Tachyon Antitelephone Paradox Calculator

Introduction

The tachyon antitelephone is one of the clearest thought experiments for showing why faster-than-light communication would be so disruptive to ordinary ideas about cause and effect. In everyday life, causes come before consequences. You press a button, then a machine starts. You send a message, then someone receives it. Special relativity preserves that order for signals that move at or below the speed of light. The trouble begins only when we imagine a messenger that can outrun light itself. This calculator explores that trouble in a concrete way by turning the standard antitelephone setup into numbers you can inspect.

In the scenario modeled here, Alice and Bob move apart at a constant relative speed. Alice sends a hypothetical tachyon signal toward Bob. Because the signal is assumed to travel faster than light, the event of Bob receiving it can become spacelike separated from the event of Alice sending it. Once that happens, different inertial observers can disagree about which event happened first. That disagreement is not a bug in relativity; it is a normal feature of spacelike separation. The paradox appears only when the spacelike connection is used to carry information. If Bob can receive the message and immediately send a reply tachyon back, Alice may receive the reply before she sent the original message.

This page does not claim that tachyons are real. No experimental evidence supports controllable faster-than-light particles, and most physicists treat the antitelephone as a warning sign rather than a blueprint for communication technology. Still, the thought experiment is valuable because it makes the geometry of spacetime easier to understand. By entering a distance, a relative speed, and a tachyon speed factor, you can see when the timeline remains ordinary and when it flips into a causality-violating loop.

How to Use

The calculator takes three inputs. The first is the initial separation $D$, measured in light-years. This is the distance between Alice and Bob at the moment Alice sends the first signal, as measured in Alice's frame. The second is the relative velocity $v$ expressed as a fraction of the speed of light, so a value of 0.5 means Bob is moving away at half the speed of light. The third is the tachyon speed factor $u/c$, also written here simply as $u$ in units where the speed of light is 1. Any value greater than 1 represents faster-than-light travel.

After entering values, select Compute Timeline. The results area reports three times. First, it shows when Bob receives Alice's outgoing signal in Alice's frame. Second, it shows the time coordinate of that same reception event in Bob's frame. Third, it shows when Alice receives Bob's reply, again measured in Alice's frame. If the final time is negative, the calculator flags a paradox because Alice receives the reply before the original send event at time zero.

To get meaningful results, use a positive distance, choose a relative speed strictly between 0 and 1, and set the tachyon speed above 1. The calculator also requires $u>v$ so that the outgoing signal can actually catch Bob in the chosen geometry. Since the page uses light-years for distance and units of $c$ for speed, the reported times come out in years. That makes the interpretation especially intuitive: a result of 2.5 means 2.5 years in the relevant frame.

If you are experimenting, try changing only one variable at a time. Increasing the separation changes the scale of the times but not the basic logic. Increasing the relative speed tends to strengthen the relativity-of-simultaneity effect. Increasing the tachyon speed makes the signal more strongly spacelike and can make paradoxical outcomes easier to produce. The sample table below the explanation is filled automatically so you can compare a few preset cases before trying your own values.

Formula

The outgoing signal and Bob's motion are described in Alice's frame. If Alice sends the tachyon at time zero from the origin, the signal follows $x=ut$ while Bob follows $x=D+vt$. Setting those equal gives the reception time in Alice's frame:

Formula: t_r = D / (u − v)

$t_r=\frac{D}{u-v}$

The reception position is then $x_r=u{t}_r$. To convert that event into Bob's frame, the calculator uses the Lorentz transformation. With units chosen so that $c=1$, the time coordinate becomes ${t_r}^{\prime }=\gamma t_r-v{x}_r$ where $\gamma =\frac{1}{\sqrt{1-v^2}}$.

If Bob immediately sends a reply tachyon back toward Alice with the same superluminal speed, the calculator uses the same algebraic structure as the original script to compute the return timing. In the implementation on this page, the reply time in Bob's frame is

Formula: t_a = (u t_r^′) / (γ(u − v))

$t_a=\frac{u{t_r}^{\prime }}{\gamma (u-v)}$

and the script then converts that quantity back to Alice's frame. The important interpretive point is simple even if the algebra looks unfamiliar: once the transformed reception time in Bob's frame becomes negative, the return leg can also become negative in Alice's frame. That is the hallmark of the antitelephone. A negative final result does not mean the arithmetic failed. It means the chosen parameters imply a closed causal loop in this hypothetical faster-than-light model.

