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Curvature invariants

  1. Feb 2, 2009 #1
    According to wiki under Kretschmann scalar-
    While Riemann curvature tensor is proportional to tidal forces [itex](\Delta g=2Gm/r^3)[/itex], in some models of rotating (and charged) black holes, K is considered to diverge at the Cauchy horizon while tidal forces remain finite-
    source- Nature of the singularity in charged or rotating black holes

    For static black holes-

    [tex]\tag{1}C=\frac{48 M^2}{r^6}[/tex]

    which remains finite at 2M and can be loosely translated for rotating black holes to-

    [tex]\tag{2}C(r,a)=\frac{12 (M+\sqrt{M^2-a^2})^2}{r^6}[/tex]

    another source has-

    [tex]\tag{3}R_{abcd}R^{abcd}=\frac{48M^2(r^2-a^2cos^2\theta)[(r^2+a^2cos^2\theta)^2-16r^2a^2cos^2\theta]}{(r^2+a^2cos^2\theta)^6}[/tex]

    yet neither seem relative to the Cauchy horizon (i.e. none seem to diverge at the CH while tidal forces remain finite). Is there an equation for K that takes into the account that the Cauchy horizon is a null singularity and that M/r^3 remains finite even though [itex]R_{abcd}R^{abcd}[/itex] diverges? Also, what are the units for K? are they simply geometric or can they be multiplied by anything (such as c^2 or G/c^2) and recognisable as SI units?

    Steve

    (1)http://members.tripod.com/~albert51/cool.html
    (2)http://members.tripod.com/~albert51/bhole.htm
    (3)'The Kerr spacetime: A brief introduction' http://arxiv.org/abs/0706.0622v3 page 7

    ________________________________

    UPDATE-
    I did find this paper which shows a curvature scalar equation by A. Ori relative to rotating black holes-

    http://arxiv.org/PS_cache/gr-qc/pdf/0304/0304052v2.pdf page 7
     
    Last edited: Feb 2, 2009
  2. jcsd
  3. Feb 2, 2009 #2

    Mentz114

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    Gold Member

    Your (1) is for the Schwarzschild space-time which does not have an inner horizon. It's only singular at r=0. The dimensions of the quadratic invariants is L-4.
     
  4. Feb 3, 2009 #3
    Thanks for the reply, Mentz. I'm aware that (1) is for Schwarzschild spacetime; when I stated that 'neither seem relative to the Cauchy horizon', I was referring to equations (2) and (3) in respect that they didn't seem to diverge at the CH (unless, of course, r=0 is at the CH but this would imply that the tidal forces would diverge also). There is an equation in the I.D. Novikov paper (page 7) which looks like it might diverge at the weak/null singularity at the CH while tidal forces remain finite.
     
    Last edited: Feb 3, 2009
  5. Feb 6, 2009 #4
    From The Universality of Mass Inflation Inside Black Holes by J. S. F. Chan

    equation 2.87-

    [tex]R_{abcd}R^{abcd}=\frac{8}{r^8}\left[6m_2^{\ 2}(v_2)r^2-12m_2^{\ 2}(v_2)q^2r+7q^4\right][/tex]

    equation 2.63-

    [tex]r_-=M\left(1-\sqrt{1-\frac{q^2}{M^2}}\right)[/tex]

    q here represents charge but it's often considered synonymous with the spin parameter a (the equation above for r- is identical when calculating the inner Cauchy horizon for rotating black holes) and Reissner-Nordström metric is sometimes used to establish principles that can be applied to both charged and rotating black holes. So the equation could be re-written as-

    [tex]R_{abcd}R^{abcd}=\frac{8}{r^8}\left[6m_2^{\ 2}(v_2)r^2-12m_2^{\ 2}(v_2)a^2r+7a^4\right][/tex]

    According to the source-

    '..This implies that the spacetime has a scalar curvature singularity at the Cauchy horizon induced by the divergence of the inner mass parameter m2.'

    which implies that there is a variable equation for m2 which causes it to blow up at r-. The one equation I found was 2.66-

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

    where '..[itex]\Delta m[/itex] is the mass energy of the outgoing null shell S. ..the mass parameter m2 increases without bound when S approaches the Cauchy horizon.'

