1. Jul 18, 2009

### Wangf

Help on my thought experiment. We know the further galaxy, the faster it runs apart from us.

Let's say there is a galaxy (assuming the laws of physics are the same there) running away from us at 99% of the speed of light. Let's say at the SAME time, the people in that galaxy doing the same thing with us on earth. The same thing can be: "start running an atomic clock". The people in that galaxy, according to General relativity, will know the atom clock on earth must run slower than theirs because the earth within milky way galaxy is speeding away at 99% light speed. We on earth will conclude the same that the atomic clock must run slower in the other galaxy speeding away from us at 99% light speed.

We can also imagine a judge in exact middle point from the two galaxies and tell each side to start and stop the atomic clock and measure the time at exact the same time, because the judge is at exact middle point, and each galaxy is speeding away from him at exact time maybe 49.5% of light speed. (Maybe there is no need for such judge. if there are ways to ensure people on both sides start and stop the same time atomic clocks.)

So now my puzzles, are the atomic clocks will measure same time? According to the General Relativity, each clock shall run slower against the other clock. if the atomic clocks measure the same time, then the earth people passed the same time as the other galaxy people, although General Relativity tells the other galaxy people must pass time slower than us as they are moving at 99% of light speed.

2. Jul 19, 2009

### DaveC426913

How will these galaxies and the judge communicate with each other? They have to come back to each other to compare notes. To do so, one or both will have to reverse direction. And that will change the rate of passage of time that is observed since they are now speeding toward each other.

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/assignments/03_rel_sim/index.html [Broken]

Last edited by a moderator: May 4, 2017
3. Jul 19, 2009

### Wangf

the judge is at exact middle point. So he can beam a laser light to each galaxy, the laser light will arrive each galaxy same time, and tell them to start the clock, and via the same method, the judge will tell each galaxy to stop the clock and measure the passage of their time, and beam back to the judge.

i think the judge will see both galaxy pass the same amount of time. But will relativity say the galaxy pass different time against each other?

4. Jul 20, 2009

### Ich

1: yes 2: no
You have to establish what "at the same time" means. If this is done symetrically, via the judge, no time dilation occurs.

5. Jul 20, 2009

### Fredrik

Staff Emeritus
This is true, but you should be aware that this is due to the expansion of the universe, and what this means.

Sounds like you think that the speed can't be >c, but it can, since the galaxies aren't really moving. It's just the space between them that's expanding. (There are lots of threads about this in the cosmology forum. I suggest you do a search).

You're talking about a time coordinate, so you must specify a coordinate system. I'm going to assume that we're talking about the coordinate system in which observers who measure the background radiation to be isotropic are at constant spatial coordinates, and the time coordinate agrees with measurements made by clocks carried by these observers.

This isn't true if the speed is due to the expansion of the universe. It's non-trivial to compare clocks at different locations (not just to do it in a real experiment, but also to define what it means for one clock to be slower than the other), but these clocks are ticking at the same rate in at least one important sense: They both agree with the time coordinate of the coordinate system mentioned above.

Sounds like you should replace the galaxies in your thought experiment with two identical rockets who take off from the judge at one specific event, at the same speed, in opposite directions. In that case, you're right that the judge would conclude that both rockets measure the same amount of time. What one rocket would say about the other depends on what coordinate systems we choose to think of as the rockets' points of view. The usual choice is the co-moving inertial frames. The time coordinates of these two coordinate systems don't agree. If you consider two events that have the same spatial coordinates in one of these frames, the difference between their time coordinates in that frame will be greater than the difference between the time coordinates of the same two events in the other frame.

Last edited: Jul 20, 2009
6. Jul 20, 2009

### Ich

In such coordinates, no galaxy is "running away from us", as the OP demanded.
In the coordinates that you intended to use, there is trivially - by definition - no time dilation, because simultaneity is the same for all comoving observers.
In coordinates where statements like "the atom clock on earth must run slower than theirs because the earth within milky way galaxy is speeding away at 99% light speed" are founded, you are thrown back on classical SR paradoxes.

