Cosmological Red Shift in a Perfectly Reflecting Box

[vanhees71]

This is true if you adopt standard FRW coordinates. But what you are calling "the CMBR frame" (by which you appear to mean standard FRW coordinates) is not what people usually think of as a "reference frame" since different comoving observers are not at rest relative to each other.

I indeed mean standard FRW coordinates, and I consider an "observer at rest" wrt. the corresponding reference frame if there world line is given by $(\chi,\vartheta,\varphi)=\text{const}$. Such an observer sees the CMBR as a homogeneous isotropic Planck spectrum, and that's what's usually called the rest frame of a thermal bath of photons.

 

[PeterDonis]

I consider an "observer at rest" wrt. the corresponding reference frame if there world line is given by $(\chi,\vartheta,\varphi)=\text{const}$.

And this usage of "at rest" is different from the usual meaning of that term, since comoving observers "at rest" by this definition are not at rest relative to each other; they are moving apart. Which is why you can't just help yourself to this definition of "at rest" and not expect it to cause confusion. Particularly not in a discussion which is explicitly considering a box whose walls are at rest relative to each other (and which therefore cannot all be comoving).

 

[anuttarasammyak]

And this usage of "at rest" is different from the usual meaning of that term, since comoving observers "at rest" by this definition are not at rest relative to each other; they are moving apart.

They become apart but we may say they keep rest in the sense that each of them observe all the bodies in inertial motion around them keep losing speed, i.e. proper speed or momentum, i.e. proper momentum though they keep zero proper momentum and stay at rest at constant $(\chi,\theta,\phi)$.

As an illustration of difficulty of "moving", let us see the north and south poles of an inflating sphere shell. They become apart on the 2D shell but how can we define their relative motion on the shell?

 

[timmdeeg]

They become apart on the 2D shell but how can we define their relative motion on the shell?

Why not just say their relative motion is given by the increase of their proper distance per proper time unit?

 

[vanhees71]

Sure, one has to clearly say what one means with "at rest". The definition I'm used to from our standard cosmology lecture (and which I considered the standard meaning in cosmlogy) where usually "at rest relative to each other" means the world lines defined by constant spatial standard FLRW coordinates (i.e., $\chi$ or $r$, $\vartheta$, $\varphi$), i.e., the congruence of comoving "dust particles". It's clear that a "box" is bound by (dominantly em.) forces and thus the different parts of the box are not all co-moving.

 

[anuttarasammyak]

Why not just say their relative motion is given by the increase of their proper distance per proper time unit?

A distance, say $l_0$ increases to $l$ with time t is mentioned
$$l=\frac{a}{a_0} l_0$$
$$\frac{l-l_0}{t-t_0}=\frac{\frac{a}{a_0}-1}{t-t_0}l_0=[\dot{a}(t_0) + \frac{1}{2}\ddot{a}(t_0)(t-t_0)+...]\frac{l_0}{a_0}=V$$
I named it V and it has dimension of $LT^{-1}$ but I hesitate to call it "velocity" or "motion".

As for red shift well described in #77,
$$\omega a = \omega_0 a_0$$
$$\frac{\omega_0-\omega}{\omega_0}=1-\frac{a_0}{a}$$
In interpretation that same effect comes from Doppler effect in IFR the recessional velocity v is
$$\frac{\omega_0-\omega}{\omega_0}=\frac{v}{c}$$
Equating these two equatins
$$\frac{v}{c}=1-\frac{a_0}{a}=1-\frac{a_0}{a_0+\dot{a}(t_0) (t-t_0)+ \frac{1}{2}\ddot{a}(t_0)(t-t_0)^2+...}$$

I think v is not real velocity which takes place but a parameter in interpretation "as if" it is due to Doppler effect in IFR. As an example, in Doppler effect in IFR v is velocity of emitter wrt receiver at the time of emission $t=t_0$, however, I cannot read in the above equation for what time v is defined.

 

[timmdeeg]

I think v is not real velocity which takes place but a parameter in interpretation "as if" it is due to Doppler effect in IFR. As an example, Doppler effect in IFR, v is velocity of emitter at the time of emission $t=t_0$, however, I cannot read when v is defined in the above equation.

One can argue with the special relativistic Doppler formula, as Bunn & Hogg do in this paper, equation (6):

https://arxiv.org/pdf/0808.1081.pdf
Equation (6) can be derived by a straightforward calculation using the rules for parallel transport, but the derivation is easier if we recast the statement in more physical terms ...

 

[anuttarasammyak]

In the paper, the accumulation of many infinitesimal Doppler shifts, is a good alternative idea to understand the red shifts. Thank you.

