Etherington Distance Duality

Etherington reciprocity is a geometric statement about exchanging the two vertices of a null beam. Distance duality is the observational consequence obtained only after photon propagation and photon-number conservation are added. For each resolved image these steps give $D_G=(1+z)D_A$ and $D_L=(1+z)D_G=(1+z)^2D_A$, where $D_G$ is the source-solid-angle, or galaxy-area, distance. The proof below uses the Jacobi equation and states where caustics, unresolved images, and opacity change the interpretation. It assumes neither FLRW symmetry nor Einstein’s equations.

Required background. Affine and Variational Geodesics supplies the Geodesics and Jacobi Fields capability used for the endpoint Jacobi maps and their affine evolution.

Helpful background. The Einstein Equation supplies the Einstein Equations, although the theorem does not require them; Relativistic Matter and Compact Objects supplies the Relativistic Matter and Compact Objects context for emitters and absorbers; Relativistic Cosmology orients the redshift and distance conventions; and The Wave Equation and Characteristics supplies the Waves, Green Functions, and Geometric Optics limit in which a ray bundle and photon flux are defined.

Two endpoint area distances

Let a source event $s$ and observer event $o$ be joined by one chosen null geodesic. Parallel-transport an orthonormal screen basis along it. A transverse Jacobi field has components $\boldsymbol\xi$ satisfying

$$\ddot{\boldsymbol\xi} =\mathcal T(\lambda)\boldsymbol\xi, \qquad \mathcal T^T=\mathcal T,$$

where a dot denotes affine differentiation and the symmetric optical tidal matrix is built from spacetime curvature. For a pencil with vertex at $o$, the physical Jacobi map $\mathcal D_{s\leftarrow o}$ converts an observed angle into source-plane separation. For the reversed pencil, $\mathcal D_{o\leftarrow s}$ converts a source-frame angle into observer-plane separation. Define

$$D_A^2 =\left|\det\mathcal D_{s\leftarrow o}\right| =\frac{dA_s}{d\Omega_o}, \qquad D_G^2 =\left|\det\mathcal D_{o\leftarrow s}\right| =\frac{dA_o}{d\Omega_s}.$$

The areas are measured in the local screens orthogonal to the relevant four-velocity and ray direction. The two definitions need not be equal: angles at the endpoints use different photon frequencies. Etherington’s original distance construction and redshift factor appear in Etherington 1932, Part III §§14–15, printed pp. 50–63, especially formulas (51)–(54) (official archive).

Wronskian proof of reciprocity

Let $\mathcal F(\lambda,\lambda_1)$ be the affine fundamental Jacobi matrix with vertex at $\lambda_1$:

$$\mathcal F(\lambda_1,\lambda_1)=0, \qquad \partial_\lambda\mathcal F(\lambda_1,\lambda_1)=I.$$

For any two matrix solutions $\mathcal X$ and $\mathcal Y$, define their Wronskian

$$\mathcal W[\mathcal X,\mathcal Y] =\mathcal X^T\dot{\mathcal Y} -\dot{\mathcal X}^{\,T}\mathcal Y.$$

Symmetry of the optical tidal matrix gives

$$\dot{\mathcal W} =\mathcal X^T\mathcal T\mathcal Y -(\mathcal T\mathcal X)^T\mathcal Y =0.$$

Choose $\mathcal X(\lambda)=\mathcal F(\lambda,\lambda_o)$ and $\mathcal Y(\lambda)=\mathcal F(\lambda,\lambda_s)$. Evaluating the same constant Wronskian at the two endpoints yields

$$-\mathcal F(\lambda_o,\lambda_s) =\mathcal F^T(\lambda_s,\lambda_o),$$

or

$$\boxed{ \mathcal F(\lambda_s,\lambda_o) =-\mathcal F^T(\lambda_o,\lambda_s) }.$$

The minus sign records the reversed affine orientation and disappears from the absolute area determinant. Choose $k^a$ future-directed from source to observer, so the frequency measured by a future-directed observer is

$$\omega=-u\cdot k, \qquad 1+z=\frac{\omega_s}{\omega_o},$$

and is positive. An angular separation is the affine transverse derivative divided by the endpoint frequency, $\delta\boldsymbol\theta=\delta\dot{\boldsymbol\xi}/\omega$; equivalently, $\delta\dot{\boldsymbol\xi}=\omega\delta\boldsymbol\theta$. Thus, up to the chosen relative orientation of the two sky bases,

$$\mathcal D_{s\leftarrow o} =\omega_o\mathcal F(\lambda_s,\lambda_o), \qquad \mathcal D_{o\leftarrow s} =\omega_s\mathcal F(\lambda_o,\lambda_s),$$

and the physical endpoint maps satisfy

$$\mathcal D_{s\leftarrow o} =-\frac{\omega_o}{\omega_s} \mathcal D_{o\leftarrow s}^{\,T}.$$

Taking the square root of the absolute two-dimensional determinants proves

$$\boxed{D_G=(1+z)D_A}.$$

This is reciprocity. Its ingredients are a metric spacetime with Levi-Civita geometric-optics propagation, a selected source–observer null geodesic, well-defined endpoint screens, and a nondegenerate Jacobi map or a signed continuation through its caustics. The modern light-bundle equation, screen construction, and symmetric tidal matrix are given in Bartelmann and Schneider 2001, §3.2.1, printed pp. 52–53, eqs. (3.21)–(3.26) (Open PDF). Perlick states the same endpoint Jacobi relation and follows it through the distance definitions in Perlick 2004, updated arXiv version 2010, §2.3, printed pp. 16–17, eqs. (28) and (34)–(35), and §2.4, printed pp. 19–22, eqs. (39)–(48) (Open PDF).