7. Jul 20, 2009

This is a simple problem in Special Relativity with three inertial reference frames instead of the usual two. The equations and theory to explain what is happening are easily found in many places. There is no need to refer to the General Theory for your solution because you are clearly just stating your interest in velocity contributions to the problem and are not stating any concern with gravitational influences, metric expansion of space, etc. General Relativity treatment of this problem would take a book to explain. Special Relativity explanations for the problem you state would take one page.

8. Jul 20, 2009

### ibcnunabit

They will only be simultaneous from the judge's frame of reference, and the clocks will only be in synch from the judge's frame of reference, because of the given that he is (and remains) halfway between them, and he initiates the timing and does the measuring (but they will still be slower than HIS OWN clocks, equally). To either galaxy the other will be slower. In any other frames, anything goes. Depending on the frame, either can be faster or slower, they will not be seen as starting simultaneously, etc. Even the judge's knowing that he is halfway between them with any accuracy would be a challenge, probably depending on multiple iterations of sending a signal for them to return, and measuring the time difference between them. We can look down upon a diagram and "see" both galaxies running in synch, but we aren't privy to any such Godlike frame of reference.

9. Jul 20, 2009

### Fredrik

Staff Emeritus
Yes, that was my point. It's clear that the OP's question is based on a simple misunderstanding.

What coordinate system would that be?.

10. Jul 21, 2009

### Ich

Riemann normal coordinates, for example, or a suitable extension to larger distances.
No, it's based on standard SR coordinates, which are appropriate in some neighbourhood of any point. There are substantial deviations at v=.99 c, of course, but the principle still holds.

11. Jul 21, 2009

### Fredrik

Staff Emeritus
He also said "We know the further galaxy, the faster it runs apart from us". There's no point in saying that unless he means that the velocity is due to the expansion of the universe.

12. Jul 21, 2009

### Ich

Yes. That changes nothing: there are two (or more) objects in relative motion, and from each object's viewpoint, the other ones are time dilated. Gravitation changes the picture quantitatively, but not qualitatively (as long as there are no issues with the topology of the universe).
I know that from the cosmology forum, and from some papers, one gets the notion that expansion is not motion. I can't help it.

13. Jul 21, 2009

### Wangf

My question now is if the people in the above two galaxies, grow a tree (assuming all other conditions are the same for tree growth), instead of measuring the atomic clock, then what will happen?

According to Relativity, each galaxy will see time run slow in the other galaxy, so the tree in other galaxy will grow shorter.

The judge, at the exact straight line middle point, can send beam light signals (which will arrive at each galaxy same time) to each side to start grow the trees, to measure the tree heights, and the galaxies beam back the tree height info via light signals to the judge.

The judge will see the heights of the trees from the two galaxies are the same. The judge can even send such info to both sides, then both side shall be surprised to know the tree heights are the same, because according the Relativity, the tree in the other galaxy must grow shorter/slower!

My question is again, how come the trees can be both relatively shorter to each other (from views of one galaxy against the other galaxy), and being the same height at the same time (from the judge's view)?

14. Jul 22, 2009

### Fredrik

Staff Emeritus
I don't think that's right. There's no time dilation between the local inertial frames of two distant galaxies in a FLRW spacetime. The two frames assign the same time coordinate to any event in spacetime. (I'm assuming that we would choose their origins to be events that are simultaneous in FLRW coordinates).

I'm pretty sure this is incorrect. You should change your thought experiment to be about two rockets that are close to each other and moving apart fast.

If we replace "galaxy" with "rocket", the answer is just that that they disagree about which events are simultaneous. I recommend that you study spacetime diagrams. These things are hard to explain in words, but are really easy to understand when you draw a diagram.

Last edited: Jul 22, 2009
15. Jul 22, 2009

### Ich

Definitely not. The coordinates you have in mind are not standard inertial coordinates. Some metric components deviate linearly from the minkowski ones, the speed of light is position- and direction dependent and so on. If you fix that - go to normal coordinates - you lose the global synchronization of cosmic time.
Why do you think that galaxies behave differently than rockets?
There is some misinformation concerning FRW spacetimes, but they can - of course, like any other spacetime - be approximated by local inertial frames. These are in relative motion, therefore there is time dilation.

