Why Rotation Matters
In reality, conservation of angular momentum ensures that collapsing stars retain their spin. As the star collapses into a black hole, this spin becomes extreme. While the Schwarzschild metric assumes zero angular momentum (J=0), the Kerr metric describes a black hole with mass M and angular momentum J. This addition breaks the perfect spherical symmetry of the Schwarzschild case, replacing it with axial symmetry around the rotation axis.
The Kerr Metric Parameters
The Kerr metric is defined by two parameters: mass \(M\) and specific angular momentum \(a = J/Mc\). The line element \(ds^2\) becomes significantly more complex than the Schwarzschild form, incorporating cross-terms like \(dt d\phi\) that signify the mixing of time and azimuthal angle due to rotation.
Schwarzschild vs. Kerr
Comparing the structural differences between the two primary solutions.
Schwarzschild (Static)
- Angular momentum \(J = 0\)
- Spherical symmetry
- Single event horizon at \(r_s = 2GM/c^2\)
- Point singularity at \(r = 0\)
Kerr (Rotating)
- Angular momentum \(J \neq 0\)
- Axial symmetry
- Two horizons (outer and inner)
- Ring singularity