## Stephen Hawking hopes that time travel is impossible, and hopes there’s a law of physics against it. But isn’t there already one?

Question

Why would he need to hope there is a law against time travel, when quite clearly the Lorentz Gamma Factor in Special Relativity, quite obviously rules out backwards time travel? If you can go faster than light, the time travel would have to happen in imaginary space, not in real space.

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## Answer ( 1 )

What we can conclude from special relativity is that the gamma factor doesn’t allow us to treat a transformation from a slower than light frame to a faster than light frame as a valid Lorentz transformation because it does not preserve the invariance of the spacetime interval.

The spacetime interval (considering just one space dimension for simplicity) is invariant under a valid Lorentz transformation. In common mathematical terms, the following ds^2 must remain the same for all valid frames:

ds^2 = dx^2 – dt^2

The above equation holds true for all slower than light inertial frames and all faster than light inertial frames, but does NOT hold true for a transformation between slower and faster than light frames. What happens is the sign of ds^2 changes sign and therefore doesn’t remain invariant under a transformation between slower and faster than light. All this says is that the Lorentz transformation is not valid in this case because it doesn’t preserve a required invariance. Note that the impact of faster than light on the imaginary Lorentz factor is to just change the signs of dt^2 and dx^2 since (i^2)*1 = -1 and (i^2)*-1 =1