← Mathematical compendium

Published equation contexts

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

Why this formula appears here

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.…

Read the full article-specific guide →

Read the representative guide

UinU_{\mathrm{in}}

Symbol U_in

UiU_in is part of the quantity the equation computes from the expression on the right.

Read this term in its guide →
gg

Symbol g

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

Read this term in its guide →

How to interpret it

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

Research cited beside this formula

Learn the mathematics behind it

Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

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.

Equation 11 · Evolutionary Physics

The Bit Comes Back Before the Bearing

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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.…

Meanings in this article

  • gg: an angle: dimensionless, periodic, and physically read as “which way the compass points” relative to the hole’s spin axis.
Equation guide → · Article →