# Mass-inflation in rotating/charged black holes

1. Feb 7, 2009

### 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 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 $m_{exp}$?

Last edited: Feb 8, 2009
2. Feb 26, 2009

### stevebd1

Re: Mass-inflation

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" [Broken]

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

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

Last edited by a moderator: May 4, 2017
3. Feb 27, 2009

### 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 realised 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?

Last edited: Feb 28, 2009
4. Mar 2, 2009

### stevebd1

Re: Mass-inflation

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

Last edited by a moderator: Apr 24, 2017