# What happens if $a^2 > M^2$ in Kerr metric?

(Boyer-Lindquist coordinates and $$c = G =1$$ taken)

As I know, line element in Kerr metric $$d s^2 = - \left( 1 - \frac{2Mr}{\rho^2} \right) d t^2 - \frac{4 M a r \sin^2 \theta}{\rho^2} d \phi d t + \frac{\rho^2}{\Delta} d r^2 + \rho^2 d \theta^2 + \left( r^2 + a^2 + \frac{2 M r a^2 \sin^2 \theta}{\rho^2} \right) \sin^2 \theta d \phi^2$$ where $$a = \frac{J}{M}, \rho^2 = r^2 + a^2 \cos^2 \theta, \Delta = r^2 - 2 M r + a^2$$ suggests that causality breaks where $$r = M \pm \sqrt{M^2 - a^2}$$.

How should I handle when $$a^2 > M^2$$ so $$r$$ is complex?

For example, the Sun's mass, $$M_\odot \approx 1.5 \text{km}$$ while it's angular momentum, $$J_\odot \approx 4.7 * 10^{19} \text{km}^2 > M_\odot ^2$$.

Will black holes with angular momentum bigger than mass squared not form an event horizon?

• As an aside: $4 M a r sin^2 \theta$ would look better as $4 M a r \sin^2 \theta$. Just using \sin instead of sin. Commented Mar 1 at 13:33
• Editted. Thanks for the advise. @JosBergervoet Commented Mar 1 at 13:35
• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Commented Mar 1 at 13:59
• BTW, the Sun's Schwarzschild radius, is ~2.95325007702 km Commented Mar 1 at 18:23

If the mass is in the form of a material body like the earth where the spin parameter is $$\rm a=J c/(G M)=890 M$$, it would have to lose some angular momentum until $$\rm a before it could collapse into a black hole, otherwise it can't since the centrifugal repulsion is larger than the centripetal attraction.
If the mass is in the form of a singularity that would be a naked singularity, and if it is an elementary particles like the electron where $$\rm a$$ is many orders of magnitudes larger than $$\rm M$$ and $$\rm r=0$$ as well the close field should be gravitationally repulsive.
According to Kerr and Penrose, bodies with $$a\ge m$$ can not exist in our universe.
They are mathematical objects with naked singularities (ring of diameter $$a$$) and are not black holes (no event horizon).