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Equation 11 · Part 3 · The Bit Comes Back Before the Bearing

Symbol g

V Uin(g)  =  [Urad(g)⊗Urem(g)] V.V\,U_{\mathrm{in}}(g) \;=\; \big[U_{\mathrm{rad}}(g)\otimes U_{\mathrm{rem}}(g)\big]\,V.
gg

What this part means

an angle: dimensionless, periodic, and physically read as “which way the compass points” relative to the hole’s spin axis.

Its job in the formula

g is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Where the article explains it

g is an angle: dimensionless, periodic, and physically read as “which way the compass points” relative to the hole’s spin axis.

The passage around this formula

The hole’s evaporation is modeled, as it is throughout this line of work, as an isometry V from the interior-plus-infalling-system Hilbert space to a radiation-plus-remaining-interior Hilbert space. What makes the conservation law bite is that V must commute with the group action on both sides: V Uin(g)  =  [Urad(g)⊗Urem(g)] VV\,U_{\mathrm{in}}(g) \;=\; \big[U_{\mathrm{rad}}(g)\otimes U_{\mathrm{rem}}(g)\big]\,V. Every symbol here is a unitary representation of the same rotation on a different Hilbert space — dimensionless linear operators, none of them observables in their own right. The equation says the hole’s internal dynamics cannot tell which way “up” was chosen before the compass fell in: rotate the input, and the output rotates the same way, split between radiation and remainder.…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

Open the illustrated variables: a letter stands for a value guide →

See this notation across published equations →

Sources cited in the surrounding passage

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