# Understanding Etherington's reciprocity theorem

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## Main Question or Discussion Point

Hi,

Etherington't reciprocity theorem states that distances measured by angular separation and by luminosity differ. My question is which one (if any of them) is the actual distance. I can understand they might differ in an expanding universe, but there's still a physical distance in such one, so... yeah, which one is it?

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George Jones
Staff Emeritus
Gold Member
Hi,

Etherington't reciprocity theorem states that distances measured by angular separation and by luminosity differ. My question is which one (if any of them) is the actual distance. I can understand they might differ in an expanding universe, but there's still a physical distance in such one, so... yeah, which one is it?
No, in the curved expanding Friedmann-Lemaitre-Robertson-Walker spacetimes, there are many different definitions of distances, and no single one of these definitions is THE definition of distance. See "Distance Measures in Cosmlogy" by Hogg,

http://arxiv.org/abs/astro-ph/9905116

See Baumann's notes,

http://www.damtp.cam.ac.uk/user/db275/Cosmology/Lectures.pdf

for nice treatments of the distance measures mentioned in your post.

• voila
Thanks for the notes, George, they are helpful. However, if physical distance is not uniquely defined, when we talk about "physical volume" in other context (see text below equation 3.3.80 of Baumann notes, your second link), what are we talking about?

Edit: the line element of spacial coordinates in FRW metric is the one in equation 1.1.13 (Baumann notes again). So this is what we understand by a physical distance, right? How does this relate to the luminosity distance and angular diameter distance?

Last edited:
George Jones
Staff Emeritus
Gold Member
I am going to use the coordinates which give the line element 1.1.14, which is the same as 1.2.65, the line element at the start of section 1.2.3 Distances.

The volume below 3.3.80 is proper volume (also see below 1.3.78), which is based on yet another definition of distance, proper distance (but I am unwilling to call this the "physical distance", because there is no practical way measuring it).

If cosmological time is fixed (held constant) at some time $t_0$ ("now") ($dt = 0$), then a 3-dimesional spatial hypersurface results. Then, an infinitesimal spatial "cube" formed with sides that have infinitesimal proper lengths $a\left(t_0 \right) d\chi$, $a\left(t_0 \right) S_k d\theta$, and $a\left(t_0 \right) S_k \sin \theta d\phi$ has infinitesimal volume

$$dV = a\left(t_0 \right)^3 S_k^2 \sin \theta d\chi d\theta d\phi.$$

As is standard practice, Baumann has normalized the scale factor so $\left(t_0 \right) = 1$.

Suppose light we receive now at $t_0$ was emitted at $t_1 < t_0$, as in Figure 1.7. Then,

$$D = a\left(t_1 \right) S_k \delta \theta = d_A \delta \theta$$

is proper distance in the spatial hypersurface that results when time is fixed at the emission time $t_1$. Compare to the proper length of second side of the above "cube", with $t_0$ replaced by $t_1$.

• voila
PAllen
2019 Award
I hope you are aware that even in special relativity, distance is observer dependent, and no observer is considered preferred. In GR, distances are coordinate dependent, even when computed using the metric. The reason is related to that is SR: different slicings (foliations) of spacetime into space by time produce different distances. To be concrete, one is normally talking about distances between things with history - between world lines. Different coordinates will pair different events on the world lines as belonging to the same spatial slice; further the induced spatial metric on a slice depends on the foliation.

Upshot: asking for true distance is asking for the 'one true coordinate system', which is totally at odds with coordinate invariance of physics in GR.

• voila
I may not have been clear. What I intuitively look for when asking for "physical distance" is the proper length between the two events at a fixed time. I know this is not measurable, but is what I can conceive as distance for the spatial hypersurface at a fixed time.

So George discusses the angular diameter distance, which turns out to be the proper distance in the spatial hypersurface when time is fixed at the emission time t1. In this sense, what would be the luminosity distance be?

Also, thanks for the warning, PAllen, but I am aware of that. I guess I didn't ask the question properly. Hope it's more clear now.

PAllen
2019 Award
I may not have been clear. What I intuitively look for when asking for "physical distance" is the proper length between the two events at a fixed time. I know this is not measurable, but is what I can conceive as distance for the spatial hypersurface at a fixed time.

So George discusses the angular diameter distance, which turns out to be the proper distance in the spatial hypersurface when time is fixed at the emission time t1. In this sense, what would be the luminosity distance be?

Also, thanks for the warning, PAllen, but I am aware of that. I guess I didn't ask the question properly. Hope it's more clear now.
Not only isn't in measurable but it has no fixed mathematical definition. There is no spatial hypersurface at a fixed time except as a feature of a particular coordinate system - which is purely a convention. Of course I realize you mean proper distance along a time slice in standard cosmological coordinates, which is fine to talk about. Just try not to think of it as 'physical distance' let alone 'true distance'. Note, for example, if you mathematically idealized the process of extending rulers out from a given comoving observer (defined by Fermi-Normal coordinates based on that world line), you would get a completely different proper distance measured along a slice of constant time. To be even more specific, imagine a universe (not ours) with very low density. Then the slices of constant time in standard cosmological coordinates (those the FLRW metric is expressed in) are strongly hyperbolic in geometry. Meanwhile, the slices of constant time in Fermi-Normal coordinates would be almost Euclidean. Thus completely different spatial geometry between these different notions of constant time slice.

Yes, I know proper distance in the spatial hypersurface depends on the coordinate system. I wasn't discussing that; I was looking for an understanding of angular distance and luminosity distance as proper distances of spatial hypersurfaces (implicitly, for my coordinate system).

So angular distance is already discussed as proper distance on the hypersurface for the fixed time when light was emitted. What about luminosity distance?

George Jones
Staff Emeritus
Gold Member
There is no spatial hypersurface at a fixed time except as a feature of a particular coordinate system - which is purely a convention.
In the Friedmann-Lemaitre-Robertson-Walker standard cosmological models that have beed discussed in this thread, special spatial hypersurfaces are picked out in a coordinate-invariant manner, as the span of the six spacelike (clearly not all linearly independent) Killing vectors that express the spatial homogeneity and isotropy of the standard cosmological models. These hypersurfaces are orthogonal the timelike conformal Killing vector $a\left(t\right) \partial / \partial t$.

PAllen
In the Friedmann-Lemaitre-Robertson-Walker standard cosmological models that have beed discussed in this thread, special spatial hypersurfaces are picked out in a coordinate-invariant manner, as the span of the six spacelike (clearly not all linearly independent) Killing vectors that express the spatial homogeneity and isotropy of the standard cosmological models. These hypersurfaces are orthogonal the timelike conformal Killing vector $a\left(t\right) \partial / \partial t$.