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Equation 31 · Part 8 · The Atlas That Refuses to Close

Numerator: |mathbf S|

ρM=∣S∣Mc.\rho_M = \frac{|\mathbf S|}{Mc}.
∣S∣|\mathbf S|

What this part means

The complete quantity above the fraction bar.

Its job in the formula

|mathbf S| occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

The scale that answer is measured against is fixed entirely by the body itself. For a system of total mass M and spin angular momentum of magnitude |S\mathbf S| — the part of JμνJ^{\mu\nu} left over once the orbital piece is subtracted — define ρM=∣S∣Mc\rho_M = \frac{|\mathbf S|}{Mc}. ρM\rho_M has units of length by inspection, angular momentum divided by momentum, and it is built from nothing but the two Poincaré-invariant labels, mass and spin, that every observer agrees on regardless of which chart they read a position off of. It will turn out to be the radius of the region within which every well-behaved chart in this atlas has to stay.

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the article section

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