Kerr metric

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. g = − 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 {egin{aligned}g=-c^{2}d au ^{2}=&-left(1-{frac {r_{s}r}{Sigma }} ight)c^{2}dt^{2}+{frac {Sigma }{Delta }}dr^{2}+Sigma d heta ^{2}\&+left(r^{2}+a^{2}+{frac {r_{s}ra^{2}}{Sigma }}sin ^{2} heta ight)sin ^{2} heta dphi ^{2}-{frac {2r_{s}rasin ^{2} heta }{Sigma }},c,dt,dphi end{aligned}}}     (1) x = r 2 + a 2 sin ⁡ θ cos ⁡ ϕ {displaystyle {x}={sqrt {r^{2}+a^{2}}}sin heta cos phi }     (2) y = r 2 + a 2 sin ⁡ θ sin ⁡ ϕ {displaystyle {y}={sqrt {r^{2}+a^{2}}}sin heta sin phi }     (3) z = r cos ⁡ θ , {displaystyle {z}=rcos heta ,}     (4) r s = 2 G M c 2 , {displaystyle r_{s}={frac {2GM}{c^{2}}},}     (5) a = J M c , {displaystyle a={frac {J}{Mc}},}     (6) Σ = r 2 + a 2 cos 2 ⁡ θ , {displaystyle Sigma =r^{2}+a^{2}cos ^{2} heta ,}     (7) Δ = r 2 − r s r + a 2 . {displaystyle Delta =r^{2}-r_{s}r+a^{2}.}     (8) 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 gmathop {longrightarrow } limits _{M o 0}-c^{2}dt^{2}+{frac {Sigma }{r^{2}+a^{2}}}dr^{2}+Sigma d heta ^{2}+left(r^{2}+a^{2} ight)sin ^{2} heta dphi ^{2}}     (9) g μ ν = η μ ν + f k μ k ν {displaystyle g_{mu u }=eta _{mu u }+fk_{mu }k_{ u }!}     (10) f = G r 2 r 4 + a 2 z 2 [ 2 M r ] {displaystyle f={frac {Gr^{2}}{r^{4}+a^{2}z^{2}}}left}     (11) 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}} ight)}     (12) k 0 = 1. {displaystyle k_{0}=1.!}     (13) 1 = x 2 + y 2 r 2 + a 2 + z 2 r 2 {displaystyle 1={frac {x^{2}+y^{2}}{r^{2}+a^{2}}}+{frac {z^{2}}{r^{2}}}}     (14) 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 {egin{aligned}g^{mu u }{frac {partial }{partial {x^{mu }}}}{frac {partial }{partial {x^{ u }}}}=&{frac {1}{c^{2}Delta }}left(r^{2}+a^{2}+{frac {r_{s}ra^{2}}{Sigma }}sin ^{2} heta ight)left({frac {partial }{partial {t}}} ight)^{2}+{frac {2r_{s}ra}{cSigma Delta }}{frac {partial }{partial {phi }}}{frac {partial }{partial {t}}}\&-{frac {1}{Delta sin ^{2} heta }}left(1-{frac {r_{s}r}{Sigma }} ight)left({frac {partial }{partial {phi }}} ight)^{2}-{frac {Delta }{Sigma }}left({frac {partial }{partial {r}}} ight)^{2}-{frac {1}{Sigma }}left({frac {partial }{partial { heta }}} ight)^{2}end{aligned}}}     (15) 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 au ^{2}=left(g_{tt}-{frac {g_{tphi }^{2}}{g_{phi phi }}} ight)dt^{2}+g_{rr}dr^{2}+g_{ heta heta }d heta ^{2}+g_{phi phi }left(dphi +{frac {g_{tphi }}{g_{phi phi }}}dt ight)^{2}.}     (16) 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_{tphi }}{g_{phi phi }}}={frac {r_{s}rac}{Sigma left(r^{2}+a^{2} ight)+r_{s}ra^{2}sin ^{2} heta }}.}     (17) 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. 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.:69-81 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. 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. 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, Mc2. The LIGO experiment that first detected gravitational waves, announced in 2016, also provided the first direct observation of a pair of Kerr black holes. 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–Penrose formalism (also known as the spin–coefficient formalism), which can be found in Newman&Adamo 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, heta ,phi } are standard spherical coordinate system, which are equivalent to the cartesian coordinates