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| {{General relativity|cTopic=[[Exact solutions in general relativity|Solutions]]}}
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| The '''Kerr [[metric tensor|metric]]''' (or '''Kerr vacuum''') describes the geometry of empty [[spacetime]] around a rotating uncharged axially-symmetric black hole with a spherical event horizon. The Kerr metric is an [[exact solutions in general relativity|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 of the [[Schwarzschild metric]], which was discovered by [[Karl Schwarzschild]] in 1916 and which describes 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:
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| {| class="wikitable" style="margin: 1em auto"
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| | Non-rotating (''J'' = 0)
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| | Rotating (''J'' ≠ 0)
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| |-
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| | Uncharged (''Q'' = 0)
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| | [[Schwarzschild metric|Schwarzschild]]
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| | Kerr
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| |-
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| | Charged (''Q'' ≠ 0)
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| | [[Reissner–Nordström metric|Reissner–Nordström]]
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| | [[Kerr–Newman metric|Kerr–Newman]]
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| |}
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| where ''Q'' represents the body's [[electric charge]] and ''J'' represents its spin [[angular momentum]].
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| According to the Kerr metric, such rotating black-holes should exhibit [[frame dragging]], an unusual prediction of general relativity. Measurement of this frame dragging effect was a major goal of 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 curvature of spacetime associated with rotating bodies. At close enough distances, all objects — even [[light]] itself — ''must'' rotate with the black-hole; the region where this holds is called the [[ergosphere]].
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| Rotating black holes have surfaces where the metric appears to have a [[gravitational singularity|singularity]]; 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 "radius of no return" also called the "event horizon"; objects passing through this radius can never again communicate with the world outside that radius. However, neither surface is a true singularity, since their apparent singularity can be eliminated in a different [[coordinate system]]. Objects between these two horizons must co-rotate with the rotating body, as noted above; this feature can be used to extract energy from a rotating black hole, up to its [[invariant mass]] energy, ''Mc''<sup>2</sup>.
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| ==Mathematical form==
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| The Kerr metric<ref name=kerr_1963>{{cite journal |last=Kerr |first=Roy P. | authorlink=Roy Kerr | year=1963 | title=Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics | journal=Physical Review Letters | volume=11 | issue=5 | pages=237–238 | doi=10.1103/PhysRevLett.11.237 | bibcode=1963PhRvL..11..237K | url=http://prola.aps.org/abstract/PRL/v11/i5/p237_1}}</ref><ref>{{cite book | last1=Landau | first1=L. D. | authorlink=Lev Landau | last2=Lifshitz |first2=E. M. |authorlink2=Evgeny Lifshitz | year=1975 | title=The Classical Theory of Fields (Course of Theoretical Physics, Vol. 2) | edition=revised 4th English | publisher=Pergamon Press | location=New York | isbn=978-0-08-018176-9 | pages=321–330}}</ref> describes the geometry of [[spacetime]] in the vicinity of a mass ''M'' rotating with [[angular momentum]] ''J''
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| :{{NumBlk|:|<math>\begin{align} c^{2} d\tau^{2} | |
| = & \left( 1 - \frac{r_{s} r}{\rho^{2}} \right) c^{2} dt^{2} - \frac{\rho^{2}}{\Delta} dr^{2} - \rho^{2} d\theta^{2} \\
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| & - \left( r^{2} + \alpha^{2} + \frac{r_{s} r \alpha^{2}}{\rho^{2}} \sin^{2} \theta \right) \sin^{2} \theta \ d\phi^{2}
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| + \frac{2r_{s} r\alpha \sin^{2} \theta }{\rho^{2}} \, c \, dt \, d\phi \end{align}</math>|{{EquationRef|1}}}}
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| where the coordinates <math>r, \theta, \phi</math> are standard [[spherical coordinate system]], and ''r''<sub>''s''</sub> is the [[Schwarzschild metric|Schwarzschild radius]]
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| :{{NumBlk|:|<math>r_{s} = \frac{2GM}{c^{2}}</math>|{{EquationRef|2}}}}
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| and where the length-scales α, ρ and Δ have been introduced for brevity
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| :{{NumBlk|:|<math>\alpha = \frac{J}{Mc}</math>|{{EquationRef|3}}}}
