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Equation 13 · Part 2 · The Atlas That Refuses to Close

Symbol u

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}.
uu

What this part means

u is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Its job in the formula

u is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

The passage around this formula

…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 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…

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Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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Sources cited in the article section

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