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General relativity
G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}\,
Einstein field equations
Introduction to...
Mathematical formulation of...
Solutions
Schwarzschild
Reissner-Nordström · Gödel
Kerr · Kerr-Newman
Kasner · Milne · Robertson-Walker
pp-wave · ADM · BSSN
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In general relativity, the Kerr metric (or Kerr vacuum) describes the geometry of spacetime around a rotating massive body. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the General relativity (GR is a Theory of Gravitation that was developed by Albert Einstein between 1907 and 1915 The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying Albert Einstein 's theory of General In General relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside In Physics and Astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space which corresponds to the gravitational The Gödel metric is an exact solution of the Einstein field equations in which the Stress-energy tensor contains two terms the first representing the The Kerr-Newman metric is a solution of Einstein's General relativity field equation that describes the spacetime geometry in the region surrounding a charged The Kasner metric is an exact solution to Einstein 's theory of General relativity. The Milne model was a special relativistic cosmological model proposed by Edward Arthur Milne. In General relativity, the pp-wave spacetimes, or pp-waves for short are an important family of Exact solutions of Einstein's field equation The ADM Formalism developed by Arnowitt, Deser and Misner is a Hamiltonian formulation of General relativity. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS According to this metric, such rotating bodies should exhibit frame dragging, an unusual prediction of general relativity; measurement of this frame dragging effect is a major goal of the Gravity Probe B experiment. Albert Einstein 's theory of General relativity predicts that rotating bodies drag Spacetime around themselves in a phenomenon referred to as frame-dragging Albert Einstein 's theory of General relativity predicts that rotating bodies drag Spacetime around themselves in a phenomenon referred to as frame-dragging Gravity Probe B ( GP-B) is a Satellite -based mission which launched in 2004 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 the curvature of spacetime associated with rotating bodies. At close enough distances, all objects — even light itself — must rotate with the body; the region where this holds is called the ergosphere. Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 The ergosphere is a region located outside a Rotating black hole.

The Kerr metric is often used to describe rotating black holes, which exhibit even more exotic phenomena. Black hole#Major features of rotating black holes A rotating black hole is a Black hole that possesses Angular momentum. Such black holes have two surfaces where the metric appears to have a singularity; the size and shape of these surfaces depends on the black hole's mass and angular momentum. A gravitational singularity (sometimes spacetime singularity) is approximately a place where quantities which are used to measure the Gravitational field become Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position The outer surface encloses the ergosphere and has a shape similar to a flattened sphere. The inner surface is spherical and marks the "radius of no return"; 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. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point 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, Mc2. Even stranger phenomena can be observed within the innermost region of this spacetime, such as some forms of time travel. For example, the Kerr metric permits closed, time-like loops in which a band of travellers returns to the same place after moving for a finite time by their own clock; however, they return to the same place and time, as seen by an outside observer.

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. In General relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the 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, perfectly spherical, and non-rotating body. In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside Karl Schwarzschild ( October 9, 1873 - May 11, 1916) was a German Jewish Physicist and Astronomer. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS The corresponding solution for a charged, spherical, non-rotating body, the Reissner-Nordström metric, was discovered shortly after (1916-1918). In Physics and Astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space which corresponds to the gravitational However, the exact solution for an uncharged, rotating body, the Kerr metric, remained unsolved until 1963, when it was discovered by Roy Kerr. Year 1963 ( MCMLXIII) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar. Roy Patrick Kerr (born 1934 is a New Zealander Mathematician who is best known for discovering the Kerr vacuum, an exact solution to the The natural extension to a charged, rotating body, the Kerr-Newman metric, was discovered shortly afterwards in 1965. The Kerr-Newman metric is a solution of Einstein's General relativity field equation that describes the spacetime geometry in the region surrounding a charged These four related solutions may be summarized by the following table:

