# Time Dilation vs Differential Aging

1. Jan 13, 2013

### arindamsinha

This topic is to clarify about related but possibly different terminologies.

Based on a number of posts in the forum, including some in response to my posts, I am getting the feeling that there is a difference between the terminologies time dilation and differential aging, as stated below.

I am trying to confirm a few things around this based on the understanding I have got:
• Is Time dilation a coordinate/observer dependent effect which includes the Doppler effect (apparent aging of a distant body based on Doppler shift of light signals) as well as differential aging (actual difference in the rate of clock ticks which is invariant and not coordinate dependent)?
• Would it be correct to say that differential aging and relative time dilation can be considered to be the same thing? (Meaning, once the 'apparent' Doppler part (i.e. symmetrical part) of time dilation is taken out, what remains is the "differential aging" or "relative time dilation")?
• Are the terms time dilation and differential aging treated to be so unambiguously different in accepted/standard scientific literature? (Reason I ask this is I seem to find quite a few references to "time dilation" when the actual phenomenon being talked about seems to be actually "differential aging")?

2. Jan 13, 2013

### Mentz114

Differential ageing is measured in proper times, so it is observer independent. Time dilation refers to the factor that must be applied when converting clock times between IRFs. So it is a coordinate effect. The Doppler shift between observers is coordinate independent also.

That is how I understand these terms. The terms are being used interchangeably by some posters, wrongly in my opinion.

I don't know what 'relative time dilation' is. Did you invent that ? Is there 'non-relative' time dilation ?

Last edited: Jan 13, 2013
3. Jan 13, 2013

### Staff: Mentor

Time dilation, at least in my understanding, is the ratio dt/dτ where t is coordinate time and τ is proper time. I don't think that it has anything directly to do with the Doppler effect nor with receiving or sending signals to a distant observer.

I don't know what "relative time dilation" is as opposed to just "time dilation".

I agree. The literature is not consistent in this terminology. In the special case where you have a twin scenario (i.e. a pair of objects that are each at two events but take different paths between those events) the differential aging can be calculated simply by integrating the time dilation for each twin along their respective paths. So they are closely related.

4. Jan 13, 2013

### tom.stoer

Look at two observers A and B starting at the same space-time point X, moving along two arbitrary (timelike) curves CA and CB and meeting again at space-time point Y. At Y they will compare their proper times (invariant properties w/o frame dependence) τA and τB elapsed when moving along the two curves.

To calculate the proper time for path i = A,B one introduces an inertial frame S with coordinate time t. The expression then reads

$$\tau_i = \int_{C_i} d\tau = \int_0^T dt\,\sqrt{1-v_i^2(t)}$$

Note that there are two effects regarding "time differences":
a) difference between proper times τA and τB
b) difference between proper times τi and coordinate time T

Last edited: Jan 13, 2013
5. Jan 13, 2013

### tom.stoer

I think SR and GR have been written down in German language - but there is no German translation for 'Differential Aging'. Is it a problem invented by Einstein after he moved to Princeton? Or has it been invented by the Americans to challenge the Germans?

6. Jan 13, 2013

### Mentz114

I thought Einstein called it 'the clock paradox'. I don't know the German for that.

7. Jan 13, 2013

### ghwellsjr

I agree with the previous posters but let me add:

If we didn't have the theory of Special Relativity, we wouldn't have Time Dilation but we would still have Differential Aging and the Doppler effect.

Differential Aging is a very simple concept. Take two clocks. They each keep track of their own Proper Time. You start by bringing them together and keeping track of the Proper Time on each of them. Note that when they are brought together, we don't care "where" that is or even if they have a relative speed between them. Then we separate them. We don't care how they travel or how they accelerate but eventually they must be brought back together again, not necessarily at the same location as the first time and again we don't care if there is a relative speed between them. We note their Proper Times again. We calculate how much Proper Time has progressed on each clock which is how much each clock has aged. Whatever difference there is between these two times is what we call their Differential Aging. So Differential Aging is a measurable phenomenon that happens whether or not we understand why it happens or whether or not we have any explanation or theory to account for it.

