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RE: Mass-inflation in rotating/charged black holes

From page 42

'Evolution Problems of General Relativity' by Jakob Hansen

www.nbi.ku.dk/english/research/phd_theses/phd_theses_2005/2005/jakob_hansen.pdf/

The above is also mentioned in-

'Developments in General Relativity: Black Hole Singularity and Beyond' by I.D.Novikov

http://arxiv.org/PS_cache/gr-qc/pdf/0304/0304052v2.pdf

[itex]r_0[/itex] appears relative to be Cauchy horizon. Any idea how to establish equations/quantities for [itex]m_{con}[/itex] and [itex]m_{exp}[/itex]?

From page 42

'Evolution Problems of General Relativity' by Jakob Hansen

www.nbi.ku.dk/english/research/phd_theses/phd_theses_2005/2005/jakob_hansen.pdf/

The mechanism responsible for the mass inflation can be understood by considering a concentric pair of thin spherical shells moving with the speed of light (e.g. are made of photons) in an empty spacetime without a black hole.

One shell of mass [itex]m_{con}[/itex] contracts, while the other shell of mass [itex]m_{exp}[/itex] expands. The contracting shell, which initially has a radius greater than the expanding one, does not create any gravitational effects inside it, so the expanding shell does not feel the existence of the external shell. On the other hand, the contracting shell moves in the gravitational field of the expanding one. The mutual potential of the gravitational energy of the shells acts as a debit (binding energy) on the gravitational mass energy of the external contracting shell. Before the crossing of the shells, the total mass of both of them, measured by an observer outside both shells, is equal to [itex]m_{con}+m_{exp}[/itex] and is constant because the debit of the absolute increase of the negative potential energy is exactly balanced by the increase of the positive energies of photons blueshifted in the gravitational field of the internal sphere.

When the shells cross one another, at radius [itex]r_0[/itex], the debit is transferred from the contracting shell to the expanding one, but the blueshift of the photons in the contracting shell will not be affected. As a result, the masses of both spheres change. The increase of mass [itex]m_{con}[/itex] is called mass inflation. An exact calculation shows that the new masses [itex]m'_{con}[/itex] and [itex]m'_{exp}[/itex] are:

[tex]m'_{con}=m_{con}+\frac{2m_{con}m_{exp}}{\epsilon}\ \ \ \ \ \ \ m'_{exp}=m_{exp}-\frac{2m_{con}m_{exp}}{\epsilon}[/tex]

where [itex]\epsilon\equiv(r_0-2m_{exp})[/itex] (with units [itex]G=c=1[/itex]). The total mass-energy is, of course, conserved: [itex]m'_{con} + m'_{exp} = m_{con} + m_{exp}[/itex]. If [itex]\epsilon[/itex] is small (the encounter is just outside the horizon of [itex]m_{exp}[/itex]), the inflation of mass of [itex]m_{con}[/itex] can become arbitrary large. It is important to notice that a necessary requirement for mass inflation is that in- and outgoing fluxes are present simultaneously.

The above is also mentioned in-

'Developments in General Relativity: Black Hole Singularity and Beyond' by I.D.Novikov

http://arxiv.org/PS_cache/gr-qc/pdf/0304/0304052v2.pdf

[itex]r_0[/itex] appears relative to be Cauchy horizon. Any idea how to establish equations/quantities for [itex]m_{con}[/itex] and [itex]m_{exp}[/itex]?

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