Question about time dilation

I think you have faster and slower the wrong way round there.

What you are describing is the Doppler effect. It would happen in a Newtonian universe too. It's the cause of the pitch change of a siren when a police car passes you.

However, in relativity, once you have corrected for the speed of motion of the source, you find that you still have the clocks running slow. That is "time dilation" at work, and happens whether the spaceship is coming towards you or going away.

There is nothing wrong with your reasoning. You will see clocks ticking slower on ships going away from you and faster on those coming towards you. But correct for the varying travel times of the light, and in a relativistic universe you will always find that the other fellow's clocks are running slow.

 

Note that the time delay caused by the finite speed of light when making observations is NOT the basis of time dilation. When thinking about these problems you must factor out any time lag. One way to do this is to think about a Reference Frame, rather than a single observer. A Reference Frame can consist of many observers (all at rest with respect to each other) with synchonised clocks and who will agree about length measurements.

In your experiment, you could consider two observers at rest with respect to the Earth, who observe the ships clock as it moves past them. In this reference frame, the ship may take 1s, say, to travel from one observer to the next. Each observer notes the time on the ship's clock. The second observer will find that the ship's clock has moved forward by less than 1s, and conclude that time is running slower on the ship than in the "at rest" reference frame.

 

The effect you're talking about is called the relativistic Doppler shift. If a clock is moving toward you, you will see it running faster, because the delay for light to travel from the clock to you keeps getting shorter. If a clock is moving away from you, you will see it running slower. Neither of these effects are relativistic time dilation. Relativistic time dilation is the computed rate of the moving clock after taking into account the travel time for light.

 

If you only have one observer, then he/she must take into account the time lag. The clock reading you see as a clock moves towards you very quickly is partly explained by the movement of the clock coming towards you (speeding up the reading) and partly explained by time dilation (always slowing down).

I would strongly recommend trying to understand time dilation without the time lag first. Once you understand the time dilation between reference frames, you can learn about the relativistic Doppler effect.

One thing you might like to look up is the Transverse Doppler effect. This is a Doppler effect of motion at right-angles to your observation and is caused purely by time dilation.