My twin galaxy puzzle Please help

[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.

 

[DaveC426913]

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.

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

 

[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?

 

[Ich]

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?

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.

 

[Fredrik]

We know the further galaxy, the faster it runs apart from us.

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

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.

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).

Let's say at the SAME time, the people in that galaxy doing the same thing with us on earth.

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.

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.

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.

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?

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.

 

[Ich]

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.

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.

 

[Ironhead]

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.

 

[ibcnunabit]

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.

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.

 

[Fredrik]

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.

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

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.

What coordinate system would that be?.

 

[Ich]

What coordinate system would that be?.

Riemann normal coordinates, for example, or a suitable extension to larger distances.

It's clear that the OP's question is based on a simple misunderstanding.

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.

 

[Fredrik]

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.

 

[Ich]

There's no point in saying that unless he means that the velocity is due to the expansion of the universe.

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.

 

[Wangf]

thanks for all the answers.

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)?

 

[Fredrik]

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.

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).

According to Relativity, each galaxy will see time run slow in the other galaxy,

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.

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)?

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.

 

[Ich]

The two frames assign the same time coordinate to any event in spacetime.

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.

If we replace "galaxy" with "rocket", the answer is just that that they disagree about which events are simultaneous.

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.

 

[DaveC426913]

Why do you think that galaxies behave differently than rockets?

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.

 

[Fredrik]

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.

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.

 

[RUTA]

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.

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.

 

[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."

 

[Ich]

In the meantime, I recommend that you think this through yourself.

That's exactly what I did, and what lead me to the viewpoint I'm presenting.

In particular, I think you should think about the fact that a curve of constant FLRW time is a geodesic...

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.

...and about the significance of geodesics in the construction of a "local inertial frame" or "Riemannian normal coordinates"

Given that t' = const. is a geodesic, that's exactly what I'm talking about.

Are those two exactly the same thing? I still haven't quite figured that out

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.

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.

Please be assured that I fully appreciate and respect your knowledge in GR, and that I didn't intend to express something different.

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.

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.

 

[RUTA]

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.

I don't know what you mean by "time dilation" in the case of galaxies. I understand the term to concern time differences measured between the same two events as related by Lorentz transformation, i.e., two frames in relative motion in the same M4. The galaxies occupy different M4 regions, so successive events along the worldline of one galaxy cannot be related via Lorentz transformation to the frame of the other galaxy. What do you mean by "time dilation" in the GR case?

 

[Ich]

What do you mean by "time dilation" in the GR case?

You can establish a natural, minkowski-like coordinate system centered at any point in spacetime (Riemann normal coordinates). Its coordinates reflect as closely as possible the usual operational definition of SR - especially the Einstein synchronization convention, which is essential.
In these coordinates, a neighbouring galaxy has a slanted worldline (i.e. velocity), and events separated by a certain proper time on it are separated by a larger coordinate time.

 

[RUTA]

You can establish a natural, minkowski-like coordinate system centered at any point in spacetime (Riemann normal coordinates). Its coordinates reflect as closely as possible the usual operational definition of SR - especially the Einstein synchronization convention, which is essential. In these coordinates, a neighbouring galaxy has a slanted worldline (i.e. velocity), and events separated by a certain proper time on it are separated by a larger coordinate time.

As a coordinate choice it in no way justifies the use of Lorentz transformations and any resulting notion of "time dilation" is therefore idiosyncratic. That is, someone else could choose another coordinate system (e.g., the usual co-moving system) and claim a totally different amount of time dilation (none in the case of the co-moving system). I don't think "time dilation" is a well-defined concept in GR except in its locally flat frames (where SR applies). I think that was the point Fredrik was making in an earlier post.

 

[Ich]

As a coordinate choice it in no way justifies the use of Lorentz transformations and any resulting notion of "time dilation" is therefore idiosyncratic.

As the best possible approximation of Minkowski coordinates, it justifies the use of Lorentz transformations within the domain of applicability. Further, I don't see how time dilation is tied to Lorentz transformations.

That is, someone else could choose another coordinate system (e.g., the usual co-moving system) and claim a totally different amount of time dilation

That would really be stupid, giving up the concept of absolute time dilation and introducing it as a coordinate dependent effect. That would lead to a number of obvious contradictions, like A claiming B to be dilated, and B claiming A to be dilated, and both being right. If we used such a concept, we'd have at least three threads per week explaining such nonsense to newbies.

 

[RUTA]

As the best possible approximation of Minkowski coordinates, it justifies the use of Lorentz transformations within the domain of applicability. Further, I don't see how time dilation is tied to Lorentz transformations.

That would really be stupid, giving up the concept of absolute time dilation and introducing it as a coordinate dependent effect. That would lead to a number of obvious contradictions, like A claiming B to be dilated, and B claiming A to be dilated, and both being right. If we used such a concept, we'd have at least three threads per week explaining such nonsense to newbies.

