Cosmic Censorship Hypothesis - Example

Example

The Kerr Metric, corresponding to a black hole of mass and angular momentum, can be used to derive the effective potential for particle orbits restricted to the equator (as defined by rotation). This potential looks like:

where is the coordinate radius, and are the test-particle's conserved energy and angular momentum respectively (constructed from the killing vectors).

To preserve cosmic censorship, the black hole is restricted to the case of . For there to exist an event horizon around the singularity, the requirement: must be satisfied:. This amounts to the angular momentum of the black hole being constrained to below a critical value, outside of which the horizon would disappear.

The following thought experiment is reproduced from Hartle's Gravity:

Imagine specifically trying to violate the censorship conjecture. This could be done by somehow imparting an angular momentum upon the black hole, making it exceed the critical value (assume it starts infinitesimally below it). This could be done by sending a particle of angular momentum . Because this particle has angular momentum, it can only be captured by the black hole if the maximum potential of the black hole is less than .

Solving the above effective potential equation for the maximum under the given conditions results in a maximum potential of exactly ! Testing other values shows that no particle with enough angular momentum to violate the censorship conjecture would be able to enter the black hole, because they have too much angular momentum to fall in.