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We first define the notion of Wald gravitational entropy, defined in a more general setting than the Bekenstein–Hawking one. Then we define Sen’s entropy function formalism, in a general gravity theory, that defines the entropy function, whose minimization at the event horizon gives the entropy. Finally, we define the effective potential of the event horizon, defined in a theory with scalars, and show that the horizon is an attractor for the equations of motion involving scalars.
For extremal black holes, we have the attractor mechanism, originally defined in the context of N = 2 supergravity. This is then interpreted and described in the Sen’s entropy function formalism. The attractor mechanism exists also in five-dimensional gauged supergravity, and by embedding it in string theory, we can relate it to holography and the AdS/CFT correspondence.
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