Lorentz transformations make time dilation unambiguous in the context of a pair of events in a single M4 frame. Time dilation is ambiguous in the context you propose, thus your only response to an alternative to your idiosyncratic definition of time dilation between distinct M4 frames is to say it's "stupid." You may have a very interesting and compelling view of time dilation in this context, but ad hominems won't advance your view.

 

[Ich]

Care to read what I actually wrote?

 

[Fredrik]

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.

I've been busy the last few days, so I haven't been able to really think about this until now. I don't find this easy at all, maybe because I haven't done any GR calculations in a long time. I still don't know if you're right, but I have realized that I didn't have a good reason to think that a geodesic in a hypersurface of constant FLRW time must be a geodesic in spacetime, so I will at least have to admit that you might be. If I was wrong about the geodesics, then many of the other things I said are almost certainly wrong too, in particular the claim that the local inertial frames of the galaxies agree about time.

I have verified that your metric is a FLRW metric. You just set k=-1, ρ=0, Λ=0 in the first of the Friedmann equations, and we immediately see that $\dot a$ is a constant. (I'll call it A below). From this we get $\ddot a=0$, and when we use this in the second Friedmann equation, we get p=0. So we're talking about a universe that's completely empty at all times, and about a spacetime with line element $ds^2=-dt^2+A^2t^2d\Omega^2$, where $d\Omega^2=d\psi^2+\sinh^2\psi\left(d\theta^2+\sin^2\theta\ d\phi^2\right)$ is the line element of a unit hyperboloid (a 3-dimensional manifold with constant negative curvature). I don't see how to proceed from here. Is the "r" in your change of variables the $\psi$ in my version of the line element? (I got the line element from Wald, page 95, eq. 5.1.9).

It seems more natural to just apply the definition of a geodesic to a "spatial geodesic", i.e. to a curve in a hypersurface of constant FLRW time that's a geodesic in the metric induced on that hypersurface by the metric of spacetime. We could settle this by checking if the spacetime metric parallel transports the tangent vector of such a curve. I haven't done that calculation yet, but I might try it later.

I would still be surprised to find that spatial geodesics aren't spacetime geodesics. I've been visualizing an expanding universe (with positive curvature) as a sequence of concentric spheres (2-dimensional since I'm too dumb to visualize 3-spheres). Time is represented by the distance from the center in this image. I've been assuming that a great circle in one of the spheres (a geodesic in space) is a geodesic in spacetime. You're saying it's not. If you're right, then I'd like to know what an actual geodesic that's tangent to a sphere at some point looks like? Is it a straight line? That would be surprising because most timelike and all null geodesics are not straight lines in this image. E.g. a null geodesic has to intersect each sphere at a 45° angle, so it would be a curved path. On the other hand, the world line of a galaxy is a straight line, so maybe a spacelike geodesic can be too. I mean, we're talking about the other extreme end of the range of geodesics that exist in this geometry. The world line of a galaxy is a geodesic representing zero velocity in FLRW coordinates, and the spacelike geodesic we're talking about represents infinite velocity in FLRW coordinates. But...uhh...that doesn't work, I think. If we take a tangent to a sphere and extend it, it would intersect some of the larger spheres at an angle >45°, which means that it goes from being spacelike, to null, to timelike. Can a geodesic do that? I don't think so, because at the point where its tangent is null, it's also tangent to a null geodesic. So the straight line can't be a geodesic at that point.

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?

You may be right about the geodesics, and therefore also about the galaxies disagreeing about time, but this logic is flawed. You can't just transfer the results from a theory that assumes that there's no gravity to a GR scenario where the cause of what we're talking about is a large-scale gravitational effect.

 

[Fredrik]

You can establish a natural, minkowski-like coordinate system centered at any point in spacetime (Riemann normal coordinates). Its coordinates reflect as closely as possible the usual operational definition of SR - especially the Einstein synchronization convention, which is essential.

I still don't fully understand these coordinates, but one thing I do understand is that they either say that the coordinate speed of light grows with (spatial) distance from that event, or that the difference between the time coordinates of any two events is the same as in FLRW coordinates (which would make the two galaxies agree about time). So if there's time dilation, your coordinate system also says that the speed of light is >c in the other galaxy.

In these coordinates, a neighbouring galaxy has a slanted worldline (i.e. velocity), and events separated by a certain proper time on it are separated by a larger coordinate time.

How do you justify the part after the "and"? I agree with the first part, but you can't just do a Lorentz transformation here.

As the best possible approximation of Minkowski coordinates, it justifies the use of Lorentz transformations within the domain of applicability. Further, I don't see how time dilation is tied to Lorentz transformations.

