Kretschmann Scalar for Rotating Black Holes: Curvature Invariant

[stevebd1]

According to wiki under http://en.wikipedia.org/wiki/Kretschmann_scalar" -

..The Kretschmann invariant is

$$K=R_{abcd}R^{abcd}$$

where $R_{abcd}$ is the Riemann curvature tensor. Because it is a sum of squares of tensor components, this is a quadratic invariant..

While Riemann curvature tensor is proportional to tidal forces $(\Delta g=2Gm/r^3)$, in some models of rotating (and charged) black holes, K is considered to diverge at the Cauchy horizon while tidal forces remain finite-

..the mass function (qualitatively $R_{\alpha\beta\gamma\delta} \propto M/r^3$) diverges at the Cauchy horizon (mass inflation). However, Ori showed both for RN and Kerr that the metric perturbations are finite (even though $R_{\mu\upsilon\rho\sigma}R^{\mu\upsilon\rho\sigma}$ diverges) so that an observer would not be destroyed by tidal forces (the tidal distortion would be finite) and could survive passage through the CH..

source- http://relativity.livingreviews.org/open?pubNo=lrr-2002-1&page=node5.html"

For static black holes-

$$\tag{1}C=\frac{48 M^2}{r^6}$$

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

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

another source has-

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

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 $R_{abcd}R^{abcd}$ 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

 

[Mentz114]

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.

 

[stevebd1]

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 http://arxiv.org/PS_cache/gr-qc/pdf/0304/0304052v2.pdf" (page 7) which looks like it might diverge at the weak/null singularity at the CH while tidal forces remain finite.

 

[stevebd1]

From http://uwspace.uwaterloo.ca/bitstream/10012/225/1/NQ30593.pdf" by J. S. F. Chan

equation 2.87-

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

equation 2.63-

$$r_-=M\left(1-\sqrt{1-\frac{q^2}{M^2}}\right)$$

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-

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

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-

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

where '..$\Delta m$ 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

 

[stevebd1]

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.'

 

[stevebd1]

According to various sources-

$\Delta m$ is the mass-energy of the outgoing null shell $S$

$\delta m$ is the mass-energy of the radiation influx

According to this source, http://arxiv.org/PS_cache/gr-qc/pdf/9507/9507047v1.pdf" -

$$m=m_0+\delta m(v)\ \ \ \ \ \delta m=\frac{A}{v^{p-1}}\ \ \ \ \ v\rightarrow+\infty$$

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

$A$ from another paper appears to be '..a constant that depends on the geodesic’s constants of motion..', http://arxiv.org/PS_cache/gr-qc/pdf/9711/9711032v1.pdf" -

$$A=\frac{1}{2}\frac{r_+}{r_-}\left(\frac{r_+}{r_-}+\frac{r_-}{r_+}\right)$$

where $r_+$ is the (outer) event horizon and $r_-$ is the (inner) Cauchy horizon.According to the source in post #3, http://uwspace.uwaterloo.ca/bitstream/10012/225/1/NQ30593.pdf"-

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

where

$$m_1(v_1)=M-\delta m(v_1)$$

$$\delta m(v)=\frac{h}{v^{p-1}}$$

where $h$ is simply described as an 'arbitary' constant and '..the integer exponent $p$ determines the decay rate of ingoing radiation and is greater than or equal to 12 for gravitational radiation..'$v$ is described as the advancing time and appears relative to-

$$v=t+\rho$$

where

$$\rho=\int\frac{dr}{N_{in}(r)}$$

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

$$N_{in}(v,r)=1-\frac{2m_{in}(v)}{r}+\frac{q^2}{r^2}$$

for rotating black holes-

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

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

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

According to the source in post #3, http://uwspace.uwaterloo.ca/bitstream/10012/225/1/NQ30593.pdf"-

$$m_{in}(v)=m_i\left(-\text{exp}(-\kappa_2\,v)\right)$$

$$\bar{m}_{out}(u)=m_o\left(-\text{exp}(-\kappa_1\,u)\right)$$

where $\bar{m}_{out}(u)=m_{out}(-u)$ and $\kappa_1$ is the Killing surface gravity at the Cauchy Horizon relative to $M_1$ (influx) and $\kappa_1$ is the Killing surface at the Cauchy horizon relative to $M_2$ (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

$$\kappa_-=\frac{\sqrt{M^2-q^2}}{r_-^2}$$

for a rotating black hole

$$\kappa_-=\frac{r_+-r_-}{2(r_-^2+a^2)}$$

$m_{in}$ and $m_{out}$ seem synonymous to the quantities $m_{exp}$ and $m_{con}$ from https://www.physicsforums.com/showthread.php?t=290542" 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 http://en.wikipedia.org/wiki/Cauchy_horizon" -

..Waves traveling in Misner space approaching a Cauchy horizon would receive a frequency boost proportional to-

$$B_\nu \propto \sqrt{\frac{1+v}{1-v}\ }$$

each time they pass through the identification world line. As this happens infinitely many times while approaching the horizon, the stress-energy tensor diverges at the horizon. Presumably, this prevents spacetime from developing closed time-like curves that would otherwise be feasible...

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

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

________________________________________________

UPDATE-

From http://arxiv.org/PS_cache/gr-qc/pdf/9403/9403019v1.pdf" (which was co-written by W. Israel), page 3-

For pure inflow (first treated by Hiscock) $T_{ab}$ is lightlike and the source term of the wave equation vanishes: $m$ remains bounded. It becomes a function $m(v)$ of advanced time only. To reproduce the fallout from a radiative tail, it should take the asymptotic form

$$\tag{4}m(v) = m_0-av^{-(p-1)}\ \ \ \ \ \ (v\rightarrow\infty)$$

where $m_0, a$ are constants. Observers falling toward CH at radius $r_0=m_0-(m_0^2-e^2)^\frac{1}{2}$ and infinite $v$ see an exponentially blueshifted energy flux $\sim(a/r^2)v^{-p}e^{2\kappa_0\,v}$ (where $\kappa_0=(m_0^2-e^2)^\frac{1}{2}/r_0^2$ is the inner "surface gravity"). But this has little effect on the geometry: the Weyl curvature scalar

$$\tag{5}-\Psi_2=(1/2)C^{\theta\varphi}_{\ \ \theta\varphi}=(m-e^2/r)r^{-3}$$

remains bounded at CH.

This state of affairs changes radically when one turns on a simultaneous outflux. In (the wave equation), $T_{ab}T^{ab}$ now functions as an exponentially divergent source near CH. The mass function and Weyl curvature diverge like

$$\tag{6}m(v,r)\sim v^{-p}e^{\kappa_0\,v}\ \ \ \ \ \ (v\rightarrow\infty, r<r_+)$$

near CH. This bizarre phenomenon has been dubbed "mass inflation". (It is not detectable externally: outside the event horizon EH $(r>r_+)$, $m$ stays bounded and resembles $(4)$ at late times.)

 

[stevebd1]

$\Delta m$ is the mass-energy of the outgoing null shell $S$

$\delta m$ is the mass-energy of the radiation influxA summary of information from post #6-

$$\] \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_+)\\ \[$$

where

$$M=m_0,\ \ \ a=1,\ \ \ p=12,\ \ \ m(v,r)=\Delta m(v)$$Cauchy horizon Killing surface gravity-

$$\kappa_0=\frac{r_+-r_-}{2(r_-^2+a^2)}$$


equation for m₂(v)-

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

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 $v$, 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)-

$$v=\frac{1}{\beta^2}+\beta^2$$

where

$$\beta=\sqrt{\frac{(r_+-r_-)}{(r-r_-)}}$$

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.