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In the prescribed synchronous coordinates, the Christoffel symbols satisfyConsequently the four-acceleration of $u={\partial }_t$ vanishes. For a spatial index $i$, $u_i=0$, so the spatial component of the relativistic Euler equation becomesOn a connected coordinate neighbourhood,A nonzero pressure is allowed, but its spatial gradient must vanish if the fluid's four-velocity follows these freely falling coordinate lines. No division by $p$ or $\rho +p$ is needed.

For relativistic dust, $p=0$, the relativistic Euler equation gives $\rho a_b=0$. The nowhere-vanishing density therefore makes the four-acceleration zero: $u$ is a timelike geodesic vector field. Let $\alpha =u^{♭}$ be the differential one-form obtained by the musical isomorphism. Its exterior derivative satisfiesHere ${\iota }_u$ is the interior product. By hypersurface orthogonality and the Frobenius theorem, $\alpha \wedge d\alpha =0$. Taking the interior product and using $\alpha (u)=-1$ givesThus $\boxed{d{u}^{♭}=0}$. By the Poincaré lemma, locally $\alpha =-dt$, and then $u(t)=-\alpha (u)=1$. Choose spatial coordinate functions $x^i$ on one level set of $t$ and extend them constantly along the integral curves of a vector field $u$, using its smooth flow. This constructs $u={\partial }_t$. Since $dt({\partial }_i)=0$,The metric on the orthogonal complement of a timelike vector is positive-definite, so the synchronous coordinates for irrotational dust haveThis construction is local; crossing geodesics can prevent these synchronous coordinates from covering a larger region.

A fundamental observer follows the mean cosmic matter flow and is at rest with respect to the homogeneous cosmological frame. Cosmological isotropy permits no preferred spatial acceleration, rotation vector or anisotropic expansion for this flow: the congruence is geodesic, irrotational and shear-free, with isotropic expansion. Its orthogonal homogeneous spatial slices define simultaneity. Choosing fixed spatial labels along these worldlines and their common proper time as $t$ gives synchronous coordinates, $g_{tt}=c^2$ and $g_{ti}=0$.

In the Friedmann-Lemaître-Robertson-Walker metric, $t$ is this cosmic proper time and $\chi$ is radial comoving coordinate. On a constant-$t$ radial slice, the proper distance in cosmology from $0$ to a fixed ${\chi }_0$ is $D(t)=a(t){\chi }_0$. ThereforeThis is the Hubble parameter. The metric describes expansion when $\dot{a}>0$; its form alone also permits static or contracting histories, so expansion additionally requires that sign of the scale factor derivative.

For a radial photon, the metric components relevant to its motion are $g_{tt}=c^2$ and $g_{\chi \chi }=-a^2(t)$, both independent of $\chi$. The angular momentum components vanish. The covariant-momentum form of the geodesic equation therefore givesOnly this radial contraction is needed: angular metric coefficients can depend on $\chi$, but multiply zero angular components. The conserved quantity is the lower-index $p_1=p_{\chi }$, rather than the contravariant component $p^1$.

For outward travel put $P=-p_{\chi }>0$. The null condition $c^2(p^t{)}^2-a^2(p^{\chi }{)}^2=0$ givesHere the time coordinate is $t$, rather than $ct$. A comoving observer has $u^t=1$ and measures photon energy $E=p_{\mu }{u}^{\mu }=p_t$. Thus energy and frequency decrease as $a^{-1}$, giving the cosmological redshiftThe reciprocal relation for wavelengths follows from their common local propagation speed $c$.

Finally center the spatial coordinates on the receiving comoving observer. At the source, the transverse spatial line element is $dl=a(t_E)S_K(\chi )d\theta$. For a short transverse rod and a small observed angle $\Delta \theta$,Thus the angular diameter distance is $d_A=a(t_E)S_K(\chi )$. Writing the reception epoch as $t_0=t_R$ gives the displayed formula in the question. The length is the proper transverse length at emission; $t_0$ cannot be replaced by an unrelated epoch while retaining that expression.

In synchronous coordinates, fixed spatial-coordinate worldlines are timelike geodesics, and $t$ is their proper time. The metric has unit lapse function and zero shift vector in time units consistent with the displayed factor $c^2$. Indeed $g_{tt}=c^2$ and $g_{ti}=0$ give ${\Gamma }^{\mu }{}_{tt}=0$. Locally, an irrotational freely falling congruence permits this construction before crossings of the coordinate worldlines. Comoving observers in a Friedmann-Lemaître-Robertson-Walker metric provide the standard cosmological example.

For relativistic dust with nowhere-zero density, stress-energy conservation implies zero four-acceleration. If its normalized four-velocity $u$ has hypersurface orthogonality, then $u^{♭}\wedge d{u}^{♭}=0$ and ${\iota }_{u}d{u}^{♭}=0$. The interior product of the first identity gives $d{u}^{♭}=0$. The Poincaré lemma supplies $u^{♭}=-dt$ locally; spatial coordinate functions constant along the smooth flow of $u$ give $u={\partial }_t$, $g_{tt}=-1$ and $g_{ti}=0$. The resulting synchronous coordinates describe the dust until the local coordinate construction fails.