# Is Grav & SR Time dilation additive?

1. Nov 24, 2014

### mcjosep

Just wondering if you are observing someone from a far out distance and they are in a gravitational field going at a high speed would the time dilation from their speed add on to the gravitational time dilation?

2. Nov 24, 2014

### Matterwave

You can, in some instances, split the effects and say "X amount of time dilation came from gravitational time dilation" and "Y amount of time dilation came from the object's speed"; but GR encompasses both gravitational effects as well as SR effects, so when you are calculating the time-dilation, it's I think easier if you just calculate the total time dilation factor directly without trying to split the effects up.

In more complicated situations, you might not be able to just add the effects like you are suggesting. Esp if the gravitational field is not static/stationary.

3. Nov 24, 2014

### Staff: Mentor

In some simple cases they are approximately multiplicative. However, even then it is an approximation and it doesn't work in more complicated spacetime. I prefer not to split them up.

4. Nov 25, 2014

### harrylin

That's an interesting though, but it does not sound convincing to me, as upon a short reflection, I can't find the problem.
It is possible to synchronize clocks on Earth with geostationary clocks in space by correcting for rotation and gravitational time dilation; these clocks can in turn be synchronized to more distant clocks that co-move with the Earth, with negligible gravitational time dilation. Is there a problem with a Lorentz transformation to those clocks from a distant reference system that is moving relative to the Earth?

Thanks!

5. Nov 25, 2014

### PAllen

You can do this precisely to the extent that gravity adds linearly (to the desired precision) and thus can be represented by a scalar potential. That is, no body is too massive, and relative motion between massive bodies is small compared to c.

6. Nov 25, 2014

### harrylin

OK, I see that there is an issue with the casual expression "add the effects"; and I had not thought about the title of this thread. The equations obviously require a multiplication of the time dilations and not a literal addition, although this is approximately correct for terms close to 1. It appears to me that that is all there is to it, as the OPs' question concerns a single gravitational field.
Thus, my question to Matterwave remains.

Last edited: Nov 25, 2014
7. Nov 25, 2014

### PAllen

Maybe I'm not understanding your question. Matterwave stated that separations and simple ways of combining effects break down for significantly non-stationary situations. Non-stationary implies motion of stress/energy beyond rotation of an ideal body. Thus, if by 'single gravitational field' you mean a single massive body, possibly rotating, and reasonably idealized (no wild density variations, axial symmetry), then matterwave's caveat doesn't apply. The most important real world case it does apply is closely orbiting bodies of sufficient mass (e.g. compact binary (neutron) stars).

8. Nov 25, 2014

### Jonathan Scott

Note that there are situations such as time dilation within a rotating space station that can either be analysed using SR from outside based on velocity or in terms of the "effective gravitational potential" caused by the acceleration as seen from within the station. Both calculations give the same result, but they represent different ways of looking at the same situation, so they don't add.

9. Nov 25, 2014

### pervect

Staff Emeritus
My understanding of the definition of time dilation is that it is the ratio of proper time to coordinate time. I haven't found a textbook reference to check and confirm this definition, but I haven't seen the definition fail fail either.

If we assume the definition is good, then time dilation doesn'the have any direct physical significance except under certain situations. One of these situations is when you have a sort of symmetry called a "timelike Killing vector"' which is present whenever you have a static or stationary metric, as Matterweave mentioned. In these cases, when your coordinate time reflects the underlying physical symmetry, time dilation has some physical significance.

Otherwise, time dilation depends on the particular coordinate system you use. More precisely, it depends on what notion of simultaneity you adopt. It should be well known that simultaneity is observer dependent in special relativity, thus different observers have different notions of simultaneity. This implies that they have differnt notions of time dilation, so the concept of time dilation, like the concept of simultaneity, is observer dependent. It seems that this point causes a lot of oblique arguments, it seems people resist understanding that simultaneity is not absolute, and can't really deal with the consequences. But I don't want to get off track, just point out that time dilation in general is observer dependen't, and that a widespread lingering belief in absolute Newtonian time is an obstacle to understanding.