# Gravity's Effect on Time

1. Jun 15, 2011

### Gravitons

Im having a hard time understanding gravitational time dilation. I have two examples I've heard(from shows on the science channel): a theoretical experiment is you leave earth in a space ship and orbit outside the event horizon of a black hole for 1 year, its possible that when you come back to earth its been 10 years or even 100 years on earth when its been only 1 year for you. This shows that as gravity increases time slows down for the person next to the black hole because of the black holes large mass? But this contradicts the fact that when astronauts orbit the earth in the space station they are technically, however small the measurement, younger than us on earth, so they age slower? Because the farther away from earth you are the less stronger gravity is? So this is saying that the less gravity the more time slows, which contradicts the black hole experiment that says more gravity has this effect on time. Can someone please explain this relationship.

2. Jun 15, 2011

### WannabeNewton

The astronauts age relatively slower because they are orbiting the earth at high speeds not because they are in a stronger gravitational field. The earth's gravitational field is too weak to cause any kind of significant time dilation.

3. Jun 15, 2011

### Gravitons

Oh thankyou i understand now. So does that mean the theoretical experiment about the black hole is true?

4. Jun 15, 2011

### WannabeNewton

I don't know the exact number but yes the premise is very true.

5. Jun 15, 2011

### Oldfart

I read somewhere on PF that atomic clocks on earth can detect a height difference of as little as three feet.

Last edited: Jun 15, 2011
6. Jun 15, 2011

### bcrowell

Staff Emeritus
This is incorrect. The gravitational and kinematic time dilation effects are of comparable orders of magnitude for satellites in earth orbit. For satellites in low earth orbit (which includes almost all crewed activity), the kinematic effect dominates. But for satellites in higher orbits (e.g., GPS and geosynchronous satellites), the gravitational effect dominates.

-Ben

7. Jun 15, 2011

### PAllen

The black hole case would combine two sources of differential aging. Consider there is a black hole a light year away, that is stationary relative to sun (say, by doppler definition). Ship A goes near the black hole (not too close), circles a while, and comes back to earth. Ship B goes a light year in the opposite direction, circles similarly* and comes back to earth. Both pilots will have aged less than their earth colleagues, but A pilot will have aged less than B pilot.

*Similarly: Near the black hole, measure speed of A relative to a momentarily coincident (NOT comoving) radial free fall frame (with zero instantaneous radial velocity compared to a coincident stationary observer). In the opposite direction, measure speed (in presumably empty region), relative to sun. The aim is to measure speeds from local Minkowski frames.

Last edited: Jun 15, 2011
8. Jun 15, 2011

### pervect

Staff Emeritus
One interesting side-note. You can't really get much differential aging due to "gravitational time dilation" if you insist on orbiting a black hole, because the photon sphere is at r=3M, and the "gravitational time dilation there" is sqrt(2M/r) = sqrt(2/3).

While you could get a long time dilation orbiting a black hole near the photon sphere, this would come from your velocity more than the gravitational effect.

If you consider traveling deep in a black hole's gravity well using rocket thrust, rather than orbiting it, you can get large time dilations due to the gravitational effect.

The gravitational effect is not due to "gravity", but is better (though someone loosely) described as being due to "gravitational potential". For instance, you'll have a maximum value of time dilation due to gravity on Earth at its center, even thought the gravity there is zero.

The "gravitational time dilation" is really due to curvature of space-time, but as long as you only consider static observers it's not too terribly misleading to use coordinate time as your base for time and think of physical clocks as ticking faster or slower than coordinate clocks. Pretty much everyone does this, I think.

Philosophically, though, there's a lot to be said for thinking of the physical clocks as being the "more real" clocks, in which case they always tick at 1 second per second. With this point of view, it's the coordinate clocks that are "off" due to the effects of curvature.

9. Jun 15, 2011

### WannabeNewton

My apologies.

10. Jun 15, 2011

### yuiop

Just to make it clear, WN is correct that astronauts (usually in low orbit) would age slower than people on the ground, but erred a bit when he said Earth's gravitational time dilation effect is insignificant, because gravitational time dilation can dominate at higher orbits. However if you were to be charitable, you could interpret WN to mean gravitational time dilation is less significant than kinematic time dilation, in low orbits where astronauts usually work.

11. Jun 15, 2011

### PAllen

Right, which is why I proposed circling with a rocket, rather than orbiting. That way you can make either components (speed, gravity) towards differential aging a large as you want.

One question is that I get sqrt(1/3) for the innermost unstable circular orbit, sqrt(2/3) for the innermost stable circular orbit. It is the former where the orbital velocity approaches lightspeed. I could have goofed, so let me know.

12. Jun 15, 2011

### yuiop

I am not sure about the philosophical desirability of this this interpretation. Let us say we have a clock (A) very far from a gravitational source that remains stationary and call this the coordinate clock.

Now let us lower another clock (B) close to the gravitational source such that clock A seems to be ticking 10 times faster than clock B from B's point of view and clock B appears to be ticking 10 times slower from A's point of view. We interpret this as clock A speeding up by a factor of 10 while clock B continues to tick at 1 second per second. (clock B undergoes no change).

