# Kerr metric
The **Kerr metric** is the exact solution of the vacuum [Einstein field equations](https://www.edgepedia.org/einstein-field-equations) that describes the spacetime around an isolated, spinning, uncharged black hole in general relativity. Within a general-relativistic framework, it is the exact description of such a black hole, and the structure of the spacetime around a single black hole is characterized solely by the black hole's mass \( M \) and angular momentum \( J \).[1](https://www.annualreviews.org/content/journals/10.1146/annurev-astro-112420-035022) It was found by the mathematician Roy P. Kerr and published in *Physical Review Letters* in 1963 under the title "Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics"; the paper was received on 26 July 1963 and published on 1 September 1963, with Kerr affiliated with the University of Texas, Austin, and the Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.[2](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.11.237)
| Key fact | Detail |
|---|---|
| Solution | Exact vacuum Einstein solution for an isolated, spinning, uncharged black hole, characterized only by mass \( M \) and angular momentum \( J \)[1](https://www.annualreviews.org/content/journals/10.1146/annurev-astro-112420-035022) |
| Publication | Roy P. Kerr, *Phys. Rev. Lett.* **11**, 237, published 1 September 1963[2](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.11.237) |
| Spin parameter | Dimensionless spin \( a \) built from \( M \) and \( J \); horizons at \( r_{\pm} = m \pm \sqrt{m^2 - a^2} \)[1](https://www.annualreviews.org/content/journals/10.1146/annurev-astro-112420-035022) • [3](https://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/kerr.html) |
| Ergosphere | Region between the outer horizon and the static limit surface where the time direction is spacelike; energy can be extracted there by the Penrose process[3](https://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/kerr.html) |
| Singularity | A ring of radius \( a \) in the equatorial plane, hidden behind the horizons when \( |a| \le m \)[3](https://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/kerr.html) |
| No-hair content | Mass and spin are the first two multipole moments and completely determine all higher-order moments of the Kerr spacetime[4](https://arxiv.org/abs/1512.03818) |
| Uniqueness | Kerr black holes are unique within the connected, non-degenerate, analytic class of regular vacuum black holes[5](https://arxiv.org/abs/0806.0016) |
## The line element
In [Boyer–Lindquist coordinates](https://www.edgepedia.org/boyer-lindquist-coordinates), which are the most straightforward generalization of spherical polar coordinates to this spacetime, the line element reads[1](https://www.annualreviews.org/content/journals/10.1146/annurev-astro-112420-035022)
\[ ds^2 = - \frac{\Delta}{\rho^2} \left( dt - a \sin^2\theta \, d\phi \right)^2 + \frac{\sin^2\theta}{\rho^2} \left( (r^2 + a^2) \, d\phi - a \, dt \right)^2 + \frac{\rho^2}{\Delta} \, dr^2 + \rho^2 \, d\theta^2 \]
with
\[ \rho^2 = r^2 + a^2 \cos^2\theta, \qquad \Delta = r^2 - 2mr + a^2 \]
where \( a \) is the Kerr parameter, a constant interpretable as the angular momentum per unit mass of the black hole.[3](https://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/kerr.html) • [13](https://www.giss.nasa.gov/pubs/abs/co01300e.html) When \( a = 0 \) the line element reduces to the Schwarzschild solution for a non-rotating black hole. The geometry is not spherically symmetric but axisymmetric, with two Killing vectors, one in the \( \phi \) direction and one in the \( t \) direction, plus the discrete symmetry of reversing both \( t \) and \( \phi \).[3](https://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/kerr.html) The rotation also produces cross terms containing \( d\phi \, dt \) in the line element, so there are no hypersurfaces orthogonal to the timelike Killing vector; such a spacetime is stationary rather than static.[3](https://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/kerr.html)
## Horizons and the ergosphere
**Two characteristic surfaces.** Horizons occur where \( \Delta = 0 \), that is at
\[ r_{\pm} = m \pm \sqrt{m^2 - a^2} \]
while the static limit surfaces, where the \( t \) direction changes from timelike to spacelike (where \( g_{tt} = 0 \)), occur at
\[ R_{\pm} = m \pm \sqrt{m^2 - a^2 \cos^2\theta} \]
