$$u^\mu u_\mu = -1$$

Equation 18 · Evolutionary Physics

The Atlas That Refuses to Close

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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.

Meanings in this article

• $u^\mu$: the four-velocity of the observer whose notion of simultaneity $\Sigma$ represents.

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