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In a talk The enigma of black hole horizons, (at 24:37), it is said that

"Raychaudhuri equation implies, if the flux into H is positive, area increases and horizon is spacelike".

How increase in area implies spacelike horizon? Similarly, how constant area implies null and decrease in area implies timelike horizon?

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    $\begingroup$ Note that he is not talking about the event horizon, but about the quasi-local horizon (whose definition is quite technical). Also note that a surface just outside the event horizon (called the stretched horizon) is timelike while the one just below the event horizon is spacelike. Your question would be answered if the quasi-local horizon dynamically lags behind the movement of the event horizon. To see if this is the case, just check the definition. If the area is constant, both horizons coincide as null. $\endgroup$
    – safesphere
    Commented Feb 10 at 16:50
  • $\begingroup$ As a side note, this lecture includes popular misconceptions. He says that an event horizon can be in your living room, but a quasi-local horizon cannot. This is not true. The event horizon of a black hole cannot be in your living room (I won’t go into details here, but it is a misconception) while the quasi-local Rindler horizon can pass through your living room just fine. $\endgroup$
    – safesphere
    Commented Feb 10 at 17:13

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