= Solution
\b[Spatial geometry.] An embedding of the <three-sphere> is
$$
w=A\cos\chi,\quad z=A\sin\chi\cos\theta,\quad
x=A\sin\chi\sin\theta\cos\phi,\quad y=A\sin\chi\sin\theta\sin\phi.
$$
Taking differentials and collecting terms gives the induced <metric tensor>
$$
d\ell^2=A^2d\chi^2+A^2\sin^2\chi\,(d\theta^2+\sin^2\theta\,d\phi^2).
$$
Put $\bar r=A\sin\chi$. Then $d\bar r=A\cos\chi\,d\chi$, so
$$
\boxed{d\ell^2=\frac{d\bar r^2}{1-\bar r^2/A^2}+\bar r^2d\Omega^2},\qquad d\Omega^2=d\theta^2+\sin^2\theta\,d\phi^2.
$$
The <areal radius> $\bar r$ is not a global radial coordinate on the entire <three-sphere>: its maximum occurs at the equator and two hemispheres share each smaller value. Use separate charts there; the coordinate singularity at $\bar r=A$ is not a physical singularity.
For <Euclidean space> and <hyperbolic space> of curvature $-1/A^2$, respectively,
$$
\boxed{d\ell_0^2=d\bar r^2+\bar r^2d\Omega^2,\qquad
d\ell_-^2=\frac{d\bar r^2}{1+\bar r^2/A^2}+\bar r^2d\Omega^2}.
$$
For the latter, $\bar r=A\sinh\chi$ gives $d\ell_-^2=A^2[d\chi^2+\sinh^2\chi\,d\Omega^2]$. In all three cases the induced two-metric at fixed $\bar r=\bar R$ is $\bar R^2d\Omega^2$. Its <surface area> is therefore
$$
\boxed{\mathcal A(\bar R)=\int_0^{2\pi}\!d\phi\int_0^\pi\!\bar R^2\sin\theta\,d\theta=4\pi\bar R^2}.
$$
This equality compares the same areal radius, not the same proper radial distance. For a proper radial distance $\ell$, the corresponding areas are $4\pi A^2\sin^2(\ell/A)$, $4\pi\ell^2$, and $4\pi A^2\sinh^2(\ell/A)$.
\b[Homogeneity, isotropy and redshift.] The <FRW metric> has constant-time spatial slices of constant curvature $k/a^2(t)$, with $k=+1,0,-1$ selecting the three geometries above. Their isometries act transitively on points and isotropically about every point, implementing the <cosmological principle>. Multiplication by one common time-dependent <scale factor> preserves these properties. With the source's $k=\pm1$ normalization, $r$ is dimensionless and $a$ has units of length; using a dimensionless <scale factor> instead rescales the spatial curvature parameter. The origin of the spherical coordinates is arbitrary; it does not identify a physical centre. Cosmic time is the <proper time> of the comoving observers, and the absence of time-space cross terms makes their worldlines orthogonal to the homogeneous slices.
For a radial <null geodesic> between comoving endpoints,
$$
\int_{t_E}^{t_0}\frac{c\,dt}{a(t)}=\int_0^R\frac{dr}{\sqrt{1-kr^2}}.
$$
The next wave crest travels between the same endpoints. Subtracting the two path integrals, for wave periods short compared with the expansion time, gives $\delta t_0/a_0=\delta t_E/a_E$. Thus the <cosmological redshift> is
$$
\boxed{1+z=\frac{\lambda_0}{\lambda_E}=\frac{\nu_E}{\nu_0}=\frac{a_0}{a_E}}.
$$
Both observers' clocks are cosmic proper-time clocks. Peculiar velocities and local gravitational-potential shifts have been left out of this homogeneous calculation.
\b[Angular size and brightness.] For a galaxy small compared with its distance, the <angular diameter distance> is $D_A=a_ER$ and its angular diameter is
$$
\boxed{\Delta\theta\simeq\frac d{a_ER}=\frac{(1+z)d}{a_0R}}.
$$
Here the emitted proper diameter is measured transverse to the line of sight; radial proper distance is a different integral when $k\ne0$. Ignoring absorption, lensing caustics and intrinsic evolution, photon-number conservation spreads an isotropic-equivalent luminosity across the sphere $4\pi a_0^2R^2$. Each <photon> loses energy by $1+z$, and its arrival interval is stretched by the same factor. The received bolometric flux is
$$
F=\frac{L}{4\pi a_0^2R^2(1+z)^2},\qquad D_L=(1+z)a_0R=(1+z)^2D_A.
$$
The disc's proper projected area is $\mathcal A_d=\pi d^2/4$, and its observed <solid angle> is $\Omega_d\simeq\mathcal A_d/D_A^2$. Therefore its <surface brightness> is
$$
\boxed{I_{\rm obs}=\frac F{\Omega_d}=\frac{L}{4\pi\mathcal A_d}(1+z)^{-4}=\frac{L}{\pi^2d^2}(1+z)^{-4}}.
$$
This is <cosmological surface-brightness dimming>. The normalization uses $L$ as isotropic-equivalent luminosity along the viewing direction; another disc emission law changes that normalization but not the redshift power. This is a bolometric result. In a specified observed frequency band the emitted spectrum is sampled at a shifted frequency and requires a spectral correction.
The distinction in <telescope aperture and resolved surface brightness> matters here. A larger <telescope> collects more <photons> and improves <angular resolution>, but cannot restore the incoming radiance lost by cosmological dimming. For aperture diameter $D$ and a diffraction resolution element of <solid angle> $\Omega_{\rm res}\sim(\lambda/D)^2$, the collected power from uniform resolved brightness scales as
$$
I_{\rm obs}\,A_{\rm tel}\Omega_{\rm res}\propto I_{\rm obs}\lambda^2,
$$
independent of $D$. This is consistent with <conservation of etendue>. It explains why increasing aperture while keeping every gain in resolution does not increase counts per native resolution element from a smooth extended source. It does \b[not] make larger telescopes useless: at a fixed angular bin the counts grow as $D^2$, so a photon-noise-limited or uniform-sky-noise-limited <signal-to-noise ratio> grows roughly as $D\sqrt{t_{\rm exp}}$. They also resolve smaller structures and improve point-source sensitivity. Aperture, exposure and angular binning help measure a dim galaxy, without changing its intrinsic $(1+z)^{-4}$ brightness penalty.
\Image[/past-exam-of-the-general-relativity-and-cosmology-course-of-the-university-of-oxford/2016-surface-brightness-and-aperture.png]
{title=Cosmological surface-brightness dimming and the distinct aperture scalings of fixed angular bins and diffraction elements}
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