The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which makes exact solutions very difficult to find.
Overview The Kerr metric is a generalization to a rotating body of the Schwarzschild metric, discovered by Karl Schwarzschild in 1915, which described the geometry of spacetime around an uncharged, spherically symmetric, and non-rotating body. The corresponding solution for a charged, spherical, non-rotating body, the Reissner–Nordström metric, was discovered soon afterwards (1916–1918). However, the exact solution for an uncharged, rotating black hole, the Kerr metric, remained unsolved until 1963, when it was discovered by Roy Kerr. The natural extension to a charged, rotating black hole, the Kerr–Newman metric, was discovered shortly thereafter in 1965. These four related solutions may be summarized by the following table, where Q represents the body's electric charge and J represents its spin angular momentum:
According to the Kerr metric, a rotating body should exhibit frame-dragging (also known as Lense–Thirring precession), a distinctive prediction of general relativity. The first measurement of this frame dragging effect was done in 2011 by the Gravity Probe B experiment. Roughly speaking, this effect predicts that objects coming close to a rotating mass will be entrained to participate in its rotation, not because of any applied force or torque that can be felt, but rather because of the swirling curvature of spacetime itself associated with rotating bodies. In the case of a rotating black hole, at close enough distances, all objects – even light – must rotate with the black hole; the region where this holds is called the ergosphere. The light from distant sources can travel around the event horizon several times (if close enough); creating multiple images of the same object. To a distant viewer, the apparent perpendicular distance between images decreases at a factor of e2π (about 535). However, fast spinning black holes have less distance between multiplicity images. Rotating black holes have surfaces where the metric seems to have apparent singularities; the size and shape of these surfaces depends on the black hole's mass and angular momentum. The outer surface encloses the ergosphere and has a shape similar to a flattened sphere. The inner surface marks the event horizon; objects passing into the interior of this horizon can never again communicate with the world outside that horizon. However, neither surface is a true singularity, since their apparent singularity can be eliminated in a different coordinate system. A similar situation obtains when considering the Schwarzschild metric which also appears to result in a singularity at r = r s {\displaystyle r=r_{\text{s}}} dividing the space above and below r s {\displaystyle r_{\text{s}}} into two disconnected patches; using a different coordinate transformation one can then relate the extended external patch to the inner patch (see Schwarzschild metric § Singularities and black holes) – such a coordinate transformation eliminates the apparent singularity where the inner and outer surfaces meet. Objects between these two surfaces must co-rotate with the rotating black hole, as noted above; this feature can in principle be used to extract energy from a rotating black hole, up to its invariant mass energy, M c 2 {\displaystyle Mc^{2}} . The LIGO experiment that first detected gravitational waves, announced in 2016, also provided the first direct observation of a pair of Kerr black holes.
Metric The Kerr metric is commonly expressed in one of two forms, the Boyer–Lindquist form and the Kerr–Schild form. It can be readily derived from the Schwarzschild metric, using the Newman–Janis algorithm by Newman–Penrose formalism (also known as the spin–coefficient formalism), Ernst equation, or Ellipsoid coordinate transformation.
Boyer–Lindquist coordinates
The Kerr metric describes the geometry of spacetime in the vicinity of a mass M {\displaystyle M} rotating with angular momentum J {\displaystyle J} . The metric (or equivalently its line element for proper time) in Boyer–Lindquist coordinates is
where the coordinates r , θ , ϕ {\displaystyle r,\theta ,\phi } are standard oblate spheroidal coordinates, which are equivalent to the cartesian coordinates
where r s {\displaystyle r_{\text{s}}} is the Schwarzschild radius
and where for brevity, the length scales a , Σ {\displaystyle a,\Sigma } and Δ {\displaystyle \Delta } have been introduced as
A key feature to note in the above metric is the cross-term d t d ϕ {\displaystyle dt\,d\phi } . This implies that there is coupling between time and motion in the plane of rotation that disappears when the black hole's angular momentum goes to zero. In the non-relativistic limit where M {\displaystyle M} (or, equivalently, r s {\displaystyle r_{\text{s}}} ) goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates
Kerr–Schild coordinates The Kerr metric can be expressed in "Kerr–Schild" form, using a particular set of Cartesian coordinates as follows. These solutions were proposed by Kerr and Schild in 1965.