    I'm currently working my way through the chapter to establish how exactly m2 is calculated but if this is apparent to anyone, I'd appreciate the feedback.

    Steve
     
    Last edited: Feb 7, 2009
  6. Feb 10, 2009 #5
    This is a relatively old paper but looks interesting-

    http://arxiv.org/abs/gr-qc/9701058
    'Mass Inflation in Quantum Gravity' by Ichiro Oda

    'Using the canonical formalism for spherically symmetric black hole inside the apparent horizon we investigate the mass inflation in the Reissner-Nordström black hole in the framework of quantum gravity. It is shown that like in classical gravity the combination of the effects of the influx coming from the past null infinity and the outflux backscattered by the black hole's curvature causes the mass inflation even in quantum gravity. The results indicate that the effects of quantum gravity neither alter the classical picture of the mass inflation nor prevent the formation of the mass inflation singularity.'
     
  7. Feb 11, 2009 #6
    According to various sources-

    [itex]\Delta m[/itex] is the mass-energy of the outgoing null shell [itex]S[/itex]

    [itex]\delta m[/itex] is the mass-energy of the radiation influx

    According to this source, link-

    [tex]m=m_0+\delta m(v)\ \ \ \ \ \delta m=\frac{A}{v^{p-1}}\ \ \ \ \ v\rightarrow+\infty[/tex]

    where '..The rate of contraction is fully determined by Price’s power law damping of the radiative tail [itex]~1/v^{(p-1)}[/itex] where [itex]p\geq11[/itex]..' and [itex]m_0[/itex] is presumably mass as observed from infinity.

    [itex]A[/itex] from another paper appears to be '..a constant that depends on the geodesic’s constants of motion..', link -

    [tex]A=\frac{1}{2}\frac{r_+}{r_-}\left(\frac{r_+}{r_-}+\frac{r_-}{r_+}\right)[/tex]

    where [itex]r_+[/itex] is the (outer) event horizon and [itex]r_-[/itex] is the (inner) Cauchy horizon.


    According to the source in post #3, link-

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

    where

    [tex]m_1(v_1)=M-\delta m(v_1)[/tex]

    [tex]\delta m(v)=\frac{h}{v^{p-1}}[/tex]

    where [itex]h[/itex] is simply described as an 'arbitary' constant and '..the integer exponent [itex]p[/itex] determines the decay rate of ingoing radiation and is greater than or equal to 12 for gravitational radiation..'


    [itex]v[/itex] is described as the advancing time and appears relative to-

    [tex]v=t+\rho[/tex]

    where

    [tex]\rho=\int\frac{dr}{N_{in}(r)}[/tex]

    where [itex]N_{in}=g_{tt}[/itex] relative to [itex]m_{in}[/itex]. For example, for charged black holes, [itex]N_{in}[/itex] would be-

    [tex]N_{in}(v,r)=1-\frac{2m_{in}(v)}{r}+\frac{q^2}{r^2}[/tex]

    for rotating black holes-

    [tex]N_{in}(v,r)=1-\frac{2m_{in}(v)r}{\rho^2}\ \ \ \ \ \rho^2=r^2+a^2cos^2\theta[/tex]

    which is derivative of the Vaidya solution which '..represents a Schwarzschild black hole being irradiated by ingoing radiation..'

    [itex]u[/itex] is described as the retarded time, [itex]u=t-\rho[/itex] where [itex]N_{out}=g_{tt}[/itex] relative to [itex]m_{out}[/itex] and presumably contributes to the quantity [itex]\Delta m[/itex] also. Both [itex]v[/itex] and [itex]u[/itex] also seem derivative of Kruskal-Szekeres & Eddington-Finkelstein coordinates.

    According to the source in post #3, link-

    [tex]m_{in}(v)=m_i\left(-\text{exp}(-\kappa_2\,v)\right)[/tex]

    [tex]\bar{m}_{out}(u)=m_o\left(-\text{exp}(-\kappa_1\,u)\right)[/tex]

    where [itex]\bar{m}_{out}(u)=m_{out}(-u)[/itex] and [itex]\kappa_1[/itex] is the Killing surface gravity at the Cauchy Horizon relative to [itex]M_1[/itex] (influx) and [itex]\kappa_1[/itex] is the Killing surface at the Cauchy horizon relative to [itex]M_2[/itex] (outflux) (for a full break-down, see pages 34-38 of the above link)

    Killing surface gravity at the Cauchy horizon for a charged black hole

    [tex]\kappa_-=\frac{\sqrt{M^2-q^2}}{r_-^2}[/tex]

    for a rotating black hole

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

    [itex]m_{in}[/itex] and [itex]m_{out}[/itex] seem synonymous to the quantities [itex]m_{exp}[/itex] and [itex]m_{con}[/itex] from this thread which implies divergence at the (inner) Cauchy horizon.

