Kerr metric

The Kerr metric describes the shape of space around a spinning black hole that has no charge. It is an exact answer to the equations of general relativity, which explain how gravity works in space. These equations are very hard to solve.

Model of the spacetime surrounding a near-extremal Kerr black hole.

Black holes are places in space where gravity is so strong that even light cannot escape. The Kerr metric helps scientists understand how space behaves near a rotating black hole. It shows that the space around a spinning black hole is not perfectly round but is shaped by the black hole’s spin.

This idea is important for studying black holes and how they affect the universe around them. It was named after the physicist who found this solution, and it remains a key part of learning about black holes today.

Overview

The Kerr metric is a way to describe the space around a spinning black hole. It was discovered in 1963 by Roy Kerr. Before this, people only had descriptions for black holes that did not spin.

A spinning black hole can pull nearby objects to move with it, even without touching them. This is because the space around the black hole is twisted by its spin. Light from far away can loop around the black hole many times, making multiple images of the same object seem closer together.

	Non-rotating (J = 0)	Rotating (any J)
Uncharged (Q = 0)	Schwarzschild (1915)	Kerr (1963)
Charged (any Q)	Reissner–Nordström (1916–1918)	Kerr–Newman (1965)

Metric

The Kerr metric can be written in two main ways: the Boyer–Lindquist form and the Kerr–Schild form. It is related to the Schwarzschild metric, which describes the space around a non-moving object.

The Kerr metric helps us understand the space around a spinning object. One special part of this metric shows a link between time and movement in the direction the object spins. When the object’s spin is very small, this link disappears.

Main article: Boyer–Lindquist coordinates

d s 2 = − c 2 d τ 2 = − ( 1 − r s r Σ ) c 2 d t 2 + Σ Δ d r 2 + Σ d θ 2 + ( r 2 + a 2 + r s r a 2 Σ sin 2 ⁡ θ ) sin 2 ⁡ θ d ϕ 2 − 2 r s r a sin 2 ⁡ θ Σ c d t d ϕ {\displaystyle {\begin{aligned}ds^{2}&=-c^{2}d\tau ^{2}\\&=-\left(1-{\frac {r_{\text{s}}r}{\Sigma }}\right)c^{2}dt^{2}+{\frac {\Sigma }{\Delta }}dr^{2}+\Sigma d\theta ^{2}+\left(r^{2}+a^{2}+{\frac {r_{\text{s}}ra^{2}}{\Sigma }}\sin ^{2}\theta \right)\sin ^{2}\theta \ d\phi ^{2}-{\frac {2r_{\text{s}}ra\sin ^{2}\theta }{\Sigma }}c\,dt\,d\phi \end{aligned}}}

x = r 2 + a 2 sin ⁡ θ cos ⁡ ϕ {\displaystyle x={\sqrt {r^{2}+a^{2}}}\sin \theta \cos \phi }

y = r 2 + a 2 sin ⁡ θ sin ⁡ ϕ {\displaystyle y={\sqrt {r^{2}+a^{2}}}\sin \theta \sin \phi }

z = r cos ⁡ θ , {\displaystyle z=r\cos \theta ,}

r s = 2 G M c 2 , {\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}},}

a = J M c , {\displaystyle a={\frac {J}{Mc}},}

Σ = r 2 + a 2 cos 2 ⁡ θ , {\displaystyle \Sigma =r^{2}+a^{2}\cos ^{2}\theta ,}

Δ = r 2 − r s r + a 2 . {\displaystyle \Delta =r^{2}-r_{\text{s}}r+a^{2}.}

g ⟶ M → 0 − c 2 d t 2 + Σ r 2 + a 2 d r 2 + Σ d θ 2 + ( r 2 + a 2 ) sin 2 ⁡ θ d ϕ 2 {\displaystyle g\mathop {\longrightarrow } _{M\to 0}-c^{2}dt^{2}+{\frac {\Sigma }{r^{2}+a^{2}}}dr^{2}+\Sigma d\theta ^{2}+\left(r^{2}+a^{2}\right)\sin ^{2}\theta d\phi ^{2}}

g μ ν = η μ ν + f k μ k ν {\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+fk_{\mu }k_{\nu }\!}

f = 2 G M r 3 r 4 + a 2 z 2 {\displaystyle f={\frac {2GMr^{3}}{r^{4}+a^{2}z^{2}}}}

k = ( k x , k y , k z ) = ( r x + a y r 2 + a 2 , r y − a x r 2 + a 2 , z r ) {\displaystyle \mathbf {k} =(k_{x},k_{y},k_{z})=\left({\frac {rx+ay}{r^{2}+a^{2}}},{\frac {ry-ax}{r^{2}+a^{2}}},{\frac {z}{r}}\right)}

k 0 = 1. {\displaystyle k_{0}=1.\!}

x 2 + y 2 r 2 + a 2 + z 2 r 2 = 1 {\displaystyle {\frac {x^{2}+y^{2}}{r^{2}+a^{2}}}+{\frac {z^{2}}{r^{2}}}=1}

