Distance measures

2.4 Distance measures

In this section we summarize various distance measures that are defined in an arbitrary spacetime. Some of them are directly related to observable quantities with relevance for lensing. The material of this section makes use of the results on infinitesimally thin bundles which are summarized in Section 2.3. All of the distance measures to be discussed refer to a past-oriented lightlike geodesic $λ$ from an observation event $pO$ to an emission event $pS$ (see Figure 4

). Some of them depend on the 4-velocity $UO$ of the observer at $pO$ and/or on the 4-velocity $US$ of the light source at $pS$ . If a vector field $U$ with $g(U, U ) = − 1$ is distinguished on $ℳ$ , we can choose for the observer an integral curve of $U$ and for the light sources all other integral curves of $U$ . Then each of the distance measures becomes a function of the observational coordinates $(s,Ψ, Θ, τ)$ (recall Section 2.1).

Figure 4: Past-oriented lightlike geodesic $λ$ from an observation event $pO$ to an emission event $pS$ . $γO$ is the worldline of the observer, $γS$ is the worldline of the light source. $UO$ is the 4-velocity of the observer at $p O$ and $U S$ is the 4-velocity of the light source at $p S$ .

Affine distance.
There is a unique affine parametrization $s ↦−→ λ (s)$ for each lightlike geodesic through the observation event $pO$ such that $λ(0) = pO$ and $( ) ˙ g λ(0),UO = 1$ . Then the affine parameter $s$ itself can be viewed as a distance measure. This affine distance has the desirable features that it increases monotonously along each ray and that it coincides in an infinitesimal neighborhood of $p O$ with Euclidean distance in the rest system of $U O$ . The affine distance depends on the 4-velocity $UO$ of the observer but not on the 4-velocity $US$ of the light source. It is a mathematically very convenient notion, but it is not an observable. (It can be operationally realized in terms of an observer field whose 4-velocities are parallel along the ray. Then the affine distance results by integration if each observer measures the length of an infinitesimally short part of the ray in his rest system. However, in view of astronomical situations this is a purely theoretical construction.) The notion of affine distance was introduced by Kermack, McCrea, and Whittaker [179 ].

Travel time.
As an alternative distance measure one can use the travel time. This requires the choice of a time function, i.e., of a function $t$ that slices the spacetime into spacelike hypersurfaces $t = constant$ . (Such a time function globally exists if and only if the spacetime is stably causal; see, e.g., [153 ], p. 198.) The travel time is equal to $t(pO) − t(pS)$ , for each $pS$ on the past light cone of $pO$ . In other words, the intersection of the light cone with a hypersurface $t = constant$ determines events of equal travel time; we call these intersections “instantaneous wave fronts” (recall Section 2.2). Examples of instantaneous wave fronts are shown in Figures 13 , 18 , 19 , 27 , and 28 . The travel time increases monotonously along each ray. Clearly, it depends neither on the 4-velocity $UO$ of the observer nor on the 4-velocity $US$ of the light source. Note that the travel time has a unique value at each point of $pO$ ’s past light cone, even at events that can be reached by two different rays from $pO$ . Near $pO$ the travel time coincides with Euclidean distance in the observer’s rest system only if $UO$ is perpendicular to the hypersurface $t = constant$ with $dt(UO ) = 1$ . (The latter equation is true if along the observer’s world line the time function $t$ coincides with proper time.) The travel time is not directly observable. However, travel time differences are observable in multiple-imaging situations if the intrinsic luminosity of the light source is time-dependent. To illustrate this, think of a light source that flashes at a particular instant. If the flash reaches the observer’s wordline along two different rays, the proper time difference $Δ τO$ of the two arrival events is directly measurable. For a time function $t$ that along the observer’s worldline coincides with proper time, this observed time delay $Δ τ O$ gives the difference in travel time for the two rays. In view of applications, the measurement of time delays is of great relevance for quasar lensing. For the double quasar 0957+561 the observed time delay $Δ τO$ is about 417 days (see, e.g., [275 ], p. 149).