The region between the outer horizon at \( r = r_+ \) and the static limit surface at \( R_+ \) is the ergosphere, which is physically accessible from outside the horizon but in which the \( t \) direction is spacelike.[3](https://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/kerr.html) Because energy inside the ergosphere carries what Penrose described as the wrong sign, a Kerr black hole can have energy extracted from it: objects dropped from craft moving through that region can be jettisoned in a way that harvests energy, the scheme first suggested by Penrose.[3](https://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/kerr.html) The ergosphere touches the event horizon at the poles and is oblate at the equator.[6](https://indico.cern.ch/event/1146409/contributions/4906269/attachments/2469205/4235627/Summer2022.pdf)
**The ring singularity.** The singularity occurs where \( \rho = 0 \), which is a ring of radius \( a \) in the equatorial plane. The metric also admits areas where \( r < 0 \), and closed timelike curves are found within them; moreover, when \( |a| > m > 0 \), one can travel through the ring separating the \( r < 0 \) and \( r > 0 \) areas without encountering the singularity.[3](https://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/kerr.html) For \( |a| > m \), no horizon can form and the ring singularity of the Kerr solution lies exposed; although the expression still solves the vacuum equations exactly, such a superextremal spacetime does not qualify as a black hole.[7](https://blackhole.org/black-holes/kerr/)
## Multipole structure and the no-hair theorem
In general relativity, the no-hair theorem implies that a black hole is fully characterized by its mass and spin alone, with the Kerr metric providing its description; these two quantities form the leading multipole moments of the Kerr spacetime, fixing every moment of higher order.[4](https://arxiv.org/abs/1512.03818) The mass multipole moments are nonzero only for even \( l \), and the current multipole moments are nonzero only for odd \( l \).[4](https://arxiv.org/abs/1512.03818) This locking of the moments makes the theorem quantitatively testable: a promising strategy is to measure at least three multipole moments of a black hole's spacetime and check the consistency relation.[4](https://arxiv.org/abs/1512.03818) In the Kerr-Newman geometry, obtained by applying the same construction to the Reissner-Nordström solution so as to include charge, a black hole is completely characterized by its mass, charge, and angular velocity; in John Wheeler's words, "A black hole has no hair."[3](https://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/kerr.html)
## Uniqueness
The Kerr metric's central role in general relativity rests partly on uniqueness theorems. Chruściel and Costa proved uniqueness of the Kerr black holes within the connected, non-degenerate, analytic class of regular vacuum black holes.[5](https://arxiv.org/abs/0806.0016) For the extremal case, it has been proved that the only four-dimensional, stationary, rotating, asymptotically flat (analytic) vacuum black hole with a single degenerate horizon is the extremal Kerr solution, with a similar theorem for extremal Kerr-Newman.[8](https://arxiv.org/abs/0906.2367) A geometrical characterization by Marc Mars shows that an asymptotically flat, stationary, vacuum spacetime whose [Killing form](https://www.edgepedia.org/killing-form) is an eigenvector of the self-dual [Weyl tensor](https://www.edgepedia.org/weyl-tensor) must be locally isometric to Kerr.[9](https://arxiv.org/abs/gr-qc/0004018) In five and higher dimensions the landscape is richer: the Myers-Perry black holes are higher-dimensional generalizations of the Kerr metric with spherical-horizon topology, but black rings and other solutions also exist, so the four-dimensional uniqueness picture does not carry over directly.[10](https://pmc.ncbi.nlm.nih.gov/articles/PMC5255892/)
