Gravitational time dilation, how much?

[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?

 

[Isaac0427]

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

 

[Erland]

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

?
What's vacuum got to do with it?

 

[PeterDonis]

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.

 

[Isaac0427]

?
What's vacuum got to do with it?

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

 

[PeterDonis]

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

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.

 

[Isaac0427]

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.

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

 

[PeroK]

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?

You mean "how much younger?"

 

[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.

 

[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.

 

[Jilang]

There are two factors at play here. One is due to gravitational effects and one due to the relative motion.
http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

 

[PeterDonis]

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

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

 

[Isaac0427]

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

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

 

[PeterDonis]

The shortest possible path between 2 points is a straight line, however you can't have those in non-Euclidean/Minkowski spacetime.

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

The shortest possible path in Euclidean space will always be shorter than in curved space, and time travels in the shortest possible path.

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.

When spacetime is bent, time moves slower, that's just geometry.

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. )

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.

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.

 

[Erland]

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.

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?

 

[PeterDonis]

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

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

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

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.

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.

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/

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

To my knowledge, this is the only practical application of relativity theory to date

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

 

[Isaac0427]

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

I was not talking about geodesics, I was talking about straight lines in Euclidean space, which do not exist in curved spaces.

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.

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.

It's not that simple. (Also, this contradicts your previous post, where you said that gravity makes time go faster. "Gravity" means "spacetime is bent".)

I just addressed that in my previous post.

 

[PeterDonis]

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.

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.

 

[Isaac0427]

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?

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

 

[PeterDonis]

I was talking about straight lines in Euclidean space, which do not exist in curved spaces.

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.

A geodesic in curved space is longer than its linear equivalent in flat space

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.

I was referring to the flow of time not the dimension of time.

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.

 

[PeterDonis]

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?

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.

 

[PeterDonis]

theoretically the direction does matter (only if the observer is moving).

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.

 

[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
$$
\sqrt{1 - \frac{3 G M}{c^2 r}}=\sqrt{1 - \frac{3v^2}{c^2}}.
$$

 

[russ_watters]

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.

Wow, that's amazing[ly idiotic]! Hadn't heard about that.

 

[PeterDonis]

Erland, yes, your simplification for an object in a free-fall orbit in post #23 is correct.

 

[Isaac0427]

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.

No, I was saying that for relative velocity direction can matter. If object A is traveling to the left at 5 m/s and object B is also traveling to the left at 5 m/s there is no relative velocity between the two objects. If both are traveling at 5 m/s but in opposite directions the relative velocity is 10 m/s.

 

[Vanadium 50]

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.

That is true. The key observation is that the guy standing on Earth is not in orbit. For objects in orbit, the effect is monotonic.

Isaac. it helps to learn first, teach second. That works much better than in the other order.

 

[Isaac0427]

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.

But, if you compare 2 spaces that exist at the same time, a very curved one near a massive object and a flatter one near a more or less empty space, the time interval will be smaller in the less curved space. If you were to draw a "line" that represents time in both the flatter and curvier spaces of points 5 time intervals apart, the "line" would be longer near the massive object, or the more curvy spacetime. Correct?

 

[Isaac0427]

And I am talking about in terms of proper time.

 

[PeterDonis]

I was saying that for relative velocity direction can matter.

All of what you are saying is already taken into account in the definition of $v$ in the formula I gave. But you will notice that $v$ itself, as a vector, does not appear in that formula; only $v^2$, the square of its magnitude, does. So, as I said, the direction of $v$ does not directly affect the time dilation for the scenario being discussed in this thread.

if you compare 2 spaces that exist at the same time, a very curved one near a massive object and a flatter one near a more or less empty space, the time interval will be smaller in the less curved space. If you were to draw a "line" that represents time in both the flatter and curvier spaces of points 5 time intervals apart, the "line" would be longer near the massive object, or the more curvy spacetime. Correct?

As you state this, it is too vague and imprecise for me to say whether it is correct or incorrect. For example:

You say "compare 2 spaces that exist at the same time". How is this comparison to be done? What actual physical measurements would you make, and how would you compare their results, in order to determine that "the time interval will be smaller in the less curved space"? (I suspect, btw, that if you actually went through the steps to unpack this, you would find that you had it backwards: the time interval is larger in the less curved space. But that's just a suspicion based on what I think you mean. The only way to know for sure is for you to take the time to specify the details.)

You say "draw a line that represents time in both the flatter and curvier spaces of points 5 time intervals apart". How would you draw these lines? Between what pairs of points? And what physical measurements would you make in order to determine the "lengths" of these lines? (One hint: when you say "points 5 time intervals apart", what does "5 time intervals apart" mean, in terms of the lengths of the lines?)

I strongly recommend that you take a step back and think carefully about what you are saying. You are throwing out vague statements that, if they were made precise in the right way, might be relevant; but you need to take the time and make the effort to make them precise in the right way. If that means you need to stop posting for a while and take the time to learn the details of the math of GR, so much the better.

 

[baudrunner]

I think your problem concerns your understanding of the word relativity, and the theory too, by the way. All effects of relativity are noted by the observer from a static frame of reference. So from our perspective, an object traveling at near-light speed has near-infinite mass. But for someone in that object, his observations are from his static frame of reference. He is not infinitely heavy and time moves normally for him as far as he is concerned. He will however, note a measurable difference in the aging rate of those who were observing him from their own static frame of reference when he returns to them. That's relativity. Yes, time progresses slower for fast moving objects. It's a fact.

relativity: A state of dependence in which the significance of one entity is solely dependent on that of another.