### 7.3 Trapping, isolated, and dynamical horizons

Hayward [83] introduced the important concept of a “trapping horizon” (roughly speaking an apparent
horizon world-tube where the expansion becomes negative if the surface is deformed in the inward null
direction) along with several useful variants. Ashtekar, Beetle, and Fairhurst [16], and Ashtekar and
Krishnan [18] later defined the related concepts of an “isolated horizon”, essentially an apparent horizon
world-tube which is null, and a “dynamical horizon”, essentially an apparent horizon world-tube which is
spacelike.
These world-tubes obey a variety of local and global conservation laws, and have many
applications in analyzing numerically-computed spacetimes. See the references cited above and also
Dreyer et al. [63], Ashtekar and Krishnan [19, 20], Gourgoulhon and Jaramillo [76], Booth [36], and
Schnetter, Krishnan, and Beyer [137] for further discussions, including applications to numerical
relativity.