Non-rotating (J = 0) Rotating (J ≠ 0)
Uncharged (Q = 0) Schwarzschild Kerr
Charged (Q ≠ 0) Reissner-Nordström Kerr-Newman

where Q represents the body's electric charge and J represents its spin angular momentum. In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside In Physics and Astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space which corresponds to the gravitational The Kerr-Newman metric is a solution of Einstein's General relativity field equation that describes the spacetime geometry in the region surrounding a charged Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position

Contents

Mathematical form

The Kerr metric[1][2] describes the geometry of spacetime in the vicinity of a mass M rotating with angular momentum J

c^{2} d\tau^{2} = \left( 1 - \frac{r_{s} r}{\rho^{2}} \right) c^{2} dt^{2} - \frac{\rho^{2}}{\Lambda^{2}} dr^{2} - \rho^{2} d\theta^{2} -
\left( r^{2} + \alpha^{2} + \frac{r_{s} r \alpha^{2}}{\rho^{2}} \sin^{2} \theta \right) \sin^{2} \theta \ d\phi^{2} + \frac{2r_{s} r\alpha \sin^{2} \theta }{\rho^{2}} \, c \, dt \, d\phi

where the coordinates r,θ,φ are standard spherical coordinate system, and rs is the Schwarzschild radius

r_{s} = \frac{2GM}{c^{2}}

and where the length-scales α, ρ and Λ have been introduced for brevity

\alpha = \frac{J}{Mc}
\ \rho^{2} = r^{2} + \alpha^{2} \cos^{2} \theta
\ \Lambda^{2} = r^{2} - r_{s} r + \alpha^{2}

In the non-relativistic limit where M (or, equivalently, rs) goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates

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}

which are equivalent to the Boyer-Lindquist coordinates[3]

{x} = \sqrt {r^2 + \alpha^2} \sin\theta\cos\phi
{y} = \sqrt {r^2 + \alpha^2} \sin\theta\sin\phi
{z} = r \cos\theta \quad

Frame dragging

We may re-write the Kerr metric in the following form

c^{2} d\tau^{2} = \left( g_{tt} - \frac{g_{t\phi}^{2}}{g_{\phi\phi}} \right) dt^{2} + 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}.

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

\Omega = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{r_{s} \alpha r}{\rho^{2} \left( r^{2} + \alpha^{2} \right) + r_{s} \alpha^{2} r \sin^{2}\theta}.

Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is frame-dragging, which has been observed experimentally. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside Oblate spheroidal coordinates are a three-dimensional orthogonal Coordinate system that results from rotating the two-dimensional elliptic coordinate system A generalization of the coordinates used for the metric of a Schwarzschild black hole that can be used to express the metric of a Kerr black hole. In Spherical coordinates, colatitude is the Complementary angle of the Latitude, i Albert Einstein 's theory of General relativity predicts that rotating bodies drag Spacetime around themselves in a phenomenon referred to as frame-dragging

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 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.
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. In General relativity, an event horizon is a boundary in Spacetime, an area surrounding a Black hole or a Wormhole, inside which events cannot An oblate Spheroid is a rotationally symmetric Ellipsoid having a polar axis shorter than the diameter of the equatorial circle whose plane The ergosphere lies between these two surfaces; within this volume, the purely temporal component gtt is negative, i. The ergosphere is a region located outside a Rotating black hole. 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.

Important surfaces

The Kerr metric has two surfaces on which it appears to be singular. The inner surface corresponds to a spherical event horizon similar to that observed in the Schwarzschild metric; this occurs where the purely radial component grr of the metric goes to infinity. In General relativity, an event horizon is a boundary in Spacetime, an area surrounding a Black hole or a Wormhole, inside which events cannot In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside Solving the quadratic equation 1/grr = 0 yields the solution

r_\mathit{inner} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2}}}{2}

Another singularity occurs where the purely temporal component gtt of the metric changes sign from positive to negative. Again solving a quadratic equation gtt=0 yields the solution

r_\mathit{outer} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2} \cos^{2}\theta}}{2}

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. The ergosphere is a region located outside a Rotating black hole. 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).