These comments are pretty strange. The "domain of applicability" should be a region of spacetime in which curvature is negligible, but this scenario is specifically about a large region of spacetime where the effect we're talking about is caused by the curvature.

You've been arguing that what we know about inertial frames in SR (i.e. about Lorentz transformations) should make it more or less obvious that there's time dilation between these galaxies, so I find it odd that you're now downplaying the importance of Lorentz transformations. If Lorentz transformations isn't what made you conclude that there's time dilation between these galaxies, then what did?

That would really be stupid, giving up the concept of absolute time dilation and introducing it as a coordinate dependent effect. That would lead to a number of obvious contradictions, like A claiming B to be dilated, and B claiming A to be dilated, and both being right. If we used such a concept, we'd have at least three threads per week explaining such nonsense to newbies.

Was the "rolleyes" meant to indicate that you were joking? In SR, the situation is of course that A can claim that B is time dilated while B is claiming that A is time dilated, and they're both right. (We don't even have to consider non-standard frames. This happens even when we only consider their co-moving global inertial frames). And we do have three threads per week (or at least three per month) explaining this "nonsense" to newbies.

 

[Fredrik]

ad hominems

I like your posts, but I can't resist pointing out that you're confusing "ad hominems" with insults. At least you spelled it right. An ad hominem is an attempt to argue that the other guy is wrong because of what he is, so if someone just calls you stupid, it's not an ad hominem. If someone says "you're wrong because you're stupid" or "you can't understand that I'm right because you don't have children yourself", that's an ad hominem.

 

[Ich]

Hi Fredrik, RUTA

Was the "rolleyes" meant to indicate that you were joking?

Of course.
Sorry, English is not my first language and I thought it is obvious that I'm simply describing SR.
From the context it should also be clear (I thought) that I'm adressing RUTA's concerns regarding the (in my "proposal") coordinate-dependence of time dilation by calling it (my "proposal", not RUTA's or RUTA himself) "stupid" and (attemptedly) humorously pointing out that it is the very nature of time dilation to be coordinate-dependent.
I should better use [ tongue in cheek ] [ /tongue in cheek ] tags in the future. Sorry for the inconvenience.
Still, I'm mortally offended that Fredrik really asks whether I'm serious or not.

 

[RUTA]

I like your posts, but I can't resist pointing out that you're confusing "ad hominems" with insults. At least you spelled it right. An ad hominem is an attempt to argue that the other guy is wrong because of what he is, so if someone just calls you stupid, it's not an ad hominem. If someone says "you're wrong because you're stupid" or "you can't understand that I'm right because you don't have children yourself", that's an ad hominem.

Is there a spell check on this site? I still haven't figured out how to get multiple quotes, either

Here's what dictionary.com has to say about usage for "ad hominem:"

As the principal meaning of the preposition ad suggests, the homo of ad hominem was originally the person to whom an argument was addressed, not its subject. The phrase denoted an argument designed to appeal to the listener's emotions rather than to reason, as in the sentence The Republicans' evocation of pity for the small farmer struggling to maintain his property is a purely ad hominem argument for reducing inheritance taxes. This usage appears to be waning; only 37 percent of the Usage Panel finds this sentence acceptable. The phrase now chiefly describes an argument based on the failings of an adversary rather than on the merits of the case: Ad hominem attacks on one's opponent are a tried-and-true strategy for people who have a case that is weak. Ninety percent of the Panel finds this sentence acceptable. The expression now also has a looser use in referring to any personal attack, whether or not it is part of an argument, as in It isn't in the best interests of the nation for the press to attack him in this personal, ad hominem way. This use is acceptable to 65 percent of the Panel.

Telling someone their idea is "just stupid" is an appeal to emotions, not reason, and therefore constitutes an ad hominem, at least in some quarters.

 

[Fredrik]

Still, I'm mortally offended that Fredrik really asks whether I'm serious or not.

Then I humbly beg your forgiveness. Yes, it sure sounded like a joke about SR and the actual situation here at PF. If you had expressed yourself that way in a PM to me, I would have assumed that it was a joke, but the thing is, you wrote it in a public forum that's read by people who aren't regulars here, and to them your comment probably didn't look like a joke. That's what confused me. I was thinking that if I had made a joke like that, I whould have made it clear to the non-regulars too. I wouldn't want them to believe that I have failed to understand one of the most basic facts about SR.

Any new thoughts about the geodesics?

 

[Fredrik]

Is there a spell check on this site? I still haven't figured out how to get multiple quotes, either

I don't think the site has a spell check, but some browsers do. You can definitely get Firefox to check the spelling for you. Multiple quotes are easy. Just click the "multi quote" button of all the posts you want to reply to (the button should change color; you can click it again to unselect that post) and then click "new reply" at the bottom of the page. Or you can click "multi quote" of all but the last of the posts you want to reply to, and then click the "quote" button for the last post you want to reply to. Then you will of course have to edit the irrelevant stuff out so that it's clear what you're replying to.

