradoxes Revisited. Phys.Lett. A245 (1998) 19-25 http://arxiv.org/abs/quant-ph/9810031