B Cosmological Red Shift in a Perfectly Reflecting Box

[PeterDonis]

proper distance between world lines involves specifying a pairing of events. The Fermi-normal convention is one. Matching cosmological time is another.

Agreed.

the following demonstrates that Fermi coordinates are global in many FLRW spacetimes

I'll take a look; that seems obviously wrong to me since my understanding is that Fermi normal coordinates should only work in a finite-sized "world tube" in any curved spacetime, but there might be something I'm missing.

 

[PeterDonis]

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

After reading through the paper, I don't think the claimed demonstration is correct.

First, a key preliminary observation: the surfaces of constant FRW coordinate time are everywhere orthogonal to the worldlines of comoving observers. That means that, if we take the set of spacelike geodesics orthogonal to a comoving worldline at some event, and extend them indefinitely, they will span a surface of constant FRW coordinate time. It also means that the "comoving proper distance", i.e., the geodesic distance within a surface of constant FRW coordinate time between two comoving worldlines, is also the spacetime geodesic distance--i.e., it is what we would get if we did the "naive" construction of picking the spacelike geodesic orthogonal to the first worldline and pointing in the direction of the second worldline, and extending it to the second worldline, and measuring arc length along it.

What does this mean in terms of the construction given in the paper? Unfortunately, it appears to me to mean that the construction goes wrong right at the very start. Equation (10) in the paper says that $\dot{t} \neq 0$ for a spacelike geodesic orthogonal to a comoving worldline. But that is saying that such a geodesic does not lie in a spacelike surface of constant FRW coordinate time. That seems obviously wrong. The 4-velocity of a comoving worldline is $(1, 0)$ everywhere. Since the metric is diagonal, any vector orthogonal to $(1, 0)$ cannot have any $t$ component, i.e., it must have $\dot{t} = 0$.

Unless I'm missing something, this invalidates the claim made in the paper.

 

[PAllen]

After reading through the paper, I don't think the claimed demonstration is correct.

I believe you are missing several things. Before substantive comments, I will make an argument by authority - by definition, not substantive. The referenced paper has been cited two dozen times by a wide range of authors, with no suggestion there is a problem with the paper. Both the paper itself, and many of the papers that cite it are published in peer reviewed journals.

First, a key preliminary observation: the surfaces of constant FRW coordinate time are everywhere orthogonal to the worldlines of comoving observers. That means that, if we take the set of spacelike geodesics orthogonal to a comoving worldline at some event, and extend them indefinitely, they will span a surface of constant FRW coordinate time.

The first statement is true, the second is false. The constant cosmological time surfaces are orthogonal to each comoving world line but geodesics in these surfaces are not spacetime geodesics. Pretty much the rest of your arguments fall down because of this mistake. The simplest demonstration is using the Milne case of linear expansion (but the same is true of all FLRW solutions, and this fact is rather well known). The hyperbolic surfaces are orthogonal to every comoving word line, but they obviously do not consist of spacetime geodesics. Instead, the spacetime geodesics orthogonal to a given comoving world line are constant time lines in the Minkowski coordinates based on that world line, that intersect the 'big bang' (light cone bounding the Milne solution). This captures the whole essence of the general demonstration in the paper - the cosmological constant time slices are all unbounded, while the Fermi slices are all bounded, yet the latter are still a global foliation.

Note, it is in some sense obvious that Fermi constant time surfaces cannot be orthogonal to any comoving observer except the defining one. This is because, in Fermi-normal coordinates, every comoving observer other than the origin is moving, therefore not orthogonal to spacelike geodesics orthogonal to the defining world line, Instead, each comoving observer defines a different set of constant Fermi-time surfaces.

It also means that the "comoving proper distance", i.e., the geodesic distance within a surface of constant FRW coordinate time between two comoving worldlines, is also the spacetime geodesic distance--i.e., it is what we would get if we did the "naive" construction of picking the spacelike geodesic orthogonal to the first worldline and pointing in the direction of the second worldline, and extending it to the second worldline, and measuring arc length along it.

Simply false, as noted above.

What does this mean in terms of the construction given in the paper? Unfortunately, it appears to me to mean that the construction goes wrong right at the very start. Equation (10) in the paper says that $\dot{t} \neq 0$ for a spacelike geodesic orthogonal to a comoving worldline. But that is saying that such a geodesic does not lie in a spacelike surface of constant FRW coordinate time. That seems obviously wrong. The 4-velocity of a comoving worldline is $(1, 0)$ everywhere. Since the metric is diagonal, any vector orthogonal to $(1, 0)$ cannot have any $t$ component, i.e., it must have $\dot{t} = 0$.