@Wangf: Fredrik is right, without diagrams you have no chance of understanding.

16. Jul 22, 2009

### DaveC426913

I think the galaxies may be adding in a confounding factor of expansion of the universe. Not that that can't be factored in, but it would be best to eliminate as many variables and confusion factors as possible.

17. Jul 22, 2009

### Fredrik

Staff Emeritus
I haven't checked all the math yet, so I can't prove it right now, but I really think you're making a mistake. I'll post it later if I can figure out the details. In the meantime, I recommend that you think this through yourself. In particular, I think you should think about the fact that a curve of constant FLRW time is a geodesic, and about the significance of geodesics in the construction of a "local inertial frame" or "Riemannian normal coordinates". (Are those two exactly the same thing? I still haven't quite figured that out).

It's obviously not the case that "galaxies behave differently than rockets", so I'd appreciate if you don't suggest that I've said something crazy like that. We're talking about two specific rockets that are experiencing time dilation because they used their engines to boost their motion to other geodesics than the ones they were following before, and we're talking about two galaxies that are on the same geodesics the whole time.

Last edited: Jul 22, 2009
18. Jul 22, 2009

### RUTA

The difference I thought you were trying to point out was that in the GR cosmology the galaxies don't occupy the same M4 frame while the rockets do (rockets are an SR problem not GR). Lorentz transformations (SR) are done to relate coordinates for an event between two systems in same M4 frame ("rotation" in flat spacetime). In GR cosmology each galaxy resides in its own region of flat spacetime (GR spacetime manifold is locally flat, SR applies in these locally flat regions), so an event in galaxy A's region can't be related via Lorentz transformation to galaxy B, because that event doesn't reside in galaxy B's M4 frame. Sorry, if that wasn't your point.

19. Jul 22, 2009

### RUTA

As Fredrik (I thk) said earlier, in FRW cosmology spacetime can be foliated by spatial surfaces of homogeniety. A global time coordinate is assigned to each surface according to proper time lapsed by observers at rest wrt to the surface at their location (these observers are therefore called "co-moving" observers). A surface of time T is generally what people mean by "the universe" at time T. A good analogy for these surfaces and co-moving observers is pennies glued onto a balloon. The pennies don't move relative to the balloon but they do move away from one another as the balloon expands. Anyway, time ticks the same for all co-moving observers regardless of their relative motion per the expansion of "the universe."

20. Jul 24, 2009

### Ich

That's exactly what I did, and what lead me to the viewpoint I'm presenting.
That's not a fact, that's wrong. You can convince yourself easily:
Consider the special FRW case a = const.*t (empty universe).
$$ds^2=dt^2-a(t)^2\, dr^2 + \ldots$$
Check that the transformation
$$t' &= &t\, \cosh (r)\\$$
$$x &= & t\, \sinh (r)\\$$
brings the metric to minkowski form
$$ds^2=dt'^2-dx^2 + \ldots$$
where t=const. denotes a hyperbola, while geodesics are straight lines.
Given that t' = const. is a geodesic, that's exactly what I'm talking about.
The former does not fix the coordinates you use -an inertial frame may be expressed e.g. in polar coordinates (or FRW coordinates, for that matter)-, while the latter is what I'd call a standard inertial frame.
Please be assured that I fully appreciate and respect your knowledge in GR, and that I didn't intend to express something different.
You know as good as I that time dilation arises because both, rockets or galaxies or whatever, are on different geodesics, and that it does not matter how they came to be there. That's what I wanted to convey: given that both galaxies see the other as moving away, according to every sensible operational definition of relative motion, how could there not be time dilation?
Some cosmologists say that "expansion is not motion" or something to that effect. Don't take such claims seriously, at least on small and short scales, expansion is motion and nothing else.