a FL spacetime with pressureless matter and a cosmological constant, $a(T)$ is completely determined from the Friedmann equation. A spherical hole of comoving radius $\chi{_{\text{h}}}$, which contains a constant mass $M =4\pi\rho a^3 f_K^3(\chi{_{\text{h}}})/3$ at its center, and whose geometry is described by the Kottler metric, can then be inserted anywhere. The resulting spacetime geometry is an exact solution of the Einstein field equations.

By construction, the clump inside the hole does not backreact on the surrounding FL region. It follows that many such holes can be inserted, as long as they do not overlap. Note that if two holes do not overlap initially, then they will never do so, despite the expansion of the universe, because their boundaries are comoving.

Propagation of light {#sec:geometric_optics}
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Light rays
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The past light cone of a given observer is a constant phase hypersurface $w =\,$const. Its normal vector $k_\mu \equiv \partial_\mu w$ (the wave four-vector) is a null vector satisfying the geodesic equation, and whose integral curves (light rays) are irrotational: $$\label{nullgeod}
k^\mu k_\mu = 0, \quad k^\nu \nabla_\nu k_\mu = 0, \quad \nabla_{[\mu}k_{\nu]}=0.$$ For an emitter and an observer with respective four-velocities $u{_{\text{em}}}^\mu$ and $u{_{\text{obs}}}^\mu$, we define the redshift by 