# Gravitational time dilation, how much?

1. Jan 22, 2016

### Erland

OK, I could probably find the answer of this simple question somewhere, but...

If an astronaut stays on space station, in a weightless state, for 30 years, how much older does he/she become compared to a person who stays on the Earth all the time?

I think it is about one second. Am I approximately right or totally wrong?

2. Jan 22, 2016

### Isaac0427

I am pretty sure that is incorrect. If the person were in a vacuum, time would move faster.

3. Jan 22, 2016

### Erland

???
What's vacuum got to do with it?

4. Jan 22, 2016

### Staff: Mentor

The general formula for the time dilation of an object moving with velocity $v$ at radius $r$ in Schwarzschild spacetime (which is not an exact description of the scenario, but is a good enough approximation for here), relative to an observer at rest at infinity, is

$$\frac{d\tau}{dt} = \sqrt{1 - \frac{2 G M}{c^2 r} - \frac{v^2}{c^2}}$$

To find the relative rate of time flow of an observer on the space station, compared to an observer on the Earth's surface, you just need to plug in the appropriate values of $v$ and $r$ for the two cases and take the ratio of the results.

5. Jan 22, 2016

### Isaac0427

The same conditions of gravity. If you are in a vacuum, you are weightless. I should have made that more clear.

6. Jan 22, 2016

### Staff: Mentor

That still doesn't have anything to do with the time dilation factor. Just being weightless doesn't affect your rate of time flow compared to something else. It depends on where you are weightless (i.e., at what altitude in the Earth's gravity field), and how fast you are moving, relative to something else.

7. Jan 22, 2016

### Isaac0427

Ok, my point was just that gravity makes time go faster in general relativity.

8. Jan 22, 2016

### PeroK

You mean "how much younger?"

9. Jan 22, 2016

### Isaac0427

I know my other posts weren't great. This is how I think of it:
The shortest possible path between 2 points is a straight line, however you can't have those in non-Euclidean/Minkowski spacetime. The shortest possible path in Euclidean space will always be shorter than in curved space, and time travels in the shortest possible path. I'm sure from your question that you are aware that gravity bends both space and time. When spacetime is bent, time moves slower, that's just geometry. By your question, you seem to be implying that gravity is low, so spacetime is fairly flat. If that is what you were implying, time would be slowe on earth.

10. Jan 22, 2016

### baudrunner

the faster you go, the slower time is for you.

the greater the gravitational force that you are subjected to, the slower time is for you.

I'm pretty sure that they used trial and error to determine just how much correction needed to be made to the onboard clocks on the GPS satellites in order to improve their accuracy. To my knowledge, this is the only practical application of relativity theory to date, but could that data be used to make accurate calculations of time dilation? Someone somewhere out there in the greater world can, I'm sure.

11. Jan 22, 2016

### Jilang

12. Jan 22, 2016

### Staff: Mentor

No, it doesn't. It's more complicated than that.

13. Jan 22, 2016

### Isaac0427

Oh sorry, I meant slower. I explained that in my other post, and I do understand how complicated it is.

14. Jan 22, 2016

### Staff: Mentor

Yes, you can. They're called "geodesics".

This is not correct. What you are calling "the shortest possible path in Euclidean space" does not exist if spacetime is curved, i.e., in the presence of gravity. Also, time does not "travel". Time is a dimension, not a thing.

It's not that simple. (I see that you corrected your previous post, but the answer I just gave still stands--it's not that simple. )

It's not that simple either. There are two effects involved, one due to relative motion and one due to altitude. (The link Jilang posted in post #11, and the equation I gave in post #4, make this clear.)

I can see that you are interested in this subject, which is good. But please take some time to learn the details.

15. Jan 22, 2016

### Erland

Thank you!

Plugging in the values for the International Space Station (ISS) (https://en.wikipedia.org/wiki/International_Space_Station), and a person at the equator on the Earth, I find that the astronaut will age 0.27 seconds less than the person on the Earth in 30 years.
I didn't expect this. I thought that the astronaut should age more than the person on the Earth. The reason is that the high velocity of the space station makes the term $\frac{v^2}{c^2}$ dominate over the "gravitation" term in the formula, so it is actually SR-type time dilation that becomes most significant.

But is this really correct? Is it just the size of the velocity that matters and not its direction? Would the time dilation be the same for an object at the same location as the space station and the same speed but moving straightly towards or away from the Earth?

