4492:
6016:, the inner and outer photon spheres degenerate, so that there is only one photon sphere at a single radius. The greater the spin of a black hole, the farther from each other the inner and outer photon spheres move. A beam of light traveling in a direction opposite to the spin of the black hole will circularly orbit the hole at the outer photon sphere. A beam of light traveling in the same direction as the black hole's spin will circularly orbit at the inner photon sphere. Orbiting geodesics with some angular momentum perpendicular to the axis of rotation of the black hole will orbit on photon spheres between these two extremes. Because the spacetime is rotating, such orbits exhibit a precession, since there is a shift in the
9363:
701:
4025:
5006:, and has been tested experimentally. Qualitatively, frame-dragging can be viewed as the gravitational analog of electromagnetic induction. An "ice skater", in orbit over the equator and rotationally at rest with respect to the stars, extends her arms. The arm extended toward the black hole will be torqued spinward. The arm extended away from the black hole will be torqued anti-spinward. She will therefore be rotationally sped up, in a counter-rotating sense to the black hole. This is the opposite of what happens in everyday experience. If she is already rotating at a certain speed when she extends her arms, inertial effects and frame-dragging effects will balance and her spin will not change. Due to the
4487:{\displaystyle g^{\mu \nu }{\frac {\partial }{\partial x^{\mu }}}{\frac {\partial }{\partial x^{\nu }}}=-{\frac {1}{c^{2}\Delta }}\left(r^{2}+a^{2}+{\frac {r_{\text{s}}ra^{2}}{\Sigma }}\sin ^{2}\theta \right)\left({\frac {\partial }{\partial t}}\right)^{2}-{\frac {2r_{\text{s}}ra}{c\Sigma \Delta }}{\frac {\partial }{\partial \phi }}{\frac {\partial }{\partial {t}}}+{\frac {1}{\Delta \sin ^{2}\theta }}\left(1-{\frac {r_{\text{s}}r}{\Sigma }}\right)\left({\frac {\partial }{\partial \phi }}\right)^{2}+{\frac {\Delta }{\Sigma }}\left({\frac {\partial }{\partial r}}\right)^{2}+{\frac {1}{\Sigma }}\left({\frac {\partial }{\partial \theta }}\right)^{2}}
6081:
11213:
6985:
31:
1573:
6618:
13194:
12043:
714:
8603:
1205:
5027:
12053:
9349:
6980:{\displaystyle {\begin{aligned}\Sigma {\frac {dr}{d\lambda }}&=\pm {\sqrt {R(r)}}\\\Sigma {\frac {d\theta }{d\lambda }}&=\pm {\sqrt {\Theta (\theta )}}\\\Sigma {\frac {d\phi }{d\lambda }}&=-\left(aE-{\frac {L_{z}}{\sin ^{2}\theta }}\right)+{\frac {a}{\Delta }}P(r)\\\Sigma {\frac {dt}{d\lambda }}&=-a\left(aE\sin ^{2}\theta -L_{z}\right)+{\frac {r^{2}+a^{2}}{\Delta }}P(r)\end{aligned}}}
5878:). 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
9286:, 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 (These are the relativistic analog of oblate spheroidal balls with nonzero mass and angular momentum but vanishing higher multipole moments). However, the exterior of the
8773:, 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. This interior solution is not likely to be physical and considered as a purely mathematical artefact.
4780:
10348:
1568:{\displaystyle {\begin{aligned}ds^{2}&=-c^{2}d\tau ^{2}\\&=-\left(1-{\frac {r_{\text{s}}r}{\Sigma }}\right)c^{2}dt^{2}+{\frac {\Sigma }{\Delta }}dr^{2}+\Sigma d\theta ^{2}+\left(r^{2}+a^{2}+{\frac {r_{\text{s}}ra^{2}}{\Sigma }}\sin ^{2}\theta \right)\sin ^{2}\theta \ d\phi ^{2}-{\frac {2r_{\text{s}}ra\sin ^{2}\theta }{\Sigma }}c\,dt\,d\phi \end{aligned}}}
8769:
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
8459:. 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.
8452:(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.
5010:, 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
2547:
1081:) â such a coordinate transformation eliminates the apparent singularity where the inner and outer surfaces meet. Objects between these two surfaces must co-rotate with the rotating black hole, as noted above; this feature can in principle be used to extract energy from a rotating black hole, up to its
8075:
3958:
5978:
Note that the inner Kerr geometry is unstable with regard to perturbations in the interior region. This instability means that although the Kerr metric is axis-symmetric, a black hole created through gravitational collapse may not be so. This instability also implies that many of the features of the
4527:
8768:
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
4984:
978:
experiment. Roughly speaking, this effect predicts that objects coming close to a rotating mass will be entrained to participate in its rotation, not because of any applied force or torque that can be felt, but rather because of the swirling curvature of spacetime itself associated with rotating
6562:
8776:
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. This is related to the idea of
7135:
2969:
5863:, 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.