## Observational role
**Spin measurements.** Because the Kerr metric is fully specified by mass and spin, measuring a black hole's spin is equivalent to measuring its Kerr parameters. Two well-understood electromagnetic techniques, X-ray reflection spectroscopy and thermal continuum fitting, measure the spins of black holes accreting at moderate rates, and many accreting stellar-mass black holes in X-ray binaries and many accreting supermassive black holes are found to be rapidly spinning.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-astro-112420-035022)
**Gravitational waves.** The product of a binary black hole coalescence is a rotating black hole whose spacetime is perturbed with respect to the Kerr metric. This ringdown stage can be treated with linear perturbation theory, and the dynamics are described by a series of decaying quasi-normal modes as the spacetime relaxes toward the Kerr metric; the frequencies and decay rates of the modes are completely determined by the mass and spin of the final black hole, so detecting several modes would test the no-hair theorem through the expected consistency relation between the frequencies.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-astro-112420-035022) The inspiral waveform, meanwhile, is affected by the magnitudes and directions of the premerger black holes' spins. Observatories detecting gravitational waves have made an important discovery: black holes that are about to merge in binary systems rotate slowly, showing a typical dimensionless spin parameter of \( a_{*} \sim 0.1\text{–}0.2 \), while 90% have \( a_{*} \lesssim 0.6 \).[11](https://www.sciencedirect.com/science/article/pii/S1387647325000259) The stark difference in spins between the black hole [X-ray binary](https://www.edgepedia.org/x-ray-binary) and the binary black hole populations shows a diversity of formation mechanisms.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-astro-112420-035022)
**Strong-field tests.** [General relativity](https://www.edgepedia.org/general-relativity) has been widely tested in weak gravitational fields but stands largely untested in the strong-field regime, and potential deviations from the Kerr metric affecting higher-order multipole moments are one route to testing it. Sagittarius A*, the compact object at the Galactic center, is a prime target: near-infrared monitoring of orbiting stars, timing of radio pulsars near the Galactic center, and [Event Horizon Telescope](https://www.edgepedia.org/event-horizon-telescope) imaging of its shadow can all probe its Kerr nature.[4](https://arxiv.org/abs/1512.03818) Parametric deviations from the Kerr metric have been constructed for such tests, though many are valid only for small spin values or small perturbations, or contain unphysical regions of space.[12](https://arxiv.org/abs/1210.0483)
## References
1. [Christopher S. Reynolds (2021). Observational Constraints on Black Hole Spin. Annual Review of Astronomy and Astrophysics 59:117–154.](https://www.annualreviews.org/content/journals/10.1146/annurev-astro-112420-035022)
2. [Roy P. Kerr (1963). Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics. Phys. Rev. Lett. 11, 237.](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.11.237)
3. [Rotating Black Holes, Geometry of General Relativity (Oregon State University course notes)](https://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/kerr.html)
4. [Tim Johannsen (2016). Sgr A* and General Relativity. Class. Quantum Grav. 33, 113001.](https://arxiv.org/abs/1512.03818)
5. [Piotr T. Chruściel, João Lopes Costa (2008). On uniqueness of stationary vacuum black holes.](https://arxiv.org/abs/0806.0016)
6. [Ervin Kafexhiu. Introduction to Black Hole Astrophysics, Data, Particle, Astroparticle School (2022).](https://indico.cern.ch/event/1146409/contributions/4906269/attachments/2469205/4235627/Summer2022.pdf)
7. [Kerr Black Hole, blackhole.org](https://blackhole.org/black-holes/kerr/)
8. [Aaron J. Amsel, Gary T. Horowitz, Donald Marolf, Matthew M. Roberts (2009). Uniqueness of Extremal Kerr and Kerr-Newman Black Holes.](https://arxiv.org/abs/0906.2367)
9. [Marc Mars (2000). Uniqueness properties of the Kerr metric.](https://arxiv.org/abs/gr-qc/0004018)
10. [Stationary Black Holes: Uniqueness and Beyond (Living Reviews in Relativity)](https://pmc.ncbi.nlm.nih.gov/articles/PMC5255892/)
11. [Spins of black holes in X-ray binaries and the tension with the gravitational wave measurements (2025).](https://www.sciencedirect.com/science/article/pii/S1387647325000259)
12. [A Metric for Testing the Nature of Black Holes (2012).](https://arxiv.org/abs/1210.0483)
13. [GISS Publications: Cohen 1968: Angular momentum and the Kerr metric](https://www.giss.nasa.gov/pubs/abs/co01300e.html)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › General relativity and curved spacetime › Exact solutions and spacetime metrics › Rotating and charged metrics › Kerr metric*
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