Comoving Observers Concepts cosmology

[binbagsss]

I have read 'comoving observers are a special set of freely-falling observers' . I have the following definitions:

Comoving Frame: "defined at a time t is the inertial frame in which the accelerated observer is instantaneously at rest at t=t₀. (Thus the term 'comoving frame' actually refers to a different frame for each t)". has
$dx^i =0$.

I'm unsure which 'special set' they are - I believe a freely-falling observe is one that follows the geodesics establised by the space-time curvature of any bodies whose path it may across. In addition to following these geodesics, I believe it will have motion due to the expansion of space-time.

And I believe a comoving observer moves with the expansion of the universe, and has $x^{i}$ a constant. So that any relative motion between 2 comoving observers is solely due to the expansion of space-time itself. I'm really struggling to tie this with the definition of a freely-falling observer , if we have a different frame for each t, then in each frame the observer would only have motion due to the expansion of space and would not follow a geodesic - but once you piece all frames together they would follow the geodesics?Also just to clarify some definitions, I have peculiar velocity - the velocity of an object as measured by a comoving observer. Am I correct in thinking apparent velocity = peculiar velocity + velocity due to expansion of space-time.Thanks very much !

 

[WannabeNewton]

Comoving Frame: "defined at a time t is the inertial frame in which the accelerated observer is instantaneously at rest at t=t₀. (Thus the term 'comoving frame' actually refers to a different frame for each t)". has
$dx^i =0$.

This is not what comoving refers to in the cosmological context. What you're talking about is an instantaneously comoving local inertial frame. This local frame can be momentarily attached to a single observer in the family of comoving observers and of course that observer will be at rest in it but the neighboring comoving observers will not be at rest in this frame, they will be receding. In the cosmological context we have a comoving coordinate system, not a comoving frame-there do not exist global comoving frames in curved space-times. A comoving coordinate system (which need not be rigid, such as in this case) is one in which the entire family of comoving observers is at rest at each instant $t$ i.e. they are at fixed coordinate positions on the global simultaneity surfaces of the family.

I'm unsure which 'special set' they are - I believe a freely-falling observe is one that follows the geodesics establised by the space-time curvature of any bodies whose path it may across. In addition to following these geodesics, I believe it will have motion due to the expansion of space-time.

They are special because they admit a foliation of space-time into a one parameter family of spacelike hypersurfaces orthogonal to their worldlines. This means each instant $t$ of coordinate time corresponds (up to the scale factor) to an instant of proper time on the clock carried by each of these observers. That is, they defined a preferred slicing of space-time, on top of being a geodesic congruence.

I'm really struggling to tie this with the definition of a freely-falling observer , if we have a different frame for each t, then in each frame the observer would only have motion due to the expansion of space and would not follow a geodesic - but once you piece all frames together they would follow the geodesics?

I honestly cannot make any sense of what you're saying here but none of the above has anything to do with the definition of a freely falling observer. Rather it just has to do with comoving coordinate systems. A congruence of freely falling observers is still just a 4-velocity field $u^{\alpha}(x^{\beta})$ such that $u^{\alpha}\nabla_{\alpha}u^{\beta} = 0$.

 

[Nugatory]

Comoving Frame: "defined at a time t is the local inertial frame i

Either add the word "local" as I did above, or you must limit this definition only to flat spacetime in special relativity.

 

[PeterDonis]

I believe a comoving observer moves with the expansion of the universe, and has $x^{i}$ a constant. So that any relative motion between 2 comoving observers is solely due to the expansion of space-time itself.

This is one way of looking at it, yes. Another way of picking out comoving observers is to note that they, and only they, always see the universe as homogeneous and isotropic. For example, a comoving observer, and only a comoving observer, would see the CMBR as isotropic (same intensity in all directions).

I'm really struggling to tie this with the definition of a freely-falling observer , if we have a different frame for each t, then in each frame the observer would only have motion due to the expansion of space and would not follow a geodesic

Yes, the observer does follow a geodesic; in an expanding universe, the observers whose only motion is due to the expansion of space are following geodesics; they are in free fall, and don't have to fire rockets or anything else to maintain their trajectory. The fact that this family of observers are all freely falling and yet move away from each other is one way of stating what "the universe is expanding" means, physically.

 

[binbagsss]

Yes, the observer does follow a geodesic; in an expanding universe, the observers whose only motion is due to the expansion of space are following geodesics; they are in free fall, and don't have to fire rockets or anything else to maintain their trajectory. The fact that this family of observers are all freely falling and yet move away from each other is one way of stating what "the universe is expanding" means, physically.

Okay thanks. But still what is the difference between a general freely-falling observer and a co-moving observer?

 

[binbagsss]

They are special because they admit a foliation of space-time into a one parameter family of spacelike hypersurfaces orthogonal to their worldlines. This means each instant tt of coordinate time corresponds (up to the scale factor) to an instant of proper time on the clock carried by each of these observers. That is, they defined a preferred slicing of space-time, on top of being a geodesic congruence.

Thanks very much !

On the bold part, I see how we get the coordinate time to be the proper time; proper time is defined by $dl^2=-d\tau^2$, where $l$ is the line element, and by definition of comoving observer $ds^2=0$, so from the FRW metric we have $dt^2=d\tau^2$*; but where does up to the scale factor come from? Is this the scale factor of the FRW metric? Integrating * will yield the integration constant which can be set to $0$.

 

[Matterwave]

Okay thanks. But still what is the difference between a general freely-falling observer and a co-moving observer?

A co-moving observer in the FRW space time is a freely-falling observer who sees the universe as isotropic and homogeneous. A general freely-falling observer may have some velocity in some direction and no longer see an isotropic universe.

 

[binbagsss]

A co-moving observer in the FRW space time is a freely-falling observer who sees the universe as isotropic and homogeneous. A general freely-falling observer may have some velocity in some direction and no longer see an isotropic universe.

I see thank you. And so would a free-falling observer still view the universe as homogenous?

 

[WannabeNewton]

Is this the scale factor of the FRW metric?

Sorry I was talking about comoving irrotational observer congruences in general space-times. In FRW you're entirely correct that proper time of a comoving observer and coordinate time in the comoving coordinates are exactly the same; this is actually a special class of comoving coordinates called synchronous coordinates.

 

[Matterwave]

I see thank you. And so would a free-falling observer still view the universe as homogenous?

Some of them, the co-moving ones, will, and others will see for example one side of the CMB being redshifted while the other side is blueshifted and therefore no longer isotropic (same in every direction). Isotropy or anisotropy is what an observer can actually see. Homogeneity is not something that we can even "see" as an observer, unless we are able to visit all the other points in space and take a look there. Homogeneity is assumed and is one result of the Copernican principle (that we are not in a special location in the universe), it is not something that an observer "views" (unlike isotropy). That is how I understand the problem anyways, I may be wrong in this, and I invite others to correct me if I am wrong.

EDIT: It was not clear, but what I was trying to get at is such an observer who observes an anisotropy (e.g. us, to a small degree) might conclude that he is merely moving w.r.t. the "background matter" and might still be able to use the Copernican principle to conclude homogeneity based on the idea that once they gauge out their own motion, the universe could be spatially homogenous.