- Kerr metric
In

general relativity , the**Kerr metric**(or**Kerr vacuum**) describes the geometry ofspacetime around a rotating massive body. According to this metric, such rotating bodies should exhibitframe dragging , an unusual prediction of general relativity; measurement of thisframe dragging effect is a major goal of theGravity 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 the curvature of spacetime associated with rotating bodies. At close enough distances, all objects — evenlight itself — "must" rotate with the body; the region where this holds is called theergosphere .The Kerr metric is often used to describe

rotating black hole s, which exhibit even more exotic phenomena. 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'smass andangular momentum . 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 differentcoordinate 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 itsinvariant mass energy, "Mc"^{2}. 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. The Kerr metric is a generalization of theSchwarzschild metric , which was discovered byKarl Schwarzschild in 1916 and which describes the geometry ofspacetime around an uncharged, perfectly spherical, and non-rotating body. The corresponding solution for a "charged", spherical, non-rotating body, theReissner-Nordström metric , was discovered shortly after (1916-1918). However, the exact solution for an uncharged, "rotating" body, the**Kerr metric**, remained unsolved until1963 , when it was discovered byRoy Kerr . The natural extension to a charged, rotating body, theKerr-Newman metric , was discovered shortly afterwards 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 spinangular momentum .**Mathematical form**:$c^\{2\}\; d\; au^\{2\}\; =\; left(\; 1\; -\; frac\{r\_\{s\}\; r\}\{\; ho^\{2\; ight)\; c^\{2\}\; dt^\{2\}\; -\; frac\{\; ho^\{2\{Lambda^\{2\; dr^\{2\}\; -\; ho^\{2\}\; d\; heta^\{2\}\; -$::::$left(\; r^\{2\}\; +\; alpha^\{2\}\; +\; frac\{r\_\{s\}\; r\; alpha^\{2\{\; ho^\{2\; sin^\{2\}\; heta\; ight)\; sin^\{2\}\; heta\; dphi^\{2\}\; +\; frac\{2r\_\{s\}\; ralpha\; sin^\{2\}\; heta\; \}\{\; ho^\{2\; ,\; c\; ,\; dt\; ,\; dphi$

where the coordinates $r,\; heta,\; phi$ are standard

spherical coordinate system , and "r"_{"s"}is the Schwarzschild radius:$r\_\{s\}\; =\; frac\{2GM\}\{c^\{2$

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

:$alpha\; =\; frac\{J\}\{Mc\}$

:$ho^\{2\}\; =\; r^\{2\}\; +\; alpha^\{2\}\; cos^\{2\}\; heta$

:$Lambda^\{2\}\; =\; r^\{2\}\; -\; r\_\{s\}\; r\; +\; alpha^\{2\}$

In the non-relativistic limit where "M" (or, equivalently, "r"

_{"s"}) goes to zero, the Kerr metric becomes the orthogonal metric for theoblate spheroidal coordinates :$c^\{2\}\; d\; au^\{2\}\; =\; c^\{2\}\; dt^\{2\}\; -\; frac\{\; ho^\{2\{r^\{2\}\; +\; alpha^\{2\; dr^\{2\}\; -\; ho^\{2\}\; d\; heta^\{2\}-\; left(\; r^\{2\}\; +\; alpha^\{2\}\; ight)\; sin^\{2\}\; heta\; dphi^\{2\}$

which are equivalent to the

Boyer-Lindquist coordinates [*cite journal | last = Boyer | first = RH | coauthors = Lindquist RW | year = 1967 | title = Maximal Analytic Extension of the Kerr Metric | journal = J. Math. Phys. | volume = 8 | pages = 265–281 | doi = 10.1063/1.1705193*]:$\{x\}\; =\; sqrt\; \{r^2\; +\; alpha^2\}\; sin\; hetacosphi$:$\{y\}\; =\; sqrt\; \{r^2\; +\; alpha^2\}\; sin\; hetasinphi$:$\{z\}\; =\; r\; cos\; heta\; quad$

**Frame dragging**We may re-write the Kerr metric in the following form

:$c^\{2\}\; d\; au^\{2\}\; =\; left(\; g\_\{tt\}\; -\; frac\{g\_\{tphi\}^\{2\{g\_\{phiphi\; ight)\; dt^\{2\}+\; g\_\{rr\}\; dr^\{2\}\; +\; g\_\{\; heta\; heta\}\; d\; heta^\{2\}\; +\; g\_\{phiphi\}\; left(\; dphi\; +\; frac\{g\_\{tphi\{g\_\{phiphi\; dt\; ight)^\{2\}.$

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:$Omega\; =\; -frac\{g\_\{tphi\{g\_\{phiphi\; =\; frac\{r\_\{s\}\; alpha\; r\}\{\; ho^\{2\}\; left(\; r^\{2\}\; +\; alpha^\{2\}\; ight)\; +\; r\_\{s\}\; alpha^\{2\}\; r\; sin^\{2\}\; heta\}.$

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.**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 theSchwarzschild metric ; this occurs where the purely radial component "g_{rr}" of the metric goes to infinity. Solving the quadratic equation 1/"g"_{"rr"}= 0 yields the solution:$r\_mathit\{inner\}\; =\; frac\{r\_\{s\}\; +\; sqrt\{r\_\{s\}^\{2\}\; -\; 4alpha^\{2\}\{2\}$

Another singularity occurs where the purely temporal component "g

_{tt}" of the metric changes sign from positive to negative. Again solving a quadratic equation "g_{tt}"=0 yields the solution:$r\_mathit\{outer\}\; =\; frac\{r\_\{s\}\; +\; sqrt\{r\_\{s\}^\{2\}\; -\; 4alpha^\{2\}\; cos^\{2\}\; heta\{2\}$

