- #176
PeterDonis
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JohnNemo said:I hope this is also correct
If you're going to talk about Doppler shift, you probably should also address the case where the observers are moving towards each other, which means they will see each other blueshifted--"running fast" instead of "running slow"--and time dilation only appears after this is corrected for light travel time.
Also, as my phrasing just now indicates, I'm not sure that describing the case you describe as "correcting for the redshifting effect" is a good way to describe it. It might help to look at the math. If the two observers are moving away from each other, the relativistic Doppler factor is
$$
D = \sqrt{\frac{1 - \beta}{1 + \beta}} = \frac{1}{\gamma} \frac{1}{1 + \beta}
$$
where ##\gamma## is the usual relativistic factor ##1 / \sqrt{1 - \beta^2}##, so ##1 / \gamma## is the time dilation factor. Notice that the Doppler factor is a smaller number than the time dilation factor; in other words, each observer actually sees (through their telescope, say) the other's clock running more slowly than the time dilation factor alone would imply. This extra factor is because as the observers move apart, the light travel time between them increases.
If the two observers are moving towards each other, the relativistic Doppler factor is
$$
D = \sqrt{\frac{1 + \beta}{1 - \beta}} = \frac{1}{\gamma} \frac{1}{1 - \beta}
$$
Here the Doppler factor is a larger number than the time dilation factor, and in fact is greater than ##1##, indicating a blueshift, not a redshift. The "correction" here, to get the time dilation factor less than ##1##, so that each observer "observes" the other's clock to be running slow (when they actually see, as in through their telescope, the other's clock running fast), is due to the light travel time between the observers decreasing as they move towards each other.
My own preference is actually to take the Doppler factor as primary, since that's what is actually directly observed. The "correction" to get the time dilation factor that appears in many of the common formulas can then be explained as being due to light travel time, as I did above. Trying to explain it the other way is possible, but might be more likely to cause confusion.