A moving particle experiences a positive proper time along its worldline, its path through spacetime. In relativity, proper time is Time measured by a single Clock between events that occur at the same place as the clock In physics the world line of an object is the unique path of that object as it travels through 4- Dimensional Spacetime. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS However, this is impossible within the ergosphere, where gtt 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.

As with the event horizon in the Schwarzschild metric the apparent singularities at rinner and router are an illusion created by the choice of coordinates (i. In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside e. , they are coordinate singularities). In fact, the space-time can be smoothly continued through them by an appropriate choice of coordinates.

Ergosphere and the Penrose process

Main article: Penrose process

A black hole in general is surrounded by a spherical surface, the event horizon situated at the Schwarzschild radius (for a nonrotating black hole), where the escape velocity is equal to the velocity of light. The Penrose process (also called Penrose mechanism) is a process theorised by Roger Penrose wherein energy can be extracted from a Rotating black hole. In General relativity, an event horizon is a boundary in Spacetime, an area surrounding a Black hole or a Wormhole, inside which events cannot The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a characteristic Radius associated with every Mass. 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 the Schwarzschild radius but there is an additional surface outside the Schwarzschild radius named the "ergosurface" given by (r − GM)2 = G2M2 − J2cos2θ 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. A generalization of the coordinates used for the metric of a Schwarzschild black hole that can be used to express the metric of a Kerr black hole. 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 sphere 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. Sir Roger Penrose, PhD, OM, FRS (born 8 August 1931) is an English Mathematical physicist and Emeritus Year 1969 ( MCMLXIX) was a Common year starting on Wednesday (link will display full calendar of the Gregorian calendar. 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. Gamma-ray bursts ( GRB s are the most luminous electromagnetic events occurring in the Universe since the Big Bang.

Gradient operator

Since even a direct check on Kerr metric involves cumbersome calculations, there would be perhaps a very good idea to introduce in here also the contravariant components gik of the metric tensor. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space These are shown below in the expression for the square of the four-gradient operator:

g^{\mu\nu}\frac{\partial}{\partial{x^{\mu}}}\frac{\partial}{\partial{x^{\nu}}} = \frac{\Lambda^{2}}{\rho^{2}}\left(\frac{\partial}{\partial{r}}\right)^{2} + \frac{1}{\rho^{2}}\left(\frac{\partial}{\partial{\theta}}\right)^{2} - \frac{2r_{s}r\alpha}{c\rho^{2}\Lambda^{2}}\frac{\partial}{\partial{\phi}}\frac{\partial}{\partial{t}} -
\frac{1}{c^{2}\Lambda^{2}}\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{1}{\Lambda^{2}\sin^{2}\theta}\left(1 - \frac{r_{s}r}{\rho^{2}}\right)\left(\frac{\partial}{\partial{\phi}}\right)^{2}

Features of the Kerr vacuum

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 horizons, Cauchy horizons, closed timelike curves, and a ring-shaped curvature singularity. The four-gradient is the Four-vector generalization of the Gradient: \partial_\alpha \ \stackrel{\mathrm{def}}{=}\ \left(\frac{1}{c} \frac{\partial}{\partial In Mathematics, an operator is a function which operates on (or modifies another function An asymptotically flat spacetime is a Lorentzian manifold in which roughly speaking the curvature vanishes at large distances from some region so that at large distances The ergosphere is a region located outside a Rotating black hole. In General relativity, an event horizon is a boundary in Spacetime, an area surrounding a Black hole or a Wormhole, inside which events cannot In Physics, a Cauchy horizon is a Light-like boundary of the domain of validity of a Cauchy problem (a particular Boundary value problem of the In a Lorentzian manifold, a closed timelike curve (CTC is a Worldline of a material particle in Spacetime that is "closed" returning to its The geodesic equation can be solved exactly in closed form. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces In addition to two Killing vector fields (corresponding to time translation and axisymmetry), the Kerr vacuum admits a remarkable Killing tensor. In Mathematics, a Killing vector field, named after Wilhelm Killing, is a Vector field on a Riemannian manifold (or Pseudo-Riemannian manifold In Mathematics, the Killing form, named after Wilhelm Killing, is a Symmetric bilinear form that plays a basic role in the theories of Lie groups 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. In Differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is the Traceless component of the Riemann curvature tensor. In Differential geometry and Theoretical physics, the Petrov classification describes the possible algebraic symmetries of the Weyl tensor at The global structure is known. Spacetime topology, the topological structure of Spacetime, is a subject studied primarily in General relativity. Topologically, the homotopy type of the Kerr spacetime can be simply characterized as a line with circles attached at each integer point. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical

While the Kerr vacuum is an exact axis-symmetric solution to Einstein's field equations, the solution is probably not stable in the interior region of the black hole (Penrose, 1968). The stable interior solution is probably not axis-symmetric. The instability of the Kerr metric in the interior region implies that many of the features of the Kerr vacuum described above would probably not be present in a black hole that came into being through gravitational collapse.

A surface on which light can orbit a black hole is called a photon sphere. The Kerr solution has two photon spheres, an inner and an outer one. A photon sphere is a Spherical region of space where Gravity is strong enough that Photons of light are forced to travel in orbits The greater the spin of the black hole is, the farther from each other the photon spheres move. A beam of light travelling 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 travelling in the same direction as the black hole's spin will circularly orbit at the inner photon sphere.

Overextreme Kerr solutions

The location of the event horizon is determined by the largest root of Delta = 0. When M < a, 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. In General relativity, a naked singularity is a Gravitational singularity without an Event horizon. [4]

Kerr black holes as wormholes

Although the Kerr solution appears to be singular at the roots of Δ = 0, these are actually coordinate singularities, and, with an appropriate choice of new coordinates, the Kerr solution can be smoothly extended through the values of r 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 thrugh the event horizon, the r coordinate now behaves like a time coordinate, so it must decrease until the curve passes through the Cauchy horizon.

The region beyond the Cauchy horizon has several surprising features. The r 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 at r = 0 is a 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.

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 (Penrose 1968).

Relation to other exact solutions

The Kerr vacuum is a particular example of a stationary axially symmetric vacuum solution to the Einstein field equation. In General relativity, a Spacetime is said to be stationary if it admits a global nowhere zero Timelike Killing vector field. A vacuum solution is a solution of a Field equation in which the sources of the field are taken to be identically zero The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the The family of all stationary axially symmetric vacuum solutions to the Einstein field equation are the Ernst vacuums.

The Kerr solution is also related to various non-vacuum solutions which model black holes. For example, the 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. The Kerr-Newman metric is a solution of Einstein's General relativity field equation that describes the spacetime geometry in the region surrounding a charged

The special case a = 0 of the Kerr metric yields the Schwarzschild metric, which models a nonrotating black hole which is static and spherically symmetric, in the Schwarzschild coordinates. In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside In General relativity, a Spacetime is said to be static if it admits a global nowhere zero Timelike Hypersurface orthogonal Killing In the theory of Lorentzian manifolds Spherically symmetric spacetimes admit a family of nested round spheres. (In this case, every Geroch moment but the mass vanishes. )

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 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. Spacetime topology, the topological structure of Spacetime, is a subject studied primarily in General relativity. In General relativity, a gravitational plane wave is a special class of a vacuum Pp-wave spacetime, and may be defined in terms of Brinkmann coordinates

Multipole moments

Each asymptotically flat Ernst vacuum can be characterized by giving the infinite sequence of relativistic multipole moments, the first two of which can be interpreted as the mass and angular momentum of the source of the field. An asymptotically flat spacetime is a Lorentzian manifold in which roughly speaking the curvature vanishes at large distances from some region so that at large distances Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position 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

M_n = M \, (i \, \alpha)^n

Thus, the special case of the Schwarzschild vacuum (α=0) gives the "monopole point source" of general relativity. In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside A point source is a single identifiable localized source of something

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 moments. Multipole moments are the Coefficients of a Series expansion of a Potential due to continuous or discrete sources (e 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

a_0 = M, \; \; a_1 = 0, \; \; a_2 = M \, \left( \frac{M^2}{3} - \alpha^2 \right)

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.