 

[PeterDonis]

They become apart but we may say they keep rest in the sense that each of them observe all the bodies in inertial motion around them keep losing speed, i.e. proper speed or momentum, i.e. proper momentum though they keep zero proper momentum and stay at rest at constant $(\chi,\theta,\phi)$.

I have no idea what you mean by this.

As an illustration of difficulty of "moving", let us see the north and south poles of an inflating sphere shell. They become apart on the 2D shell but how can we define their relative motion on the shell?

I have no idea what point you are trying to make here.

 

[PeterDonis]

The definition I'm used to from our standard cosmology lecture (and which I considered the standard meaning in cosmlogy) where usually "at rest relative to each other" means the world lines defined by constant spatial standard FLRW coordinates

No, that is not what at rest relative to each other (my emphasis) means, even in cosmology. That's why I used that specific term, with the specific qualifier I just bolded; because that specific term means, specifically, at rest in the ordinary lay person's sense--the proper distance between the objects stays constant. If we wanted an explicit physical test for this, we would have the two objects exchange repeated round-trip light signals; they are at rest relative to each other if and only if the round-trip light travel time stays constant.

The obvious conflict between "at rest relative to each other" (which, as I just noted, matches the ordinary lay person's understanding of what "at rest" means) and "at rest in FRW coordinates" is why I do not think using the term "at rest" to mean "at rest in FRW coordinates" is a good idea. Particularly, as I said, in this thread, where the discussion is explicitly focused on a case where not all objects are comoving. The walls of the box are at rest relative to each other, even though they are not at rest in FRW coordinates.

 

[PeterDonis]

One can argue with the special relativistic Doppler formula, as Bunn & Hogg do in this paper, equation (6)

Equation (6) in the paper is the SR Doppler formula; the paper says so explicitly. Why do you think the paper is "arguing with" the SR Doppler formula?

 

[PAllen]

Equation (6) in the paper is the SR Doppler formula; the paper says so explicitly. Why do you think the paper is "arguing with" the SR Doppler formula?

I definitely think @timmdeeg meant "with" in the sense of "using".

 

[timmdeeg]

I definitely think @timmdeeg meant "with" in the sense of "using".

Yes, indeed, thanks for assisting!

 

[timmdeeg]

That's why I used that specific term, with the specific qualifier I just bolded; because that specific term means, specifically, at rest in the ordinary lay person's sense--the proper distance between the objects stays constant.

But I think a lay persons who understands the meaning of "the proper distance between the objects stays constant" refers this to flat Minkowski spacetime and not to the expanding universe. Regarding the latter he refers "at rest" relative to the CMB or relative to the isotropic universe. A lay person who doesn't have this very basic knowledge doesn't understand the meaning of "the proper distance between the objects stays constant" and doesn't even think about the meaning of "at rest".

 

[jbriggs444]

But I think a lay persons who understands the meaning of "the proper distance between the objects stays constant" refers this to flat Minkowski spacetime and not to the expanding universe.

I consider myself a layperson. I would most certainly understand the phrase "the proper distance between the objects stays constant" as an attempt to keep coordinates and even the geometry of an expanding universe out of it.

Though yes, one can see that outside of flat Minkowski spacetime, proper distance might become a fuzzier concept as the separation between two objects increases. Two sides of a small box maintained at a fixed proper separation in a large expanding universe would seem to qualify as adequately local so that the concept is not fuzzy enough to worry about.

 

[PAllen]

But I think a lay persons who understands the meaning of "the proper distance between the objects stays constant" refers this to flat Minkowski spacetime and not to the expanding universe. Regarding the latter he refers "at rest" relative to the CMB or relative to the isotropic universe. A lay person who doesn't have this very basic knowledge doesn't understand the meaning of "the proper distance between the objects stays constant" and doesn't even think about the meaning of "at rest".

Actually there are accepted technical terms for both senses of distance, and therefore 'rest', in cosmology. One refers to the comoving distance versus the Fermi distance. Either one is a proper distance in the sense of being a geodesic distance. One is computed using hypersurfaces of constant cosmological time, the other using a Fermi-normal slice based on a particular comoving observer. Unfortunately, in cosmology, it is common to allow unqualified proper distance to be the comoving distance.

 

[jbriggs444]

Actually there are accepted technical terms for both senses of distance, and therefore 'rest', in cosmology. One refers to the comoving distance versus the Fermi distance. Either one is a proper distance in the sense of being a geodesic distance. One is computed using hypersurfaces of constant cosmological time, the other using a Fermi-normal slice based on a particular comoving observer. Unfortunately, in cosmology, it is common to allow unqualified proper distance to be the comoving distance.

