Faster‑Than‑Light Signals and the Causality Trap

The tachyon antitelephone is a celebrated thought experiment that dramatizes the problems raised by hypothetical faster‑than‑light (FTL) particles. Tachyons are imagined entities that always move faster than light. While none have been observed and most physicists doubt their existence, playing with their properties offers a window into the deep structure of spacetime. If tachyons were real and could be manipulated, they would enable communication into the past. The resulting paradoxes challenge our notions of cause and effect and highlight the protective role that relativity plays in keeping timelines consistent. This calculator lets you tinker with the geometry of one such paradox. By specifying the initial separation between two observers, their relative speed, and the tachyon's superluminal factor, you can see how a message from one observer might loop back to arrive before it was sent.

Imagine two experimenters, Alice and Bob, drifting apart in their spaceships. In Alice’s frame, Bob heads off along the positive x‑axis at a constant fraction $v$ of light speed. When they have separated by a distance $D$, Alice impulsively fires a tachyon beam toward Bob. Because the beam travels with speed $u$ satisfying $u>c$, it eventually catches up. The catch is that Bob is in motion, so determining when and where the signal reaches him requires tracking both the tachyon and Bob’s trajectory. In Alice’s rest frame the signal follows $x=ut$ while Bob follows $x=D+\mathrm{vt}$. Setting these equal yields the reception time $\mathrm{t\_r}=\frac{D}{u-v}$. Even for modest separations the superluminal beam can outrun Bob quickly, so $\mathrm{t\_r}$ is often much less than the light‑time $\frac{D}{c}$.

The real mischief begins when we view the same event from Bob’s perspective. Special relativity links Alice’s time and space coordinates to Bob’s via the Lorentz transformation. If the beam reaches Bob at Alice’s coordinates $\left(\mathrm{t\_r},\mathrm{x\_r}\right)$, then in Bob’s frame the corresponding time is $\mathrm{t\_r\text{'}}=\gamma \mathrm{t\_r}-v\mathrm{x\_r}$ where $\gamma$ is the familiar factor $\frac{1}{\sqrt{1-v^2}}$. Because $\mathrm{x\_r}$ can be large for a fast tachyon, the term $v\mathrm{x\_r}$ may outweigh $\mathrm{t\_r}$, driving $\mathrm{t\_r\text{'}}$ negative. In that case Bob concludes that the tachyon arrived before Alice launched it! Relativity alone does not forbid this: simultaneity is relative, and spacelike separated events can be seen in opposite time orders by different observers. Normally this is harmless because no signal connects the events. Tachyons, however, provide just such a connection, opening the door to contradictions.

To escalate the paradox, Bob can immediately send a reply tachyon back toward Alice. If his tachyon travels at the same speed $u$ but in the negative x direction, we can repeat the calculation in Bob’s frame to find when it intercepts Alice. The result, after another Lorentz transformation, is an arrival time in Alice’s frame given by $\mathrm{t\_a}=\frac{u}{\mathrm{t\_r\text{'}}}/\gamma (u-v)$. Remarkably, if $\mathrm{t\_r\text{'}}$ was negative then $\mathrm{t\_a}$ will be negative as well: Alice receives Bob’s response before she sent the initial message. With suitable choices of parameters, the interval can be arbitrarily large. Alice could even be warned about disasters days before deciding to start the exchange. The loop thus allows information to propagate into the past, a flagrant violation of causality.

Our calculator exposes these time loops numerically. Enter a separation of ten light‑years, a relative speed of $0.5c$, and a tachyon speed four times the speed of light. The reception time in Alice’s frame is about 2.86 years. However, in Bob’s frame the signal arrives roughly 3.3 years before the send event. Bob dutifully replies, and thanks to the geometry of the situation Alice intercepts the response a few years before she ever pressed “send.” The display flags this with a negative arrival time. By adjusting the speed parameters you can map out when the paradox appears. If the tachyon speed approaches infinity, any nonzero relative velocity produces causal issues. With slower tachyons the paradox requires a sufficiently fast relative motion.

One might wonder whether tachyons could be tamed by imposing a rule that they cannot transmit useful information. But the antitelephone uses only simple linear propagation, so banning communication would require forbidding any tachyon interaction. Another escape route posits that tachyons always move forward in time in their own frame, so they cannot be controlled to send backward‑propagating messages. Yet this merely shifts the problem: any coordinated use of tachyons by two parties with relative motion leads to inconsistencies. The prevailing view is that the universe prevents such particles from existing, preserving causality by decree.

The paradox also illuminates how special relativity intertwines space and time. Causality is preserved for lightlike and timelike signals because no observer can transform them into reversed temporal order. Spacelike signals, by contrast, have no invariant temporal direction. Superluminal communication therefore makes causality frame‑dependent: different observers disagree on which event causes which. Relativity does not forbid such signals outright; rather, their absence appears to be an empirical fact necessary for a sensible universe. The tachyon antitelephone is a kind of reductio ad absurdum demonstrating that if even a single superluminal messenger existed, logical contradictions would proliferate.

Beyond its pedagogical value, the thought experiment has inspired creative fiction and speculative physics alike. Science‑fiction authors have used tachyon phones to enable time travel plots, often invoking paradoxes akin to the one quantified here. In theoretical circles, discussions of tachyons appear in string theory and quantum field theory, but there the term refers to instabilities rather than real faster‑than‑light particles. Our calculator does not aspire to settle these debates. Instead it provides a sandbox for exploring spacetime geometry. By crunching the algebra for arbitrary speeds and distances, it reveals that causality violations are not fringe curiosities but an inevitable outcome of superluminal signaling in relativity.

The table below summarizes a few illustrative scenarios. Each row lists the chosen velocities along with the computed times. Negative values for Alice’s receive time indicate paradoxical arrivals before the signal was sent.

v/c	u/c	D (ly)	tr (A yrs)	t'r (B yrs)	ta (A yrs)
0.5	4	10
0.3	2	5
0.8	10	20

In the first case the paradox is evident. The second scenario, with a more modest tachyon speed, still returns a negative time. The third pushes both parameters near their extremes, generating a dramatic backward arrival exceeding a decade. Such numerical experiments drive home why physicists view FTL signaling with deep suspicion. Even in the absence of grandparent‑killing antics, the mere ability to send information into the past undermines determinism and renders cause and effect ambiguous.

Ultimately the antitelephone encourages humility about the assumptions underpinning physics. Relativity emerged not from abstract musings but from resolving contradictions in electromagnetism and mechanics. By probing new hypothetical contradictions, we can test the limits of our theories. If someday a consistent theory of quantum gravity permits controlled superluminal effects without paradox, it will require rewriting foundational notions of time. Until then, the tachyon antitelephone serves as a cautionary tale: tinker with the speed limit of the universe, and causality unravels.