From reciprocity to luminosity duality

Suppose the source emits isotropically with bolometric luminosity $L_s$. The power launched into a small source solid angle is $L_s\,d\Omega_s/(4\pi)$. At the observer, the corresponding beam occupies $dA_o=D_G^2d\Omega_s$. Each photon loses an energy factor $(1+z)^{-1}$, and the arrival interval is dilated by another factor $1+z$. If photons remain in the beam and their number is conserved,

$$F_o =\frac{L_s}{4\pi D_G^2(1+z)^2}.$$

Using $F_o=L_s/(4\pi D_L^2)$ gives

$$D_L=(1+z)D_G, \qquad \boxed{D_L=(1+z)^2D_A}.$$

Photon conservation is not part of geometric reciprocity. It is the extra hypothesis that turns reciprocity into luminosity duality. Likewise, a finite-band flux needs a spectral model and bandpass correction; the clean statement above is bolometric.

Caustics, multiple images, and failed hypotheses

At a conjugate point, $\det\mathcal D=0$. Across a generic simple caustic, one eigenvalue changes sign and image parity flips. At a multiplicity-two caustic both eigenvalues can cross, so parity need not flip. The matrix reciprocity relation remains valid under signed, parity-aware continuation, although the positive scalar distance alone discards that information. Exactly at the caustic, geometric optics predicts a singular magnification; finite wavelength and finite source size regularize the observation. Perlick’s area distance $|D_+D_-|^{1/2}$ and signed magnification make this distinction explicit in Perlick 2004, updated arXiv version 2010, §2.4, printed p. 20, eqs. (40)–(42) (Open PDF).

A simple check takes $\mathcal T=\operatorname{diag}(-\kappa^2,0)$ along a segment of affine length $\Delta\lambda$. Then

$$\mathcal F =\operatorname{diag} \left[ \frac{\sin(\kappa\Delta\lambda)}{\kappa}, \Delta\lambda \right].$$

One eigenvalue vanishes at $\Delta\lambda=\pi/\kappa$ and changes sign afterward, yet the reversed fundamental matrix is still minus the transpose. Reciprocity therefore survives the parity change; an unsigned area description merely hides it.

When several null geodesics connect the same events, apply the theorem to each resolved image. For a steady unresolved source with no differential absorption, an aperture containing every image, and mutually incoherent images—or time/frequency averaging that removes interference—intensities add, so an effective combined luminosity distance obeys $D_{L,\rm eff}^{-2}=\sum_iD_{L,i}^{-2}$. There is generally no corresponding single image area whose $D_A$ can be inserted into the theorem. If a path delay is comparable with the coherence time, amplitudes can interfere and this inverse-square sum fails. Variability, finite observing windows, and image-dependent extinction make the aggregation still more explicitly observational.

Gray opacity supplies a counterexample to distance duality without touching reciprocity. If a beam has optical depth $\tau$, then

$$F_{\rm obs}=e^{-\tau}F_{\rm geom}, \qquad D_{L,\rm inferred} =e^{\tau/2}(1+z)^2D_A.$$

Absorption, photon conversion, or scattering out of an otherwise unchanged metric ray bundle preserves Jacobi reciprocity but breaks the flux step to $D_L$. Nonmetric propagation can instead spoil the symmetric tidal/Jacobi system and invalidate reciprocity itself. Homogeneity, isotropy, uniqueness of the image, and Einstein’s equations were never used in the metric proof.

FLRW check

For the FLRW background,

$$D_A=\frac{S_K(\chi)}{1+z}, \qquad D_G=S_K(\chi), \qquad D_L=(1+z)S_K(\chi).$$

Hence both $D_G=(1+z)D_A$ and $D_L=(1+z)^2D_A$ follow immediately. This independent check also fixes which endpoint frequency belongs in the reciprocity factor.

Exercises

Show directly that opacity can mimic a violation of distance duality while leaving the Jacobi reciprocity relation unchanged. Express the usual duality-test parameter

$$\eta(z)\equiv \frac{D_{L,\rm inferred}}{(1+z)^2D_A}$$

in terms of optical depth.

Solution

The geometric Jacobi maps depend on the metric and remain unchanged when photons are merely removed from an otherwise identical beam. Thus $D_G=(1+z)D_A$ still holds. Attenuation changes the observed flux to $F_{\rm obs}=e^{-\tau}F_{\rm geom}$. Since luminosity distance is inferred from $F\propto D_L^{-2}$,

$$D_{L,\rm inferred} =e^{\tau/2}D_{L,\rm geom} =e^{\tau/2}(1+z)^2D_A.$$

Therefore

$$\boxed{\eta(z)=e^{\tau(z)/2}}.$$

For small optical depth, $\eta=1+\tau/2+O(\tau^2)$. The deviation diagnoses a failure of the photon-flux hypotheses, not a failure of the geometric Wronskian identity.

References

• Bartelmann, Matthias, and Peter Schneider. “Weak Gravitational Lensing.” Physics Reports 340 (2001): 291–472. DOI; Open PDF.
• Etherington, I. M. H. On Relativistic Cosmology and on the Definition of Distance in General Relativity. PhD thesis, University of Edinburgh, 1932. Official archive.
• Perlick, Volker. “Gravitational Lensing from a Spacetime Perspective.” Living Reviews in Relativity 7 (2004): article 9; updated arXiv:1010.3416v1, 17 October 2010. DOI; Open PDF.