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| :{{NumBlk|:|<math>\rho^{2} = r^{2} + \alpha^{2} \cos^{2} \theta</math>|{{EquationRef|4}}}}
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| :{{NumBlk|:|<math>\Delta = r^{2} - r_{s} r + \alpha^{2}</math>|{{EquationRef|5}}}}
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| In the non-relativistic limit where ''M'' (or, equivalently, ''r''<sub>''s''</sub>) goes to zero, the Kerr metric becomes the orthogonal metric for the [[oblate spheroidal coordinates]]
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| :{{NumBlk|:|<math>c^{2} d\tau^{2} = c^{2} dt^{2} - \frac{\rho^{2}}{r^{2} + \alpha^{2}} dr^{2} - \rho^{2} d\theta^{2}- \left( r^{2} + \alpha^{2} \right) \sin^{2}\theta d\phi^{2} </math>|{{EquationRef|6}}}} | |
| which are equivalent to the [[Boyer-Lindquist coordinates]]<ref>{{cite journal | last1=Boyer | first1=Robert H. | last2=Lindquist | first2=Richard W. | year=1967 | title=Maximal Analytic Extension of the Kerr Metric | journal=J. Math. Phys. | volume=8 | issue=2 | pages=265–281 | doi=10.1063/1.1705193 | bibcode=1967JMP.....8..265B}}</ref>
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| :{{NumBlk|:|<math>{x} = \sqrt {r^2 + \alpha^2} \sin\theta\cos\phi</math>|{{EquationRef|7}}}}
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| :{{NumBlk|:|<math>{y} = \sqrt {r^2 + \alpha^2} \sin\theta\sin\phi</math>|{{EquationRef|8}}}}
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| :{{NumBlk|:|<math>{z} = r \cos\theta</math>|{{EquationRef|9}}}}
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| ==Gradient operator==
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| Since even a direct check on the Kerr metric involves cumbersome calculations, the [[Covariance and contravariance of vectors|contravariant]] components <math>g^{ik}</math> of the [[metric tensor]] are shown below in the expression for the square of the [[four-gradient]] [[Differential operator|operator]]:
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| :{{NumBlk|:|<math>\begin{align}
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| g^{\mu\nu}\frac{\partial}{\partial{x^{\mu}}}\frac{\partial}{\partial{x^{\nu}}} =
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| & \frac{1}{c^{2}\Delta}\left(r^{2} + \alpha^{2} + \frac{r_{s}r\alpha^{2}}{\rho^{2}}\sin^{2}\theta\right)\left(\frac{\partial}{\partial{t}}\right)^{2} + \frac{2r_{s}r\alpha}{c\rho^{2}\Delta}\frac{\partial}{\partial{\phi}}\frac{\partial}{\partial{t}} \\
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| & - \frac{1}{\Delta\sin^{2}\theta}\left(1 - \frac{r_{s}r}{\rho^{2}}\right)\left(\frac{\partial}{\partial{\phi}}\right)^{2} - \frac{\Delta}{\rho^{2}}\left(\frac{\partial}{\partial{r}}\right)^{2} - \frac{1}{\rho^{2}}\left(\frac{\partial}{\partial{\theta}}\right)^{2} \end{align}</math>|{{EquationRef|10}}}}
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| ==Frame dragging==
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| We may rewrite the Kerr metric ({{EquationRef|1}}) in the following form:
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| :{{NumBlk|:|<math>c^{2} d\tau^{2} = \left( g_{tt} - \frac{g_{t\phi}^{2}}{g_{\phi\phi}} \right) dt^{2}
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| + g_{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}.</math>|{{EquationRef|11}}}}
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| 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]].
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| :{{NumBlk|:|<math> \Omega = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{r_{s} r \alpha c}{\rho^{2} \left( r^{2} + \alpha^{2} \right) + r_{s} r \alpha^{2} \sin^{2}\theta}. </math>|{{EquationRef|12}}}}
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| 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.<ref>{{cite journal
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| | last = Will
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| | first = Clifford M.
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| | authorlink = Clifford Martin Will
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| |date=May 2011
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| | title =Finally, results from Gravity Probe B
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| | journal = Physics
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| | volume = 4
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| | page = 43
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| | publisher = American Physical Society
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| | doi = 10.1103/Physics.4.43
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| | accessdate =
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| | url =http://link.aps.org/doi/10.1103/Physics.4.43
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| |arxiv = 1106.1198 |bibcode = 2011PhyOJ...4...43W }}</ref>
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| 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|Principle of Equivalence]] 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 be also be interpreted through [[Mach's principle]].
