# Is SR and GR Time Dilation the same thing?

1. Nov 9, 2011

For instance you have time dilation in special relativity which is said to be due to something moving faster than something else relative to it.

You also have the time dilation in GR where time speeds up the further away you go from a body such as the Sun, Earth, etc.

Now could it really all be due to the same thing? Is time slowing down for an object closer to the Sun the same overall process for time slowing down for something moving fast.

Could it be that time dilation for both SR (moving faster in this case) and GR (gravity in this case) are due to the same thing?

2. Nov 9, 2011

### PAllen

There are similarities and differences.

Some similarities:

- clocks slow and redshift go hand in hand (really the same thing).
- clock seems locally normal - other clocks seem off

Some differences:

- time dilation between inertially moving observers is symmetric; each sees the other slow. For gravitational time dilation, it is not symmetric: surface guy thinks mountain guy is fast; mountain guy things surface guy is slow.

However, there is a way to correlate gravitational time dilation to kinematic time dilation: A free fall observer will see two static clocks accelerating such that their relative speed is not identical (distance between them is shrinking); and for this free fall observer, their relative motion explains their difference in clock speed.

3. Nov 9, 2011

### tom.stoer

In a sense they are the same thing. Proper time along a curve C in spacetime is calculated according to

$$\tau = \int_C d\tau$$

Now compare two curves C and C' both connecting two points A and B in spacetime, and calculate the difference for proper times tau and tau' measured along C and C', respectively

$$\Delta\tau_{A\to B} = \int_{C_{A\to B}} d\tau - \int_{C^\prime_{A\to B}} d\tau$$

All these formulas are valid for both SR and GR and for arbitrary timelike curves. The difference arises only when looking at specific curves i.e. specific experiments
case 1) a geodesic C ('twin on earth') and a curve C' deformed by acceleration ('the twin in the spaceship')
case 2) a geodesic C ('a satellite orbiting the earth') and a curve C' with non-constant radius measuring the difference in gravitational potential

4. Nov 9, 2011

### pervect

Staff Emeritus
Consider an accelerating elevator or rocket.

If you use inertial coordinates, the metric tensor everywhere is Lorentzian, and you explain all observed time dilations as due to velocity.

If you use Rindler coordinates, which are natural coordinates for the accelerated observer in which the accelerating observer is always at the origin, the metric coefficients are not Lorentzian, and you explain some time dilation as being due to velocity, and other time dilation as being due to "gravitational potential".

In particular, you see clocks at the nose of the rocket ticking faster than those at the tail, even when the nose and tail are "at relative rest" in the sense of having a constant round-trip travel time for light signals.

So it's clear from the example hat the division of time dilation into "velocity" and "gravity" parts depends on your choice of coordinates.

5. Nov 9, 2011

### Passionflower

Hos does the free faller measure that the distance is shrinking?
Could you demonstrate with mathematics that that is the case or provide a reference in the literature?

A twin on Earth does not travel on a geodesic.

Last edited: Nov 9, 2011
6. Nov 10, 2011

### tom.stoer

Correct, but usually in the example for the twin paradox you neglect this tiny effect; let's assume the twin stays at the center of the earth ;-)

7. Nov 10, 2011

### Fredrik

Staff Emeritus
According to GR, the reason why clocks on different floors of the same building have (slightly) different ticking rates is that the internal forces in the building are making the floors (and therefore the clocks) accelerate differently.

This effect exists in SR too. Consider a rocket that's initially at rest in some inertial coordinate system, and then accelerates gently to a speed where relativistic effects are noticeable. In the inertial coordinate system where the rocket started out at rest, the rocket is now shorter than before by a factor of gamma. This means that the rear must have had a larger acceleration than the front!

Clocks don't measure coordinate time. They measure proper time, i.e. the integral of $\sqrt{dt^2-dx^2-dy^2-dz^2}$ along the curve that represents their motion. Constant velocity would mean dx=dy=dz=0. Any deviation from that will make the proper time smaller, because of the minus signs. If you understand that, it shouldn't be too hard to believe that if two clocks have different accelerations, the one that accelerates less (the one in the front of the rocket or the one on the higher floor of the building) will measure a larger proper time.

