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?