Mass-inflation in rotating/charged black holes

[stevebd1]

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 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 $m_{con}$ contracts, while the other shell of mass $m_{exp}$ 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 $m_{con}+m_{exp}$ 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 $r_0$, 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 $m_{con}$ is called mass inflation. An exact calculation shows that the new masses $m'_{con}$ and $m'_{exp}$ are:

$$m'_{con}=m_{con}+\frac{2m_{con}m_{exp}}{\epsilon}\ \ \ \ \ \ \ m'_{exp}=m_{exp}-\frac{2m_{con}m_{exp}}{\epsilon}$$​

where $\epsilon\equiv(r_0-2m_{exp})$ (with units $G=c=1$). The total mass-energy is, of course, conserved: $m'_{con} + m'_{exp} = m_{con} + m_{exp}$. If $\epsilon$ is small (the encounter is just outside the horizon of $m_{exp}$), the inflation of mass of $m_{con}$ 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

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

 

[stevebd1]

Regarding mass-inflation at the Cauchy horizon in a BH-

$$\] \textit{m}_2(v)=m_1(v)+\Delta m(v,r)\\ \\ m_1(v)=M-\delta m(v)\\ \\ \delta m(v)=hv^{-(p-1)}\ \ \ \ \ \ \ (v\rightarrow\infty)\\ \\ \Delta m(v,r)=v^{-p}e^{\kappa_0\,v}\ \ \ \ \ \ \ (v\rightarrow\infty,\ r<r_+) \[$$

where $\delta m$ is the mass-energy of the radiation influx, $\Delta m$ is the mass-energy of the outgoing null shell $S$ (r<r₊), h is a constant regarding the nature of the gravitational source, $\kappa_0$ is the Cauchy horizon Killing surface gravity, p determines the decay rate of ingoing radiation (≥11 for gravitational radiation) and v is ingoing null coordinates.

Producing a final equation for m₂(v)-

$$m_2(v)=M-hv^{-(p-1)}+v^{-p}e^{\kappa_0\,v}$$

which according to one source '..Δm diverges when v goes to infinity because the exponential growth is dominant compared to the inverse power attenuation of $1/v^p$. In other words, when the shell $S$ approaches the Cauchy horizon (v→∞), its mass increases without bound..'

Another source also states that m₁(v) tends to M at infinity and at the Cauchy horizon.

The one area I'm having an issue with is establishing the ingoing null coordinates.

While I'm familiar with dv in Eddington-Finkelstein coordinates, I'm not too familiar with the following expressions-

$$v=r*+t$$

$$u=r*-t$$

according to http://arxiv.org/PS_cache/gr-qc/pdf/0103/0103012v1.pdf"

$$\frac{dr}{dr^*}=\frac{(r-r_+)(r-r_-)}{(r^2+a^2)}$$

and '..The event horizon and the IH correspond to $u=-\infty$ and $v=\infty$, respectively..'

I understand the 'tortoise' coordinate, $r^*$ but I'm not exactly clear on what the actual quantity of t is. Is it advancing time? I've also seen it sometimes expressed as ct implying it might be multiplied by c.

While it seems clear that v tends to infinity at the Cauchy horizon, what is the quantity of v at the outer event horizon and at large radii? (based on $1/v^p$, ensuring that m₁(v) tends to M at infinity and at the Cauchy horizon). dr/dr* suggests that r* approaches ∞ at both r₊ and r_-, how is this accommodated in v without Δm blowing up at r₊?

Any insight would be appreciated.Sources-

'Oscillatory null singularity inside realistic spinning black holes' by Amos Ori
http://arxiv.org/PS_cache/gr-qc/pdf/0103/0103012v1.pdf

'Structure of the Inner Singularity of a Spherical Black Hole' by A. Bonanno, S. Droz, W. Israel and S.M. Morsink
http://arxiv.org/PS_cache/gr-qc/pdf/9403/9403019v1.pdf

Eddington-Finkelstein coordinates
http://en.wikipedia.org/wiki/Eddington-Finkelstein_coordinates

 

[stevebd1]

ingoing null coordinates

Having looked a bit more into the properties of ingoing null coordinates and v, I went back to the basics and looked at a static black hole where-

$$r^\star=r+2M\,\ln\left|\frac{r}{2M}-1\right|$$

where $r^\star$ is the tortoise coordinate

which is synonymous with-

$$dr^\star=\frac{r}{r-2M}\,dr$$

and

$$dv=dt+dr^\star$$

When looking at Minkowski space, I realized that $v=t+r$ produces a 45 degree line across the graph implying that r=t in flat space so it might be fair to say that t is the same quantity (if not the same units) as r, would this be correct?

When the tortoise coordinate is incorporated, things appear to get interesting. While $r^\star$ at large r is approx. equal to t (producing the ~45 degree line), the closer we get to 2M, the more $r^\star$ tends to negative infinity and the more 'vertical' v becomes. This is in conjunction with the statement on the wiki page '..The tortoise coordinate approaches −∞ as r approaches the Schwarzschild radius r=2M..' but based on t being equal to r, this also implies that v will tend to −∞ at 2M also, flipping from positive to negative on the way.In respect of rotating black holes, the change in the tortoise coordinate is-

$$dr^\star=\frac{r^2+a^2}{\Delta}\,dr$$

where $\Delta=r^2-2Mr+a^2$ which is equivalent to $(r-r_+)(r-r_-)$ from post #2

Again, $v=t+r^\star$ and $u=t-r^\star$ where '..the event horizon and the IH correspond to u=−∞ and v=∞, respectively..'.

Does anyone know how to derive the 'natural logarithm' equation for $r^\star$ in respect of a rotating black hole (as with the static black hole) in order to establish v for a rotating BH?

 

[stevebd1]

tortoise coordinate for Kerr black hole-

$$r^\star(r)=r+\frac{2Mr_+}{r_+-r_-} \ln\left|\frac{r-r_+}{2M}\right|-\frac{2Mr_-}{r_+-r_-} \ln\left|\frac{r-r_-}{2M}\right|$$

which is derived from $dr^\star/dr=(r^2+a^2)/\Delta$ and reduces to the static solution when a/M=0.

source-
http://web.mit.edu/sahughes/www/Papers/h20000415.pdf page 11

while for Minkowski spacetime $v=t+r\equiv2r$, for curved spacetime $v=t+r^\star \neq 2r$

$v=t+r^\star$ provides v=∞ at large radii, v=-∞ at r₊, v=∞ at r_- and finite at r=0; the opposite applies for $u=t-r^\star$ (i.e. u=-∞ at large radii, u=∞ at r₊, etc.).

'..the event horizon and the IH correspond to u=−∞ and v=∞, respectively..' seems to imply that while the above equation provides a positive quantity for u at r₊, the sign is reversed when calculating Δm in some mass-inflation models, which is in part explained by the following '..since the outflow stream is inside the black hole, we must switch the sign of u in order to have an increasing mass parameter, $\bar{m}_{out}(u)=m_{out}(-u)$..'. This means that in the following equation-

$$m_2(v_2)=m_1(v_1)+\Delta m(v_2)$$

v₁=v and v₂ can be in some way related to -u which would be synonymous with v being the ingoing radiation flux, -u being the outgoing null shell and m₂ being the resulting mass-inflation at the Cauchy horizon.

This source provides a slightly different equation for the Kerr tortoise coordinate (which also provides a solution for a/M=1)-
http://arxiv.org/PS_cache/gr-qc/pdf/0103/0103054v1.pdf page 5