8. Nov 10, 2011

### zonde

This seems wrong.
http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Clock_Hypothesis" states that acceleration does not affect clock rate and it is experimentally verified.

On the same line. Clocks that are orbiting gravitating body should be affected by gravitational time dilation just the same.

It seems more reasonable to say that body is undergoing gravitational acceleration because clock rates are different at different heights.

Last edited by a moderator: Apr 26, 2017
9. Nov 10, 2011

### Fredrik

Staff Emeritus
I'm not saying that the properties of clocks depends on the acceleration. I'm saying that what a clock measures is a coordinate-independent property of the curve in spacetime that describes its motion. The world lines of two clocks that are glued to opposite ends of a solid rod that's accelerated gently to relativistic speeds are significantly different. The world line of the clock in the rear is more curved than the world line of the clock in the front. Perhaps I should also have mentioned that this implies that it has a higher speed in an inertial coordinate system where the rod starts out at rest.

The term "gravitational time dilation" usually refers to the different rates of clocks held at different heights above some fixed position on the ground of a non-rotating planet or star. I certainly don't expect a clock at the center of the Earth to remain synchronized with a clock in orbit just because they both move as described by geodesics, but I'm not sure I would like to describe the different rates as "gravitational time dilation".

In my opinion, the term "time dilation" is only useful in very specific scenarios. In more complicated situations, it's better to drop the term and just talk about the proper time of the relevant timelike curves.

So instead of thinking that the geometry determines what numbers a clock displays, I should be thinking that the numbers displayed by clocks determine the geometry?

Last edited by a moderator: Apr 26, 2017
10. Nov 10, 2011

### A.T.

Time dilation is not just one clock rate, which is somehow locally affected by acceleration. It is the ratio of two clock rates. Two clocks at rest in an accelerated frame (with spatial separation along the acceleration direction) will have different clock rates. This is how acceleration "generates" gravitational time dilation.

Last edited by a moderator: Apr 26, 2017
11. Nov 10, 2011

### zonde

"Curve in spacetime" is description of your choice for physical reality. I somehow feel uncomfortable with the statement that clock measures "description" of physical reality.

We have physical fact - two clocks tick at different rates. How would you name (not describe) this physical fact?

Is your question like - does theory determines physical reality or physical reality determines theory?
Then of course later.

But I was trying to say something different. It was something about causal relationship between two physical facts.

Look when Einstein was trying to come up with some description for gravity he had one physical fact about gravity - objects accelerate near gravitating body. He deduced that in order for object to be free of internal stress as it accelerates it should "tick" slower in the part that is closer to gravitating body. The two things (gravitational acceleration and gravitational time dilation) are related - if we see one thing there should be the other one too.
We have experimentally verified fact that clocks tick slower when they are closer to gravitating body. So we have two physical facts.
Now we can talk about causal relationship between two things. And I am saying that gravitational time dilation causes gravitational acceleration.

12. Nov 10, 2011

### zonde

Object that is stationary in respect to gravitating body is not undergoing continuous length contraction that would be the case for ordinary accelerated body.
Because there is continuous length contraction in the case of ordinary accelerated body it's rear and front is moving at different speeds and have different clock rates. Just like Fredrik was saying.
There is no such thing for stationary body in gravitation field. Unless of course you want to propose that spacetime somehow "moves" at different speeds at different gravitational potentials.

13. Nov 10, 2011

### PAllen

The mathematics is essentially identical to the standard accelerating rocket in SR. To a free falling observer, with locally Minkowski frame, two reasonably nearby stationary clocks (say one 100 meters higher than the other) are simply accelerating clocks maintaining constant distance from the point of view of e.g. the lower clock. So, just look up the SR case and you have the math. This fact about the GR case is extremely well known.

14. Nov 10, 2011

### Passionflower

Measuring distance in curved spacetime has complications that do not exist when measuring distance in flat spacetime.

I take it you will not demonstrate it with formulas perhaps because it is too simple and instead you leave it as an exercise for me to find out this obvious thing?

15. Nov 10, 2011

### Fredrik

Staff Emeritus
It's only a choice of what theory to use to answer questions about motion. In SR and GR, statements about motion are statements about curves in spacetime.

My statement about clocks and curves is part of the definition of both SR and GR. A theory of physics can't be defined by mathematics alone. The physics is in the statements that tell us how to interpret the mathematics as predictions about results of experiments. That's the sort of statement I made.

