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I'm studying properties of Kerr spacetimes and a lot of fuss is made about area of BH. It is defined to be integral of area element on event horizon $r=r_+$, $t=const.$ where $r_+$ is radial coordinate of outer event horizon. In what way this can be interpreted as actual area of anything? What kind of measuring process would yield as a result this area?

What bothers me is the result we get depends on foliation of spacetime we choose, doesn't it?

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The area of an event horizon is an invariant, so it doesn't matter what coordinate system you use to calculate it.

The area is of physical interest because it's proportional to the entropy of the black hole, and in a second law kind of way that means the area cannot decrease.

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