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| [[Image:Ergosphere.svg|thumb|left|300px|The two surfaces on which the Kerr metric appears to have singularities; the inner surface is the spherical [[event horizon]], whereas the outer surface is an [[oblate spheroid]]. The ergosphere lies between these two surfaces; within this volume, the purely temporal component ''g<sub>tt</sub>'' 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.]]
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| ==Important surfaces==
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| The Kerr metric has two physical relevant surfaces on which it appears to be singular. The inner surface corresponds to an [[event horizon]] similar to that observed in the [[Schwarzschild metric]]; this occurs where the purely radial component ''g<sub>rr</sub>'' of the metric goes to infinity. Solving the quadratic equation 1/''g''<sub>''rr''</sub> = 0 yields the solution:
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| :<math>r_\mathit{inner} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2}}}{2}</math> | |
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| Another singularity occurs where the purely temporal component ''g<sub>tt</sub>'' of the metric changes sign from positive to negative. Again solving a quadratic equation ''g<sub>tt</sub>''=0 yields the solution:
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| :<math>r_\mathit{outer} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2} \cos^{2}\theta}}{2}</math>
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| Due to the cos<sup>2</sup>θ 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]]. There are two other solutions to these quadratic equations, but they lie within the event horizon, where the Kerr metric is not used, since it has unphysical properties (see below).
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| A moving particle experiences a positive [[proper time]] along its [[worldline]], its path through [[spacetime]]. However, this is impossible within the ergosphere, where ''g<sub>tt</sub>'' is negative, unless the particle is co-rotating with the interior mass ''M'' with an angular speed at least of Ω. Thus, no particle can rotate opposite to the central mass within the ergosphere.
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| As with the event horizon in the [[Schwarzschild metric]] the apparent singularities at r<sub>inner</sub> and r<sub>outer</sub> are an illusion created by the choice of coordinates (i.e., they are [[coordinate singularities]]). In fact, the space-time can be smoothly continued through them by an appropriate choice of coordinates.
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| ==Ergosphere and the Penrose process==
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| {{main|Penrose process}}
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| 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''.
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| <!--- [[File:Ergosphere.svg|thumb|right|250px|ergosphere of a rotating black hole|Two important surfaces around a rotating black hole. The inner sphere is the static limit (the event horizon). It is the inner boundary of a region called the [[ergoregion | ergosphere]]. The oblate spheroidal surface, touching the event horizon at the poles, is the outer boundary of the ergosphere. Within the ergosphere a particle is forced (dragging of space and time) to rotate and may gain energy at the cost of the rotational energy of the black hole ([[Penrose process]]).]] -->
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| 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 <math>(r-GM)^{2} = G^{2}M^{2}-J^{2}\cos^{2}\theta</math> 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.
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| 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 burster|gamma ray bursts]].
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| ==Features of the Kerr vacuum==
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| The Kerr vacuum 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 horizon]]s, [[Cauchy horizon]]s, [[closed timelike curve]]s, 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 vacuum 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 classification|Petrov type]] '''D'''. The [[global spacetime structure|global structure]] is known. Topologically, the [[homotopy type]] of the Kerr spacetime can be simply characterized as a line with circles attached at each integer point.
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| Note that the Kerr vacuum is unstable with regards 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 vacuum described above would also probably not be present in such a black hole.
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| A surface on which light can orbit a black hole is called a photon sphere. The Kerr solution has infinitely many [[photon sphere]]s, lying between an inner one and an outer one. In the nonrotating, Schwarzschild solution, with α=0, the inner and outer photon spheres degenerate, so that all the photons sphere occur at the same radius. The greater the spin of the black hole is, 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 space-time is rotating, such orbits exhibit a precession, since there is a shift in the <math>\phi \,</math> variable after completing one period in the <math>\theta \,</math> variable.
