Antitelefon

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 presented a thought experiment of how faster-than-light signals can lead to a paradox of causality, which was described by Arnold Sommerfeld in 1910 as a means "to telegraph into the past". The same thought experiment was described by Richard Chace Tolman in 1917, 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 path

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

Einstein and Tolman considered the following thought experiment: Imagine a material rod of length L' with endpoints A and B. Let a force be applied at A whose action propagates with velocity w' towards B, measured in the rest frame of the rod. So in this frame, the event at A is the cause of the event at B. However, in a reference frame moving with relative velocity -v, the action propagates (according to the relativistic Velocity-addition formula) with velocity

${\frac {w'-v}{1-{\frac {w'v}{c^{2}}}}}$

thus it arrives at B at time

$T=L{\frac {1-{\frac {w'v}{c^{2}}}}{w'-v}}$

It can be easily shown that if w' > c (where c is the speed of light), one can always choose certain values of v in order to make T negative. In other words, the effect arises before the cause in this frame. Einstein concluded, that this result contains in his view no logical contradiction, however, it contradicts the totality of our experience so that the impossibility of w' > c is sufficiently proven.

Two-way path

A more common variation of this thought experiment is to sent back the signal to the sender (a similar one was given by David Bohm). 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 $a$$c$ 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) 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. wrote about such paradoxes in general:

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

See also

• Grandfather paradox
• Ansible