Due to the cos

^{2}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 theergosphere . 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 itsworldline , its path throughspacetime . However, this is impossible within the ergosphere, where "g_{tt}" 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 r_{inner}and r_{outer}are an illusion created by the choice of coordinates (i.e., they arecoordinate singularities ). In fact, the space-time can be smoothly continued through them by an appropriate choice of coordinates.**Ergosphere and the Penrose process**A black hole in general is surrounded by a spherical surface, the

event horizon situated at theSchwarzschild 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 the Schwarzschild radius but there is an additional surface outside the Schwarzschild radius named the "ergosurface" given by $(r-GM)^\{2\}\; =\; G^\{2\}M^\{2\}-J^\{2\}cos^\{2\}\; heta$ 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 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 in1969 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 of the Kerr vacuum**The Kerr vacuum exhibits many noteworthy features: the

maximal analytic extension includes a sequence ofasymptotically flat exterior regions, each associated with anergosphere ,stationary limit surfaces ,event horizon s,Cauchy horizon s,closed timelike curve s, and a ring-shapedcurvature singularity . Thegeodesic equation can be solved exactly in closed form. In addition to twoKilling vector fields (corresponding to "time translation" and "axisymmetry"), the Kerr vacuum admits a remarkableKilling tensor . There is a pair ofprincipal null congruences (one "ingoing" and one "outgoing"). TheWeyl tensor isalgebraically special , in fact it hasPetrov type **D**. The global structure is known. Topologically, thehomotopy type of the Kerr spacetime can be simply characterized as a line with circles attached at each integer point.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 sphere s, an inner and an outer one. 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 larger root of $Lambda=0$. 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 .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 | page = 375]**Kerr black holes as wormholes**Although the Kerr solution appears to be singular at the roots of $Lambda=0$, 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 theEinstein field equation . The family of all stationary axially symmetric vacuum solutions to the Einstein field equation are theErnst 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 special case $a=0$ of the Kerr metric yields the

Schwarzschild metric , which models a "nonrotating" black hole which is static andspherically symmetric , in theSchwarzschild coordinates . (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 theChandrasekhar/Ferrari CPW vacuum , an example of acolliding 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 suitablegravitational plane waves .**Multipole moments**Each

asymptotically flat Ernst vacuum can be characterized by giving the infinite sequence ofrelativistic multipole moments , the first two of which can be interpreted as themass andangular 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:$M\_n\; =\; M\; ,\; (i\; ,\; alpha)^n$Thus, the special case of the Schwarzschild vacuum (α=0) gives the "monopolepoint source " of general relativity."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:$a\_0\; =\; M,\; ;\; ;\; a\_1\; =\; 0,\; ;\; ;\; a\_2\; =\; M\; ,\; left(\; frac\{M^2\}\{3\}\; -\; alpha^2\; ight)$In particular, we see that the Schwarzschild vacuum has nonzero second order Weyl moment, corresponding to the fact that the "Weyl monopole" is theChazy-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 ofmomentum 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.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:

*the isolated mass monopole source with "zero" angular momentum is the "Schwarzschild vacuum" family (one parameter),

*the isolated mass monopole source with "radial" angular momentum is the "Taub-NUT vacuum " family (two parameters; not quite asymptotically flat),

*the isolated mass monopole source with "axial" angular momentum is the "Kerr vacuum" family (two parameters).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 aneutron star --- or theEarth . This works out very nicely for the non-rotating case, where we can match the Schwarzschild vacuum exterior to aSchwarzschild fluid interior, and indeed to more generalstatic 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, theWahlquist 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 theNeugebauer/Meinel disk , an exactdust solution which models a rotating thin disk, approaches in a limiting case the "a"="M" Kerr vacuum.**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:::::$S\; =\; -E\_\{0\}t\; +\; Lphi\; +\; S\_\{r\}(r)\; +\; S\_\{\; heta\}(\; heta)$

where $E\_\{0\}$, m, and L are the conserved

energy , therest mass and the component of theangular 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:::$left(frac\{dS\_\{\; heta\{d\; heta\}\; ight)^\{2\}\; +\; left(aE\_\{0\}sin\; heta\; -\; frac\{L\}\{sin\; heta\}\; ight)^\{2\}\; +\; a^\{2\}m^\{2\}cos^\{2\}\; heta\; =\; K$::$Deltaleft(frac\{dS\_\{r\{dr\}\; ight)^\{2\}\; -\; frac\{1\}\{Delta\}left\; [left(r^\{2\}\; +\; a^\{2\}\; ight)E\_\{0\}\; -\; aL\; ight]\; ^\{2\}\; +\; m^\{2\}r^\{2\}\; =\; -K$

where K is a new arbitrary constant. 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:::$\{frac\{partial\{S\{partial\{E\_\{0\; =\; const$::$\{frac\{partial\{S\{partial\{L\}\; =\; const$::$\{frac\{partial\{S\{partial\{K\}\; =\; const$

**ee also***

Schwarzschild metric

*Kerr-Newman metric

*Reissner-Nordström metric

*Spin-flip **References***

*

* "See chapter 19" for a readable introduction at the advanced undergraduate level.

* "See chapters 6--10" for a very thorough study at the advanced graduate level.

* "See chapter 13" for the Chandrasekhar/Ferrari CPW model.

* "See chapter 7".

*

*cite arXiv |author=Perez, Alejandro; and Moreschi, 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.

*cite journal |author=Sotiriou, Thomas P.; and Apostolatos, 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 [

*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).*

*

*Wikimedia Foundation.
2010.*

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