8923:
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 geometry were computed by Hansen; they turn out to be
7808:
5035:
5500:
9026:
arise from treating a certain metric function (formally corresponding to
Newtonian gravitational potential) which appears the WeylâPapapetrou chart for the Ernst family of all stationary axisymmetric vacuum solutions using the standard euclidean scalar
7813:
2356:
8328:
9185:
of the source. These are multi-indexed quantities whose suitably symmetrized and 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.
5764:
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
5252:
8175:
5613:
is negative, i.e., acts like a purely spatial metric component. Consequently, particles within this ergosphere must co-rotate with the inner mass, if they are to retain their time-like character. A moving particle experiences a positive
9153:
4775:{\displaystyle c^{2}d\tau ^{2}=\left(g_{tt}-{\frac {g_{t\phi }^{2}}{g_{\phi \phi }}}\right)dt^{2}+g_{\mathrm {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}.}
1023:; objects passing into the interior of this horizon can never again communicate with the world outside that horizon. However, neither surface is a true singularity, since their apparent singularity can be eliminated in a different
6411:
3743:
9281:
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
8894:
of this CPW solution is quite different from that of the Kerr geometry, 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
4822:
7349:
6073:
3219:
8585:), 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
5585:
7689:
102:
9031:. They are distinct from the moments computed by Hansen, above. 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
6199:
2662:
3502:
8398:
7439:
6993:
2786:
1788:
1702:
5363:
2763:
7220:
5854:
2152:
6623:
1210:
3314:
2235:
1953:
8883:
8228:
5395:
8079:
8538:
7555:
2069:
1104:
The Kerr metric is commonly expressed in one of two forms, the BoyerâLindquist form and the KerrâSchild form. It can be readily derived from the
Schwarzschild metric, using the
1844:
6368:
6084:
Another trajectory of a test mass around a spinning (Kerr) black hole. Unlike orbits around a
Schwarzschild black hole, the orbit is not confined to a single plane, but will
1623:
3733:
3643:
2542:{\displaystyle g\mathop {\longrightarrow } _{M\to 0}-c^{2}dt^{2}+{\frac {\Sigma }{r^{2}+a^{2}}}dr^{2}+\Sigma d\theta ^{2}+\left(r^{2}+a^{2}\right)\sin ^{2}\theta d\phi ^{2}}
1064:
9287:
8581:
2341:
1887:
946:
898:
10960:
Gives the relativistic multipole moments for the Ernst vacuums (plus the electromagnetic and gravitational relativistic multipole moments for the charged generalization).
10449:
Bardeen, James M.; Press, William H.; Teukolsky, Saul A. (1972). "Rotating Black Holes: Locally
Nonrotating Frames, Energy Extraction, and Scalar Synchrotron Radiation".
9478:
Melia, Fulvio (2009). "Cracking the
Einstein code: relativity and the birth of black hole physics, with an Afterword by Roy Kerr", Princeton University Press, Princeton,
8977:
6228:
3588:
11938:
8815:
5738:
3104:
3021:
2283:
9333:
8681:
8493:
7469:
6328:
5014:
system with the black hole being the sun gear, the ice skater being a planetary gear and the outside universe being the ring gear. This can also be interpreted through
1997:
7493:
7374:
5163:
7658:
7586:
6611:
6060:
5677:
5295:
2021:
9159:
4002:
12897:
7517:
6287:
6036:
472:
6128:
5772:
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
9012:
8847:
8428:
6012:
3536:
3369:
9209:
9045:
8764:
8737:
8709:
8221:
8197:
7682:
7634:
7610:
6400:
6256:
5653:
5594:
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
5106:
5082:
5058:
3698:
3671:
3612:
3560:
3339:
2310:
1181:
1154:
8887:
8070:{\displaystyle {\ddot {x}}+i{\ddot {y}}=4iMa{\frac {r}{\Sigma ^{2}}}W\left-M(x+iy)\left({\frac {4r^{2}}{\Sigma }}-1\right){\frac {C-a^{2}W^{2}}{r\Sigma ^{2}}}}
3953:{\displaystyle 2M_{\rm {irr}}^{2}=M^{2}+{\sqrt {M^{4}-J^{2}c^{2}/G^{2}}}\Longrightarrow M^{2}=M_{\rm {irr}}^{2}+{\frac {J^{2}c^{2}}{4M_{\rm {irr}}^{2}G^{2}}}.}
9265:
but if the solution is held to be valid only outside some compact region (subject to certain restrictions), in principle, it should be able to be used as an
3401:
10393:
4979:{\displaystyle \Omega =-{\frac {g_{t\phi }}{g_{\phi \phi }}}={\frac {r_{\text{s}}rac}{\Sigma \left(r^{2}+a^{2}\right)+r_{\text{s}}ra^{2}\sin ^{2}\theta }}.}
2287:. This implies that there is coupling between time and motion in the plane of rotation that disappears when the black hole's angular momentum goes to zero.
