A Exact meaning of the mass M in the Kerr metric event horizon formula?

[Scott92]

TL;DR Summary

Trying to determine whether the mass M in the kerr metric event horizon formula is the irreducible mass or the total mass-energy.

Posting this as I have so far not been able to find a straightforward answer to the following question. The formula for the outer event horizon of a kerr black hole is given by the following equation:

$$r_+ = \frac{GM}{c^2}\left(1+\sqrt{1-\frac{J^2c^2}{M^4G^2}}\right)$$
Where $J$ is the angular momentum of the black hole. My question is this: does the mass $M$ in this equation correspond to the irreducible mass of the black hole? Or does it correspond to the total mass equivalent of the black hole when the rotational mass-energy is included?

Mathematically speaking, does $M = M_{irr}$? Or does $M = \sqrt{M_{irr}^2 + \frac{J^2c^2}{4M_{irr}^2G^2}}$?

I should also stress that, despite my best efforts, I've yet to come across a textbook that answers this explicitly, with the distinction seemingly being taken for granted. Even when talking to others informally about this, there does not seem to be a consensus (there are already answers/comments stating opposing viewpoints in a Physics StackExchange thread that I made). So if anyone has an answer, a source would be greatly appreciated if possible, thanks!

 

[PeterDonis]

does the mass $M$ in this equation correspond to the irreducible mass of the black hole?

No. That should be obvious from the formula you give later on, which you will find in most treatments of Kerr spacetime, and which gives a relationship between $M$ and $M_{irr}$.

does it correspond to the total mass equivalent of the black hole when the rotational mass-energy is included?

"Rotational mass-energy" is not a well-defined concept for Kerr spacetime, because it is a vacuum solution so there is no matter "rotating" whose rotational kinetic energy could be assessed.

The physical meaning of $M$ can be read off from the limit of the metric as $r \to \infty$, which tells you that $M$ is the Keplerian mass of the hole as measured by an observer at rest at infinity. (This also tells you that $M$ cannot be the same thing as the irreducible mass, which has a different physical meaning.)

I've yet to come across a textbook that answers this explicitly, with the distinction seemingly being taken for granted.

When a textbook explicitly gives you two different symbols, $M$ and $M_{irr}$, it does take it for granted that you will understand that those symbols refer to two different quantities. Particularly when an explicit formula is given relating the two quantities that makes it obvious that they are not equal. It will also expect you to understand that it knows which of the two quantities it means in any particular formula, such as the formula for the Kerr metric, so you can take it for granted that if $M$ appears in the metric, it's not a mistake, they didn't mean to say $M_{irr}$ and just forget to include the subscript.

 

[PeterDonis]

if anyone has an answer, a source would be greatly appreciated

Irreducible mass is discussed in pretty much every GR textbook I've seen that discusses Kerr spacetime at all. See, for example, Wald section 12.4 or Misner, Thorne & Wheeler Box 33.4.

 

[Scott92]

When a textbook explicitly gives you two different symbols, $M$ and $M_{irr}$, it does take it for granted that you will understand that those symbols refer to two different quantities. Particularly when an explicit formula is given relating the two quantities that makes it obvious that they are not equal. It will also expect you to understand that it knows which of the two quantities it means in any particular formula, such as the formula for the Kerr metric, so you can take it for granted that if $M$ appears in the metric, it's not a mistake, they didn't mean to say $M_{irr}$ and just forget to include the subscript.

I did not get both of these formulas from the same source. I merely came across the event horizon formula with the symbol M simply labelled as "the mass of the black hole". I then independently learnt of the formula for the irreducible mass from another source. So yes, the distinction was entirely obvious to me - I just had yet to encounter an explicit explanation from a single source that clarified which mass was the one referred to in the event horizon formula.

Anyways, I appreciate the answer since you actually provided a corresponding source, so thanks for that.

 

[PeterDonis]

I did not get both of these formulas from the same source.

You will, though, if you look at the sources I provided. Plenty of other textbook treatments will also give both of them.