Kerr Quantum...
Black holes are predictions of Albert Einstein's theory of general relativity. In particular, they occur in the Schwarzschild Metric, one of the earliest and simplest solutions to Einstein's equations, found by Karl Schwarzschild in 1915. This solution describes the curvature of spacetime in the volume of a non-rotating (static) and spherically symmetric object:
Schwarzschild Metric:
ds^2 = - \left(1 - \frac{2M}{r} \right) dt^2 + \left(1 - \frac{2M}{r} \right)^{-1} dr^2 + r^2 d \Omega^2
Solid angle standard element:
d \Omega^2 = d \theta^2 + \cos^2 \theta d \phi^2
The neutron degenerancy mass is called the Oppenheimer Mass Limit for a neutron star.
Oppenheimer Mass Limit:
M_o = 3.2M_\odot
The radial solution for a spherically symmetric Oppenheimer BH sphere horizon is:
r_o = \sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}}
The radial Schwarzschild Solution for a spherically symmetric non-rotating gravitational BH photon sphere is:
r_s = \frac{3 G M_o}{2c^2}
radial criterion:
r_o = r_s
Schwarzschild-Oppenheimer Density Solution for spherically symmetric non-rotating gravitational Schwarzschild-Oppenheimer BH photon sphere:
\rho_o = \left( \frac{2c^6}{9 \pi G^3 M_o^2} \right)
The radial Schwarzschild Solution for a spherically symmetric non-rotating gravitational BH event horizon is:
r_s = \frac{2 G M_o}{c^2}
\sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}} = \frac{2 G M_o}{c^2}
Schwarzschild-Oppenheimer Density Solution for spherically symmetric non-rotating gravitational Schwarzschild-Oppenheimer BH:
\rho_o = \left( \frac{3c^6}{32 \pi G^3 M_o^2} \right)
The BH singularity surface and volume geometry is IDENTICAL to its BH event horizon surface and volume geometry and has a radius equal to Planck's Radius:
r_p = \sqrt{ \frac{ \hbar G}{c^3}}
QM/Classical GR criterion shutdown:
radial criterion:
r_s = r_p
Schwarzschild-Oppenheimer Density Solution for spherically symmetric non-rotating gravitational Schwarzschild-Oppenheimer Singularity:
\sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}} = \sqrt{ \frac{ \hbar G}{c^3}}
\rho_o = \frac{3M_o}{4 \pi} \left( \frac{c^3}{ \hbar G} \right)^{3/2}
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Schwarzschild-Oppenheimer BHs do not exist.
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The Kerr Metric is a metric discovered in 1963 which is an exact solution to the Einstein field equations. It describes the geometry of spacetime around a rotating black hole:
The Boyer-Lindquist form of the line element:
\inline{ds^2 = \rho \left( \frac{dr^2}{\Delta} + d \theta^2 \right) + \left( r^2 + a^2 \right) \sin^2 \theta d \phi^2 - dt^2 + \frac{2mr}{\rho^2} \left(a \sin^2 \theta d \phi - dt \right)^2}
\rho^2 = r^2 + a^2 \cos^2 \theta
\Delta = r^2 - 2mr + a^2
m - black hole mass
a - angular velocity, (as measured by a distant observer).
Note that r does not agree with the radial coordinate of the Schwarzschild Solution, except asymptotically.
For rotating black holes, the event horizon is predicted have an oblate spheroid (ellipsoid):
The radial solution for a oblate spheroid Oppenheimer BH sphere horizon is:
r_o^3 = r_a^2 r_c
r_o = \sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}} = \sqrt[3]{r_a^2 r_c}
Kerr radial solution for rotating oblate spheroid event horizon:
r_s = \frac{G M_o}{c^2}
r_o = r_s
\sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}} = \frac{G M_o}{c^2}
Kerr-Oppenheimer Density Solution for rotating oblate spheroid gravitational Kerr-Oppenheimer BH:
\rho_o = \left( \frac{3c^6}{4 \pi G^3 M_o^2} \right)
Kerr radial solution for rotating oblate spheroid Kerr Singularity:
r_p = \sqrt{ \frac{ \hbar G}{c^3}}
r_o = r_p
\sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}} = \sqrt{ \frac{ \hbar G}{c^3}}
Kerr-Oppenheimer Density Solution for rotating oblate spheroid gravitational Kerr-Oppenheimer Singularity:
\rho_o = \frac{3M_o}{4 \pi} \left( \frac{c^3}{ \hbar G} \right)^{3/2}
Kerr-Oppenheimer BHs exist.
BH Singularity Densities are extremely dense, however their densities are NOT infinite.
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BH Singularity Density infinities do NOT exist.
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Based upon the Orion1 Equasions:
What is the 'density value' for a Kerr-Oppenheimer Singularity?
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Reference:
http://www.space.com/scienceastronomy/white_hole_030917.html
http://en.wikipedia.org/wiki/Black_hole
http://www.maths.soton.ac.uk/relativity/GRExplorer/singularities/singtheorems.htm
http://imagine.gsfc.nasa.gov/docs/science/know_l2/black_holes.html
http://www.gothosenterprises.com/black_holes/static_black_holes.html
http://en.wikipedia.org/wiki/Kerr_metric
http://mathworld.wolfram.com/OblateSpheroid.html
