Can a Maximally Rotating Black Hole be Defined by the Kerr Metric?

[Imax]

Is it possible to define a maximally rotating black hole? Could it be:

$$4\alpha^2 = r_s^2$$

(i.e. http://en.wikipedia.org/wiki/Kerr_metric, inner event horizon)

 

[stevebd1]

Is it possible to define a maximally rotating black hole? Could it be:

$$4\alpha^2 = r_s^2$$

(i.e. http://en.wikipedia.org/wiki/Kerr_metric, inner event horizon)

Hello Imax

This can be reduced to a=M or a/M=1 where r_s=2M.

 

[Imax]

Hi steve:

Having problems with LaTex. see attached

 

[stevebd1]

It's also worth noting that when charge is included, J_max becomes-

$$J_{max}=M^2\sqrt{1-\frac{Q^2}{M^2}}$$

which means the following should also apply-

$$Q_{max}\equiv M\sqrt{1-\frac{a^2}{M^2}}$$

The above can reduce (for a maximal BH) to-

$a^2+Q^2=M^2$

where $M=Gm/c^2,\ a=j/mc$ and $Q=C\sqrt(G k_e)/c^2$

where M and Q are mass and charge in geometric units and m and C are the SI units respectively, a is the spin parameter (normally J is used for both geometric and SI units for angular momentum but for some clarity I've used j to represent SI units and J to represent geometric units).

where there's no charge-

$$J_{max}=M^2$$

for a non-maximal, non-charged rotating black hole-

$$J=Ma$$

(while wiki are happy to use $\alpha$ to represent the spin parameter, this could get confusing later on when using the redshift or reduction factor in Kerr metric which is more commonly represented by $\alpha$ also).

Event horizons for a black hole with both spin and charge (Kerr-Newman) is represented by-

$$r_{\pm}=M \pm \sqrt{M^2-Q^2-a^2}$$

 

[Imax]

(while wiki are happy to use $\alpha$ to represent the spin parameter, this could get confusing later on when using the redshift or reduction factor in Kerr metric which is more commonly represented by $\alpha$ also).

So it would be less confusing if I used the symbole $a$? What would be a common symbole (good symbole) to use for the dimensionless spin parameter:

$$\frac{cJ}{GM^2}$$

I've seen $a_{*}$ and $\chi$.

 

[stevebd1]

What would be a common symbole (good symbole) to use for the dimensionless spin parameter:

$$\frac{cJ}{GM^2}$$

I've seen $a_{*}$ and $\chi$.

$a_{*}$ or $a^*$ appear to be used the most to represent a/M though I've also seen $\bar{a}$.

 

[Imax]

Having problems with LaTex. see attached

I think I may have solved my problem with Latex. This is what was in the attachement:

$$4\alpha^2=4\left ( \frac{J}{Mc} \right )^2=4\frac{J^2}{M^2c^2}=r_s^2=\left ( \frac{2GM}{c^2} \right )^2=4\frac{G^2M^2}{c^4}$$

Isolating J gives the maximum angular momentum as:

$$J_{max}=\frac{GM^2}{c}$$

And also limits a to:

$$\alpha_{max}=\frac{J_{max}}{Mc}=\frac{1}{Mc}\left ( \frac{GM^2}{c} \right )=\frac{GM}{c^2}=\frac{1}{2}r_s$$

Seems like hitting the preview button too many times is not a good idea.

 

[Imax]

The angular momentum J for any black hole should be between 0 and $J_{max}$, so, for any black hole, J can be defined as some fraction of the maximum:

$$J=a_*J_{max}$$

$$0\leq a_*\leq 1$$

with $a_*$ a dimensionless spin parameter:

$$a_*=\frac{J}{J_{max}}=J\frac{1}{J_{max}}=\frac{cJ}{GM^2}$$

The value $a_*=0$ corresponds to a Schwarzschild black hole and $a_*=1$ to an extreme Kerr black hole. According to this equation, the value of $a$ for any black hole is:

$$a=\frac{J}{Mc}=a_*\frac{J_{max}}{Mc}=\frac{a_*}{2}r_s$$

If, for any black hole, the radius $r$ can be expressed as a multiple of $r_s$ then

$$r=nr_s$$

$$n=\frac{r}{r_s}$$

$$n\geq 1$$

Substituting $r$ with $nr_s$ and $a$ with

$$\frac{a_*}{2}r_s$$

can simplify (??) some equations.

 

[Imax]

As an example, according to Wiki, the Kerr Metric is equivalent to a co-rotating reference frame that rotates with angular speed $\Omega$, and this angular speed depends on both the radius $r$ and the colatitude $\theta$:

$$\Omega =- \frac{g_{t\phi}}{g_{\phi \phi}}=\frac{r_sarc}{\rho^2(r^2+a^2)+r_sa^2 r \sin^2\theta }$$

$$\rho ^2=r^2+a^2\cos^2\theta$$

Substituting $r$ with $nr_s$ and $a$ with

$$\frac{a_*}{2}r_s$$

gives, after about a page of math, something like:

$$\Omega =\frac{c}{r_s}\left ( \frac{8na_*}{16n^4+4n^2a_*^2+4n^2a_*^2\cos^2\theta+4na_*^2\sin^2\theta+a_*^4\cos^2\theta } \right )$$

Or

$$\Omega =\frac{c}{r_s}p(n,a_*,\theta)$$

The angular speed is given by the speed of light divided by the Schwarzschild radius times a polynomial $p(n,a_*,\theta)$ which is a dimensionless scale factor given by:

$$p(n,a_*,\theta)=\frac{8na_*}{16n^4+4n^2a_*^2+4n^2a_*^2\cos^2\theta+4na_*^2\sin^2\theta+a_*^4\cos^2\theta }$$

$$n=\frac{r}{r_s}=\frac{rc^2}{2GM}\geq 1$$

$$a_*=\frac{J}{J_{max}}=\frac{Jc}{GM^2}\leq 1$$

 

[Cold Winter]

?

Are you trying to suggest that a singularity would ease to exist if its radial velocity was approx. 15% of "c"?

 

[Imax]

?

Are you trying to suggest that a singularity would ease to exist if its radial velocity was approx. 15% of "c"?

If $J>J_{max}$ then the event horizon becomes imaginary with components of
$\sqrt{-1}$. The event horizon could disappear, leaving a naked singularity o=).

Where did 15% of c come from?