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Equation 13 · Part 14 · The Atlas That Refuses to Close

Starting index or lower bound: Sigma

Xμ(u,Σ)=∫Σxμ Tαβuα dΣβ∫ΣTαβuα dΣβ.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}.
Σ\Sigma

What this part means

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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.

The passage around this formula

Naming the slice is the first entry in this atlas’s notation. Let Σ\Sigma be a spacelike hypersurface and uμu^\mu the four-velocity of the observer whose notion of simultaneity Σ\Sigma represents, with Σ\Sigma taken orthogonal to u . Replace the discrete sum with the body’s stress-energy tensor TαβT^{\alpha\beta} and define the energy centroid Xμ(u,Σ)=∫Σxμ Tαβuα dΣβ∫ΣTαβuα dΣβX^\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

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

Open the illustrated sums and products: repeat an operation over an index guide →

See this notation across published equations →

Sources cited in the article section

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