# What is irreducible mass

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

Irreducible mass is the energy that cannot be extracted from a black hole via classic processes. For instant, static (Schwarzschild) black holes with no rotation or electrical charge have 100% irreducible mass while Kerr, Kerr-Newman and Reissner–Nordström black holes have <100% irreducible mass.

'The rotational energy and the Coulomb energy are extractable by physical means such as the Penrose process, the superradiance (analogous to stimulated emission in atomic physics) or electrodynamical processes, while the irreducible part cannot be lowered by classical (e.g. non quantum) processes.'

Equations

The total mass-energy of a black hole is-

$$M^2=\frac{J^2}{4M_{ir}^{2}}+\left(\frac{Q^2}{4M_{ir}}+M_{ir}\right)^2$$

where

$$M_{ir}=\frac{1}{2}\sqrt{\left(M+\sqrt{M^2-Q^2-a^2}\right)^2+a^2}$$

where J is angular momentum $(aM)$, Q is electrical charge, a is the spin parameter and M is the gravitational radius $(M=Gm/c^2)$.

The first term (J) is rotational energy, the second term (Q) is coulomb energy and the third term (Mir) is irreducible energy.

The irreducible part cannot be lowered by classical (e.g. non-quantum) processes and can only be lost through Hawking radiation. As high as 29% of a black holes total mass can be extracted by the first process and up to 50% for the second process (but realistically, charged black holes probably only exist in theory or are very short lived as they would probably neutralise quickly after forming).

Maximum spin $J=M^2$, maximum electrical charge $Q=M$, maximum spin parameter $a=M$

when both charge and spin are present in a black hole, $a^2+Q^2\leq M^2$ must apply-

$$J_{max}=M^2\sqrt{1-\frac{Q^2}{M^2}}$$

which means the following should also apply-

$$Q_{max}\equiv M\sqrt{1-\frac{a^2}{M^2}}$$

The total mass of a black hole is analogous with the first law of black hole thermodynamics.

Extended explanation

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