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 complex equations of general relativity, which describe how gravity works in space. These equations are very hard to solve because they are not straightforward.

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 happens because the space around the black hole is twisted by its spin. Light from far away can also 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 holds more energy than a non-spinning black hole of the same size. This extra energy comes from the spinning motion itself. Because of how energy and mass are linked in physics, this extra energy makes the spinning black hole appear heavier.

The spinning black hole's total mass includes both 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 weigh more than a slower one, even if they started with the same basic mass.

Wave operator

Checking the Kerr metric can be very complicated. 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 can be shown in a special way that helps us understand how space and time behave near a spinning black hole. This shows that a reference frame moving with a certain speed depends on where it is — both how far away and in what direction.

Because of this, an object that is not moving will start to turn because of the black hole’s spin. This effect is called frame-dragging and has been tested. Think of it like this: an ice skater orbiting above the equator and not moving with the stars will feel pulled to spin in 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 close up. One way to picture this is with gears, where the black hole is like the center gear, the skater is a middle gear, and everything else is like the outer gear. This idea relates to how some scientists think about motion and gravity.

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 describes the space around a spinning black hole. There are two important surfaces around this black hole. The inner surface is like a boundary called an event horizon. This is where the space-time stretches out endlessly.

There is also an outer surface called the ergosphere. This area touches the inner surface at the poles but is flattened elsewhere. Inside the ergosphere, particles must move in the same direction as the black hole's spin. This is because the space-time itself is dragged along with the rotation. The edge of the ergosphere is not a point of sudden change but can be smoothly passed through with the right coordinates.

Ergosphere and the Penrose process

Main article: Penrose process

A black hole is surrounded by a surface called the event horizon, where nothing can escape 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 itself is dragged around by the black hole’s spin so fast that anything inside must move with it.

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

Features

The Kerr geometry describes the space around a spinning black hole. It has several important features, including regions like ergospheres, event horizons, and singularities. Light can orbit the black hole in specific paths called photon spheres. The geometry allows for exact solutions to the equations that describe gravity, making it a key topic in the study of black holes.

Symmetries

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

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

Overextreme Kerr solutions

The event horizon of a black hole is found where a certain value becomes zero. When the black hole's spin is too high compared to its size, this value never becomes zero. In such cases, there is no event horizon, meaning 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 seems to have points where it breaks down, but these are just problems with how we look at it. By choosing new ways to describe these points, we can smoothly continue the solution through them. The larger of these special points marks where the event horizon is, and the smaller marks where the Cauchy horizon is. A path can start outside the black hole and go through the event horizon. After passing through, the way we measure distance changes, and it must keep decreasing until the path goes through the Cauchy horizon.

Anti-universe region

The Kerr metric describes the space around a rotating black hole and can be extended past its inner edge. When moving past this inner edge, a special area called a ring appears. Unlike a black hole without rotation, this ring is not a single point but a circle in the middle area.

Beyond this ring, in areas 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 and understood.

In this new universe, some parts of space and time act in strange ways, allowing for paths where time might seem to go backward. These paths are very unusual and make scientists wonder about how time and cause-and-effect work in such extreme places. 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, solving 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, which 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 called relativistic multipole moments. The first two of these 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, which is like a simple point source in the theory of space and time.

Open problems

The Kerr geometry is often used to describe a rotating black hole, but it could also represent the space around other rotating objects like neutron stars or even Earth. This works well for non-rotating objects, where we can match the space outside with the space inside very neatly. However, matching a rotating object’s inside space with the Kerr geometry has been very challenging.

One idea, the Wahlquist fluid, once seemed promising but doesn’t work either. Right now, we only have approximate solutions for slowly rotating objects. There are also special cases, like the Neugebauer–Meinel disk, a model of a thin, rotating disk, which can closely match the Kerr geometry in certain situations.