Reading comprehension check... So from a layman's perspective, if we have a box that is slewing in free fall through the universe at a high rate of speed as measured against co-moving coordinates, we have a choice of talking about a Fermi-normal "proper distance" between the sides of the box that bugs on those box sides might measure and a co-moving "proper distance" between the sides of the box that bugs floating at rest in co-moving coordinates might measure?

Yes, my intuitive notion of proper distance (and hence "at rest relative to") is the one measured by bugs crawling on the walls.

 

[timmdeeg]

Actually there are accepted technical terms for both senses of distance, and therefore 'rest', in cosmology. One refers to the comoving distance versus the Fermi distance.

Supposed we understand "at rest" as a place where we view the universe isotropic, then this definition is coordinate independent, right?

Either one is a proper distance in the sense of being a geodesic distance. One is computed using hypersurfaces of constant cosmological time, the other using a Fermi-normal slice based on a particular comoving observer.

Could one say proper distance in Fermi-normal coordinates is the distance between a comoving observer (who sees the universe isotropic) and another observer who is moving away (doesn't see the universe isotropic)?

 

[jbriggs444]

Supposed we understand "at rest" as a place where we view the universe isotropic, then this definition is coordinate independent, right?

Yes, you can define a standard of rest in this manner without referring to any particular pre-existing coordinate system. By doing so you have established a good part of what would go into setting up a co-moving coordinate system.

Could one say proper distance in Fermi-normal coordinates is the distance between a comoving observer (who sees the universe isotropic) and another observer who is moving away (doesn't see the universe isotropic)?

You are trying to measure the distance between two world lines that are not even close to being parallel? When are you making the measurement? That is, between which two events are you trying to evaluate a distance?

As I understand things, "Fermi normal" means that you pick an event on the world line over here. You pick out the event (hopefully unique!) that is simultaneous on the world line over there. And you measure the space-time interval on a (hopefully unique!) geodesic between those two events. The key is that "simultaneous" is picked out according to the tangent inertial frame on the world line over here. [I don't entirely grok the definition of Fermi normal coordinates, but I think that's the 'gist. The bit about Fermi-Walker transport whizzed past a bit].

Meanwhile, the "comoving" definition means that you pick an event on the world line over here. You pick out the event that is simultaneous on the world line over there. And you measure the space-time interval on the geodesic between those two events. The key is that "simultaneous" is picked out according to the co-moving foliation.

Seems reasonable anyway. Hope I got it right.

 

[PAllen]

Reading comprehension check... So from a layman's perspective, if we have a box that is slewing in free fall through the universe at a high rate of speed as measured against co-moving coordinates, we have a choice of talking about a Fermi-normal "proper distance" between the sides of the box that bugs on those box sides might measure and a co-moving "proper distance" between the sides of the box that bugs floating at rest in co-moving coordinates might measure?

Yes, my intuitive notion of proper distance (and hence "at rest relative to") is the one measured by bugs crawling on the walls.

Fermi distance would be the one to use for a bound object. It is the closest GR generalization of distance in Minkowski coordinates, and further, in SR, corresponds to distance along a Born rigid body.

Freely comoving box sides would have growing Fermi distance, as well as growing comoving distance. In commonly considered cosmologies, comoving distance would be larger than Fermi distance, leading to a larger value for proper distance over proper time between comoving galaxies. For example, if expansion is linear, comoving distance over time grows without bound as galaxies are farther and farther away, while Fermi distance over time remains less than c.

 

[PAllen]

Supposed we understand "at rest" as a place where we view the universe isotropic, then this definition is coordinate independent, right?Could one say proper distance in Fermi-normal coordinates is the distance between a comoving observer (who sees the universe isotropic) and another observer who is moving away (doesn't see the universe isotropic)?

Yes, that definition of rest is coordinate independent, but note the incongruity that galaxies thus at rest have rapidly growing comoving distance.

Fermi distance is defined relative to some world line, most commonly an inertial one. It implies a simulaneity surface which is different from constant cosmological time, so Fermi distance between two world lines will pair up different events, as well as compute distance within a differently shaped surface than comoving distance.

A world line of constant Fermi distance from a comoving world line would have inward peculiar velocity as well as slowly shrinking comoving distance (in the most common cosmologies).

 

[Ibix]

Yes, that definition of rest is coordinate independent, but note the incongruity that galaxies thus at rest have rapidly growing comoving distance.