Also, when you reply to a post using the quote button, you'll see the quote tags in your own post. You can copy and paste them to break the quote up into pieces. You can even nest them:

You suck.

No, you suck.

Try to keep the number in the opening quote tag. It's used to create a link back to the post you're quoting. (Ich might want to take that advice too).

Here's what dictionary.com has to say about usage for "ad hominem:"

Interesting. Maybe this is one of those times when so many people use a word wrong that it quickly becomes the standard way of using it, like "atheist" and "agnostic". "Atheist" is supposed to mean "someone who isn't a theist" (where a "theist" is someone who believes there's at least one god). That definition implies that every infant is an atheist, but the word is used to describe someone who believes that no gods exist. "Agnostic" is supposed to mean "someone who thinks it's impossible to obtain knowledge about God", but people use it to describe people who aren't sure what they believe.

 

[Ich]

Then I humbly beg your forgiveness.

Granted, thank you!

Any new thoughts about the geodesics?

Many, I will answer your interesting posts as soon as I can. But I don't have much time these days, maybe tomorrow.

 

[Ich]

I have verified that your metric is a FLRW metric ... I don't see how to proceed from here. Is the "r" in your change of variables the psi in my version of the line element?

Yes, r == psi. I simply use A=1, which means dimensionless r and a scalefactor with units of time or length.
Proceed with the total derivative
$$dt' = \frac{\partial t'}{\partial t}dt + \frac{\partial t'}{\partial r} dr = cosh(r)dt + t\, sinh(r) dr$$
and so on, insert in the metric, sort it out and find the minkowski metric. It's similar to Rindler coordinates with x and t swapped.

It seems more natural to just apply the definition of a geodesic to a "spatial geodesic"...I would still be surprised to find that spatial geodesics aren't spacetime geodesics.

Spatial geodesics generally have not much physical meaning. In this case, it's clear that the curved spatial geodesics can't be spacetime geodesics.

I've been visualizing an expanding universe (with positive curvature) as a sequence of concentric spheres (2-dimensional since I'm too dumb to visualize 3-spheres). Time is represented by the distance from the center in this image.

That's not a FRW universe, since you have constant $$\dot a$$. I don't know if this visualization is helpful, the best starting point is imho the empty milne model above, since it allows you to see the deviations of FRW coordinates from standard SR ones even if there is no gravitation.

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?

You may be right about the geodesics, and therefore also about the galaxies disagreeing about time, but this logic is flawed. You can't just transfer the results from a theory that assumes that there's no gravity to a GR scenario where the cause of what we're talking about is a large-scale gravitational effect.

The cause is inflation (in the standard model), and that's long over. What we have now is an almost empty spacetime, where we can start with flat/Newtonian/postNewtonian approximations at each event, and try to look at it from this point of view.

I still don't fully understand these coordinates, but one thing I do understand is that they either say that the coordinate speed of light grows with (spatial) distance from that event, or that the difference between the time coordinates of any two events is the same as in FLRW coordinates (which would make the two galaxies agree about time).

No. Examine the empty universe example, where normal coordinates apply to all of spacetime. Neither is c!=1, nor are the time coordinates the same.

In these coordinates, a neighbouring galaxy has a slanted worldline (i.e. velocity), and events separated by a certain proper time on it are separated by a larger coordinate time.

How do you justify the part after the "and"? I agree with the first part, but you can't just do a Lorentz transformation here.

You don't have to do a transform. You have the metric, the worldline, and you can read off the time coordinates and integrate to get proper time.

As the best possible approximation of Minkowski coordinates, it justifies the use of Lorentz transformations within the domain of applicability. Further, I don't see how time dilation is tied to Lorentz transformations.

These comments are pretty strange. The "domain of applicability" should be a region of spacetime in which curvature is negligible, but this scenario is specifically about a large region of spacetime where the effect we're talking about is caused by the curvature.

No. The effect is largely caused by curved coordinates, and whether or not spacetime curvature is important is a completely different matter. But you're right that effective time dilation does not follow naturally.
Generally, since v increases linearly with distance, time dilation is second order in distance. If there is also gravity (or "antigravity" like a cosmological constant), this is also a second order correction to "clock speed". Starting with normal coordinates, both effects are exactly defined and can be added, like in https://www.physicsforums.com/showpost.php?p=1600272&postcount=8".
So I will express myself more precisely: Normal coordinates define a set of observers that are at rest wrt the origin. FRW comoving particles have relative velocity to these observers, therefore, at any point, there is time dilation wrt the respective "static" observer.
Those static observers are additionally up or down a gravitational well, therefore their clocks are generally ticking at a different rate than the one at the origin.
Both effects together give the complete picture, accurate to second order in distance.