Redshift.
In cosmology it is common to use the redshift as a distance measure. For assigning a redshift to a lightlike geodesic $λ$ that connects the observation event $p O$ on the worldline $γ O$ of the observer with the emission event $pS$ on the worldline $γS$ of the light source, one considers a neighboring lightlike geodesic that meets $γO$ at a proper time interval $ΔτO$ from $pO$ and $γS$ at a proper time interval $Δ τS$ from $pS$ . The redshift $z$ is defined as

If $λ$ is affinely parametrized with $λ(0) = pO$ and $λ(s) = pS$ , one finds that $z$ is given by

( ) g λ˙(0),UO 1 + z = -(--------)-. (37 ) g ˙λ(s),US

This general redshift formula is due to Kermack, McCrea, and Whittaker [179 ]. Their proof is based on the fact that $g(˙λ,Y )$ is a constant for all Jacobi fields $Y$ that connect $λ$ with an infinitesimally neighboring lightlike geodesic. The same proof can be found, in a more elegant form, in [41 ] and in [311

], p. 109. An alternative proof, based on variational methods, was given by Schrödinger [299 ]. Equation (37

) is in agreement with the Hamilton formalism for photons. Clearly, the redshift depends on the 4-velocity $UO$ of the observer and on the 4-velocity $US$ of the light source. If a vector field $U$ with $g(U, U ) = − 1$ has been distinguished on $ℳ$ , we may choose one integral curve of $U$ as the observer and all other integral curves of $U$ as the light sources. Then the redshift becomes a function of the observational coordinates $(s,Ψ, Θ, τ)$ . For $s → 0$ , the redshift goes to 0,

with a (generalized) Hubble parameter $h (Ψ,Θ, τ)$ that depends on spatial direction and on time. For criteria that $h$ and the higher-order coefficients are independent of $Ψ$ and $Θ$ (see [151 ]). If the redshift is known for one observer field $U$ , it can be calculated for any other $U$ , according to Equation (37

), just by adding the usual special-relativistic Doppler factors. Note that if $UO$ is given, the redshift can be made to zero along any one ray $λ$ from $pO$ by choosing the 4-velocities $Uλ(s)$ appropriately. This shows that $z$ is a reasonable distance measure only for special situations, e.g., in cosmological models with $U$ denoting the mean flow of luminous matter (“Hubble flow”). In any case, the redshift is directly observable if the light source emits identifiable spectral lines. For the calculation of Sagnac-like effects, the redshift formula (37

) can be evaluated piecewise along broken lightlike geodesics [23 ].

Angular diameter distances.
The notion of angular diameter distance is based on the intuitive idea that the farther an object is away the smaller it looks, according to the rule

The formal definition needs the results of Section 2.3 on infinitesimally thin bundles. One considers a past-oriented lightlike geodesic $s −→ λ(s)$ parametrized by affine distance, i.e., $λ (0) = pO$ and $( ˙ ) g λ (0),UO = 1$ , and along $λ$ an infinitesimally thin bundle with vertex at the observer, i.e., at $s = 0$ . Then the shape parameters $D+ (s)$ and $D − (s)$ (recall Figure 3

