Equation 13 · Part 4 · The Atlas That Refuses to Close

Symbol x^mu

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}.

x^\mu

What this part means

a spacetime coordinate, units of length.

Its job in the formula

$x^m$ u occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Full expressionSymbol x^muArticle meaning

Where the article explains it

$T^{\alpha\beta}$ is an energy-momentum density, units of energy per volume; d $\Sigma_\beta$ is a directed three-volume element on $\Sigma$ , units of volume; $u_\alpha$ is dimensionless once normalized to $u^\mu$ $u_\mu$ = -1 ; $x^\mu$ is a spacetime coordinate, units of length.

The passage around this formula

… $T^{\alpha\beta}$ is an energy-momentum density, units of energy per volume; d $\Sigma_\beta$ is a directed three-volume element on $\Sigma$ , units of volume; $u_\alpha$ is dimensionless once normalized to $u^\mu$ $u_\mu$ = -1 ; $x^\mu$ is a spacetime coordinate, units of length. The factor $T^{\alpha\beta}$ $u_\alpha$ \, d $\Sigma_\beta$ appears in numerator and denominator identically and cancels completely, leaving $X^\mu$ with units of length — a genuine spacetime point, not a bookkeeping artefact. That is the atlas’s first check,…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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See this notation across published equations →

Sources cited in the article section

[14] General Relativistic Center-of-Mass Coordinates for Composite Quantum Particles ↗

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