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Mass-inflation in rotating/charged black holes

  1. Feb 7, 2009 #1
    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]?
     
    Last edited: Feb 8, 2009
  2. jcsd
  3. Feb 26, 2009 #2
    Re: Mass-inflation

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

    [tex]
    \]
    \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_+)
    \[ [/tex]

    where [itex]\delta m[/itex] is the mass-energy of the radiation influx, [itex]\Delta m[/itex] is the mass-energy of the outgoing null shell [itex]S[/itex] (r<r+), h is a constant regarding the nature of the gravitational source, [itex]\kappa_0[/itex] 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)-

    [tex]m_2(v)=M-hv^{-(p-1)}+v^{-p}e^{\kappa_0\,v}[/tex]

    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 [itex]1/v^p[/itex]. In other words, when the shell [itex]S[/itex] 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-

    [tex]v=r*+t[/tex]

    [tex]u=r*-t[/tex]

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

    [tex]\frac{dr}{dr^*}=\frac{(r-r_+)(r-r_-)}{(r^2+a^2)}[/tex]

    and '..The event horizon and the IH correspond to [itex]u=-\infty[/itex] and [itex]v=\infty[/itex], respectively..'

    I understand the 'tortoise' coordinate, [itex]r^*[/itex] 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 [itex]1/v^p[/itex], 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
  4. Feb 27, 2009 #3
    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-

    [tex]r^\star=r+2M\,\ln\left|\frac{r}{2M}-1\right|[/tex]

    where [itex]r^\star[/itex] is the tortoise coordinate

    which is synonymous with-

    [tex]dr^\star=\frac{r}{r-2M}\,dr[/tex]

    and

    [tex]dv=dt+dr^\star[/tex]

    When looking at Minkowski space, I realised that [itex]v=t+r[/itex] 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 [itex]r^\star[/itex] at large r is approx. equal to t (producing the ~45 degree line), the closer we get to 2M, the more [itex]r^\star[/itex] 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-

    [tex]dr^\star=\frac{r^2+a^2}{\Delta}\,dr[/tex]

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

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

    Does anyone know how to derive the 'natural logarithm' equation for [itex]r^\star[/itex] 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
  5. Mar 2, 2009 #4
    Re: Mass-inflation

    tortoise coordinate for Kerr black hole-

    [tex]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|[/tex]

    which is derived from [itex]dr^\star/dr=(r^2+a^2)/\Delta[/itex] 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 [itex]v=t+r\equiv2r[/itex], for curved spacetime [itex]v=t+r^\star \neq 2r[/itex]

    [itex]v=t+r^\star[/itex] provides v=∞ at large radii, v=-∞ at r+, v=∞ at r- and finite at r=0; the opposite applies for [itex]u=t-r^\star[/itex] (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, [itex]\bar{m}_{out}(u)=m_{out}(-u)[/itex]..'. This means that in the following equation-

    [tex]m_2(v_2)=m_1(v_1)+\Delta m(v_2)[/tex]

    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
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