) satisfy the initial conditions $D (0) = 0 ±$ and $D˙ (0) = 1 ±$ . They have the following physical meaning. If the observer sees a circular image of (small) angular diameter $α$ on his or her sky, the (small but extended) light source at affine distance $s$ actually has an elliptical cross-section with extremal diameters $α |D ± (s)|$ . It is therefore reasonable to call $D+$ and $D −$ the extremal angular diameter distances. Near the vertex, $D+$ and $D−$ are monotonously increasing functions of the affine distance, $2 D ±(s) = s + 𝒪 (s )$ . Farther away from the vertex, however, they may become decreasing, so the functions $s ↦→ D+ (s)$ and $s ↦→ D − (s)$ need not be invertible. At a caustic point of multiplicity one, one of the two functions $D+$ and $D −$ changes sign; at a caustic point of multiplicity two, both change sign (recall Section 2.3). The image of a light source at affine distance $s$ is said to have even parity if $D (s)D (s) > 0 + −$ and odd parity if $D (s)D (s) < 0 + −$ . Images with odd parity show the neighborhood of the light source side-inverted in comparison to images with even parity. Clearly, $D+$ and $D −$ are reasonable distance measures only in a neighborhood of the vertex where they are monotonously increasing. However, the physical relevance of $D+$ and $D −$ lies in the fact that they relate cross-sectional diameters at the source to angular diameters at the observer, and this is always true, even beyond caustic points. $D +$ and $D −$ depend on the 4-velocity $UO$ of the observer but not on the 4-velocity $US$ of the source. This reflects the fact that the angular diameter of an image on the observer’s sky is subject to aberration whereas the cross-sectional diameter of an infinitesimally thin bundle has an invariant meaning (recall Section 2.3). Hence, if the observer’s worldline $γO$ has been specified, $D +$ and $D −$ are well-defined functions of the observational coordinates $(s,Ψ, Θ, τ)$ .

Area distance.
The area distance $Darea$ is defined according to the idea

As a formal definition for $D area$ , in terms of the extremal angular diameter distances $D +$ and $D −$ as functions of affine distance $s$ , we use the equation

$D (s)2 area$ indeed relates, for a bundle with vertex at the observer, the cross-sectional area at the source to the opening solid angle at the observer. Such a bundle has a caustic point exactly at those points where $Darea(s) = 0$ . The area distance is often called “angular diameter distance” although, as indicated by Equation (41

), the name “averaged angular diameter distance” would be more appropriate. Just as $D+$ and $D−$ , the area distance depends on the 4-velocity $UO$ of the observer but not on the 4-velocity $US$ of the light source. The area distance is observable for a light source whose true size is known (or can be reasonably estimated). It is sometimes convenient to introduce the magnification or amplification factor

The absolute value of $μ$ determines the area distance, and the sign of $μ$ determines the parity. In Minkowski spacetime, $D ± (s) = s$ and, thus, $μ(s) = 1$ . Hence, $|μ(s)| > 1$ means that a (small but extended) light source at affine distance $s$ subtends a larger solid angle on the observer’s sky than a light source of the same size at the same affine distance in Minkowski spacetime. Note that in a multiple-imaging situation the individual images may have different affine distances. Thus, the relative magnification factor of two images is not directly observable. This is an important difference to the magnification factor that is used in the quasi-Newtonian approximation formalism of lensing. The latter is defined by comparison with an “unlensed image” (see, e.g., [298

]), a notion that makes sense only if the metric is viewed as a perturbation of some “background” metric. One can derive a differential equation for the area distance (or, equivalently, for the magnification factor) as a function of affine distance in the following way. On every parameter interval where $D+D −$ has no zeros, the real part of Equation (27

) shows that the area distance is related to the expansion by

Insertion into the Sachs equation (25

) for $𝜃 = ϱ$ gives the focusing equation

Between the vertex at $s = 0$ and the first conjugate point (caustic point), $Darea$ is determined by Equation (44

) and the initial conditions

The Ricci term in Equation (44

) is non-negative if Einstein’s field equation holds and if the energy density is non-negative for all observers (“weak energy condition”). Then Equations (44

, 45

) imply that

i.e., $1 ≤ μ(s)$ , for all $s$ between the vertex at $s = 0$ and the first conjugate point. In Minkowski spacetime, Equation (46

) holds with equality. Hence, Equation (46

) says that the gravitational field has a focusing, as opposed to a defocusing, effect. This is sometimes called the focusing theorem.

Corrected luminosity distance.
The idea of defining distance measures in terms of bundle cross-sections dates back to Tolman [322 ] 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 $D ′ lum$ . 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

The redshift factor has its origin in the fact that the definition of $D ′lum$ refers to an affine parametrization adapted to $US$ , and the definition of $Darea$ refers to an affine parametrization adapted to $UO$ . While $D area$ depends on $U O$ but not on $U S$ , $D ′ lum$ depends on $U S$ but not on $U O$ .

