Equation 13 · Part 3 · The Atlas That Refuses to Close

Symbol Sigma

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}.

\Sigma

What this part means

a spacelike hypersurface and $u^\mu$ the four-velocity of the observer whose notion of simultaneity $\Sigma$ represents.

Its job in the formula

Sigma appears in the bound of this integral. The bound states where the repeated operation starts, ends, or which values it includes.

Full expressionSymbol SigmaArticle meaning

Where the article explains it

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 .

The passage around this formula

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…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the article section

[14] General Relativistic Center-of-Mass Coordinates for Composite Quantum Particles ↗

These citations provide research context; check each source for the exact claim it supports.