Again, wrong as noted above.

 

[PeterDonis]

The constant cosmological time surfaces are orthogonal to each comoving world line but geodesics in these surfaces are not spacetime geodesics.

Yes, I see your argument for the Milne case, which is the easiest since the corresponding Fermi normal coordinates (Minkowski coordinates) are obvious. I'll need to work through it for the case of FRW spacetime with nonzero density, but I can do that offline.

I'm still confused about equation (10) in the paper, though. It seems to be saying that even at $\rho = 0$ (i.e., on the chosen comoving worldline), $\dot{t} \neq 0$. But it seems to me that, on the chosen comoving worldline, the Fermi constant time surface and the FRW constant time surface should be parallel. (Your argument shows that they are not parallel off the chosen comoving worldline, but considering the Milne case shows that they are parallel on that worldline--that's the one event where the constant FRW time hyperbolas and the constant Fermi time straight lines are parallel.) So I would expect $\dot{t}$ to be proportional to $\rho$, i.e., to distance from the comoving worldline. But equation (10) in the paper doesn't seem to be saying that.

 

[PAllen]

I'm still confused about equation (10) in the paper, though. It seems to be saying that even at $\rho = 0$ (i.e., on the chosen comoving worldline), $\dot{t} \neq 0$. But it seems to me that, on the chosen comoving worldline, the Fermi constant time surface and the FRW constant time surface should be parallel. (Your argument shows that they are not parallel off the chosen comoving worldline, but considering the Milne case shows that they are parallel on that worldline--that's the one event where the constant FRW time hyperbolas and the constant Fermi time straight lines are parallel.) So I would expect $\dot{t}$ to be proportional to $\rho$, i.e., to distance from the comoving worldline. But equation (10) in the paper doesn't seem to be saying that.

Consider the next sentence after equation 10. That forces t dot to be zero on the defining world line itself.

 

[PeterDonis]

I would expect $\dot{t}$ to be proportional to $\rho$, i.e., to distance from the comoving worldline. But equation (10) in the paper doesn't seem to be saying that.

To work this out explicitly for the Milne case: the hyperbola of constant FRW coordinate time is, in Minkowski coordinates (which are also the Fermi normal coordinates),

$$
t^2 - x^2 = t_0^2
$$

where $t_0$ is the corresponding Fermi normal coordinate time on the chosen comoving worldline, which is at $x = 0$. Parameterizing this hyperbola by arc length $s$ in the standard way (the analogue of how the timelike hyperbolas are parameterized in Rindler coordinates) gives

$$
t = t_0 \cosh \frac{s}{t_0}
$$
$$
x = t_0 \sinh \frac{s}{t_0}
$$

which gives

$$
\frac{dt}{ds} = \frac{x}{t_0}
$$
$$
\frac{dx}{ds} = \frac{t}{t_0}
$$

This gives $dt / ds = 0$ on the comoving worldline, as expected. (It also gives $dx / ds = 1$ on that worldline, as expected, since on the worldline $t = t_0$.) But I don't see how it corresponds to what is being done in the paper.

 

[PeterDonis]

Consider the next sentence after equation 10. That forces t dot to be zero on the world line itself.

Only at one value of $t$, though, correct? (Since $a(t)$ only equals $a_0$ at one value of $t$.) If so, that doesn't seem right.

 

[PAllen]

Only at one value of $t$, though, correct? (Since $a(t)$ only equals $a_0$ at one value of $t$.) If so, that doesn't seem right.

But this calculation is for an orthogonal geodesic at one event on the defining world line. You re-do it for each proper time value along the defining geodesic. In each case, t dot is zero for given geodesic orthogonal to a given proper time on the defining comoving world line.

 

[PeterDonis]

You re-do it for each proper time value along the defining geodesic.

Hm, ok. I'll look at the paper more carefully tomorrow.

 

[timmdeeg]

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.

Ok, thanks. That also clarifies that the second observer moves inward as @PAllen pointed out in post #111.

But why is constant Fermi coordinate distance represented by constant proper distance? Is the definition of proper distance not restricted to the distance measured on a surface of constant time? Whereas the Fermi simultaneity surface is different from that (@PAllen #111).

I'm also not clear with following question: If one end of a long rod is comoving then the other end has a peculiar inward velocity. From the above it seems that the coordinates of the ends of this rod are described by constant Fermi normal coordinates. Is that true?

