Hi everyone, and happy new year if you happen to be reading this tomorrow. Rather than partying, I am writing up 100+ pages of astrophysics lecture notes, which I think will take infinite time as I keep getting stuck on every other line.(adsbygoogle = window.adsbygoogle || []).push({});

My current problem is with the equation for the surface area of a Kerr black hole, I'm not sure if I am failing to understand this, or have made some transcription errors in my notes. (Possibly both; my lecturer writes really fast and this material is a real stretch for me!)

We have defined the radius of the event horizon as

r_{+}= r_{g}+ √(r_{g}^{2}- a^{2})

where r_{g}is GM/c^{2}

Then we say that surface area of event horizon A is given by

A = 4∏(r_{+}^{2}+ a^{2}) (no explanation given of where this equation comes from)

and we jump to

A = 8∏r_{g}(r_{g}+ √(r_{g}^{2}+ a^{2})

I can't follow the algebra in that step, is it correct?

Then we expand that, just replacing r_{g}with GM/c^{2}to get

A = 8∏(GM/c^{2})[(GM/c^{2}) + √{(GM/c^{2})^{2}- (J/Mc)^{2}}

Now we write area as

A = 4∏(2GM_{1}/c^{2})

where M_{1}is defined as the "irreversible mass"

M_{1}= M√{(1/2)(1+√{1-(a/r_{g})^{2}}}

I can't make the algebra work to equate this expression for area in terms of irreversible mass into our original expression for area, or to get the intermediate steps to work properly. I have tried looking online for some more help, but most authors wite about "irreducible mass" and I'm not sure if that's the same thing. Unfortunately I'm not clever enough to compare alternative expressions for these equations (that I have found in textbooks) to my lecture notes, and say definitively if I have written it down correctly.

Any help on this would be much appreciated!

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# Area of event horizon and irreversible mass of Kerr black hole

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