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| ==Overextreme Kerr solutions==
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| The location of the event horizon is determined by the larger root of <math>\Delta=0</math>. When <math>{r_s / 2} < \alpha </math> (i.e. <math>G M^2 < J c </math>), there are no (real valued) solutions to this equation, and there is no event horizon. With no event horizons to hide it from the rest of the universe, the black hole ceases to be a black hole and will instead be a [[naked singularity]].<ref name="Chandrasekhar_1983">{{cite book | last = Chandrasekhar | first = S. | authorlink = Subrahmanyan Chandrasekhar | year = 1983 | title = The Mathematical Theory of Black Holes | series = International Series of Monographs on Physics | volume = 69 | pages = 375}}</ref>
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| ==Kerr black holes as wormholes==
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| {{Refimprove section|date=February 2011}}
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| <!-- Image with unknown copyright status removed: [[File:Space-Time_Diagram_for_a_Spinning_Black_Hole.gif|thumb|right|300px|A [[Penrose diagram]] for the Kerr metric: an object traveling on world-line B can emerge out of the spinning black hole.]] -->
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| Although the Kerr solution appears to be singular at the roots of Δ = 0, these are actually [[coordinate singularity|coordinate singularities]], and, with an appropriate choice of new coordinates, the Kerr solution can be smoothly extended through the values of <math>r</math> corresponding to these roots. The larger of these roots determines the location of the event horizon, and the smaller determines the location of a Cauchy horizon. A (future-directed, time-like) curve can start in the exterior and pass through the event horizon. Once having passed through the event horizon, the <math>r</math> coordinate now behaves like a time coordinate, so it must decrease until the curve passes through the Cauchy horizon.
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| The region beyond the Cauchy horizon has several surprising features. The <math>r</math> coordinate again behaves like a spatial coordinate and can vary freely. The interior region has a reflection symmetry, so that a (future-directed time-like) curve may continue along a symmetric path, which continues through a second Cauchy horizon, through a second event horizon, and out into a new exterior region which is isometric to the original exterior region of the Kerr solution. The curve could then escape to infinity in the new region or enter the future event horizon of the new exterior region and repeat the process. This second exterior is sometimes thought of as another universe. On the other hand, in the Kerr solution, the singularity is a [[Ring singularity|ring]], and the curve may pass through the center of this ring. The region beyond permits closed time-like curves. Since the trajectory of observers and particles in general relativity are described by time-like curves, it is possible for observers in this region to return to their past.
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| While it is expected that the exterior region of the Kerr solution is stable, and that all rotating black holes will eventually approach a Kerr metric, the interior region of the solution appears to be unstable, much like a pencil balanced on its point.<ref>Penrose 1968</ref> This is related to the idea of [[Cosmic censorship hypothesis|cosmic censorship]].
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| ==Relation to other exact solutions==
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| The Kerr vacuum is a particular example of a [[stationary spacetime|stationary]] [[axially symmetric spacetime|axially symmetric]] [[vacuum solution]] to the [[Einstein field equation]]. The family of all stationary axially symmetric vacuum solutions to the Einstein field equation are the [[Ernst vacuum]]s.
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| The Kerr solution is also related to various non-vacuum solutions which model black holes. For example, the [[Kerr–Newman metric|Kerr–Newman electrovacuum]] models a (rotating) black hole endowed with an electric charge, while the [[Kerr–Vaidya null dust]] models a (rotating) hole with infalling electromagnetic radiation.
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| The special case <math>\alpha = 0 \,</math> of the Kerr metric yields the [[Schwarzschild metric]], which models a ''nonrotating'' black hole which is [[static spacetime|static]] and [[spherically symmetric]], in the [[Schwarzschild coordinates]]. (In this case, every Geroch moment but the mass vanishes.) | |
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| The ''interior'' of the Kerr vacuum, or rather a portion of it, is [[locally isometric]] to the [[Chandrasekhar–Ferrari CPW vacuum]], an example of a [[colliding plane wave]] model. This is particularly interesting, because the [[global spacetime structure|global structure]] of this CPW solution is quite different from that of the Kerr vacuum, and in principle, an experimenter could hope to study the geometry of (the outer portion of) the Kerr interior by arranging the collision of two suitable [[gravitational plane waves]].
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| ==Multipole moments==
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| Each [[asymptotically flat]] Ernst vacuum can be characterized by giving the infinite sequence of relativistic [[multipole moment]]s, the first two of which can be interpreted as the [[mass]] and [[angular momentum]] of the source of the field. There are alternative formulations of relativistic multipole moments due to Hansen, Thorne, and Geroch, which turn out to agree with each other. The relativistic multipole moments of the Kerr vacuum were computed by Hansen; they turn out to be
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| :<math> M_n = M \, (i \, \alpha)^n </math>
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| Thus, the special case of the [[Schwarzschild metric|Schwarzschild vacuum]] (α=0) gives the "monopole [[point source]]" of general relativity.
