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Equation 13 · Part 6 · The Atlas That Refuses to Close

Symbol u_α

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}.
uαu_\alpha

What this part means

u_α is an input to the expression that computes the quantity on the left.

Its job in the formula

u_α is an input to the expression that computes the quantity on the left.

The passage around this formula

…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 quantity here has a declared type. TαβT^{\alpha\beta} is an energy-momentum density, units of energy per volume; dΣβ\Sigma_\beta is a directed three-volume element on Σ\Sigma , units of volume; uαu_\alpha is dimensionless once normalized to uμu^\mu uμu_\mu = -1 ; xμx^\mu is a spacetime coordinate, units of length. The factor TαβT^{\alpha\beta}uαu_\alpha\, dΣβ\Sigma_\beta appears in numerator and denominator identically and cancels completely, leaving XμX^\mu with units of length — a genuine spacetime…

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the article section

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