    The divergence at the Cauchy horizon is backed up to some extent by an equation from the wiki page for Cauchy horizon-

    I'm still not entirely clear on how to establish [itex]\Delta m[/itex] but it appears that [itex]\delta m[/itex], which falls into the black hole, reduces to zero at the Cauchy horizon while [itex]\Delta m[/itex] grows without bound (presumably [itex]\infty[/itex] at the Cauchy horizon) implying [itex]m_2(v_2)=m_1(v_1)+\Delta m(v_2)[/itex] diverges at [itex]r_-[/itex].

    I'll probably add to this as I work through the info but any feedback/insight would be appreciated.

    ________________________________________________

    UPDATE-

    From this paper (which was co-written by W. Israel), page 3-

     
  8. Feb 24, 2009 #7
    [itex]\Delta m[/itex] is the mass-energy of the outgoing null shell [itex]S[/itex]

    [itex]\delta m[/itex] is the mass-energy of the radiation influx


    A summary of information from post #6-

    [tex]
    \]
    \textit{m}_2(v)=m_1(v)+\Delta m(v)\\
    \\
    m_1(v)=M-\delta m(v)\\
    \\
    \delta m(v)=v^{-(p-1)}\\
    \\
    m(v) = m_0-av^{-(p-1)}\ \ \ \ \ \ (v\rightarrow\infty)\\
    \\
    m(v,r)=v^{-p}e^{\kappa_0\,v}\ \ \ \ \ \ (v\rightarrow\infty, r<r_+)\\
    \[
    [/tex]

    where

    [tex]M=m_0,\ \ \ a=1,\ \ \ p=12,\ \ \ m(v,r)=\Delta m(v)[/tex]


    Cauchy horizon Killing surface gravity-

    [tex]\kappa_0=\frac{r_+-r_-}{2(r_-^2+a^2)}[/tex]


    equation for m2(v)-

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

    where the second part of the equation (Δm) is only applicable inside the EH (r+→r-). m2(v) remains equal to M until within about 99.99995% of the Cauchy horizon where it blows up exponentially in a Dirac delta function manner*. Outside the event horizon EH (r>r+), m stays bounded and resembles m1(v) at late times.

    *The lower the spin, the sooner this happens; the example shown above is for a BH with a spin of a/M=0.95, for a BH with a/M=0.1, m2(v) begins to blow up at within about 92% of the Cauchy horizon (i.e. at r=r-/0.92).


    The only area I'm a bit grey on is [itex]v[/itex], creating a make-shift equation for an ingoing coordinate that tends to infinity at the Cauchy horizon and at large radii (so that m1(v) would tend to M in both cases)-

    [tex]v=\frac{1}{\beta^2}+\beta^2[/tex]

    where

    [tex]\beta=\sqrt{\frac{(r_+-r_-)}{(r-r_-)}}[/tex]

    which is zero at ∞, 1 at r+ and ∞ at r- (compared to the conventional quantity for β which is zero at ∞, 1 at r+, +1 in the shallow region of the BH, 1 at r- and zero at r=0).

    There is a suggestion that the entire space within the Cauchy horizon is light-like but Kerr metric suggests it becomes time-like again. The quantity for β becomes a negative square root at r<r- implying an 'imaginary' (or new) space inside the Cauchy horizon; if we carry on the equation with r smaller than r-, m2(v) almost instantly becomes M again on the inside of the Cauchy horizon.

    While v isn't correct, it does give some idea of the gradient to expect at the Cauchy horizon and with a bit of refinement (Δm needs to be zero at r+), m2(v) can replace M in eq. (3), post #1 to provide the curvature scalar for a rotating black hole incorporating mass inflation at the Cauchy horizon.
     
    Last edited: Feb 24, 2009
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