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| ''Warning:'' do not confuse these relativistic multipole moments with the ''Weyl multipole moments'', which arise from treating a certain metric function (formally corresponding to Newtonian gravitational potential) which appears the Weyl-Papapetrou chart for the Ernst family of all stationary axisymmetric vacuums solutions using the standard euclidean scalar [[multipole moment]]s. In a sense, the Weyl moments only (indirectly) characterize the "mass distribution" of an isolated source, and they turn out to depend only on the ''even order'' relativistic moments. In the case of solutions symmetric across the equatorial plane the ''odd order'' Weyl moments vanish. For the Kerr vacuum solutions, the first few Weyl moments are given by
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| :<math>a_0 = M, \; \; a_1 = 0, \; \; a_2 = M \, \left( \frac{M^2}{3} - \alpha^2 \right) </math>
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| In particular, we see that the Schwarzschild vacuum has nonzero second order Weyl moment, corresponding to the fact that the "Weyl monopole" is the [[Chazy–Curzon vacuum]] solution, not the Schwarzschild vacuum solution, which arises from the Newtonian potential of a certain finite length uniform density thin ''rod''.
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| In weak field general relativity, it is convenient to treat isolated sources using another type of multipole, which generalize the Weyl moments to ''mass multipole moments'' and ''momentum multipole moments'', characterizing respectively the distribution of [[mass]] and of [[momentum]] of the source. These are multi-indexed quantities whose suitably symmetrized (anti-symmetrized) parts can be related to the real and imaginary parts of the relativistic moments for the full nonlinear theory in a rather complicated manner.
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| Perez and Moreschi have given an alternative notion of "monopole solutions" by expanding the standard NP tetrad of the Ernst vacuums in powers of ''r'' (the radial coordinate in the Weyl-Papapetrou chart). According to this formulation:
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| *the isolated mass monopole source with ''zero'' angular momentum is the ''Schwarzschild vacuum'' family (one parameter),
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| *the isolated mass monopole source with ''radial'' angular momentum is the ''[[Taub–NUT vacuum]]'' family (two parameters; not quite asymptotically flat),
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| *the isolated mass monopole source with ''axial'' angular momentum is the ''Kerr vacuum'' family (two parameters).
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| In this sense, the Kerr vacuums are the simplest stationary axisymmetric asymptotically flat vacuum solutions in general relativity.
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| ==Open problems==
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| The Kerr vacuum is often used as a model of a black hole, but if we hold the solution to be valid only outside some compact region (subject to certain restrictions), in principle we should be able to use it as an [[exterior solution]] to model the gravitational field around a rotating massive object other than a black hole, such as a [[neutron star]]--- or the [[Earth]]. This works out very nicely for the non-rotating case, where we can match the Schwarzschild vacuum exterior to a [[Schwarzschild fluid]] interior, and indeed to more general [[static spherically symmetric perfect fluid]] solutions. However, the problem of finding a rotating perfect-fluid interior which can be matched to a Kerr exterior, or indeed to any asymptotically flat vacuum exterior solution, has proven very difficult. In particular, the [[Wahlquist fluid]], which was once thought to be a candidate for matching to a Kerr exterior, is now known not to admit any such matching. At present it seems that only approximate solutions modeling slowly rotating fluid balls are known. (Slowly rotating fluid balls are the relativistic analog of oblate spheroidal balls with nonzero mass and angular momentum but vanishing higher multipole moments.) However, the exterior of the [[Neugebauer–Meinel disk]], an exact [[dust solution]] which models a rotating thin disk, approaches in a limiting case the <math> \alpha = M </math> Kerr vacuum.
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| ==Trajectory equations==
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| {{double image|right|Particle trajectories around a clockwise rotating black hole.svg|250|Particle trajectories around a counter-clockwise rotating black hole.svg|250|
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| Particle trajectories in the plane <math>\theta=\pi/2 </math> around a clockwise (left) and counter-clockwise (right) rotating black hole. The black hole has mass <math>M </math> and <math>\alpha=\pm 0.9 </math>. The coordinates in the plots are <math>(\cos(\phi)r,\sin(\phi)r) </math>. The dots are separated by a proper time of <math>M</math>. The particles all start with only horizontal velocity at <math>\cos(\phi)r=-10M </math>. All particles have conserved energy <math>E=1</math>. This is the energy per unit mass the particles had infinitely far away from the black hole. Units such that <math>c=1,\ G=1 </math> have been used.}}
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| The equations of the trajectory and the time dependence for a particle in the Kerr field are as follows.