Let lower one more clock (C) even closer to the gravitational source so that from A's point of view C's clock is ticking 100 times slower and from C's point of view A's clock is ticking 100 times faster. We interpret this as clock A speeding up by a factor of 100 while clock C continues to tick at 1 second per second. (clock C undergoes no change).

Now we have agreed that neither clock B nor clock C have undergone any sort of change and yet they are ticking at different rates to each other when initially they were ticking at the same rate when they were adjacent to each other. They have changed, but they have not changed. A bit paradoxical.

Meanwhile clock A, that has remained stationary all the while, is now the one that is really ticking 10 times faster relative to B and ticking 100 times faster relative to C. Since clocks B and C have undergone no change, clock A must be doing all the changing even though it is the only clock that that has no reason to change because it has not changed its velocity or gravitational potential. If A was originally ticking at 1 second per second what is its clock rate now? Is it 10 seconds per second which is what B says it is or is it 100 seconds per second which is what C says it is? Again, a bit paradoxical. Alternatively we say all clocks tick at one second per second (in their own rest frame) independent of their relative motion or gravitational potential which makes things a lot simpler, but completely glosses over the fact that clocks are measured to run at different rates.

To me it makes a whole lot more sense to say that the tick rates of clocks B and C change because they are the clocks that have changed their location in the gravitational field. This explains why the tick rates of both B and C have changed relative to each other (from everyone's point of view). Clock A, that has undergone no change in velocity or gravitational potential undergoes no change in tick rate, because it has no reason to. I think the main reason this common sense interpretation is avoided, is because when followed to its conclusion, it implies that clocks stop ticking at the event horizon and that is a bit awkward.

13. Jun 15, 2011

### bcrowell

Staff Emeritus
Not trying to be too hard on WN. Charitability isn't one of my strong points :-)

14. Jun 15, 2011

### yuiop

The help you give people here and the educational information you provide on your website seems pretty charitable to me

15. Jun 16, 2011

### pervect

Staff Emeritus
Time for one of my favorite analogy.

You're on the surface of the Earth, and you mark off coordinates for lattitude and longitude. You have a sailing ship on the equator, and it covers 20 arc-minute of longitude in 1 hour.

You move up to closer to the north pole - and you observer it covers 40 arc-minutes of longitude in 1 hour.

If we followed the same rules that we did for time dilation we'd be talking about "distance contraction" as the ship moved further up north.

This idea wouldn't be totally wrong, at least not for navigational purposes, but aside from navigation, it'd be pretty confusing to think of your meter stick as changing as you moved north and south. One might even say it was wrong, for anything except navigational purposes. And, we don't actually do that. Nor do we say that "ships go faster near the North pole", or anything like that. We recognize that the "real" speed of the ships is the same near the poles and near the equator, that this speed is independent of the coordinates we use to describe the surface of the Earth, and that a meter is also independent of the coordinates, and that what happens is that we need to convert coordinate arc-minutes into physical meters by a conversion factor that depends on lattitude - the spatial "metric" of the Earth.

And we might well explain this phenomenon briefly just by saying that the paths we draw in order to compare the distances are curved.

In relativity, it's the concept of a static observer that allows us to compare the distant clocks at all, and it's this particular concept of time comparison, the particular concept of simultaneity used by static observers, that makes the physical clocks deeper in the gravity well appear to run slowly when measured according to "coordinate time".

16. Jun 16, 2011

### harrylin

Commenting on:
"Philosophically, [..] there's a lot to be said for thinking of the physical clocks as being the "more real" clocks, in which case they always tick at 1 second per second. [..]"
I fully agree - but hey it appears to be a matter of how one's brains are wired. :tongue2:

Pervect commented:
I don't see that example as support for "always 1 second per [nominal] second". In this example it would be "always N meters per degree longitude". And that's obviously not true. So, my brain is wired differently from yours.

Moreover the interpretation that clocks near mass tick slower than clocks far from mass (ceteris paribus), is independent of the observer and independent of the units.

Harald

17. Jun 16, 2011

### nitsuj

That is a really good analogy, requires a fair bit of background knowledge. But conceptualy really gets the main points across.

18. Jun 16, 2011

### pervect

Staff Emeritus
The interpretation that clocks near mass tick slower than clocks far away is not independent of the observer, if one includes relativistically moving observers. It is only for the special class of static observers that it really works, when there's a "special" method of comparing them that defines simultaneity.

One of the points of the analogy is that the Schwarzschild coordinates (r, theta, and phi) are just coordinates, like lattitude and longitude.

To put the coordinate description before the actual physics of what you measure with real instruments is quite literally putting DesCartes before the horse :-).

19. Jun 16, 2011

### bcrowell

Staff Emeritus
Oh...no...

20. Jun 17, 2011

### harrylin

Evidently an elaboration is necessary: I wrote ceteris paribus and that includes relativistically moving observers. Surely all observers agree that clocks tick slower near co-moving masses; this is the actual physics of what you will measure with real instruments.
Else you would get the kind of contradictions that the OP imagined. :tongue2:

21. Jun 17, 2011

### pervect

Staff Emeritus
I don't see how taking "all things equal" applies here, especially not to the point I'm trying to make about the requirement for a stationary observer to define a notion of simultaneity in order to be able to compare clocks.