Notice that k is a unit 3-vector, making the 4-vector a null vector, with respect to both g and η. Here M is the constant mass of the spinning object, η is the Minkowski tensor, and a is a constant rotational parameter of the spinning object. It is understood that the vector a → {\displaystyle {\vec {a}}} is directed along the positive z-axis. The quantity r is not the radius, but rather is implicitly defined by
Notice that the quantity r becomes the usual radius R
r → R = x 2 + y 2 + z 2 {\displaystyle r\to R={\sqrt {x^{2}+y^{2}+z^{2}}}}
when the rotational parameter a {\displaystyle a} approaches zero. In this form of solution, units are selected so that the speed of light is unity ( c = 1 {\displaystyle c=1} ). At large distances from the source (R ≫ a), these equations reduce to the Eddington–Finkelstein form of the Schwarzschild metric. In the Kerr–Schild form of the Kerr metric, the determinant of the metric tensor is everywhere equal to negative one, even near the source.
Soliton coordinates As the Kerr metric (along with the Kerr–NUT metric) is axially symmetric, it can be cast into a form to which the Belinski–Zakharov transform can be applied. This implies that the Kerr black hole has the form of a gravitational soliton.
Mass of rotational energy If the complete rotational energy E r o t = c 2 ( M − M i r r ) {\displaystyle E_{\rm {rot}}=c^{2}\left(M-M_{\rm {irr}}\right)} of a black hole is extracted, for example with the Penrose process, the remaining mass cannot shrink below the irreducible mass. Therefore, if a black hole rotates with the spin a = M {\displaystyle a=M} , its total mass-equivalent M {\displaystyle M} is higher by a factor of 2 {\displaystyle {\sqrt {2}}} in comparison with a corresponding Schwarzschild black hole where M {\displaystyle M} is equal to M irr {\displaystyle M_{\text{irr}}} . The reason for this is that in order to get a static body to spin, energy needs to be applied to the system. Because of the mass–energy equivalence this energy also has a mass-equivalent, which adds to the total mass–energy of the system, M {\displaystyle M} . The total mass equivalent M {\displaystyle M} (the gravitating mass) of the body (including its rotational energy) and its irreducible mass M irr {\displaystyle M_{\text{irr}}} are related by
2 M i r r 2 = M 2 + M 4 − J 2 c 2 / G 2 ⟹ M 2 = M i r r 2 + J 2 c 2 4 M i r r 2 G 2 . {\displaystyle 2M_{\rm {irr}}^{2}=M^{2}+{\sqrt {M^{4}-J^{2}c^{2}/G^{2}}}\Longrightarrow M^{2}=M_{\rm {irr}}^{2}+{\frac {J^{2}c^{2}}{4M_{\rm {irr}}^{2}G^{2}}}.}
Wave operator Since even a direct check on the Kerr metric involves cumbersome calculations, the contravariant components g i k {\displaystyle g^{ik}} of the metric tensor in Boyer–Lindquist coordinates are shown below in the expression for the square of the four-gradient operator:
Frame dragging The Kerr metric may be rewritten (1) in the following form:
This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radius r and the colatitude θ, where Ω is called the Killing horizon.
Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is called frame-dragging, and has been tested experimentally. Qualitatively, frame-dragging can be viewed as the gravitational analog of electromagnetic induction. An "ice skater", in orbit over the equator and rotationally at rest with respect to the stars, extends her arms. The arm extended toward the black hole will be torqued spinward. The arm extended away from the black hole will be torqued anti-spinward. She will therefore be rotationally sped up, in a counter-rotating sense to the black hole. This is the opposite of what happens in everyday experience. If she is already rotating at a certain speed when she extends her arms, inertial effects and frame-dragging effects will balance and her spin will not change. Due to the equivalence principle, gravitational effects are locally indistinguishable from inertial effects, so this rotation rate, at which when she extends her arms nothing happens, is her local reference for non-rotation. This frame is rotating with respect to the fixed stars and counter-rotating with respect to the black hole. A useful metaphor is a planetary gear system with the black hole being the sun gear, the ice skater being a planetary gear and the outside universe being the ring gear. This can also be interpreted through Mach's principle.
Important surfaces
There are several important surfaces in the Kerr metric (1). The inner surface corresponds to an event horizon similar to that observed in the Schwarzschild metric; this occurs where the purely radial component grr of the metric goes to infinity. Solving the quadratic equation 1/grr = 0 yields the solution:
r H ± = r s ± r s 2 − 4 a 2 2 {\displaystyle r_{\rm {H}}^{\pm }={\frac {r_{\text{s}}\pm {\sqrt {r_{\text{s}}^{2}-4a^{2}}}}{2}}}
which in natural units (that give G = M = c = 1 {\displaystyle G=M=c=1} ) simplifies to:
r H ± = 1 ± 1 − a 2 {\displaystyle r_{\rm {H}}^{\pm }=1\pm {\sqrt {1-a^{2}}}}
While in the Schwarzschild metric the event horizon is also the place where the purely temporal component gtt of the metric changes sign from positive to negative, in Kerr metric that happens at a different distance. Again solving a quadratic equation gtt = 0 yields the solution:
r E ± = r s ± r s 2 − 4 a 2 cos 2 θ 2 {\displaystyle r_{\rm {E}}^{\pm }={\frac {r_{\text{s}}\pm {\sqrt {r_{\text{s}}^{2}-4a^{2}\cos ^{2}\theta }}}{2}}}
or in natural units:
r E ± = 1 ± 1 − a 2 cos 2 θ {\displaystyle r_{\rm {E}}^{\pm }=1\pm {\sqrt {1-a^{2}\cos ^{2}\theta }}}
Due to the cos2θ term in the square root, this outer surface resembles a flattened sphere that touches the inner surface at the poles of the rotation axis, where the colatitude θ equals 0 or π; the space between these two surfaces is called the ergosphere. Within this volume, the purely temporal component gtt is negative, i.e., acts like a purely spatial metric component. Consequently, particles within this ergosphere must co-rotate with the inner mass, if they are to retain their time-like character. A moving particle experiences a positive proper time along its worldline, its path through spacetime. However, this is impossible within the ergosphere, where gtt is negative, unless the particle is co-rotating around the interior mass M {\displaystyle M} with an angular speed at least of Ω {\displaystyle \Omega } . Thus, no particle can move in the direction opposite to central mass's rotation within the ergosphere. As with the event horizon in the Schwarzschild metric, the apparent singularity at rH is due to the choice of coordinates (i.e., it is a coordinate singularity). In fact, the spacetime can be smoothly continued through it by an appropriate choice of coordinates. In turn, the outer boundary of the ergosphere at rE is not singular by itself even in Kerr coordinates due to non-zero d t d ϕ {\displaystyle dt\ d\phi } term.
Ergosphere and the Penrose process
A black hole in general is surrounded by a surface, called the event horizon and situated at the Schwarzschild radius for a nonrotating black hole, where the escape velocity is equal to the velocity of light. Within this surface, no observer/particle can maintain itself at a constant radius. It is forced to fall inwards, and so this is sometimes called the static limit. A rotating black hole has the same static limit at its event horizon but there is an additional surface outside the event horizon named the "ergosurface" given by
( r − M ) 2 = M 2 − J 2 cos 2 θ {\displaystyle (r-M)^{2}=M^{2}-J^{2}\cos ^{2}\theta }
in Boyer–Lindquist coordinates, which can be intuitively characterized as the sphere where "the rotational velocity of the surrounding space" is dragged along with the velocity of light. Within this sphere the dragging is greater than the speed of light, and any observer/particle is forced to co-rotate. The region outside the event horizon but inside the surface where the rotational velocity is the speed of light, is called the ergosphere (from Greek ergon meaning work). Particles falling within the ergosphere are forced to rotate faster and thereby gain energy. Because they are still outside the event horizon, they may escape the black hole. The net process is that the rotating black hole emits energetic particles at the cost of its own total energy. The possibility of extracting spin energy from a rotating black hole was first proposed by the mathematician Roger Penrose in 1969 and is thus called the Penrose process. Rotating black holes in astrophysics are a potential source of large amounts of energy and are used to explain energetic phenomena, such as gamma-ray bursts.
Features The Kerr geometry exhibits many noteworthy features: the maximal analytic extension includes a sequence of asymptotically flat exterior regions, each associated with an ergosphere, stationary limit surfaces, event horizons, Cauchy horizons, closed timelike curves, and a ring-shaped curvature singularity. The geodesic equation can be solved exactly in closed form. In addition to two Killing vector fields (corresponding to time translation and axisymmetry), the Kerr geometry admits a remarkable Killing tensor. There is a pair of principal null congruences (one ingoing and one outgoing). The Weyl tensor is algebraically special, in fact it has Petrov type D. Note that the inner Kerr geometry is unstable with regard to perturbations in the interior region. This instability means that although the Kerr metric is axis-symmetric, a black hole created through gravitational collapse may not be so. This instability also implies that many of the features of the Kerr geometry described above may not be present inside such a black hole. A surface on which light can orbit a black hole is called a photon sphere. The Kerr solution has infinitely many photon spheres, lying between an inner one and an outer one. In the nonrotating, Schwarzschild solution, with a = 0 {\displaystyle a=0} , the inner and outer photon spheres degenerate, so that there is only one photon sphere at a single radius. The greater the spin of a black hole, the farther from each other the inner and outer photon spheres move. A beam of light traveling in a direction opposite to the spin of the black hole will circularly orbit the hole at the outer photon sphere. A beam of light traveling in the same direction as the black hole's spin will circularly orbit at the inner photon sphere. Orbiting geodesics with some angular momentum perpendicular to the axis of rotation of the black hole will orbit on photon spheres between these two extremes. Because the spacetime is rotating, such orbits exhibit a precession, since there is a shift in the ϕ {\displaystyle \phi } variable after completing one period in the θ {\displaystyle \theta } variable.
Trajectory equations
The equations of motion for test particles in the Kerr spacetime are governed by four constants of motion. The first is the invariant mass μ {\displaystyle \mu } of the test particle, defined by the relation
− μ 2 = p α g α β p β , {\displaystyle -\mu ^{2}=p^{\alpha }g_{\alpha \beta }p^{\beta },}
where p α {\displaystyle p^{\alpha }} is the four-momentum of the particle. Furthermore, there are two constants of motion given by the time translation and rotation symmetries of Kerr spacetime, the energy E {\displaystyle E} , and the component of the orbital angular momentum parallel to the spin of the black hole L z {\displaystyle L_{z}} .
