# Relativistic ship and base on Earth talking

Let say humans make it possible to travel so close to the speed of light that time dilation for the ship can achieve 100 times the time of a standing still object. So one of that ships starts its journey and after a while something goes wrong and they have to call the base on Earth. The chat consists of two sentences that the pilot on board says in 1 minute. My question is why his sentences the people on the base should listen for almost two hours? To not have problems with signal shift due to ship relative motion regarding Earth let assume the ship is orbiting Earth at a large but constant distance.

• From your question, you seem to understand that time dilation is a thing and how it works. Could you be more precise on what you don't understand about the message undergoing time dilation as well? Or is the problem to understand time dilation itself? Commented Jun 17, 2022 at 7:42
• The answer depends entirely on whether the ship is traveling toward earth, away from earth, or in orbit around the earth. As currently worded, the question is ambiguous on this point. Commented Jun 17, 2022 at 20:27

## 1 Answer

Let's say there are two events:

A. The pilot starts their message B. The pilot finishes their message

In the rocket frame, which we'll call $$S^{\prime}$$, the time between those events is $$\Delta t_{AB}^{\prime}$$ as you've said in the question. To work out the time between the message being first received and ending on Earth, the $$S$$ frame, we'll need to Lorentz transform.

$$$$\Delta t_{AB} = \gamma \Delta t_{AB}^{\prime}$$$$

where

$$$$\begin{split} \gamma &= \left(1 - \frac{v^{2}}{c^{2}}\right)^{-\frac{1}{2}} &= 100 \end{split}$$$$

from the information you've given about the relative velocity of the ship and someone standing still on Earth. Therefore $$\Delta t_{AB} = 100$$min.

This effect is known as time dilation. Special relativity predicts that, in order for the speed of light to remain constant in all inertial frames, time has to run at different rates for observers moving with different velocities.

EDIT:

As WillO has pointed out in the comments of your original question and both answers given, the ambiguity as to whether the ship is orbiting or travelling towards/away from Earth is actually quite important. The answer given here is based on the assumption that the ship is orbiting Earth, however in the other two cases you must consider the relativistic Doppler effect as detailed in the answer below.

• This is correct if the ship is a) moving relative to earth and b) not changing its distance from the earthbound observer. In other words, it is correct if the ship is in orbit around a very tiny planet. I hastily downvoted, thinking that this was contradictory to the OP's assumptions, but I see now that the OP is ambiguous on this point. So I apologize for the downvote, which I seem to be unable to retract. Commented Jun 17, 2022 at 20:25
• I've edited my answer which should remove the vote lock. If you would still like to remove your downvote then you can click on the downarrow again to do so. No big deal either way though. Commented Jun 18, 2022 at 8:56
• Downvote removed. I still think it would be much better if you stated ckearly that your answer applies only to an orbiting ship around a planet small enough to treat as a point. Commented Jun 18, 2022 at 10:45
• Given your comments I've edited my answer appropriately. Between both answers given, the OP should have a reasonably complete answer to their original question (although, as Nickolas Alves points out, it sounds as though the original question stemmed from misunderstandings regarding the existence of time dilation). Commented Jun 18, 2022 at 12:50