16. Jan 22, 2016

### Staff: Mentor

This is relative because "faster" is relative. There is no absolute sense in which you are going "faster".

This is not correct. The key thing is gravitational potential, not force. For the case of an isolated, spherical (or approximately spherical) gravitating body like the Earth, "potential" means "altitude"--the lower your altitude, the slower time is for you, taking only the effects of gravity into account. You can be at a lower altitude and also experience less force--for example, someone at the center of the Earth would be at a lower altitude than someone on the Earth's surface, and would experience a slower time flow, but they would feel no gravitational force at all.

No, they didn't. They calculated what the correction would be in advance, but the bureaucrats they were reporting to weren't sure that they believed GR was correct, so they insisted on measuring the "raw" clock rates of the satellites when they were put in orbit before applying any corrections. The measured rates matched the calculations; no trial and error was needed.

The best reference I'm aware of on GPS and relativity is Neil Ashby's article here:

http://relativity.livingreviews.org/Articles/lrr-2003-1/ [Broken]

Section 5 describes what happened when the first GPS satellites were deployed and their clock rates were measured.

I assume you mean general relativity, since the practical applications of SR are numerous.

Last edited by a moderator: May 7, 2017
17. Jan 22, 2016

### Isaac0427

I was not talking about geodesics, I was talking about straight lines in Euclidean space, which do not exist in curved spaces.
That was my point. A geodesic in curved space is longer than its linear equivalent in flat space (I don't know the proper term for this). I was referring to the flow of time not the dimension of time.
I just addressed that in my previous post.

18. Jan 22, 2016

### Staff: Mentor

Yes, and "most significant" means "slows down more". In other words, for a low Earth orbit, the slowing down of time due to relative velocity (the person in the space station is moving at about 8000 meters/second whereas the person on Earth is moving at only about 450 meters/second, as viewed from a non-rotating frame centered on the Earth) is greater than the speeding up of time due to higher altitude. So the net effect is to make the person on the space station age slower.

But for a high enough orbit, this reverses--orbital velocity gets slower as the orbit gets higher, so at some point the altitude effect becomes greater and the net effect is to make the person in orbit age faster. A good exercise is to calculate the altitude at which this happens.

19. Jan 22, 2016

### Isaac0427

It's all relative velocity, so theoretically the direction does matter (only if the observer is moving).

20. Jan 22, 2016

### Staff: Mentor

Exactly, which means you can't measure them in curved spacetime because they don't exist. All that exists are geodesics of the curved geometry of spacetime. So nothing can "flow" along these Euclidean lines in a curved spacetime; they aren't there.

Also, for timelike geodesics, i.e., possible worldlines of objects like planets and people, the geodesic is the longest possible curve between two points, not the shortest.

If you consider a case where the flat space actually exists, yes, this is true. For example, the straight line through the Earth from New York to Tokyo is shorter than the great circle around the Earth's surface between the two.

But in the curved spacetime we actually live in, there is no flat space to compare to; there is no analogue of going through the Earth. So your statement here is meaningless as far as the physics of our actual curved spacetime is concerned.

By "the flow of time", I assume you mean "the proper time experienced by someone following a particular worldline in spacetime". But nobody can follow a "straight" worldline in a flat spacetime instead of a geodesic in our curved spacetime; the flat spacetime doesn't exist.

21. Jan 22, 2016

### Staff: Mentor

Only for a short time, because moving straight towards or away from the Earth would change the object's altitude. (It would also change its speed if it were in free fall with no rocket engines.)

So yes, the direction does matter, indirectly, through its effect on altitude and speed. But it doesn't matter directly--the direction of $v$ does not appear in the formula, only its (squared) magnitude.

22. Jan 22, 2016

### Staff: Mentor

I assume you meant the direction doesn't matter, only if the observer is moving. That's not quite correct; the direction doesn't matter directly, but it does indirectly through its effect on altitude and speed. See my previous post.

23. Jan 22, 2016

### Erland

Consider the factor
$$\sqrt{1 - \frac{2 G M}{c^2 r} - \frac{v^2}{c^2}}.$$
If we have an object orbiting the Earth with a speed making it weightless, such as a typical space station, we have $\frac {v^2}r=\frac{GM}{r^2}$, or $\frac {v^2}{c^2}=\frac{GM}{c^2 r}$, so the factor simplifies to
$$\sqrt{1 - \frac{3 G M}{c^2 r}}=\sqrt{1 - \frac{3v^2}{c^2}}.$$

24. Jan 22, 2016