# Tachyonic antitelephone

The tachyonic antitelephone is a hypothetical device in theoretical physics that can be used to send signals into one's own past. Such a device was first contemplated by R. C. Tolman in 1917 [R. C. Tolman, "The theory of the Relativity of Motion", (Berkeley 1917), p. 54] in a demonstration of how faster-than-light signals can lead to a paradox of causality (a.k.a. "Tolman's paradox"). The problem of detecting faster-than-light particles (a.k.a. tachyons) via causal contradictions is considered in Ref. [G. A. Benford, D. L. Book, and W. A. Newcomb, [http://link.aps.org/abstract/PRD/v2/p263 "The Tachyonic Antitelephone"] , "Physical Review" D 2, 263-5 (1970)]

ending signals into one's own past

Suppose we have a device that is capable of transmitting and receiving tachyons at a speed of $a c$ with $a>1$. Consider sending such a tachyon to a spacecraft that moves away from us in the negative x-direction with speed $v$. Let's choose the origin of the coordinates to coincide with the reception of the tachyon by the spacecraft. If the spacecraft sends a tachyon back to us then, in the rest frame of the spacecraft, the coordinates of the tachyon are given by:

:$\left(t,x\right) = \left(t,act\right)$

To find out when the particle is received by us, let's perform a Lorentz transformation to the frame S' moving in the positive x-direction with velocity v, with respect to the spacecraft. In this frame we are at rest at position $x\text{'}=L$ where $L$ is the distance the tachyon we send to the spacecraft traversed in our rest frame. The coordinates of the tachyon are given by:

:$\left(t\text{'},x\text{'}\right)=left\left(gammaleft\left(1-frac\left\{av\right\}\left\{c\right\} ight\right)t,gammaleft\left(ac-v ight\right)t ight\right)$

The tachyon is received by us when $x\text{'}=L$. This means that $t=frac\left\{L\right\}\left\{gamma\left(ac-v\right)\right\}$ and thus:

:$t\text{'}=frac\left\{c-av\right\}\left\{ac-v\right\}frac\left\{L\right\}\left\{c\right\}$

Since the tachyon we send to the spacecraft took a time of $frac\left\{L\right\}\left\{ac\right\}$ to reach it, the tachyon we receive back from the spacecraft will reach us a time:

:$T=frac\left\{L\right\}\left\{ac\right\} + t\text{'}=left \left[frac\left\{1\right\}\left\{a\right\}+frac\left\{c-av\right\}\left\{ac-v\right\} ight\right] frac\left\{L\right\}\left\{c\right\}$

later than we send it. However, if $v>frac\left\{2ac\right\}\left\{1+a^\left\{2$ then $T<0$ and we'll receive the tachyon back from the spacecraft before we have sent our tachyon to the spacecraft.

References

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