B Mass of Space (B but approaching I)

[PeterDonis]

Is it correct that the wordlines of these particles are representing a "congruence of wordlines"

Yes.

and that the rate at which the volume of this ball shrinks or grows is given by the expansion scalar?

I don't know off the top of my head if it's numerically equal, but it's related. However...

would it be correct to describe the "expansion" of the universe as corresponding to an increasing volume of comoving galaxies?

No, because there's a big difference between Baez's scenario and the universe: in Baez's scenario, you are working in the rest frame of the ball (more precisely, the rest frame of its center of mass) and the ball is very small. For the universe, there is no single "rest frame" for comoving objects and the comoving objects are not restricted to a small ball--they are spread throughout the entire universe, which is spatially infinite.

 

[phinds]

this is a coordinate dependent concept

I hear this over and over, but here's my confusion. If you have two galaxies, each X units of distance in diameter and separated by a distance of 10X, then after some amount of time they will each still be X units of distance in diameter but the distance between them will be 12X units of distance. How is this coordinate dependent? Is it somehow that their diameter is not coordinate dependent but the distance between them is?

 

[PeterDonis]

If you have two galaxies, each X units of distance in diameter and separated by a distance of 10X, then after some amount of time they will each still be X units of distance in diameter but the distance between them will be 12X units of distance. How is this coordinate dependent?

"After some amount of time" depends on the time coordinate you choose. "Separated by a distance of..." depends on the space coordinates you choose.

 

[phinds]

"Separated by a distance of..." depends on the space coordinates you choose.

OK, but how can you choose a different space coordinate for the distance across the galaxies vs the distance between the galaxies?

 

[timmdeeg]

No, because there's a big difference between Baez's scenario and the universe: in Baez's scenario, you are working in the rest frame of the ball (more precisely, the rest frame of its center of mass) and the ball is very small. For the universe, there is no single "rest frame" for comoving objects and the comoving objects are not restricted to a small ball--they are spread throughout the entire universe, which is spatially infinite.

Ah ok, so there seems no possibilty to extend Baez's ball somehow to the universe.

 

[PeterDonis]

how can you choose a different space coordinate for the distance across the galaxies vs the distance between the galaxies?

I didn't say you had to. I just said "separated by a distance of...". The point is simply that "distance" is coordinate-dependent.

 

[timmdeeg]

The point is simply that "distance" is coordinate-dependent.

While if we say "increasing distances" (standing for increasing redshifts) that's invariant, right?

 

[PeterDonis]

While if we say "increasing distances" (standing for increasing redshifts) that's invariant, right?

Increasing with respect to what?

 

[timmdeeg]

Increasing with respect to what?

increasing with respect to an arbitrary observer who measures the redshift of far away galaxies.

 

[PeterDonis]

increasing with respect to an arbitrary observer

This is too vague. With respect to what property of the observer?

 

[timmdeeg]

This is too vague. With respect to what property of the observer?

With respect to a co-moving observer.

 

[PeterDonis]

With respect to a co-moving observer.

With respect to what property of a co-moving observer? You are talking about a rate of change--"increasing". A rate of change relative to what? To put it another way, you are taking a derivative of something; with respect to what are you taking the derivative?

 

[timmdeeg]

With respect to what property of a co-moving observer? You are talking about a rate of change--"increasing". A rate of change relative to what? To put it another way, you are taking a derivative of something; with respect to what are you taking the derivative?

Hm, a rate of change relative to the proper time of the observer?

 

[PeterDonis]

a rate of change relative to the proper time of the observer?

Yes. But this will only be an invariant if it's the rate of change of an invariant. So the rate of change of the observed redshift with respect to proper time will be an invariant; but the rate of change of "distance" with respect to proper time won't be, because "distance" is not an invariant, it's coordinate-dependent.

 

[timmdeeg]

Thanks, that has been a helpful course!

 

[timmdeeg]

... but the rate of change of "distance" with respect to proper time won't be, because "distance" is not an invariant, it's coordinate-dependent.

If one talkes about the expansion of the universe without further details wouldn't then the sign of the rate of change of "distance" with respect to proper time be invariant?

 

[PeterDonis]

If one talkes about the expansion of the universe without further details wouldn't then the sign of the rate of change of "distance" with respect to proper time be invariant?

It can't be an invariant if "distance" isn't an invariant.

 

[timmdeeg]

It can't be an invariant if "distance" isn't an invariant.

So the sign of the observed rate of change of the redshift doesn't correspond to a certain sign of the rate of change of distance, that sign could still be positive or negative depending on the chosen coordinates if I understand you correctly.

 

[PeterDonis]

the sign of the observed rate of change of the redshift doesn't correspond to a certain sign of the rate of change of distance

No.

that sign could still be positive or negative depending on the chosen coordinates if I understand you correctly.

Not only due to choice of coordinates. The observed rate of change of the redshift depends on whether the expansion is accelerating or decelerating--or more precisely whether it was doing so when the light we are seeing now from a particular distant galaxy was emitted.

The observed redshift itself can be related to a "rate of change of distance", but the relationship is model dependent as well as coordinate dependent. The best way to interpret the redshift is not as telling you a speed of recession, but as telling you by what factor the universe has expanded since the light you are seeing now was emitted. But that, of course, involves the scale factor, which assumes that you have made a particular choice of coordinates (it isn't even meaningful in other coordinates).

 

[timmdeeg]

No.

Got it.

Not only due to choice of coordinates. The observed rate of change of the redshift depends on whether the expansion is accelerating or decelerating--or more precisely whether it was doing so when the light we are seeing now from a particular distant galaxy was emitted.

Ok, that I've been missing.
Thanks for your great explanation!

Quite often here in PF it comes to a point where someone explains expansion of the universe doesn't mean that space expands, it means increasing distances. This seems quite vague and with no reference to coordinate dependence. But it at least points to right direction. Would you agree to that?

 

[PeterDonis]

Quite often here in PF it comes to a point where someone explains expansion of the universe doesn't mean that space expands, it means increasing distances. This seems quite vague and with no reference to coordinate dependence. But it at least points to right direction. Would you agree to that?

Sort of. When you fill in the missing information about what "increasing distances" means, the coordinate dependence appears. So whether it "points in the right direction" depends on what your goal is.