B Proper time of the observer resting in CMB reference frame

[ongoer]

So you're measuring the redshifted appearance of the old clock rate to the current rate. That's just a measure of redshift, as we have been saying since post #4.

And the measure of this redshift is also a measure of cosmological time dilation.

 

[ongoer]

If their mechanical watch does not agree with a light clock that they are also carrying along... then the mechanical watch is broken.

Assume that it's Swiss.

 

[ongoer]

There is no such thing. "Time dilation" is defined (see my post about gravitational time dilation above) to be the ratio of the proper times between two different pairs of events that have the same coordinate time (that is, have been defined to happen "at the same time"). There's no way of doing that between someone's past and present.

Cosmological time dilation is defined as the observed redshift, and the immortal observer could have seen it changing.

 

[Ibix]

Cosmological time dilation is defined as the observed redshift, and the immortal observer could have seen it changing.

If he has sufficiently good and also co-moving mirrors, yes. He can also see no redshift if he deploys his mirrors differently (rigidly attached to him, I think).

What he's seeing is a product of the light path he uses to observe his past watch image, which isn't the case with either kinematic or gravitational time dilation. That's why simply relabelling redshift "time dilation" seems nonsensical and misleading to me.

 

[PeterDonis]

And the measure of this redshift is also a measure of cosmological time dilation.

Sure, if you define the term "cosmological time dilation" to mean "redshift". But that's not physics. That's just playing with words.

The key point you are missing is at the end of post #27 by @Nugatory. Please read that again and again until it sinks in. There is simply no way to make the comparison you would need to make to support the claim that "cosmological time dilation" refers to something "real", as opposed to just an appearance. For another quick way of stating the point: a "real" time dilation, such as gravitational time dilation between two observers at different altitudes in a gravitational field, requires round-trip light signals to measure it. But you obviously can't exchange round trip light signals with the past.

 

[Nugatory]

Assume that it's Swiss.

The better designed and built it is, the more closely it will agree with an idealized and infinitesimal light clock. It has to be that way, because the Swiss wristwatch is measuring proper time - the difference between two readings of the watch is, to limits of accuracy of the mechanism, the proper time along the wristwatch’s worldline between the readings - and a light clock is idealized and theoretically perfect device for measuring proper time.

 

[Ibix]

and a light clock is idealized and theoretically perfect device for measuring proper time.

Note that it's also just a pair of mirrors. You could actually look into it and see a reflection (of a reflection of a reflection...) of your mechanical watch 14bn years ago ticking at 1s/s. In fact, all of the reflections you could see would tick in synchrony.

 

[ongoer]

@PeterDonis Alice is falling on the event horizon of a BH and Bob is watching her. Let's fall with Alice in her reference frame. After we cross the event horizon with her, we can't compare our time dilation with Bob using a light signal. All we've got is maths. Does this mean, that the time dilation between us and Bob makes no physical sense anymore?

 

[Ibix]

Does this mean, that the time dilation between us and Bob makes no physical sense anymore?

Yes. The spacetime inside a black hole isn't a stationary spacetime, so you can't establish a definition of "only moving in space" and hence there's no way to define "correcting for movement". You can measure a redshift in one direction but not the other. You are free to define simultaneity between Alice and Bob in any way you wish, so time dilation between the two is anything you want. So it's not meaningful - it's just your choice of coordinates.

 

[Nugatory]

Does this mean, that the time dilation between us and Bob makes no physical sense anymore?

Yes, that is what it means.

 

[ongoer]

Yes. The spacetime inside a black hole isn't a stationary spacetime, so you can't establish a definition of "only moving in space" and hence there's no way to define "correcting for movement". You can measure a redshift in one direction but not the other. You are free to define simultaneity between Alice and Bob in any way you wish, so time dilation between the two is anything you want. So it's not meaningful - it's just your choice of coordinates.

This also means, that you can't assume, that the other immortal observer separated from us by the spacelike interval as a result of the expansion is at the same age as we (assuming our immortality) equal to the universe age, since we can't communicate anymore and compare our proper times. Do you agree?

 

[Nugatory]

is at the same age…

“Is at the same age” is another way of saying “has experienced the same amount of proper time between birth and now” so is meaningful only if we have a definition of “now”. And if they are not colocated, there is no common definition of “now”.

 

[ongoer]

“Is at the same age” is another way of saying “has experienced the same amount of proper time between birth and now” so is meaningful only if we have a definition of “now”. And if they are not colocated, there is no common definition of “now”.

Yes, there is. It's called the cosmological time and it's equal to the proper time of all the immortal observers resting in their cmb reference frames and the same for all of them, because it's also the age of the universe.

 

[Ibix]

This also means, that you can't assume, that the other immortal observer separated from us by the spacelike interval as a result of the expansion is at the same age as we (assuming our immortality) equal to the universe age

Well, since the universe age is defined to be the time measured by comoving observers since the big bang singularity, yes they are the same age by definition. So I don't have to assume it - you did it for me in your formulation of the question.

You could pick a different simultaneity criterion if you want, yes, but nobody ever does because the maths is much nastier.

 

[ongoer]

@Ibix you skipped the lack of possibility to communicate :) Also, tell it to @Nugatory please :)

 

[Ibix]

@Ibix you skipped the lack of possibility to communicate :)

It doesn't matter. You constrained the answer by the way you asked the question. Other answers are possible (and the lack of ability to communicate is a part of that), but not given the definition of "at the same time" that you used.

As I said - I don't have to assume it because you did it for me. If you want to relax that assumption then we can do so.

 

[PeterDonis]

Does this mean, that the time dilation between us and Bob makes no physical sense anymore?

Yes.