In general, if you have two observers A and B in relative motion, they can't agree whether A's clock is faster or whether B's clock is faster. This is true even in flat space-time, and remains true in curved space-time. This is because they have different notions of simultaneity.

Graphically, I think of it rather like the attached diagram below (which is for flat space-time) where one observer uses red lines to compare the clocks, and the other the green lines. (Add: I tried to upload this, don't see it, and it won't let me re-up).

The purpose of the stationary observer is to define the "set of lines" on the space-time diagram that define simultaneity which is a pre-requisite in order to be able to compare the clock's rate.

I've also seen a LOT of confusion in the past over this issue, related to the issue of what a relativistically infalling observer to a black hole sees. If one takes the "time slows down" argument seriously, one would think that an infalling observer sees time "slow down" just like a stationary one, and that since a stationary observer sees the future of the universe play out in a flash as, so does the infalling observer.

The actual situation is that the infalling observer does NOT see the future of the universe play out in a flash. IIRC, a detailed calculation shows that for an observer free-falling from infinity, the clock at infinity appear to be slower than his own from his (infalling) viewpoint, not faster (as judged by the doppler shift of received signals that the infalling observer receives).

That's happens because the infalling observer has a different notion of how to compare clocks (i.e a different notion of simultaneity) than the stationary observer - though one can explain it in other terms. The whole "rate of time" idea doesn't really hold up reliably when you have speeding observers. In flat spacetime, or curved, it's pretty common to see the "slow/slow" case, where both observers think the other's clock is slow.

Last edited: Jun 17, 2011
22. Jun 17, 2011

### pervect

Staff Emeritus
let's see if this gets the diagram up...

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23. Jun 17, 2011

### PAllen

Which is why I proposed to deal with the OP's questions always in terms of clocks that start and end colocated, with different paths through spacetime. The, no notion of simultaneity or distinguishing what you optically see from some interpretation imposed on it.

24. Jun 20, 2011

### harrylin

The OP asked about a clock's rate which according to all distant (=unaffected) reference systems will be less in the presence of a heavy mass than in the absence of that mass, all other things equal (in particular identical motions).
That is quite a different question from the one of this thread, and to which yuiops' answer #12 is not related. Confusing different questions leads to confusions.
Note that I don't know why yuiop thinks that clocks should stop ticking at the event horizon, but that's again a different topic.
The OP's question as I understood it is about the effect of gravity (thus "curved spacetime") on observed clock rate; that made it a straightforward and unambiguous question.
Experimenters have found it useful from the start (Hafele and Keating) to distinguish between the effects of gravitation and speed. The introduction with references of Wikipedia may be useful for the OP (http://en.wikipedia.org/wiki/Time_dilation).

Last edited: Jun 20, 2011
25. Jun 20, 2011

### yuiop

Here is an alternative analogy. Let us say we have some hour glasses filled with treacle. At room temperature they take one hour to empty, but they are temperature sensitive because the treacle get more viscous at lower temperatures. Now let us say we also have a fridge with a glass door so we can see what happens inside the fridge. We place an hour glass and observer in the fridge and superficially the clock inside the fridge appears to be ticking slower than the clock outside the fridge. The observer inside the fridge says "No!, my clock is continuing to tick at on hour per hour!". To prove this he places another treacle filled hour glass inside the fridge alongside his original one and indeed they tick at the same rate inside the fridge seemingly proving his assertion. This is not unlike the observer very close to a black hole declaring his clock continues to tick at one second per second. He can only prove this by putting another clock next to his original one, but it proves nothing because both clocks in the gravitational case are equally affected by the gravitational field just as both clocks in the fridge are equally affected by temperature. Our observer in the fridge further claims it is impossible that his clock his slowing down because that would imply that a absolute zero temperature everything would stop. "How absurd!" he exclaims. He then further claims that the clocks outside the fridge (and everywhere else in the universe) actually speed up as a result of him and his clock entering the fridge. That (as far as the fridge observer is concerned) is the only logical explanation because he has already demonstrated that his fridge clock continues to tick at a rate of one hour per hour independently of temperature. He is also completely un-swayed by the observer outside the fridge pointing out that he can prove that the clocks outside the fridge also continue to tick at a rate of one hour per hour. The observer outside the fridge starts to wonder if the cold temperatures inside the fridge are actually slowing down the brain functions of the fridge observer and lets him out to warm up.

There was also a third observer outside the fridge who could see the clocks both inside and outside the fridge and could compare their different tick rates directly. He sides with the external observer, saying he saw the fridge observers clocks slow down when they entered the fridge and he saw no change in the tick rate of the clocks that remained outside the fridge. The fridge observer (who has not fully thawed out yet) says the third observers observations do not count because the third observer is just a coordinate observer (or book-keeper) and it well known that coordinate measurements do not have physical significance. :tongue: