The Kerr Solution
The exact solution for rotating black holes — the most astrophysically relevant spacetime
In 1963, Roy Kerr discovered the exact solution to Einstein's vacuum field equations describing a rotating black hole. This was a monumental achievement: finding the Schwarzschild solution required solving ODEs, but the Kerr solution required solving the full nonlinear PDEs of general relativity with only axial (not spherical) symmetry. Every astrophysical black hole has angular momentum, making the Kerr metric the physically relevant description of real black holes in nature.
The Boyer-Lindquist Metric
The Kerr metric in Boyer-Lindquist coordinates (t, r, θ, φ) takes the following form. These coordinates are the rotating-black-hole generalization of Schwarzschild coordinates and are the most commonly used representation:
The Kerr Metric
$$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right)dt^2 - \frac{4Mar\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}\,dr^2 + \Sigma\,d\theta^2 + \frac{\mathcal{A}\sin^2\theta}{\Sigma}\,d\phi^2$$
where the auxiliary functions are:
$$\Sigma = r^2 + a^2\cos^2\theta$$
Reduces to r² at equator and r² + a² at poles
$$\Delta = r^2 - 2Mr + a^2$$
Horizons are at Δ = 0
$$\mathcal{A} = (r^2 + a^2)^2 - a^2\Delta\sin^2\theta$$
Governs the φφ-component of the metric
$$a = \frac{J}{M}$$
Spin parameter (angular momentum per unit mass)
The crucial new feature compared to Schwarzschild is the off-diagonal gtφ component, which encodes frame dragging — the rotation of spacetime itself around the black hole. The metric is stationary (∂/∂t is a Killing vector) and axisymmetric (∂/∂φ is a Killing vector), but not static: t → -t does not leave the metric invariant due to the dt dφ cross term.
The Newman-Janis Algorithm
How did Kerr find this solution? The original derivation used sophisticated algebraic techniques involving null tetrads and the Petrov classification of spacetimes (Kerr sought type D vacuum solutions). However, a remarkable alternative approach was discovered by Newman and Janis (1965): the Kerr metric can be generated from the Schwarzschild metric by a complex coordinate transformation.
Newman-Janis procedure:
- Write the Schwarzschild metric in advanced null (Eddington-Finkelstein) coordinates
- Express the metric in terms of a null tetrad (lμ, nμ, mμ, m̄μ)
- Perform a complex coordinate transformation: r → r + ia cos θ
- Require the resulting metric to be real
- Transform to Boyer-Lindquist coordinates
Remarkably, this procedure produces an exact vacuum solution. It remains somewhat mysterious why a complex transformation of a spherically symmetric solution yields the correct axially symmetric rotating solution. The algorithm also works to generate the Kerr-Newman (charged, rotating) solution from Reissner-Nordstrom.
Event Horizons
The event horizons occur where Δ = 0, i.e., where grr diverges (a coordinate singularity, just as r = 2M is for Schwarzschild). Solving the quadratic:
Kerr Horizons
$$r_\pm = M \pm \sqrt{M^2 - a^2}$$
There are two horizons:
Outer horizon r₊
The event horizon — the point of no return. Ranges from r₊ = 2M (Schwarzschild, a = 0) to r₊ = M (extremal, a = M). This is the physically relevant horizon for external observers.
Inner (Cauchy) horizon r₋
An inner horizon where predictability breaks down. Ranges from r₋ = 0 (Schwarzschild) to r₋ = M (extremal). Believed to be unstable (mass inflation instability).
Note that horizons exist only when M² ≥ a², i.e., |a| ≤ M. The angular momentum of the black hole is bounded above by its mass.
The Ergosphere
The ergosphere is the region where the time Killing vector ξμ = (∂/∂t)μbecomes spacelike. This occurs where gtt > 0:
$$g_{tt} = -\left(1 - \frac{2Mr}{\Sigma}\right) > 0 \quad \Longrightarrow \quad r < r_{\text{ergo}}(\theta)$$
Solving for the ergosphere boundary:
Ergosphere Radius
$$r_{\text{ergo}}(\theta) = M + \sqrt{M^2 - a^2\cos^2\theta}$$
Key properties of the ergosphere:
- θ-dependent: At the equator (θ = π/2): rergo = 2M (same as Schwarzschild radius). At the poles (θ = 0): rergo = r₊ (coincides with horizon).
- Region between ergosphere and horizon: The ergoregion is the volume between r₊ and rergo(θ). It is widest at the equator and vanishes at the poles.
- Physical meaning: Inside the ergosphere, no observer can remain stationary — even light must co-rotate with the black hole. However, unlike the region inside the horizon, escape is still possible.
- Ring singularity: The true singularity of Kerr is at Σ = 0, i.e., r = 0 and θ = π/2 — a ring of radius a in the equatorial plane, not a point.
Important Limiting Cases
a → 0: Schwarzschild Limit
When the spin vanishes, Σ → r², Δ → r² - 2Mr = r(r - 2M), and the cross term vanishes. The metric reduces exactly to Schwarzschild. The two horizons merge: r₊ → 2M, r₋ → 0. The ergosphere becomes the event horizon.
a = M: Extremal Kerr
The maximum spin case. The two horizons coincide at r₊ = r₋ = M. The surface gravity vanishes (κ = 0), so the Hawking temperature is zero — an extremal Kerr black hole does not radiate. The ergosphere extends from r = M to r = 2M at the equator.
$$\text{Extremal: } \quad \Delta = (r - M)^2, \qquad r_+ = r_- = M$$
a > M: Naked Singularity (Forbidden)
When a > M, the discriminant M² - a² < 0, and Δ = 0 has no real solutions — there are no horizons. The ring singularity is exposed to the entire universe. The cosmic censorship conjecture (Penrose, 1969) asserts this cannot form from realistic gravitational collapse.
M → 0: Flat Space in Oblate Coordinates
When M = 0 with a ≠ 0, the Kerr metric reduces to flat Minkowski spacetime written in oblate spheroidal coordinates — confirming the geometric meaning of the spin parameter.
$$ds^2 = -dt^2 + \frac{r^2 + a^2\cos^2\theta}{r^2 + a^2}\,dr^2 + (r^2 + a^2\cos^2\theta)\,d\theta^2 + (r^2+a^2)\sin^2\theta\,d\phi^2$$
Metric Determinant and Inverse
The determinant of the Kerr metric is surprisingly simple:
$$\sqrt{-g} = \Sigma\sin\theta$$
The inverse metric components (needed for computing Christoffel symbols) are:
$$g^{tt} = -\frac{\mathcal{A}}{\Sigma\Delta}, \qquad g^{t\phi} = -\frac{2Mar}{\Sigma\Delta}$$
$$g^{rr} = \frac{\Delta}{\Sigma}, \qquad g^{\theta\theta} = \frac{1}{\Sigma}, \qquad g^{\phi\phi} = \frac{\Delta - a^2\sin^2\theta}{\Sigma\Delta\sin^2\theta}$$