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Equation 25 · The Atlas That Refuses to Close

What does this equation mean?

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

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Inputs and operationsint_Sigma T^munu dSigma_nu
Result or conditionP^mu
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

Read it piece by piece

PμP^\mu

Symbol P^mu

PmP^mu is part of the quantity the equation computes from the expression on the right.

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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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TμνT^{\mu\nu}

Symbol T^munu

TmT^munu is an input to the expression that computes the quantity on the left.

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dd

Symbol d

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

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Σν\Sigma_\nu

Symbol Sigma_nu

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

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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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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How to interpret it

Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

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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Sources cited in the article section

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