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| In the [[Hamilton-Jacobi equation]] we write the [[action (physics)|action]] S in the form: | |
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| :<math>\ S = -E_{0}t + L\phi + S_{r}(r) + S_{\theta}(\theta)</math>
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| where <math>E_{0}</math>, m, and L are the [[conservation laws|conserved]] [[energy]], the [[rest mass]] and the component of the [[angular momentum]] (along the axis of symmetry of the field) of the particle consecutively, and carry out the separation of variables in the Hamilton Jacobi equation as follows:
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| :<math>\left(\frac{dS_{\theta}}{d\theta}\right)^{2} + \left(\alpha E_{0}\sin\theta - \frac{L}{\sin\theta}\right)^{2} + \alpha^{2}m^{2}\cos^{2}\theta = K</math>
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| :<math>\Delta\left(\frac{dS_{r}}{dr}\right)^{2} - \frac{1}{\Delta}\left[\left(r^{2} + \alpha^{2}\right)E_{0} - \alpha L\right]^{2} + m^{2}r^{2} = -K</math>
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| where K is a fourth arbitrary constant (usually called [[Carter constant|Carter's constant]]). The equation of the [[trajectory]] and the time dependence of the coordinates along the trajectory ([[Motion (physics)|motion]] equation) can be found then easily and directly from these equations:
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| :<math>{\frac{\partial{S}}{\partial{E_{0}}}} = const</math>
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| :<math>{\frac{\partial{S}}{\partial{L}}} = const</math>
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| :<math>{\frac{\partial{S}}{\partial{K}}} = const</math>
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| ==Symmetries==
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| The group of isometries of the Kerr metric is the subgroup of the ten-dimensional [[Poincaré group]] which takes the two-dimensional locus of the singularity to itself. It retains the time translations (one dimension) and rotations around its axis of rotation (one dimension). Thus it has two dimensions. Like the Poincaré group, it has four connected components: the component of the identity; the component which reverses time and longitude; the component which reflects through the equatorial plane; and the component that does both.
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| In physics, symmetries are typically associated with conserved constants of motion, in accordance with [[Noether's theorem]]. As shown above, the geodesic equations have four conserved quantities: one of which comes from the definition of a geodesic, and two of which arise from the time translation and rotation symmetry of the Kerr geometry. The fourth conserved quantity does not arise from a symmetry in the standard sense and is commonly referred to as a hidden symmetry.
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| ==See also==
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| * [[Schwarzschild metric]]
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| * [[Kerr–Newman metric]]
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| * [[Reissner–Nordström metric]]
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| * [[Spin-flip]]
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| * [[Kerr–Schild spacetime]]
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| ==References==
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| ===Notes===
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| {{reflist|1}}
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| ===Further reading===
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| *{{cite book | author=Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius & Herlt, Eduard | title=Exact Solutions of Einstein's Field Equations | location=Cambridge | publisher=Cambridge University Press | year=2003 | isbn=0-521-46136-7}}
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| *{{cite book | last1=Meinel | first1=Reinhard | authorlink=Reinhard Meinel | last2=Ansorg | first2=Marcus | last3=Kleinwachter | first3=Andreas | last4=Neugebauer | first4=Gernot | last5=Petroff |first5=David | title=Relativistic Figures of Equilibrium | location=Cambridge | publisher=Cambridge University Press | year=2008 | isbn=978-0-521-86383-4 |url=http://www.cambridge.org/us/knowledge/isbn/item6918194/?site_locale=en_US |accessdate=12 May 2013}}
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| *{{cite book | last=O'Neill |first=Barrett| title=The Geometry of Kerr Black Holes| location=Wellesley, MA | publisher=A. K. Peters| year =1995 | isbn=1-56881-019-9 }}
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| *{{cite book | last=D'Inverno |first=Ray | title=Introducing Einstein's Relativity | location=Oxford | publisher=Clarendon Press | year=1992 | isbn=0-19-859686-3}} ''See chapter 19'' for a readable introduction at the advanced undergraduate level.