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 $1∕(D ′ )2 lum$ . As the energy of each photon undergoes a redshift, the energy flux at the observer is proportional to $1∕(D )2 lum$ , where

Thus, $Dlum$ is the relevant quantity for calculating the luminosity (apparent brightness) of pointlike light sources (see Equation (52

)). For this reason $Dlum$ is called the (uncorrected) luminosity distance. The observation that the purely geometric quantity $D ′lum$ must be modified by an additional redshift factor to give the energy flux is due to Walker [342 ]. $Dlum$ depends on the 4-velocity $U O$ of the observer and of the 4-velocity $U S$ of the light source. $D lum$ and $D ′ lum$ 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 $Darea(s)$ but also $Dlum(s)$ and $D ′ (s) lum$ are of the form $s + 𝒪 (s2)$ . Thus, near the observer all three distance measures coincide with Euclidean distance in the observer’s rest space.

Parallax distance.
In an arbitrary spacetime, we fix an observation event $pO$ and the observer’s 4-velocity $UO$ . We consider a past-oriented lightlike geodesic $λ$ parametrized by affine distance, $λ(0) = pO$ and $( ) g ˙λ(0),UO = 1$ . To a light source passing through the event $λ(s)$ we assign the (averaged) parallax distance $D (s) = − 𝜃(0)−1 par$ , where $𝜃$ is the expansion of an infinitesimally thin bundle with vertex at $λ(s)$ . This definition follows [171 ]. Its relevance in view of cosmology was discussed in detail by Rosquist [288 ]. $Dpar$ can be measured by performing the standard trigonometric parallax method of elementary Euclidean geometry, with the observer at $pO$ and an assistant observer at the perimeter of the bundle, and then averaging over all possible positions of the assistant. Note that the method refers to a bundle with vertex at the light source, i.e., to light rays that leave the light source simultaneously. (Averaging is not necessary if this bundle is circular.) $Dpar$ depends on the 4-velocity of the observer but not on the 4-velocity of the light source. To within first-order approximation near the observer it coincides with affine distance (recall Equation (32 )). For the potential obervational relevance of $Dpar$ see [288 ], and [298 ], p. 509.

In view of lensing, $D+$ , $D −$ , and $Dlum$ are the most important distance measures because they are related to image distortion (see Section 2.5) and to the brightness of images (see Section 2.6). In spacetimes with many symmetries, these quantities can be explicitly calculated (see Section 4.1 for conformally flat spactimes, and Section 4.3 for spherically symmetric static spacetimes). This is impossible in a spacetime without symmetries, in particular in a realistic cosmological model with inhomogeneities (“clumpy universe”). Following Kristian and Sachs [189 ], one often uses series expansions with respect to $s$ . For statistical considerations one may work with the focusing equation in a Friedmann–Robertson–Walker spacetime with average density (see Section 4.1), or with a heuristically modified focusing equation taking clumps into account. The latter leads to the so-called Dyer–Roeder distance [86 , 87 ] which is discussed in several text-books (see, e.g., [298 ]). (For pre-Dyer–Roeder papers on optics in cosmological models with inhomogeneities, see the historical notes in [173 ].) As overdensities have a focusing and underdensities have a defocusing effect, it is widely believed (following [344 ]) that after averaging over sufficiently large angular scales the Friedmann–Robertson–Walker calculation gives the correct distance-redshift relation. However, it was argued by Ellis, Bassett, and Dunsby [99 ] that caustics produced by the lensing effect of overdensities lead to a systematic bias towards smaller angular sizes (“shrinking”). For a spherically symmetric inhomogeneity, the effect on the distance-redshift relation can be calculated analytically [231 ]. For thorough discussions of light propagation in a clumpy universe also see Pyne and Birkinshaw [284 ], and Holz and Wald [160 ].