E = − p t , {\displaystyle E=-p_{t},} and
L z = p ϕ {\displaystyle L_{z}=p_{\phi }}
Using Hamilton–Jacobi theory, Brandon Carter showed that there exists a fourth constant of motion, Q {\displaystyle Q} , now referred to as the Carter constant. It is related to the total angular momentum of the particle and is given by
Q = p θ 2 + cos 2 θ ( a 2 ( μ 2 − E 2 ) + ( L z sin θ ) 2 ) . {\displaystyle Q=p_{\theta }^{2}+\cos ^{2}\theta \left(a^{2}\left(\mu ^{2}-E^{2}\right)+\left({\frac {L_{z}}{\sin \theta }}\right)^{2}\right).}
Since there are four (independent) constants of motion for degrees of freedom, the equations of motion for a test particle in Kerr spacetime are integrable. Using these constants of motion, the trajectory equations for a test particle can be written (using natural units of G = M = c = 1 {\displaystyle G=M=c=1} ),
Σ d r d λ = ± R ( r ) Σ d θ d λ = ± Θ ( θ ) Σ d ϕ d λ = − ( a E − L z sin 2 θ ) + a Δ P ( r ) Σ d t d λ = − a ( a E sin 2 θ − L z ) + r 2 + a 2 Δ P ( r ) {\displaystyle {\begin{aligned}\Sigma {\frac {dr}{d\lambda }}&=\pm {\sqrt {R(r)}}\\\Sigma {\frac {d\theta }{d\lambda }}&=\pm {\sqrt {\Theta (\theta )}}\\\Sigma {\frac {d\phi }{d\lambda }}&=-\left(aE-{\frac {L_{z}}{\sin ^{2}\theta }}\right)+{\frac {a}{\Delta }}P(r)\\\Sigma {\frac {dt}{d\lambda }}&=-a\left(aE\sin ^{2}\theta -L_{z}\right)+{\frac {r^{2}+a^{2}}{\Delta }}P(r)\end{aligned}}}
with
Θ ( θ ) = Q − cos 2 θ ( a 2 ( μ 2 − E 2 ) + L z 2 sin 2 θ ) {\displaystyle \Theta (\theta )=Q-\cos ^{2}\theta \left(a^{2}\left(\mu ^{2}-E^{2}\right)+{\frac {L_{z}^{2}}{\sin ^{2}\theta }}\right)}
P ( r ) = E ( r 2 + a 2 ) − a L z {\displaystyle P(r)=E\left(r^{2}+a^{2}\right)-aL_{z}}
R ( r ) = P ( r ) 2 − Δ ( μ 2 r 2 + ( L z − a E ) 2 + Q ) {\displaystyle R(r)=P(r)^{2}-\Delta \left(\mu ^{2}r^{2}+(L_{z}-aE)^{2}+Q\right)}
where λ {\displaystyle \lambda } is an affine parameter such that d x α d λ = p α {\displaystyle {\frac {dx^{\alpha }}{d\lambda }}=p^{\alpha }} . In particular, when μ > 0 {\displaystyle \mu >0} the affine parameter λ {\displaystyle \lambda } , is related to the proper time τ {\displaystyle \tau } through λ = τ / μ {\displaystyle \lambda =\tau /\mu } . Because of the frame-dragging-effect, a zero-angular-momentum observer (ZAMO) is corotating with the angular velocity Ω {\displaystyle \Omega } which is defined with respect to the bookkeeper's coordinate time t {\displaystyle t} . The local velocity v {\displaystyle v} of the test-particle is measured relative to a probe corotating with Ω {\displaystyle \Omega } . The gravitational time-dilation between a ZAMO at fixed r {\displaystyle r} and a stationary observer far away from the mass is
t τ = ( a 2 + r 2 ) 2 − a 2 Δ sin 2 θ Δ Σ . {\displaystyle {\frac {t}{\tau }}={\sqrt {\frac {\left(a^{2}+r^{2}\right)^{2}-a^{2}\Delta \sin ^{2}\theta }{\Delta \ \Sigma }}}.}
In Cartesian Kerr–Schild coordinates, the equations for a photon are
x ¨ + i y ¨ = 4 i M a r Σ 2 W [ x ˙ + i y ˙ − x + i y r r ˙ ] − M ( x + i y ) ( 4 r 2 Σ − 1 ) C − a 2 W 2 r Σ 2 {\displaystyle {\ddot {x}}+i{\ddot {y}}=4iMa{\frac {r}{\Sigma ^{2}}}W\left[{\dot {x}}+i{\dot {y}}-{\frac {x+iy}{r}}{\dot {r}}\right]-M(x+iy)\left({\frac {4r^{2}}{\Sigma }}-1\right){\frac {C-a^{2}W^{2}}{r\Sigma ^{2}}}}
z ¨ = − M z (