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:

:$(t,x)\; =\; (t,act)$

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:

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

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

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

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

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

later than we send it. However, if $v>frac\{2ac\}\{1+a^\{2$ then and we'll receive the tachyon back from the spacecraft before we have sent our tachyon to the spacecraft.

References