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| *{{cite book | last=Chandrasekhar |first=S. |authorlink=Subrahmanyan Chandrasekhar | title=The Mathematical Theory of Black Holes | location=Oxford | publisher=Clarendon Press | year=1992 | isbn=0-19-850370-9 }} ''See chapters 6--10'' for a very thorough study at the advanced graduate level.
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| *{{cite book | last=Griffiths |first=J. B. | title=Colliding Plane Waves in General Relativity | location=Oxford | publisher=Oxford University Press | year=1991 | isbn=0-19-853209-1 }} ''See chapter 13'' for the Chandrasekhar/Ferrari CPW model.
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| *{{cite book | last1=Adler |first1=Ronald | last2=Bazin | first2=Maurice | last3=Schiffer | first3=Menahem | title=Introduction to General Relativity | edition=Second | location=New York | publisher=McGraw-Hill | year=1975 | isbn=0-07-000423-4}} ''See chapter 7''.
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| *{{Cite book | last=Penrose |first=R. |authorlink=Roger Penrose | title=Battelle Rencontres | editor=ed C. de Witt and J. Wheeler | publisher=W. A. Benjamin, New York | year=1968 | page=222}}
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| *{{cite arXiv | last1=Perez |first1=Alejandro |last2=Moreschi |first2=Osvaldo M. |title=Characterizing exact solutions from asymptotic physical concepts | year=2000| eprint=gr-qc/0012100 | version=27 Dec 2000 }} Characterization of three standard families of vacuum solutions as noted above.
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| *{{cite journal |last1=Sotiriou |first1=Thomas P. |last2=Apostolatos |first2=Theocharis A. | title=Corrections and Comments on the Multipole Moments of Axisymmetric Electrovacuum Spacetimes | journal=Class. Quant. Grav. | volume=21 | year=2004 | pages=5727–5733 | doi=10.1088/0264-9381/21/24/003 | arxiv=gr-qc/0407064 | bibcode=2004CQGra..21.5727S | issue=24}} [http://www.arxiv.org/abs/gr-qc/0407064 arXiv eprint] Gives the relativistic multipole moments for the Ernst vacuums (plus the electromagnetic and gravitational relativistic multipole moments for the charged generalization).
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| *{{cite journal | last=Carter | first=B. | authorlink = Brandon Carter | year = 1971 | title = Axisymmetric Black Hole Has Only Two Degrees of Freedom | journal = Physical Review Letters | volume = 26 | pages = 331–333 |doi=10.1103/PhysRevLett.26.331 | bibcode=1971PhRvL..26..331C | issue = 6}}
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| *{{cite book | last = Wald | first = R. M. | authorlink = Robert Wald | year = 1984 | title = General Relativity | publisher = The University of Chicago Press | location = Chicago | isbn = 0-226-87032-4 | pages = 312–324}}
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| *{{cite journal |author=Kerr, R. P.; and Schild, A. |year=2009 |title= Republication of: A new class of vacuum solutions of the Einstein field equations |journal=General Relativity and Gravitation |volume=41 |pages=2485–2499 |doi=10.1007/s10714-009-0857-z |issue=10|bibcode = 2009GReGr..41.2485K }}
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| *{{cite journal |last1=Krasiński |first1=Andrzej |last2=Verdaguer |first2=Enric |last3=Kerr |first3=Roy Patrick | year=2009 | title= Editorial note to: R. P. Kerr and A. Schild, A new class of vacuum solutions of the Einstein field equations |journal=General Relativity and Gravitation |volume=41 |pages=2469–2484 |doi=10.1007/s10714-009-0856-0 |issue=10 |bibcode = 2009GReGr..41.2469K }} "… This note is meant to be a guide for those readers who wish to verify all the details [of the derivation of the Kerr solution]…"
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| == External links ==
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| *[http://arxiv.org/abs/0706.0622 The Kerr spacetime-A brief introduction] by Matt Visser at arxiv.org
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| {{Black holes}}
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| {{Time travel}}
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| {{Relativity}}
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| {{DEFAULTSORT:Kerr Metric}}
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| [[Category:Exact solutions in general relativity]]
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| [[Category:Black holes]]
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| [[Category:Metric tensors]]
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| [[fr:Trou noir de Kerr#Métrique de Kerr]]
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