# Formula for send and response times of messages at fraction of light speed?

I have a ship that begins the equation traveling at 50 percent the speed of light, but decelerating constantly to finally land on a certain planet (towards which it is moving the entire time). It begins its journey 7.5 light years away from the planet, with which it is exchanging messages as rapidly as possible. I am wondering if there is a formula I can use to calculate how many messages they can send, and at what times/distances each will be sent (and received) by the time they arrive. Does this make sense? Is it even a sensible question?

• It is a sensible question. I don't remember the formula by heart but I know it. – The Square-Cube Law Dec 24 '19 at 21:09
• If the ship begins its journey 20 light years from the planet at 0.5c, with constant deceleration, it will take 80 years to arrive, not 10 — average speed is 0.25c. Relativity will change that a bit from the ship’s frame of reference, but not much — time dilation is only about 14% at 0.5c. . – Mike Scott Dec 24 '19 at 21:13
• Are they starting 20 light years away and then traveling to the planet while sending messages, or traveling away from the planet while sending messages? Just a little confused by your phrasing on where the planet is in relation to the ship. – cegfault Dec 25 '19 at 13:50
• Sounds like you are saying the ship is 20 light years from destination and initially traveling at 0.5 light years/year. So even if it never slowed down from initial velocity, from the planetary point of view it wold take 40 years to reach the planet and fly by still at 0.5 C – Jim Dec 25 '19 at 14:25

The first thing you need to keep in mind is that when two observers are moving relative to each other in a system, each one will perceive the other as experiencing dilated time. That means from your point of view, it will seem that whomever you are communicating with is experimenting slowed down time - but from their point of view, it is you who are experiencing slow time. You can find more at the wikipedia article for time dilation, under "reciprocity". While you keep at different velocities, you may disagree on the timespan between events.

That said, I think the formulas you are looking for is this. For the span of time at the other end of communication:

$$\Delta t = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$

Where $$v$$ is their speed relative to you.

If your signals are traveling at the speed of light, they will clear the space between you and the other guy in time $$t = \frac{d}{c}$$, where $$d$$ is the distance between you both. Otherwise you will need to calculate compressed space. Its formula is the opposite of the first one. Compressed space is calculated as:

$$s = \sqrt{1 - {v^2 \over c^2}}$$

That's the length the signal has to traverse if it is at subluminal speeds, then you can calculate the time it takes to arrive (as measured from its frame of reference) by using the old $$s = vt$$ formula. Notice that its own measure time may be different from the time measured by sender and recipient.

• You got your powers of two in the wrong place; I think I've fixed them now. You might make the whole thing a bit briefer by defining the Lorentz factor first, and then doing everything else in terms of that, but it doesn't really matter. – Starfish Prime Dec 24 '19 at 21:35
• @StarfishPrime actually $(v/c)^2 = (v^2)/(c^2)$. – The Square-Cube Law Dec 24 '19 at 21:48
• So it is. You did miss the $1-$ bit for length contraction though, so you had $\sqrt{ \left( \frac{v}{c} \right)^2}$ which I'm pretty certain wa wrong though ;-) – Starfish Prime Dec 24 '19 at 22:00
• @StarfishPrime ah ok! Thanks for the correction :) – The Square-Cube Law Dec 24 '19 at 22:00
• Your formulae may need further clarification. As they stand, they give the same answer for any given velocity. Additional terms for time and distance do seem to be needed. A simple enough edit. – a4android Dec 24 '19 at 22:18

You do not need relativistic formula if you do not change the reference frame.

From POV of a planet this comunication process is straightforward: first message from the ship travels 20y, when planet recieves it ship is at 20 - 0,5*20 +(1/160)*20^2 / 2 = 11,25ly distance. Reply would take +7,5y (27,5y from start) to reach the ship and so on.

From POV of ship it's a little harder since starting conditions differ (its only 17,2ly from a planet) and start point will "move" (it is "move" without move - like universe expanding) to 20 ly while this ship deaccelearates with all other points. This leads to "strange" fact - speed of light in non-inertial frame is not c. It even is not constant in time (and, for rotating frames, in space). But we can make a shortcut - take numbers from planet POV and transform them with Lorenz formula using speed of the ship at the event points as a parameter. Since accelearation is very small - 1/160 g (it easy to get using suprising coinsedence - 1ly/y^2 is a little less then 1g) we can neglet acceleration time dilation, using only speed time dilation. But we can't do this directly - we need to integrate to calculate "accumulated" dilation. So time in planet frame transform to time in ship frame like this:

$$t_{ship} = \int_{0}^{t_{planet}}\sqrt{1 - (0.5 - t/160)^2}dt$$

So from POV a ship first signal will reach the planet in about 18y, reply will arrive in about 25y from start

(total deacceleration time for the ship is about 76.5y)

BUT if you want a constant acceleration in ship frame - it would be different and a harder calculations.