Now let's add in the Doppler effect. First, let me clarify that when we talk about Doppler in this context, we are not limiting it to what is normally referred to as Doppler in other contexts where there is a shift in the frequency of the observed light from a relatively moving source which changes its observed color. There is also a shift in the observed tick rate of a relatively moving clock and that is what we are focusing on. Of course, since the relative speed is excessive, the observed image of the relatively moving clock will be Doppler shifted outside the human-visible range and so we assume that an optical instrument of some kind is available to correctly observe the time on the remote clock. Or equivalently, we can assume that a radio signal is transmitted by each clock at periodic times.

So with all that in mind, as each clock is observing the other clock, they can compare the Proper Time on their own clock to the Doppler shifted Proper Time that they see on the other clock. At the start and end of the process described earlier for Differential Aging, because they are colocated, they will be able to compare Proper Times without any Doppler shift. But as soon as they separate, there will be a Doppler shift making the comparison of Proper Times offset which they can both observe and keep track of. If they watch each others clocks all the way through until they once again are colocated, they will then see the same final times that we said they saw in the explanation of Differential Aging. So, just as in Differential Aging, the Doppler shift effect is also a measurable phenomenon that happens whether or not we understand why it happens or whether or not we have any explanation or theory to account for it.

Now we come to the theory of Special Relativity which provides a means to make sense of Differential Aging and the Doppler effect. In this theory, we establish an Inertial Reference Frame (IRF) which provides spatial and time coordinates so that we can describe the coordinate positions of our two clocks as a function of coordinate time. Note that our selection of the IRF can change completely what these coordinates are and have no bearing on the measurements that were described earlier--we just have to make sure that our theory is consistent with all the measurements.

So in the theory of Special Relativity, we can easily calculate the tick rate of the Proper Time on any clock by simply noting its coordinate speed according to the IRF. For any speed, we calculate the factor call gamma and this provides a ratio of the Coordinate Time tick rate to the Proper Time tick rate of the moving clock. This ratio is called Time Dilation and is unmeasurable and unobservable by the clocks in our scenario. Once we establish a scenario as a series of events (position/time coordinates for the clocks) according to our arbitrarily selected IRF, we can convert the coordinates of the events to any other IRF moving with respect to the first one and get a new set of coordinate speeds, coordinate times, coordinate positions and therefore a new set of Time Dilations.

Note that in some limited scenarios, such as the classic Twin Paradox, it turns out that the factor gamma is also the same factor that the Differential Aging provides and so this incorrectly leads many people to jump to the conclusion that they are the same thing.

Regarding your term "relative time dilation", I think you are trying to conflate the SR term Time Dilation with the other terms that are independent of SR and so it doesn't make any sense.

8. Jan 13, 2013

### tom.stoer

Thanks for the explanation. I think in the context of SR nearly everything is said with my post #4
a) is both measurable and calculable using SR.
b) is measurable only if there is a third observer C with proper time equal to coordinate time t, i.e. being at rest in the inertial frame S.

Everything else is a matter of definitions and wording, right?

That's not what I mean; 'clock paradox' is 'Uhrenparadoxon'. But that's missleading b/c we all know that there is no paradox at all.

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9. Jan 13, 2013

### Mentz114

I agree with that. I've just read loads of rubbish about asymmetry and acceleration by people who don't know how to calculate proper time.

I know there's no paradox. I spent my early years on this forum objecting to the use of the 'p' word. It only encourages idiots.

10. Jan 13, 2013

### tom.stoer

Great! That's why I start with the general formula and arbitrary curves.

11. Jan 13, 2013

### Mentz114

Yes, given the metric and a few postulates, it's all in the geometry. We need the LT to transform coords between events on curves to close it all off.

Too many times discussions focus on things which are coordinate dependent, when we should be talking about invariants like proper times and Doppler ratios.

ghwellsjr's post is a good explanation because it's expressed in coordinate independent quantities. Nice.

12. Jan 13, 2013

### tom.stoer

I absolutely agree!!