Mass of rotational energy

When a black hole spins, it has more energy than a black hole that does not spin. This extra energy comes from the spinning itself. Because energy and mass are linked in physics, this extra energy makes the spinning black hole look heavier.

The total mass of a spinning black hole includes its basic mass and the extra mass from its spin. The faster it spins, the more extra mass it has. This is why a very fast-spinning black hole can seem heavier than a slower one, even if they started with the same basic mass.

Wave operator

Checking the Kerr metric can be hard. The contravariant components of the metric tensor in Boyer–Lindquist coordinates are shown below in the expression for the square of the four-gradient operator:

g μ ν ∂ ∂ x μ ∂ ∂ x ν = − 1 c 2 Δ ( r 2 + a 2 + r s r a 2 Σ sin 2 ⁡ θ ) ( ∂ ∂ t ) 2 − 2 r s r a c Σ Δ ∂ ∂ ϕ ∂ ∂ t + 1 Δ sin 2 ⁡ θ ( 1 − r s r Σ ) ( ∂ ∂ ϕ ) 2 + Δ Σ ( ∂ ∂ r ) 2 + 1 Σ ( ∂ ∂ θ ) 2 {\displaystyle {\begin{aligned}g^{\mu \nu }{\frac {\partial }{\partial x^{\mu }}}{\frac {\partial }{\partial x^{\nu }}}&=-{\frac {1}{c^{2}\Delta }}\left(r^{2}+a^{2}+{\frac {r_{\text{s}}ra^{2}}{\Sigma }}\sin ^{2}\theta \right)\left({\frac {\partial }{\partial t}}\right)^{2}\\&-{\frac {2r_{\text{s}}ra}{c\Sigma \Delta }}{\frac {\partial }{\partial \phi }}{\frac {\partial }{\partial {t}}}+{\frac {1}{\Delta \sin ^{2}\theta }}\left(1-{\frac {r_{\text{s}}r}{\Sigma }}\right)\left({\frac {\partial }{\partial \phi }}\right)^{2}\\&+{\frac {\Delta }{\Sigma }}\left({\frac {\partial }{\partial r}}\right)^{2}+{\frac {1}{\Sigma }}\left({\frac {\partial }{\partial \theta }}\right)^{2}\end{aligned}}}

Frame dragging

The Kerr metric shows us how space and time act near a spinning black hole. It helps us see that an object that isn’t moving can start to turn because of the black hole’s spin. This is called frame-dragging and has been tested.

Think of it like this: an ice skater above the equator who isn’t moving with the stars will feel pulled to spin the opposite way the black hole turns. If the skater is already spinning, sometimes the forces balance out and the spin stays the same. This happens because gravity and motion can feel the same up close.

Main article: Killing horizon

Main articles: equivalence principle, Mach's principle

c 2 d τ 2 = ( g t t − g t ϕ 2 g ϕ ϕ ) d t 2 + g r r d r 2 + g θ θ d θ 2 + g ϕ ϕ ( d ϕ + g t ϕ g ϕ ϕ d t ) 2 . {\displaystyle c^{2}d\tau ^{2}=\left(g_{tt}-{\frac {g_{t\phi }^{2}}{g_{\phi \phi }}}\right)dt^{2}+g_{\mathrm {rr} }dr^{2}+g_{\theta \theta }d\theta ^{2}+g_{\phi \phi }\left(d\phi +{\frac {g_{t\phi }}{g_{\phi \phi }}}dt\right)^{2}.}

Ω = − g t ϕ g ϕ ϕ = r s r a c Σ ( r 2 + a 2 ) + r s r a 2 sin 2 ⁡ θ . {\displaystyle \Omega =-{\frac {g_{t\phi }}{g_{\phi \phi }}}={\frac {r_{\text{s}}rac}{\Sigma \left(r^{2}+a^{2}\right)+r_{\text{s}}ra^{2}\sin ^{2}\theta }}.}

Important surfaces

The Kerr metric tells us about the space around a spinning black hole. There are two important areas around this black hole.

