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Published equation contexts

Pμ=∫ΣTμν dΣνP^\mu = \int_\Sigma T^{\mu\nu}\, d\Sigma_\nu

Why this formula appears here

Xμ(u,Σ)X^\mu(u,\Sigma) is this article’s first chart. It takes two pieces of auxiliary data, an observer and a slice, and returns a point. Different choices of (u,Σ\Sigma) for the identical physical body — the identical TαβT^{\alpha\beta} field, the identical total four-momentum PμP^\mu = ∫Σ\int_\Sigma TμνT^{\mu\nu}\, dΣν\Sigma_\nu and total angular momentum JμνJ^{\mu\nu} — can and do return different points, and the question this article is built to answer precisely is how different, and under what conditions the difference means something rather than nothing.

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Σ\Sigma

Symbol Sigma

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

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Σ\Sigma

Starting index or lower bound: Sigma

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.

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Published contexts (1)

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Pμ=∫ΣTμν dΣνP^\mu = \int_\Sigma T^{\mu\nu}\, d\Sigma_\nu

Equation 25 · 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.

Xμ(u,Σ)X^\mu(u,\Sigma) is this article’s first chart. It takes two pieces of auxiliary data, an observer and a slice, and returns a point. Different choices of (u,Σ\Sigma) for the identical physical body — the identical TαβT^{\alpha\beta} field, the identical total four-momentum PμP^\mu = ∫Σ\int_\Sigma TμνT^{\mu\nu}\, dΣν\Sigma_\nu and total angular momentum JμνJ^{\mu\nu} — can and do return different points, and the question this article is built to answer precisely is how different, and under what conditions the difference means something rather than nothing.

Meanings in this article

  • Σ\Sigma: a spacelike hypersurface and uμu^\mu the four-velocity of the observer whose notion of simultaneity Σ\Sigma represents.
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