 

[PeterDonis]

we can't communicate anymore

Sure you can. You can communicate with other comoving observers in the universe--just wait long enough for the round trip light signal. You might have to wait a long time, but a round trip light signal is still possible.

The reason you can't do that if you're inside a black hole and someone else has remained outside is that it is impossible for a light signal to make a round trip. That makes a big difference.

 

[PeterDonis]

you can't assume, that the other immortal observer separated from us by the spacelike interval as a result of the expansion is at the same age as we (assuming our immortality) equal to the universe age

At the particular spacelike separated events along your worldlines that you refer to, you both have experienced the same proper time since the Big Bang. In that sense you are "the same age". But of course that is not the only possible sense of the word "age".

As for "assuming" that, well, it's not an assumption we just pulled out of thin air, it's an implication of our best current model of the universe. To confirm it, of course, you would have to exchange round trip light signals with the other comoving observer, which, as I noted in my previous post just now, might take time. But it can be done. There is nothing in principle preventing you and other comoving observers from exchanging round trip light signals and confirming whatever predictions of our best current model of the universe you are interested in.

Whereas, if you fall into a black hole and I stay outside, there is no way, even in principle, for you to ever send me a light signal telling me what things are like inside the hole. Again, that makes a big difference.

 

[Nugatory]

Yes, there is. It's called the cosmological time and it's equal to the proper time of all the immortal observers resting in their cmb reference frames and the same for all of them, because it's also the age of the universe.

Nonetheless, that definition of "now" is an arbitrary choice, albeit one that we often make because it simplifies the math. We could define "now" some other way and no physics would change, no observations would come out differently, the spacetime interval and proper times between pairs of events remains the same.

And none of that is helpful for defining time dilation along our own worldline, which seems to be what you were trying to do above.

 

[ongoer]

@PeterDonis that's not nice to pull a half of my sentence out of context and change its meaning in this way.

You don't have to communicate with anyone to know, that when you're moving with respect to the CMB and measure its dipole in your frame, then you're time dilated with respect to all the observers, who rest in their CMB reference frames even if they are separated from you by the spacelike intervals.

 

[Ibix]

You don't have to communicate with anyone to know, that when you're moving with respect to the CMB and measure its dipole in your frame, then you're time dilated with respect to all the observers, who rest in their CMB reference frames even if they are separated from you by the spacelike intervals.

That would depend on how they choose to extend their local reference frames into global coordinate systems. The obvious way to do it is using cosmological time, but then you are back to "yes everybody is the same age because I have defined it so".

 

[ongoer]

That would depend on how they choose to extend their local reference frames into global coordinate systems. The obvious way to do it is using cosmological time, but then you are back to "yes everybody is the same age because I have defined it so".

What is unreasonable in assumption that all my immortal observers are in the same age equal to the universe age, if it's homogenous on the large scale?

 

[Ibix]

What is unreasonable in assumption that all my immortal observers are in the same age equal to the universe age, if it's homogenous on the large scale?

It's a very sensible definition. However, others are possible. But as I already said, they make the maths messy.

 

[Nugatory]

What is unreasonable in assumption that all my immortal observers are in the same age equal to the universe age, if it's homogenous on the large scale?

It's tautologically and uninterestingly true that for each one their age and any given moment is equal to the proper time along their worldline (presuming these immortals have all been around from the beginning, as we have in this thread).

Now if you choose to define a surface of simultaneity as the locus of all events on those worldlines at which they are a particular age... then yes, of course they are all the same age because that's how you defined it.

 

[ongoer]

@Nugatory what's your definition of the age of the universe?

 

[PeterDonis]

@PeterDonis that's not nice to pull a half of my sentence out of context and change its meaning in this way.

Since you didn't quote anything from me, I don't know what you're referring to here.

You don't have to communicate with anyone to know, that when you're moving with respect to the CMB and measure its dipole in your frame, then you're time dilated with respect to all the observers, who rest in their CMB reference frames

No, you're not. They're time dilated with respect to you. At least that's what things look like from your rest frame. The concept of "time dilation" you're using here, due to relative motion, is frame dependent; there is no absolute sense in which either you or the comoving observers are "time dilated".

even if they are separated from you by the spacelike intervals

To even define these intervals, you have to adopt a simultaneity convention--a convention that tells you which events on different worldlines happen at the same time. And any such convention is arbitrary. Simultaneity is not a physical thing.

In any case, none of this is even relevant to the claim you are actually trying to defend, which is that there is some meaningful concept of "time dilation" between an observer now and that same observer in the distant past. Every actual example you've given has been about comparisons between different observers at events on their worldlines that are spacelike separated. But the claim you're making is about comparisons between events on the same worldline, which are timelike separated. There is simply no meaningful concept of "time dilation" that even applies to such a case.

 

[PeterDonis]

@Nugatory what's your definition of the age of the universe?

The usual definition is the proper time since the Big Bang along comoving worldlines.

 

[ongoer]

The usual definition is the proper time since the Big Bang along comoving worldlines.

It's tautologically and uninterestingly true that for each one their age and any given moment is equal to the proper time along their worldline (presuming these immortals have all been around from the beginning, as we have in this thread).

Now if you choose to define a surface of simultaneity as the locus of all events on those worldlines at which they are a particular age... then yes, of course they are all the same age because that's how you defined it.

 

[ongoer]

No, you're not. They're time dilated with respect to you. At least that's what things look like from your rest frame. The concept of "time dilation" you're using here, due to relative motion, is frame dependent; there is no absolute sense in which either you or the comoving observers are "time dilated".

You're teaching me about frame dependency even though I wrote "with respect to" every time I was talking about the different reference frames.

If they're time dilated with respect to me, then I'm also time dilated with respect to them.