The antitelephone

You stay home. Your friend flies past at a good fraction of light speed. You send a message faster than light; she replies the same way, by her own clock and her own idea of "instant". Here is when the reply lands.

you, at rest your friend, moving light your message her reply

Controls

Diagram is a spacetime diagram: position across, time up, light at 45°. Everything above the dashed line happened after you sent the message. If the reply lands below it, it arrived before you sent.

Things to notice

Faster than light is already backwards in time for someone

Set the message speed just above c and slide her speed up. In her frame your outbound message tilts until it points into her past, before it ever reaches her.

Why the tilt

The reply is the killer

One FTL message is odd but survivable. Two, exchanged between people in relative motion, deliver the answer before the question. That is the antitelephone.

Why two

There is a threshold

The reply beats the question only when her speed exceeds 2u over 1 plus u squared, with u the message speed in units of c. At ten times light speed, walking pace nearly does it.

The formula

Switching frames changes nothing

Draw it in her frame instead. Same events, same order of cause and effect for anyone on either worldline. The paradox is not an illusion of one viewpoint.

What is invariant

The one escape is a preferred frame

If "instant" meant instant in one special frame for everyone, the reply could never beat the question. That escape costs you special relativity for the FTL sector.

Register CAU-4

Every drive in the catalogue owes this bill

Warp bubbles, wormholes, hyperspace, ansibles: any of them lets you build this diagram. The register scores each on what it does about it.

Register CAU-1

What is going on

Special relativity does not say nothing can go faster than light. It says that if anything did, observers moving relative to each other would disagree about which of two events came first. For ordinary signals that never matters, because light-speed limits keep cause safely before effect in every frame. For a faster-than-light signal it matters immediately: pick the right frame and the message arrives before it was sent.

Einstein saw this in 1907 and Tolman spelled it out in 1917. In 1970 Benford, Book and Newcomb turned it into the two-signal version drawn here and gave it the name. The full entry, with citations, is CAU-1 in the register. The escape hatch, a preferred frame, is CAU-4.