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. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product 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.

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:

In this sense, the Kerr vacuums are the simplest stationary axisymmetric asymptotically flat vacuum solutions in general relativity.

Open problems

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. A neutron star is a type of remnant that can result from the Gravitational collapse of a massive Star during a Type II, Type Ib or Type EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 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. In Metric theories of gravitation, particularly General relativity, a static spherically symmetric perfect fluid solution (a term which is often abbreviated as 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 (the relativistic analog of oblate spheroidal balls with nonzero mass and angular momentum but vanishing higher multipole moments) are known. However, the exterior of the Neugebauer/Meinel disk, an exact dust solution which models a rotating thin disk, approaches in a limiting case the a=M Kerr vacuum. In General relativity, a dust solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the

The equations of the trajectory and the time dependence for a particle in the Kerr field

In the Hamilton-Jacobi equation we write the action S in the form:

\ S = -E_{0}t + L\phi + S_{r}(r) + S_{\theta}(\theta)

where E0, m, and L are the 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:

\left(\frac{dS_{\theta}}{d\theta}\right)^{2} + \left(aE_{0}\sin\theta - \frac{L}{\sin\theta}\right)^{2} + a^{2}m^{2}\cos^{2}\theta = K
\Delta\left(\frac{dS_{r}}{dr}\right)^{2} - \frac{1}{\Delta}\left[\left(r^{2} + a^{2}\right)E_{0} - aL\right]^{2} + m^{2}r^{2} = -K

where K is a new arbitrary constant. In Physics, the Hamilton–Jacobi equation (HJE is a reformulation of Classical mechanics and thus equivalent to other formulations such as Newton's laws of In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position The equation of the trajectory and the time dependence of the coordinates along the trajectory (motion equation) can be found then easily and directly from these equations:

{\frac{\partial{S}}{\partial{E_{0}}}} = const
{\frac{\partial{S}}{\partial{L}}} = const
{\frac{\partial{S}}{\partial{K}}} = const


See also

References

  1. ^ Kerr, RP (1963). Trajectory is the path a moving object follows through space The object might be a Projectile or a Satellite, for example In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside The Kerr-Newman metric is a solution of Einstein's General relativity field equation that describes the spacetime geometry in the region surrounding a charged In Physics and Astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space which corresponds to the gravitational Roy Patrick Kerr (born 1934 is a New Zealander Mathematician who is best known for discovering the Kerr vacuum, an exact solution to the "Gravitational field of a spinning mass as an example of algebraically special metrics". Physical Review Letters 11: 237–238.  
  2. ^ Landau, LD; Lifshitz, EM (1975). Lev Davidovich Landau ( Russian language: Ле́в Дави́дович Ланда́у ( January 22, 1908 &ndash April 1, 1968 The Classical Theory of Fields (Course of Theoretical Physics, Vol. 2), revised 4th English ed. , New York: Pergamon Press, pp. 321–330. ISBN 978-0-08-018176-9.  
  3. ^ Boyer, RH; Lindquist RW (1967). "Maximal Analytic Extension of the Kerr Metric". J. Math. Phys. 8: 265–281.  
  4. ^ Chandrasekhar, S. (1983). Padma Vibhushan Subrahmanyan Chandrasekhar, FRS ( Tamil: சுப்பிரமணியன் சந்திரசேகர் English ˌtʃʌndrəˈʃeɪkɑr( The Mathematical Theory of Black Holes, International Series of Monographs on Physics 69, 375.  
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