$$X^\mu(u,\Sigma) = \frac{\int_\Sigma x^\mu\, T^{\alpha\beta}u_\alpha\, d\Sigma_\beta}{\int_\Sigma T^{\alpha\beta}u_\alpha\, d\Sigma_\beta}$$

Equation 13 · 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.

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…

Meanings in this article

• $X^\mu$: the leaving.
• $\Sigma$: a spacelike hypersurface and $u^\mu$ the four-velocity of the observer whose notion of simultaneity $\Sigma$ represents.
• $x^\mu$: a spacetime coordinate, units of length.
• $T^{\alpha\beta}$: an energy-momentum density, units of energy per volume.
• $\Sigma_\beta$: a spacelike hypersurface and $u^\mu$ the four-velocity of the observer whose notion of simultaneity $\Sigma$ represents with subscript beta (a directed three-volume element on $\Sigma$ , units of volume).

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