You may have encountered statements like this one made by Sabine:

Once you can send information faster than the speed of light, you can also send it back in time. If you can send information back in time, you can create inconsistent histories, that is, you can create various different pasts, a problem commonly known as “grandfather paradox:” What happens if you travel back in time and kill your grandpa? Will Marty McFly be born if he doesn’t get his mom to dance with his dad? Exactly this problem.

This is correct. Special relativity implies that, if faster than light signaling would be true, you would be able to signal to your own past, and this can lead to paradoxes. Here I will explain how exactly this can happen. This is rather elementary special relativity stuff, but I realized there is much confusion around it. First, I never saw a precise scenario in which faster than light (FTL) signaling can be used to signal back to your own past, so I will give one. Second, I have the feeling that when people make statements like this,

- they either refer to the fact that, if an observer A sends FTL signals in her own future, for another observer B it may look like sending in back in time, in B's reference frame, as in this figure:

Orange lines represent light cones, blue represent timelike curves
(observers), red represents the proper space of an observer, and green
represents FTL signals. While the picture represents the proper space of A as
a horizontal red line, the proper space of B is oblique, due to
the Lorentz transformation (relativity of simultaneity).

The first scenario is not that
paradoxical, because observer B can always reinterpret the signal from A
to B as a signal going in his own future, from B to A. But even in this
case, we will have the problem of who actually created the message in the first place.

- or they refer to examples where the observer sends an FTL signal toward her own past, as in this figure:

Here is how FTL signaling would imply that one can signal back in time,

*using only signals sent in the future and received from the past*, with respect to the proper reference frame:
The inertial observer A accelerates away from B, then sends an FTL signal at t₀. Observer B receives it at t'₀ in his proper time, then accelerates away from observer A, then sends it back, at t'₁. Observer A receives the signal at t₋₁, where t₋₁< t₀.

So indeed FTL implies signaling back in your own past, even if FTL signals are sent only to the proper future and received only from the proper past.

Let us see how this allows paradoxes. Suppose that earlier A and B agreed on the following: if A receives the message "Yes", she sends the message "No", and if she receives "No", she sends "Yes". If B receives a signal, he just resends it without changing. Then, we have a paradox: does A send the message "Yes", or "No"? It is similar to the liar paradox, since if she sends "Yes", then she receives "Yes", so she sends "No", and so on. But it is also like grandfather's paradox, because B can send instead of a message, a killing FTL ray, to kill A or her grandfather before she was born.

So far there is no evidence of FTL signaling, except for some misunderstandings of the EPR "paradox". I don't know either of a fundamental physical law which prevents it, given that tachyonic solutions are mathematically consistent, both in special relativity, and in quantum field theory. But as we have seen, FTL would lead to time travel paradoxes.

## 2 comments:

Two fast comments:

1 Regarding the top figure and your comment:

"The first scenario is not that paradoxical, because observer B can always reinterpret the signal from A to B as a signal going in his own future, from B to A. But even in this case, we will have the problem of who actually created the message in the first place."

So your second sentence questions the first. Let me give an example which shows where (at which event) a signal originates. In fact, this example is controversial because people prefer to talk about non-local correlations, not about FTL signals, but I guess one can come up with a lot of examples serving the same purpose.

Consider an Einstein-Podolsky-Rosen-Bell type experiment. Imagine that the measurement of the polarization of a photon is carried out in A’s reference frame (at its origin). This measurement determines (instantaneously in A) the polarization of the other photon of the entangled pair at the origin of B's frame (obviously, in this example the green line in your figure should be horizontal). So the "signal" originated in A's frame (I do not know well the technical details in Aspect's experiments, for example, but believe it can be ensured that it is the measurement at A's origin that collapsed the wave function of the entangled pair of photons).

2. Regarding the bottom figure. I believe the paradox you formulated effectively rules out FTL signals. This is another example of the power of exploring the internal logic of an idea (in this case FTL) - a method behind the greatest discoveries in physics, successfully employed by Galileo, Newton, Einstein and many other great physicists, which is now the main component of a research strategy that is being developed and employed at the Minkowski Institute.

Dear Vesselin,

Thank you for your interesting comments and for the facebook discussions regarding FTL.

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