# B Special relativity and expansion of the Universe, A paradox

Tags:
1. Jul 20, 2017

### Rafeek AR

Consider two bodies A &B are moving apart with a velocity V due to the expansion of space. According to an observer in A the body B is moving away and an observer in B feels the body A is moving apart. Can some one answer in which body the time dilates and why?. ( I am specifying once again that the case is movement under expansion of space and not due to ordinary kinematical motion. If you are interested in discussing it even though you don't know the right answer, reply your thoughts.)

2. Jul 20, 2017

### shihab-kol

Time dilation occurs due to the curvature of space time and has a mathematical formula .
But if the space is expanding then both the bodies are moving away from each other.
According to my understanding of the matter, time dilation will be experienced in both as they are moving away relative to each other .

I am a novice so, my understanding may be wrong.
It is up to the more experienced members to say whether I am right or wrong.

3. Jul 20, 2017

### jbriggs444

The subject line mentions special relativity. Let us start there. In special relativity, if you ask which of two moving bodies in relative motion are time-dilated, the correct answer is "it depends". It depends on what frame of reference you choose. If you choose the frame of reference in which body A is at rest, body B's time is dilated. If you choose the frame of reference in which body B is at rest, body A is time dilated. The thing that allows both to be true is the relativity of simultaneity. In order to assess time dilation, we need to have a notion of simultaneity. In special relativity, the natural simultaneity convention is Einstein clock synchronization. Einstein clock synchonization requires one free choice: A frame of reference.

But you make it clear that you are not interested in ordinary kinematic time dilation in special relativity.

If we consider two objects that have a recession velocity due to the expansion of the universe we still need to make a choice on clock synchronization in order to determine time dilation. But the choice of synchronization convention is not quite so simple. In our universe, one can choose to use "co-moving" coordinates. This is a coordinate system in which an observer at rest sees the universe as homogenous and isotropic. In particular, such an observer will see the same cosmic microwave background radiation in all directions. In co-moving coordinates, two objects that are both at rest are not time dilated relative to one another.

In other coordinates, time dilation can, of course, be different. There are no global inertial frames of reference in general relativity. The choice of a coordinate system/frame of reference is not as simple as picking one object to treat as being at rest.

4. Jul 20, 2017

### Ibix

No - neither experiences time dilation. Both say that their clocks are ticking normally. Both say that the other fellow's clock is ticking slowly.

As @jbriggs444 notes, though, this only applies in special relativity and making a conventional choice of simultaneity criterion. In general relativity it's more complex. Making the "obvious" choice of simultaneity criterion in the cosmological case he appears to be interested in, the two clocks both regard the other as ticking normally and all of the redshift is due to the expansion of space.

5. Jul 20, 2017

### EdgarLOwen

Rafeek,

1. During the relative motion each observer sees the other's coordinate time running slower relative to the proper time of his own comoving clock. The two observers disagree.

2. However if the relative motion then stops then both observers will agree their clocks are now running at the same rate but that one observer's clock shows less actual elapsed time than the other (in the general case). The question is why is that? The answer is that the observer with less elapsed time on his clock has moved further in space thus he has moved less far in time.

Edgar L. Owen

6. Jul 20, 2017

### Rafeek AR

Lets consider your first answer." It depends". If " it depends" on either frame of reference then that fact can be used to create a new "TWIN PARADOX". Consider again that both the twins are not coming back together. A return in the journey could only be possible with the help of " local kinematics" and that seems resolve this paradox but it didn't.
lets consider the other argument that both the frames are experiencing time dilation but the very saying does unknowingly imply a "preferred reference frame " according to which that statement is correct. But then again if the preferred reference frame is placed in the universe itself, the question of " whether time dilates in the preferred reference?" does exist. So a prfered reference frame is impossible in the case of expansion of universe.
The third option is " time doesn't dilates in any frame of reference". I don't know what considerations have he took to said that. The expansion of "space" is purely spacial and so the time has to dilate.

7. Jul 20, 2017

### Bandersnatch

The point is, you can't use special relativity to describe it. SR only works in static space-times, and expanding universe is an example of a non-static space-time.

8. Jul 20, 2017

### Rafeek AR

In the case of expansion the relative motion doesn't stops. It seems absolutely relative.

9. Jul 20, 2017

### Rafeek AR

SR does works in expanding universe. What official reference do you have to prove your statement.