 

[PAllen]

But why is constant Fermi coordinate distance represented by constant proper distance? Is the definition of proper distance not restricted to the distance measured on a surface of constant time? Whereas the Fermi simultaneity surface is different from that (@PAllen #111).

Actually, this gets to a pet peeve of mine. Cosmologists have added a specialized lingo on top GR that IMO is often not helpful. Outside of cosmology, the proper distance is defined between pairs of events connected by a spacelike geodesic of the manifold (and is the invariant length of that geodesic). As I discussed with @PeterDonis , this means that to discuss proper distance between world lines (of different bodies), you must specify a pairing of events on them, interpreted as a simultaneity convention. Even in SR, this means that the proper distance between bodies is frame dependent because of the simultaneity difference - this is known as length contraction. All well and good, but cosmologists have instead commonly used proper distance to mean a distance computed along a geodesic of a surface (of constant cosmological time) that is not a geodesic of the overall spacetime, and differs from the normal GR notion of proper distance, even between that pair of events. The cosmological usage of proper distance just feels 'wrong' to anyone used to using GR for any other application.

As to why use different notions of simultaneity - well that is the essence of relativity. There is no inherent notion of simultaneity in either SR or GR. You pick a convention consistent with your purpose. Fermi simultaneity and Fermi proper distance are believed to most accurately model a bound object without getting into the details of a specific material theory and model of local gravity. It is the simplest reasonable abstraction from the full complexity. You ask about why use something other than a surface of constant time for measuring distance? Well, again, the essence of SR and GR is there is no such thing as an objective fact - a surface of constant time has no meaning beyond implementing a simultaneity convention. And the choice of a convention is based on what makes it easiest for the thing your are trying to model - a bound system or the whole universe.

[edit: in SR, there is a related notion of proper length for an object whose parts are all at mutural rest, and is the distance computed in the rest frame of the object. This is considered to be frame invariant, and can be defined without frames by saying you use a geodesic 4-orthogonal to any of the object world lines. This is unambiguous and invariant only in the case of all parts mutually at rest; else you would get different values depending on which world line you picked or which event you pick on a paritcular world line.]

I'm also not clear with following question: If one end of a long rod is comoving then the other end has a peculiar inward velocity. From the above it seems that the coordinates of the ends of this rod are described by constant Fermi normal coordinates. Is that true?

Yes.

 

[timmdeeg]

Thank you very much, this post is very enlightening. I will have to think about it.

You ask about why use something other than a surface of constant time for measuring distance? Well, again, the essence of SR and GR is there is no such thing as an objective fact - a surface of constant time has no meaning beyond implementing a simultaneity convention. And the choice of a convention is based on what makes it easiest for the thing your are trying to model - a bound system or the whole universe.

Which includes the box in an expanding universe.

 

[vanhees71]

Well, nevertheless there are measurable facts, and in GR those are expressible as local scalar quantities and of course depends on the reference frame within which the quantity is measured, e.g., the temperature of a medium in (local) thermal equilibrium is measured by definition by a thermometer comoving with the medium, i.e., in the local rest frame of the fluid cell under consideration.

The frequency of an electromagnetic wave of course also depends on the reference frame where it is measured, because it's defined by $$\omega=u^{\mu} k^{\nu} g_{\mu \nu} $$, where $u^{\mu}$ is the four-velocity of the spectrometer used to measure this frequency.

In cosmology if you consider the cosmological redshift of the radiation from a distant galaxy, usually (at least in the standard textbooks and papers on the subject) it's defined as comparing the frequency of the corresponding em. wave as measured by a (fictitious) comoving ("fundamental") observer at the place of emission with the frequency measured by a comoving observer at the place of detection. This gives the standard formula for the red shift parameter $z$, i.e., $1+z=\omega_{\text{em}}/\omega_{\text{obs}}=a(t_{\text{obs}})/a(t_{\text{em}})$. The world lines of such "fundamental observers" in standard FLRM coordinates is $(t,\chi_0,\vartheta_0,\varphi_0)$ with $(\chi_0,\vartheta_0,\varphi_0)=\text{const}$.

Of course the different pieces of the box are not comoving, because the box is held together by the usual binding forces of solids, and thus the box does not change its dimensions with the Hubble expansion and the frequencies of the em. cavity modes thus do not undergo a redshift but are determined by the boundary conditions of the cavity.