What does this equation mean?

Naming the slice is the first entry in this atlas’s notation. Let $\Sigma$ be a spacelike hypersurface and $u^\mu$ the four-velocity of the observer whose notion of simultaneity $\Sigma$ represents, with $\Sigma$ taken orthogonal to u . Replace the discrete sum with the body’s stress-energy tensor $T^{\alpha\beta}$ and define the energy centroid $X^\mu(u,\Sigma) = \frac{\int_\Sigma x^\mu\, T^{\alpha\beta}u_\alpha\, d\Sigma_\beta}{\int_\Sigma T^{\alpha\beta}u_\alpha\, d\Sigma_\beta}$. Every quantity here has a declared type. $T^{\alpha\beta}$ is an energy-momentum density, units of energy per volume; d$\Sigma_\beta$ is a directed three-volume element on $\Sigma$ , units of volume; $u_\alpha$ is dimensionless once normalized to $u^\mu$ $u_\mu$ = -1 ; $x^\mu$ is a spacetime coordinate, units of length. The factor $T^{\alpha\beta}$$u_\alpha$\, d$\Sigma_\beta$ appears in numerator and denominator identically and cancels completely, leaving $X^\mu$ with units of length — a genuine spacetime point, not a bookkeeping artefact. That is the atlas’s first check, a unit audit passed by construction rather than coincidence, because the same weighting sits on both sides of the ratio.