10. Jul 20, 2017

### Staff: Mentor

Special relativity assumes the existence of global inertial frames, which exist only in flat spacetime. In fact, that's why it's called "special" relativity; it applies to the special case of flat spacetime. Our expanding universe is not a flat spacetime, so special relativity does not apply across the entire spacetime.

11. Jul 20, 2017

### Ibix

Any relativity textbook, I should think. Certainly any general relativity textbook.

12. Jul 20, 2017

### weirdoguy

Umm, most of the books that deal with general relativity?

13. Jul 20, 2017

### sweet springs

Hi. Co-moving coordinate time of Universe is what you expect to know. All the things that stick to expanding universe share it. Best.

14. Jul 20, 2017

### Rafeek AR

If in comoving distance there exist no global reference frame as you said, then you have to resolve in which frame of reference does the time dilates. Or can you specifically solve this issue with enough mathematics? ( in GR ofcourse).

15. Jul 20, 2017

### Rafeek AR

In FLRW matrices the spacial dimensions are multiplied by a factor a(t). If this factor have to multiply uniformly along all the three spacial dimensions then the problem of time dilation do exist isn't it?. Do my understanding have something faulty in it. I would like if someone could solve it with enough mathematics.

16. Jul 20, 2017

### Staff: Mentor

In GR, the usual solution describing a homogenous and isotopic universe, the one consistent with the data, is called the FLRW metric. It is not a static metric, so there is no global gravitational time dilation. However, there is still local kinematic time dilation. Any non-comoving observer is time dilated relative to a local comoving observer, but two non-local comoving observers cannot be unambiguously compared.

17. Jul 20, 2017

### Rafeek AR

If we can't use SR to solve it, does that implies the " twin paradox" we can create with this time dilation does need to be solved using GR.If so it violates a statement of Einstein himself " twin paradoxes are actually solvable within special relativity itself".

18. Jul 20, 2017

### jbriggs444

As has been pointed out, special relativity applies in flat space-time. The twin paradox of special relativity is resolvable within special relativity. A "twin paradox" within a non-flat space time would not be resolvable within special relativity.

Note that you have not posed a "twin paradox" yet. So there is nothing yet to resolve.

19. Jul 20, 2017

### Rafeek AR

does your statement implies that any two non-local comoving observers "cannot" be compared with the help of even FLRW matrice or if we compared to know " time dilation" the FLRW matrice won't give an answer. Does that implies the " paradox" does exist or does not?.

20. Jul 20, 2017

### jbriggs444

You should first understand that the FLRW "metric" is a global feature of our expanding space-time. It will indeed resolve any "twin paradox" within our universe. It will do this by allowing one to unambiguously compute the elapsed time along any trajectory that either twin could follow (i.e. any timelike trajectory).

If you have two twins that start together, separate and follow different paths and then re-unite, this will allow you to unambiguously predict how much each twin has aged during their trips.

Note that a "metric" is not the same thing as a "frame of reference". A metric is a global feature. For any two events in space-time, the metric tells you how far apart those two events are. This terminology is made more clear in the mathematical field known as "topology". The metric measure of distance is a scalar function. It takes as input two events in space-time. It produces as output a scalar value. That value is either a positive real number (the two points have a timelike separation -- one is unambiguously after the other), an imaginary number (the two points have a spacelike separation -- you cannot get from one to the other at less than the speed of light) or zero (the two points are separated by a null interval -- a light signal could get from one to the other).

None of this requires a coordinate system to be set up. The inputs to this metric function are events.

A frame of reference or "coordinate chart" can be applied to label each event in space time with 4 dimensional coordinates (x, y, z, t). [More generally, we allow space-time to be separated into multiple patches with a different coordinate chart for each -- that's a manifold]. Given a coordinate chart, you could express the metric as a function that takes two coordinate tuples as input and produces a scalar as output. It's the same metric. It just takes coordinates as inputs instead of events.

Obviously the metric function you get for one coordinate chart could be entirely different from the metric function you get using another coordinate chart. But it turns out that the path length (and elapsed time is a path length) determined using one coordinate chart and its associated metric function will be identical to the path length determined using another coordinate chart and its associated metric function. That is to say that path length is an invariant feature of space-time. It does not depend on your frame of reference.

[Note that pretty much everything I know about general relativity, I've picked up by osmosis, lurking in these forums over the years]

Last edited: Jul 20, 2017