Worked Example

Suppose Alice and Bob are initially separated by 10 light-years, Bob moves away at $0.5c$, and the tachyon speed is $4c$. In Alice's frame, the outgoing signal catches Bob after

$t_r=\frac{10}{4-0.5}=\frac{10}{3.5}\approx 2.857$ years.

The reception position is about $4\times 2.857\approx 11.429$ light-years from Alice. When that event is transformed into Bob's frame, the time coordinate becomes negative. In plain language, Bob judges that he received the message before Alice sent it. If Bob immediately answers with another tachyon, the return signal can reach Alice at a negative time in Alice's own frame as well. The calculator reports that negative value directly, which is why the result area labels the situation as a paradox.

This example is useful because it shows that the paradox does not require absurdly extreme numbers. The relative speed is only half the speed of light, and the tachyon speed is finite rather than infinite. Yet the combination is already enough to scramble temporal order. If you lower the relative speed or bring the tachyon speed closer to light speed, the paradox may disappear. If you raise either one, the backward-time effect usually becomes stronger.

Interpreting the Results

Positive values for all reported times mean the sequence still looks forward-moving in the relevant frame assignments used by the script. That does not make faster-than-light signaling physically acceptable, but it does mean the chosen numbers do not produce a full antitelephone loop in this simplified setup. A negative value for Bob's reception time indicates that the order of send and receive has reversed in Bob's frame. A negative value for Alice's final receive time is the stronger and more dramatic outcome: the reply arrives before the original transmission event at time zero.

Because the calculator works in units where distance is measured in light-years and speed is measured in multiples of $c$, the times are numerically easy to read. If the distance doubles while the speed ratios stay the same, the times double too. If the denominator $u-v$ becomes small, the outgoing catch-up time grows because Bob is harder to overtake. If the Lorentz factor $\gamma$ grows because $v$ approaches 1, frame-dependent time ordering becomes more dramatic.

The results should be read as outputs of a thought experiment, not as predictions for a real device. The page is best understood as a spacetime geometry calculator. It helps you see how relativity handles spacelike intervals and why physicists are wary of any mechanism that would allow information to travel faster than light.

Limitations and Assumptions

This calculator intentionally simplifies the antitelephone setup. It assumes one-dimensional motion along a shared axis, instantaneous sending and replying, and a single tachyon speed used symmetrically for both directions. Real discussions in relativity can be framed in different conventions, and more elaborate derivations may track additional coordinates or use different sign conventions. The script on this page preserves the original computational behavior exactly, so the displayed numbers follow that implementation rather than an expanded physical model.

Another limitation is conceptual: tachyons are hypothetical. In modern physics, the word “tachyon” sometimes appears in advanced theory, but often as a sign of instability in a mathematical model rather than as a literal faster-than-light particle that could carry messages. This calculator is therefore not evidence that time travel is possible. It is evidence that if controllable faster-than-light signaling existed within ordinary special relativity, causality would be in serious trouble.

You should also keep units in mind. The page uses naturalized relativity units with $c=1$. That is why the velocity input is dimensionless and why a distance in light-years naturally produces a time in years. If you are used to meters and seconds, the same relationships still hold, but the arithmetic would look less tidy. The current unit choice is standard for teaching and makes the paradox easier to inspect.

Finally, the calculator does not attempt to resolve the paradox. It does not include speculative mechanisms such as chronology protection, preferred frames, signal restrictions, or exotic consistency rules. Its purpose is narrower and clearer: given the assumptions of the thought experiment, it shows when the timeline becomes self-contradictory. That is exactly why the tachyon antitelephone remains such a memorable teaching tool 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.

In the first case the paradox is evident. The second scenario uses a slower tachyon and a smaller separation, yet it can still produce a backward-time result. The third pushes the relative speed much closer to light speed and shows how quickly the transformed times can become extreme. Comparing rows helps build intuition: the paradox is not tied to one magic number, but to the broader combination of superluminal signaling and relative motion.