Isolated, slowly evolving, and dynamical trapping horizons: Geometry and mechanics from surface deformations

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We study the geometry and dynamics of both isolated and dynamical trapping horizons by considering the allowed variations of their foliating two-surfaces. This provides a common framework that may be used to consider both their possible evolutions and their deformations as well as derive the well-known flux laws. Using this framework, we unify much of what is already known about these objects as well as derive some new results. In particular we characterize and study the ``almost isolated'' trapping horizons known as slowly evolving horizons. It is for these horizons that a dynamical first law holds and this is analogous and closely related to the Hawking-Hartle formula for event horizons.