A tachyonic antitelephone is a hypothetical device in theoretical physics that could be used to send signals into one’s own past. Albert Einstein in 1907[1] presented a thought experiment of how faster-than-light signals can lead to a paradox of causality, which was described by Einstein and Arnold Sommerfeld in 1910 as a means “to telegraph into the past“.[2] The same thought experiment was described by Richard Chace Tolman in 1917[3], therefore it is also known as Tolman’s paradox.

A device capable of “telegraphing into the past” was later also called “tachyonic antitelephone” by Gregory Benford et al. According to the current understanding of physics, no such faster-than-light transfer of information is actually possible. For instance, the hypothetical tachyon particles which give the device its name do not exist even theoretically in the standard model of particle physics, due to tachyon condensation, and there is no experimental evidence that suggests that they might exist. The problem of detecting tachyons via causal contradictions was treated scientifically

One-way example

This was illustrated in 1911 by Paul Ehrenfest using a Minkowski diagram. Signals are sent in frame B1 into the opposite directions OP and ON with a velocity approaching infinity. Here, event O happens before N. However, in another frame B2, event N happens before O.[5]

Tolman used the following variation of Einstein’s thought experiment:[1][3] Imagine a distance with endpoints A and B. Let a signal be sent from A propagating with velocity a towards B. All of this is measured in an inertial frame where the endpoints are at rest. The arrival at B is given by:

\Delta t=t_{1}-t_{0}=\frac{B-A}{a}.

Here, the event at A is the cause of the event at B. However, in the inertial frame moving with relative velocity v, the time of arrival at B is given according to the Lorentz transformation:

\Delta t'=t'_{1}-t'_{0}=\frac{t_{1}-vB/c^{2}}{\sqrt{1-v^{2}/c^{2}}}-\frac{t_{0}-vA/c^{2}}{\sqrt{1-v^{2}/c^{2}}}=\frac{1-av/c^{2}}{\sqrt{1-v^{2}/c^{2}}}\Delta t.

It can be easily shown that if a > c, then certain values of v can make Δt’ negative. In other words, the effect arises before the cause in this frame. Einstein and Tolman concluded that this result contains in their view no logical contradiction; they said, however, it contradicts the totality of our experience so that the impossibility of a > c is sufficiently proven.

[edit] Two-way example

A more common variation of this thought experiment is to send back the signal to the sender (a similar one was given by David Bohm[6]). Suppose Alice (A) is on a spacecraft moving away from the Earth in the positive x-direction with a speed v, and she wants to communicate with Bob (B) back home. Assume both of them have a device that is capable of transmitting and receiving faster-than-light signals at a speed of ac with a > 1. Alice uses this device to send a message to Bob, who sends a reply back. Let us choose the origin of the coordinates of Bob’s reference frame, S, to coincide with the reception of Alice’s message to him. If Bob immediately sends a message back to Alice, then in his rest frame the coordinates of the reply signal (in natural units so that c=1) are given by:

(t,x) = (t,at)

To find out when the reply is received by Alice, we perform a Lorentz transformation to Alice’s frame S' moving in the positive x-direction with velocity v with respect to the Earth. In this frame Alice is at rest at position x' = L, where L is the distance that the signal Alice sent to Earth traversed in her rest frame. The coordinates of the reply signal are given by:

t' = \gamma \left(1 - av\right) t
x' = \gamma \left(a -  v\right) t

The reply is received by Alice when x' = L. This means that t = \tfrac{L}{\gamma(a - v)} and thus:

t' = \frac{1 - av}{a - v}L

Since the message Alice sent to Bob took a time of \tfrac{L}{a} to reach him, the message she receives back from him will reach her at time:

T = \frac{L}{a} + t' = \left(\frac{1}{a} + \frac{1 - av}{a - v}\right)L

later than she sent her message. However, if v > \tfrac{2a}{1 + a^2} then T < 0 and Alice will receive the message back from Bob before she sends her message to him in the first place.

Benford et al.[4] wrote about such paradoxes in general:

The paradoxes of backward-in-time communication are well known. Suppose A and B enter into the following agreement: A will send a message at three o’clock if and only if he does not receive one at one o’clock. B sends a message to reach A at one o’clock immediately on receiving one from A at two o’clock. Then the exchange of messages will take place if and only if it does not take place. This is a genuine paradox, a causal contradiction.

They concluded that superluminal particles such as Tachyons are therefore not allowed to convey signals.