There are two common mistakes when teaching SR:
- focussing on coordinates and their transformation
- following all historical detours

13. Jan 13, 2013

### Staff: Mentor

I agree with this one in particular. Especially Einstein's thought experiments. They always confused me more than they helped.

Last edited: Jan 13, 2013
14. Jan 13, 2013

### tom.stoer

I always stop reading whenever the 7th train, the 13th clock and the 4th frame is introduced

What was most helpful when starting with SR was the fact that 0.64 + 0.36 = 1

15. Jan 13, 2013

### ghwellsjr

You're welcome and yes, I already agreed with your post. But your math may not communicate to those who are still novices to Special Relativity which prompted my elaboration.
Even in this case, an observer at rest in the selected IRF cannot measure or observe the Time Dilation of a moving clock anymore than either of the other observers or clocks because they are all subject to the same issue of light propagation time. Of course he can make a measurement of the Doppler and calculate what the Time Dilation was (after the fact) for his rest frame, but then so can the others for their own or any other IRF, all after the fact.

The closest way to demonstrate a "measurement" of Time Dilation is to have a series of IRF-synchronized clocks at rest in the selected IRF all along the path of the moving (and maybe accelerating) clock so that the Proper Times on the moving clock can be compared to the Proper Times on successive stationary clocks but this obfuscates the whole concept of Coordinate Time which is an abstraction of time that simultaneously pervades all of the Coordinate Space. Even if you did this "measurement" for one particular IRF, you could then transform to a different IRF where there would no longer be any synchronized clocks (or stationary observers) but yet the concept of Time Dilation applies just as legitimately and equally as abstractly.

16. Jan 13, 2013

### ghwellsjr

Huh? Without coordinates and their transformation, what is left of SR?

17. Jan 13, 2013

### tom.stoer

Proper time, differences in proper times, Doppler effect, invariant masses, ...

You can resolve the twin using two curves intersecting at X and B as described in post #4 w/o referring to coordinates!

18. Jan 13, 2013

### ghwellsjr

But in post #4 you said, "one introduces an inertial frame S with coordinate time t" and proceeded to use an equation with t in it, so I'm confused.

But as I pointed out earlier, we can resolve the twin scenario without resorting to frames or coordinates or Time Dilation or anything else associated with Special Relativity, but that would not be teaching Special Relativity, it would be explaining the problem that Special Relativity addresses, that is, how do we make sense of the differing Proper Times on clocks that accelerate differently?

19. Jan 13, 2013

### tom.stoer

Yes, in order to make contact with well-known formulas. The invariant length of a curve C can be defined w/o referring to coordinates. And to explain how the 'would-be paradox is resolved' one only needs a), i.e. no coordinates.

I don't understand; this is fully addressed in SR.

20. Jan 13, 2013

### PAllen

You can define it without coordinates, and derive properties without coordinates, but I'm not clear how you could compute anything without coordinates. Is there a way?

21. Jan 13, 2013

### tom.stoer

It depends what you mean.

If observer A stays at rest whereas B moves along a circular curve with constant speed v, then one immediately finds the well-known result

$$\tau_A = T$$
$$\tau_B = \sqrt{1-v^2}\,T = \sqrt{1-v^2}\,\tau$$

Note that I haven't introduced any specific spatial coordinates. But I had to introduce at least time t, and I don't see a way to derive this result w/o using any coordinate at all. So strictly speaking my answer to your question is "no".

I would say that this holds even in GR; one can derive many results w/o specific charts, but (in the framewok of differential geometry) one always uses the existence of charts, so again strictly speaking the answer is "no".

Last edited: Jan 13, 2013
22. Jan 14, 2013

### arindamsinha

Didn't get this part. First, are you taking c = 1 in this case?

Secondly, do you actually mean the below?
$$\tau_B = \sqrt{1-v^2}\,T = \tau$$

23. Jan 14, 2013

### tom.stoer

Sorry; yes, c = 1; and
$$\tau_B = \sqrt{1-v^2}\,T = \sqrt{1-v^2}\, \tau_A$$