The inner area is called an event horizon. This is a boundary where space-time stretches out forever.

There is also an outer area called the ergosphere. This area touches the inner area at the poles but is flatter elsewhere. Inside the ergosphere, particles must move in the same way as the black hole spins. This is because space-time is pulled along with the rotation. The edge of the ergosphere is not a sudden change. It can be passed through smoothly with the right coordinates.

Ergosphere and the Penrose process

Main article: Penrose process

A black hole has a surface called the event horizon. Nothing can escape from inside this surface because it would need to move faster than light.

For a rotating black hole, there is an area outside this surface called the ergosphere. In this area, space is pulled around by the black hole’s spin. Anything inside this area must move with it.

Particles that enter the ergosphere can gain energy and sometimes escape the black hole. This idea is called the Penrose process. It was suggested by a mathematician in 1969. It shows how rotating black holes might be linked to powerful bursts of energy in space.

Features

The Kerr geometry describes the space around a spinning black hole. It has important parts, like areas called ergospheres, event horizons, and singularities. Light can move around the black hole in special paths called photon spheres. The geometry gives exact answers to the equations about gravity, which makes it important for learning about black holes.

Symmetries

The Kerr metric describes the space around a rotating black hole. It has special patterns called symmetries. These symmetries include time moving forward and rotation around the black hole's axis. They help scientists understand some unchanging features of the black hole's motion.

These symmetries are linked to important rules in physics. They mean that some quantities stay the same as objects move around the black hole. Some of these come from basic motion rules, and others come from the special symmetries of the Kerr metric.

Overextreme Kerr solutions

The event horizon of a black hole is where a special value becomes zero. If a black hole spins too fast for its size, this value never becomes zero. In this case, there is no event horizon, so the black hole cannot hide from the universe. Without an event horizon, it is no longer a black hole but becomes something called a naked singularity.

Kerr black holes as wormholes

The Kerr solution has some special points where it seems to stop working. But these are just problems with how we look at it. By using new ways to describe these points, we can keep the solution going smoothly through them. The bigger of these points shows where the event horizon is, and the smaller shows where the Cauchy horizon is. A path can start outside the black hole and go through the event horizon. After that, the way we measure distance changes, and it must keep getting smaller until the path goes through the Cauchy horizon.

Anti-universe region

The Kerr metric describes the space around a rotating black hole. If you go past the inner edge of this black hole, you find a special circle in the middle.

Beyond this circle, in places where a certain value becomes negative, scientists imagine a whole new universe. This new universe has unusual traits, such as having less than zero total mass-energy. This idea of negative mass is still being studied.

In this new universe, some parts of space and time act in strange ways. The edge where these strange time paths start is called the Cauchy horizon.

The idea of this anti-universe helps scientists explore how space, time, and gravity might behave in very extreme conditions, even though we do not know if such a place really exists.

Relation to other exact solutions

The Kerr geometry is a special example of a steady, rotating space around a black hole with no charge. It solves the complex equations of general relativity. It is part of a group known as the Ernst vacuums.

When the Kerr solution has no rotation, it becomes the Schwarzschild metric. This describes a non-moving, round black hole. Part of the Kerr geometry's inside can look like a model where two gravitational waves crash into each other, even though the overall shape is different.

Multipole moments

Each empty space around a spinning black hole can be described by a special set of numbers. These numbers are called relativistic multipole moments. The first two numbers are like the weight and the spin of the black hole. These numbers help scientists understand the shape and behavior of the space around the black hole.

There are different ways to calculate these moments, and they all agree with each other. For the Kerr black hole, these moments were figured out by a scientist named Hansen. When the spin of the black hole is zero, the space around it becomes what we call the Schwarzschild vacuum. This is like a simple point source in the theory of space and time.

Open problems

The Kerr geometry helps us imagine a rotating black hole. It can also describe space around other spinning objects, like neutron stars or Earth. For objects that do not spin, matching the space outside with the space inside is easy. But for spinning objects, it is very hard to match their inside space with the Kerr geometry.

One idea, the Wahlquist fluid, looked good at first but did not work. Right now, we only have rough answers for objects that spin very slowly. There are also special cases, like the Neugebauer–Meinel disk, which is a model of a thin, spinning disk. This can match the Kerr geometry in some situations.