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In the paper "Asymptotic Symmetries in Gravitational Theory" by R. Sachs from 1962, the author says the following:

In analyzing gravitational fields it is sometimes useful to introduce coordinates which share some properties of the coordinates $u,r,\theta$ and $\phi$. The crucial properties are: (i) the hypersurfaces $u = \text{constant}$ are everywhere tangent to the local lightcone; (ii) $r$ is the corresponding luminosity distance; (iii) the scalars $\theta$ and $\phi$ are constant along each "ray". A ray is defined as the line with tangent $k^a = -g^{ab}\partial_b u$, where $g^{ab}$ is the metric tensor.

He claims that in a neighborhood of any point on the spacetime manifold there always is such a coordinate system satisfying (i)-(iii).

I'm trying to first understand the properties. Properties (i) and (iii) are simple. The first property means that the level sets of the function $u$ are null hypersurfaces, so the coordinates are adapted to such submanifolds. The third property means that $\theta,\phi$ are constant along each generator of the surface.

But what luminosity distance really means? I have googled about it and found something about cosmology, but that's not the case here. Sachs says the construction can be done locally in any spacetime.

My initial guess was: by definition $k^a$ is the null normal to the null surfaces. Hence we know there is a second null vector field $\ell^a$ with $\ell_a k^a = -1$. My guess is that "luminosity distance" is the affine parameter along the geodesics which on the null hypersurface have tangent $\ell^a$.

But I might be totally wrong. So is my guess correct? If not, what rigorously means to say that "$r$ is the corresponding luminosity distance" in a general spacetime?

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  • $\begingroup$ That guess sounds reasonable to me. I assume if there's a light source at $r=0$, then for any $R\ge r$ the luminosity is constant on the $r=R$ surface. $\endgroup$ – PM 2Ring Jul 25 at 17:12
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Mentions of cosmology in references do not at all limit the generality of the definitions and instead provide a reason to look for distance measures between astronomical sources suitable for “general spacetimes”: when the distances are becoming cosmological there are fewer assumptions that could be made on the structure of spacetime.

Here is a source that has discussion on various distance measures, and has links to original papers:

  • Perlick, V. (2004). Gravitational lensing from a spacetime perspective. Living reviews in relativity, 7(1), 9, doi:10.12942/lrr-2004-9.

From it we have the following definitions (pp. 24–25 of the pdf):

Corrected luminosity distance.

The idea of defining distance measures in terms of bundle cross-sections dates back to Tolman [323] and Whittaker [351]. Originally, this idea was applied not to bundles with vertex at the observer but rather to bundles with vertex at the light source. The resulting analogue of the area distance is the so-called corrected luminosity distance $D^{'}_\text{lum}$. It relates, for a bundle with vertex at the light source, the cross-sectional area at the observer to the opening solid angle at the light source. Owing to Etherington’s reciprocity law (35), area distance and corrected luminosity distance are related by $$D^{'}_\text{lum}= (1 +z) D_\text{area}.\tag{47}$$ The redshift factor has its origin in the fact that the definition of $D^{'}_\text{lum}$ refers to an affine parametrization adapted to $U_\text{S}$, and the definition of $D_\text{area}$ refers to an affine parametrization adapted to $U_\text{O}$. While $D_\text{area}$ depends on $U_\text{O}$ but not on $U_\text{S}$, $D^{'}_\text{lum}$ depends on $U_\text{S}$ but not on $U_\text{O}$.

Luminosity distance.

The physical meaning of the corrected luminosity distance is most easily understood in the photon picture. For photons isotropically emitted from a light source, the percentage that hit a prescribed area at the observer is proportional to $1/(D^{′}_\text{lum})^2$. As the energy of each photon undergoes a redshift, the energy flux at the observer is proportional to $1/(D_\text{lum})^2$, where $$D_\text{lum}= (1 +z)D^{′}_\text{lum}= (1 +z)^2 D_\text{area}.\tag{48}$$ Thus, $D_\text{lum}$ is the relevant quantity for calculating the luminosity (apparent brightness) of point-like light sources (see Equation (52)). For this reason $D_\text{lum}$ is called the (uncorrected) luminosity distance. The observation that the purely geometric quantity $D^{′}_\text{lum}$ must be modified by an additional redshift factor to give the energy flux is due to Walker [342]. $D_\text{lum}$ depends on the 4-velocity $U_\text{O}$ of the observer and of the 4-velocity $U_\text{S}$ of the light source. $D_\text{lum}$ and $D^{'}_\text{lum}$ can be viewed as functions of the observational coordinates $(s,Ψ,Θ,τ)$ if a vector field $U$ with $g(U,U) =−1$ has been distinguished, one integral curve of $U$ is chosen as the observer, and the other integral curves of $U$ are chosen as the light sources. In that case Equation (38) implies that not only $D_\text{area}(s)$ but also $D_\text{lum}(s)$ and $D^{'}_\text{lum}(s)$ are of the form $s+O(s^2)$. Thus, near the observer all three distance measures coincide with Euclidean distance in the observer’s rest space.


Addition: Among the original works I would recommend looking at

  • Kristian, J., & Sachs, R. K. (1966). Observations in cosmology. The Astrophysical Journal, 143, 379, republished in a Golden Oldies series of Gen. Rel. Grav. (open access) doi:10.1007/s10714-010-1114-1, an editorial note by G. Ellis on republication.

And since R. Sachs is one of its authors, the definition in there must be the closest to the authors intent in his 1962 paper.

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