# Apparent paradox in special relativity: transmitting a message and setting off after it

I have a question about special relativity, which is set in the context of the science fiction anime Voices of a Distant Star, though it is independent of that context.

Mikako goes on a spaceship with a speed of $0.99999c$, heading towards a star system 4 light years away. In her rest frame, the journey takes a few days, but for her friend Noboru on Earth it takes approximately 4 years.

Now when Mikako arrives at the star system, she sends Noboru an e-mail at light speed, which takes another 4 years to arrive.

So 8 years after Mikako departs, Noboru gets an e-mail, inviting him to meet with Mikako.

Suppose that after sending the e-mail, Mikako immediately heads towards earth in her $0.99999c$ spaceship, while Noboru gets a similar spaceship and heads toward the direction of the star system. In a few days' rest frame time for both of them, they meet in deep space. Mikako then gives Noboru an album of pictures of the star system.

This seems like a paradox. When Noboru gets the e-mail, he is 2 light years away from Mikako. But a few days later in his reference frame, he obtains the pictures of the star system.

Why does this work? Is this not a paradox due to the fact that Noboru must wait 8 years for the e-mail to arrive? Or is it not a paradox due to relativity not working the way I expect it to work? Yet even from Noboru's $0.99999c$ perspective, Mikako's relative speed should never exceed $c$, so why, from Noboru's perspective, do the photos seem to be transmitted superluminally?

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It is incorrect to state that

When Noboru gets the e-mail, he is 2 light years away from Mikako.

If Mikako sets off at $v=(1-\epsilon)c$ right after she sends the message, she will trail the message by a very slight delay. When Noboru gets the message, Mikako will be overwhelmingly near Earth; Noboru can still fly out to meet her but he might as well stay on Earth, as their meeting would not take place on deep space, but close to Earth. (Specifically, at a fraction $\epsilon/2$ of the 8 ly distance.)

(Time goes upwards, space horizontally. Mikako in blue, Noboru in red, message in green.)

Edit in response to comment:

OK, I think I know what's confusing you. Suppose now that Mikako waits two years before departing. You state that

In that case, Mikako would be 2 ly away from Earth when Noboru gets the e-mail.

and this is where your misconception kicks in. Essentially, the problem is to do with the relativity of simultaneity: it makes no sense to speak of events that are happening elsewhere but simultaneously to Noboru getting the email.

The message is in green. The dotted lines are null trajectories for guidance. Mikako is in blue; she waits two years and then sets off at $v=(1-\epsilon)c$ towards Earth.

Noboru is in red, travelling to meet Mikako at $v=(1-\epsilon)c$ when he gets the message. The important line is the red dashed line: this is the spatial axis of Noboru's frame of reference once he starts travelling. The point where Mikako's worldline crosses this is "where Mikako is when Noboru gets the message", though I must stress that such language is misleading and should be avoided at all costs. Specifically, and to be clear,

## It makes no sense to speak of two events as simultaneous if they take place in different spatial locations.

Additionally, there's length contraction to be reckoned with. In Noboru's frame, the distance to that intersection (between Mikako's worldline and Noboru's space axis) is length contracted.

Thus, from Noboru's perspective once he sets off, he's at rest in his spaceship, Mikako is hurtling towards him at velocity $$v'=v\oplus v=\frac{v+v}{1+v^2/c^2}=2\frac{1-\epsilon}{1+(1-\epsilon)^2}c\approx(1-\epsilon^2)c,$$ and she is at a fraction of the contracted distance $d'=d/\gamma$ to the star system, which is already quite short. No wonder, then, that he meets Mikako after very little time.

More generally: before stopping and screaming "paradox!" one needs to do a full relativistic analysis of the situation. Don't put words in a character's mouth before you analyse the system from their (relativistic) point of view ;).

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What if Mikako waits 2 years before departing? In that case, Mikako would be 2 ly away from Earth when Noboru gets the e-mail. But in that case, if Noboru gets on a spaceship and heads to space, he will still meet Mikako in a very short time. – user54609 Nov 8 '13 at 18:13