This statement is only unambiguous at an event where the clocks are both present and have the same velocity. (If they tick at different rates at such an event, I would say that at least one of them is broken). In any other situation, we need a definition that tells us how to compare the ticking rates. I don't think the rates can be thought of as "physical facts".

I don't understand the question.

Reality certainly determines which theories will be successful, but once you have decided what theory you're going to use to try to answer a question, reality becomes irrelevant, and all that matters is what the theory says.

I think in most cases where it's possible to say that A is the reason for B, it makes just as much sense to say that B is the reason for A. For example, do we have conservation laws because of symmetries, or do we have symmetries because of conservation laws? However, I think a given theory usually makes one of the possibilities more "natural" than the other, in the sense that it will be much easier to explain. In SR and GR, there's a simple(ish) formula that tells you how to calculate the numbers displayed by a clock at different events on its world line, given a metric. I don't know if there's a way to input those numbers into a calculation that determines the metric. I wouldn't be surprised if there is, but I would still reject the suggestion that this would be a more accurate way to think about these things. As long as we're working with the standard formulation of SR and GR, it would at best be a more complicated way to think about these things.

Different floors in the same building accelerate by different amounts. Suppose that we pick an event A on the world line of the clock on the top floor, and draw a spacetime diagram showing what the world line looks like in a local inertial coordinate system that's comoving with the clock at A. Suppose that we do the same to the other clock, this time involving a local inertial coordinate system that's comoving with this clock at some event B. Then because the two clocks accelerate by different amounts, the two curves we draw will curve away from the time axes of these diagrams by different amounts. They will eventually have significantly different coordinate velocities in these two fixed coordinate systems.

Edit: In the special relativistic accelerating rocket scenario, we would usually draw only one spacetime diagram, but there's nothing that prevents us from drawing one for each clock. The result would be essentially the same as in the general relativistic two-clocks-on-different-floors scenario. The desynchronization of the clocks can in both cases be attributed to the coordinate velocity difference discussed above. I can't see any reason to say that we're not dealing with the same phenomenon in both cases.

Last edited: Nov 10, 2011
16. Nov 10, 2011

### Passionflower

Yes it is usually ignored.

It is interesting that if we toss the twin straight in the air (with adequate protection) and wait for him to come back he will be older than the twin who stayed on Earth while if we toss him harder so that he does not come back but he eventually returns by using some rockets he will be younger than the one who stayed on Earth.

17. Nov 10, 2011

### atyy

SR and GR time dilation are slightly different things arising from the same underlying physics.

In SR, there are global inertial frames, and the term "time dilation" refers to differences in coordinate time assignments between frames.

In GR, there are usually no global inertial frames, and the term "time dilation" refers to an experiment result that is more like the SR doppler effect.

The underlying physics in both cases is that of a spacetime metric.

18. Nov 10, 2011

### A.T.

Length contraction is frame dependent. From the perspective of an inertial (free falling) frame an object that is stationary in respect to the gravitating body (and experiences proper acceleration) can be undergoing continuous length contraction.

19. Nov 11, 2011

### zonde

Your statement was that clock measures property of the curve. It kind of implies that we are performing experiments on "curves" and clock is measurement equipment that we use in order to find some physical quantity that describes "curves". So I am wondering if you are thinking about "curves" in similar fashion as you would think about "fields".

We have to take something as a starting point to do some reasoning. In case of two distant clocks we can evaluate global situation that we observe and if it's fairly static and clocks have static configuration in respect to that global situation then we can claim that distance between clocks is not changing.
I think that's enough to remove ambiguity in relative ticking rates of clocks when we compare them by exchanging some signals.

Can not agree with that. Map is not the territory.
Established theory still does not determine physical reality. It only changes our our interpretation of physical reality.

And how do you measure metric?

On what are these two local inertial coordinate system fixed? Physical laws for a body that is moving inertially in one of those coordinate systems are not fixed in respect to anything.

20. Nov 11, 2011

### zonde

And is your choice of coordinates physically justified? Physical context remains the same from perspective of stationary body. But from perspective of falling body physical context is undergoing continuous change. You have to apply global transformations to keep things consistent.