Equation 31 · The Atlas That Refuses to Close
What does this equation mean?
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.
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
Symbol rho_M
rh is part of the quantity the equation computes from the expression on the right.
Symbol S
S occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol M
M occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol c
c occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
Denominator: Mc
The complete quantity below the fraction bar; it must be nonzero for this division.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
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 || — the part of left over once the orbital piece is subtracted — define . 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.
Sources cited in the article section
These citations give research context. Read each source to check which claims it supports.
Return to The Atlas That Refuses to Close