Addition of Cosmic Velocities: Expansion Equation & Horizons

[gerald V]

TL;DR Summary

Is the velocity resulting from the cosmic expansion simply to be added to relative velocities?

The expansion equation of the universe be $\dot{R} = f(R)$. There shall be a photon at distance $x$ from us moving directly towards us along the line of sight. Are the velocities to be simply added, so the photon has speed $f(x) -1$ relative to us (local velocity of light set unity)? This appears as plausible to me, since then the horizon would be where $f(R) = 1$, that is the photon stands still relative to us.

Now there shall be a reference object at a distance $x$ perfectly comoving with the cosmic dynamics, so it has velocity $f(x)$ relative to us (the gravitational influence of that object shall be negligible). If another object is at the same location as the reference object, but has velocity $v$ in the rest frame of the reference object in the lign of sight away from or towards us, what is the velocity of those object relative to us? Is it simply $f(x) \pm v$, the relative sign dependent on the direction of the relative motion?

As always, please forgive me if my questions are dumb or I made errors. Thank you in advance for any answer.

 

[gerald V]

Meanwhile I considered the model of a 1-dimensional circular universe (parametrized by the angle $\varphi$) resulting from embeedding in 3 flat dimensions. The line element in 3 dimensions be $\mbox{d}s^2 - \mbox{d}R^2 - R^2\mbox{d}\varphi^2$. A geodesic in 3-space $\mbox{d}s^2 = 0$ implies $R^2\dot{\varphi}^2 = 1 -\dot{R}^2$. So where $\dot{R} =1$, there is no velocity inside the universe left.
Though this appears as pretty clear, I have some problems handling the full picture (addition of arbitray velocities at arbitrary locations), in particular without the help of embedding.

 

[Orodruin]

The concept of "velocity" due to expansion is not a globally well defined concept. It is not part of an actual physical velocity, but rather an artifact of an imposed coordinate system and resulting foliation of the spacetime.

 

[gerald V]

Thank you. But I still see a problem.
The velocity of a galaxy relative to us appears to me as a quite well defined concept. The redshift tells. Combined with an assumedly precise measurement of the distance (say, by standard candles), we know enough. So, for a pair of galaxies with estimated distance and velocity relative to us, there must exist a formula to compute the relative speed and distance between those two galaxies. Eventually, these are measurable quantities for observers sitting in those galaxies.

 

[Orodruin]

The velocity of a galaxy relative to us appears to me as a quite well defined concept. The redshift tells.

It does not. Please see:
https://www.physicsforums.com/insights/coordinate-dependent-statements-expanding-universe/

 

[vanhees71]

Another nice explanation can be found here:

http://www.edu-observatory.org/physics-faq/Relativity/GR/hubble.html

 

[PAllen]

Thank you. But I still see a problem.
The velocity of a galaxy relative to us appears to me as a quite well defined concept. The redshift tells. Combined with an assumedly precise measurement of the distance (say, by standard candles), we know enough. So, for a pair of galaxies with estimated distance and velocity relative to us, there must exist a formula to compute the relative speed and distance between those two galaxies. Eventually, these are measurable quantities for observers sitting in those galaxies.

Even in special relativity distance is coordinate dependent, even between different inertial coordinates. It literally has no meaning outside of a coordinate choice, even in special relativity. This remains true in general relativity. However, in special relativity, there is an unambiguous relative velocity between distant bodies. In GR (with spacetime curvature) there is, in principle, no unambiguous notion of relative velocity. In fact, this inherent ambiguity can be used to define curvature.

Thus, redshift is an unambiguous, invariant observable. However, distance and relative velocity of a distant object are entirely coordinate dependent. Note that peculiar velocity is actually an invariant notion, in that it is the local relative velocity between a body and a colocated comoving observer, the latter being defined by symmetries of the spacetime

 

[vanhees71]

There are of course also physical, coordinate independent measures for distance in GR (like luminosity distance).

https://en.wikipedia.org/wiki/Luminosity_distance

and various other distance measures mentioned in the Wikipedia article. One must only be careful to clearly keep in mind which distance measure one is using.

Of course, anything physical is independent of the choice of the coordinates (as anything physical related with electromagnetism is gauge invariant).

 

[Orodruin]

One must only be careful to clearly keep in mind which distance measure one is using.

And even more importantly, what "distance" means in those cases.

 

[PAllen]

There are of course also physical, coordinate independent measures for distance in GR (like luminosity distance).

https://en.wikipedia.org/wiki/Luminosity_distance

and various other distance measures mentioned in the Wikipedia article. One must only be careful to clearly keep in mind which distance measure one is using.

Of course, anything physical is independent of the choice of the coordinates (as anything physical related with electromagnetism is gauge invariant).

I wouldn't call those coordinate independent. Each is a coordinate choice. The very fact that there are several, means none are invariant as I would use the term. Each has a physical definition, as do the distances in each inertial reference frame in special relativity. These latter can be defined in terms of pairs of detectors at mutual rest, such that the distance per any such pair can be computed in any coordinates, but that does not normally lead to distance being considered coordinate independent.

 

[PeterDonis]

I wouldn't call those coordinate independent. Each is a coordinate choice.

I don't think luminosity distance or angular size distance are coordinate dependent; they are just misnamed, since neither one is actually a distance. One is the apparent luminosity of the object, converted to distance units (with an assumption about the absolute luminosity of the object), and the other is the apparent angular size of the object, converted to distance units (with an assumption about the absolute size of the object).

I also don't think either one is a coordinate choice; I'm not aware of any coordinates that use either of these as coordinate distance. I'm not sure how one would construct such coordinates.

 

[PAllen]

I don't think luminosity distance or angular size distance are coordinate dependent; they are just misnamed, since neither one is actually a distance. One is the apparent luminosity of the object, converted to distance units (with an assumption about the absolute luminosity of the object), and the other is the apparent angular size of the object, converted to distance units (with an assumption about the absolute size of the object).

I also don't think either one is a coordinate choice; I'm not aware of any coordinates that use either of these as coordinate distance. I'm not sure how one would construct such coordinates.

Keeping the standard cosmological foliation, one could simply scale the position coordinates such that difference on a slice gives either of these.

An additional note is that either of these is also based on information from the past, i.e. billions of years ago for distant objects. Extrapolating to a current distance (however defined) is then very model dependent.

 

[PAllen]

Back to the original question, despite the caveats, one could simply answer the original question in terms of some definition of distance divided by 'our' time. Taken literally, this would be somewhat tricky, because Earth has non-negligible peculiar velocity. Easier is to assume a comoving observer at Earth's current position (whose proper time then matches cosmological time coordinate). Then, if the distance chosen is the non-observable comoving distance, the answer is simple because growth of comoving distance with cosmological time shares the additivity property of rapidity - so the the 'velocities' under this definition just add.

 

[pervect]

I favor the following explanation from Baez, "The Meaning of Einstein's Equation", [link1], [link2]

In general relativity, we cannot even talk about relative velocities, except for two particles at the same point of spacetime -- that is, at the same place at the same instant. The reason is that in general relativity, we take very seriously the notion that a vector is a little arrow sitting at a particular point in spacetime. To compare vectors at different points of spacetime, we must carry one over to the other. The process of carrying a vector along a path without turning or stretching it is called `parallel transport'. When spacetime is curved, the result of parallel transport from one point to another depends on the path taken! In fact, this is the very definition of what it means for spacetime to be curved. Thus it is ambiguous to ask whether two particles have the same velocity vector unless they are at the same point of spacetime.