Time dilation and 2 identical clocks

[Herman Trivilino]

there is no observer on the ship

The observer external to the ship will percieve the egg timer on the ship as ticking faster than the egg timer at rest wrt the observer

It occurs to me that you may be misunderstanding what we mean when we say that the external observer will observe that the clock on the ship is running slow compared to the time-keeping devices that are at rest relative to him.

Let's say that three minutes pass on the clock aboard the ship, and during that time an egg is cooked aboard the ship. It will be a perfectly-cooked three minute egg, and all observers, regardless of their state of motion relative to it, will observe that it did indeed take three minutes to cook it. They will all agree that three minutes went by on the timer and that that is the reason the egg came out perfectly cooked.

This is an example of what we call a relativistic invariant. It simply means that all observers, regardless of their motion relative to a device used to make a measurement, will agree on the result of the measurement. In this case the measurement is 3 minutes. But the measurement could also be of the air pressure inside the ship. If a pressure gauge inside the ships reads 101 kPa, all observers will agree that the gauge reads 101 kPa.

The reason we say that the external observer will conclude that the egg timer is running slow is because of the results of measurements that he takes with other clocks. Clocks that are at rest with respect to that external observer, and synchronized in the rest frame of that external observer, are carefully placed so that one is next to the ship when the timer starts and the other is next to the ship when the timer stops. They might show that, for example, five minutes of time passed while the egg was cooking. This would be the case if the ship were moving at a speed of $\frac{4}{5}c$ relative to your external observer.