...hence the 'space expands' description.

 

[PeterDonis]

I think a lay persons who understands the meaning of "the proper distance between the objects stays constant" refers this to flat Minkowski spacetime and not to the expanding universe. Regarding the latter he refers "at rest" relative to the CMB or relative to the isotropic universe.

I'm not so sure. "Proper distance remains constant" can be defined in any spacetime, including an expanding FRW universe.

 

[PeterDonis]

Could one say proper distance in Fermi-normal coordinates is the distance between a comoving observer (who sees the universe isotropic) and another observer who is moving away (doesn't see the universe isotropic)?

No. The second observer is not moving away from the first; the second observer is at rest relative to the first. Fermi normal coordinates centered on the first observer's worldline are constructed so that constant proper distance is represented by constant coordinate distance, so the second observer's spatial coordinates in Fermi normal coordinates would be constant.

Also, proper distance is a coordinate-independent concept. Fermi normal coordinates are simply the most convenient ones to use in the case you're discussing.

 

[PeterDonis]

if expansion is linear, comoving distance over time grows without bound as galaxies are farther and farther away, while Fermi distance over time remains less than c.

I don't understand this. In flat Minkowski spacetime, Fermi normal coordinates centered on an inertial worldline are just standard Minkowski coordinates. In those coordinates the distance between two galaxies moving apart increases without bound. It is true that this coordinate distance remains smaller than the comoving distance between the same two galaxies (since the latter is not a geodesic distance but is measured along a hyperbola of constant proper time from the origin), but that doesn't change the fact that both distances increase without bound.

 

[PAllen]

I don't understand this. In flat Minkowski spacetime, Fermi normal coordinates centered on an inertial worldline are just standard Minkowski coordinates. In those coordinates the distance between two galaxies moving apart increases without bound. It is true that this coordinate distance remains smaller than the comoving distance between the same two galaxies (since the latter is not a geodesic distance but is measured along a hyperbola of constant proper time from the origin), but that doesn't change the fact that both distances increase without bound.

“over time” meant to imply divided proper time of the reference observer. I guess this was ambiguous, but the statement of less than c for the Fermi ideally should have been a hint. Rate of change of comoving distance by proper time for a reference comoving observer is the recession rate and grows without bound as you consider galaxies further and further away. For Fermi distance, for linear expansion, you get relative velocity which is less than c.

 

[PAllen]

Also, proper distance is a coordinate-independent concept. Fermi normal coordinates are simply the most convenient ones to use in the case you're discussing.

Well, you have to specify a foliation, as normally used, especially if referring to world lines or bodies. And unfortunately many cosmology articles refer to geodesic distance within a surface of constant cosmological time as proper distance, which, of course, is totally different from Fermi proper distance.

Without a foliation, you would need to specify events rather than world lines, and then foliation would, indeed, be irrelevant (using the geodesic between two spacelike separated events).

 

[PeterDonis]

the statement of less than c for the Fermi ideally should have been a hint

That confused me because I thought you were talking about distance, not speed. For speed, yes, the "relative speed" in terms of comoving distance increases without bound, but in terms of Fermi normal distance it does not.

 

[PeterDonis]

you have to specify a foliation

For Fermi normal coordinates, which are limited to a sufficiently narrow "world tube" around the chosen worldline, the foliation is simply the spacelike segments orthogonal to the worldline at each event. "Sufficiently narrow" for the world tube then means "narrow enough that the segments don't intersect". It just happens that, for the case of a geodesic in flat spacetime, "sufficiently narrow" ends up giving you a foliation that covers the entire spacetime. But that is not the case for comoving geodesics in curved FRW spacetime (i.e., FRW spacetime with nonzero density).

 

[PAllen]

For Fermi normal coordinates, which are limited to a sufficiently narrow "world tube" around the chosen worldline, the foliation is simply the spacelike segments orthogonal to the worldline at each event. "Sufficiently narrow" for the world tube then means "narrow enough that the segments don't intersect". It just happens that, for the case of a geodesic in flat spacetime, "sufficiently narrow" ends up giving you a foliation that covers the entire spacetime. But that is not the case for comoving geodesics in curved FRW spacetime (i.e., FRW spacetime with nonzero density).

Actually, Fermi-normal coordinates cover a very large part of a general FLRW cosmology when built from a comoving observer. Be that as it may, proper distance between world lines involves specifying a pairing of events. The Fermi-normal convention is one. Matching cosmological time is another.

For example, the following demonstrates that Fermi coordinates are global in many FLRW spacetimes:

https://arxiv.org/abs/1010.0588