10112:
745:
5699:). In fact, the spacetime can be smoothly continued through it by an appropriate choice of coordinates. In turn, the outer boundary of the ergosphere at
10169:
8717:. 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
6135:
12208:
10916:
Sotiriou, Thomas P.; Apostolatos, Theocharis A. (2004). "Corrections and
Comments on the Multipole Moments of Axisymmetric Electrovacuum Spacetimes".
7226:
9158:
In particular, we see that the
Schwarzschild vacuum has nonzero second order Weyl moment, corresponding to the fact that the "Weyl monopole" is the
8332:
6236:
of the particle. Furthermore, there are two constants of motion given by the time translation and rotation symmetries of Kerr spacetime, the energy
9169:
In weak field general relativity, it is convenient to treat isolated sources using another type of multipole, which generalize the Weyl moments to
8713:
corresponding to these roots. The larger of these roots determines the location of the event horizon, and the smaller determines the location of a
6072:
1075:
into two disconnected patches; using a different coordinate transformation one can then relate the extended external patch to the inner patch (see
124:
9189:
Perez and
Moreschi have given an alternative notion of "monopole solutions" by expanding the standard NP tetrad of the Ernst vacuums in powers of
6566:
Since there are four (independent) constants of motion for degrees of freedom, the equations of motion for a test particle in Kerr spacetime are
3120:
13216:
8804:
6557:{\displaystyle Q=p_{\theta }^{2}+\cos ^{2}\theta \left(a^{2}\left(\mu ^{2}-E^{2}\right)+\left({\frac {L_{z}}{\sin \theta }}\right)^{2}\right).}
5511:
5886:. Rotating black holes in astrophysics are a potential source of large amounts of energy and are used to explain energetic phenomena, such as
5379:
of the metric changes sign from positive to negative, in Kerr metric that happens at a different distance. Again solving a quadratic equation
11294:
37:
3392:
In the KerrâSchild form of the Kerr metric, the determinant of the metric tensor is everywhere equal to negative one, even near the source.
11299:
9162:
solution, not the
Schwarzschild vacuum solution, which arises from the Newtonian potential of a certain finite length uniform density thin
5034:
329:
12944:
10116:
13179:
12018:
11675:
9278:
2594:
682:
7130:{\displaystyle \Theta (\theta )=Q-\cos ^{2}\theta \left(a^{2}\left(\mu ^{2}-E^{2}\right)+{\frac {L_{z}^{2}}{\sin ^{2}\theta }}\right)}
2964:{\displaystyle \mathbf {k} =(k_{x},k_{y},k_{z})=\left({\frac {rx+ay}{r^{2}+a^{2}}},{\frac {ry-ax}{r^{2}+a^{2}}},{\frac {z}{r}}\right)}
12968:
12714:
10073:
9253:
In this sense, the Kerr vacuums are the simplest stationary axisymmetric asymptotically flat vacuum solutions in general relativity.
3425:
492:
334:
10363:
The LIGO Scientific
Collaboration; the Virgo Collaboration (2007). "The Kerr spacetime: A brief introduction". p. 13, below eq. 52.
5002:
Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is called
11613:
11439:
11339:
10538:
Riazuelo, Alain (December 2020). "Seeing Relativity -- III. Journeying within the Kerr metric toward the negative gravity region".
7387:
3969:
1725:
1639:
1093:
738:
447:
9273:, or the Earth. This works out very nicely for the non-rotating case, where the Schwarzschild vacuum exterior can be matched to a
7803:{\displaystyle {\frac {t}{\tau }}={\sqrt {\frac {\left(a^{2}+r^{2}\right)^{2}-a^{2}\Delta \sin ^{2}\theta }{\Delta \ \Sigma }}}.}
4798:
This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radius
12750:
12704:
10484:
Frolov, Andrei V.; Frolov, Valeri P. (2014). "Rigidly rotating zero-angular-momentum observer surfaces in the Kerr spacetime".
10186:
Bhat, Manjiri; Dhurandhar, Sanjeev; Dadhich, Naresh (1985). "Energetics of the KerrâNewman black hole by the penrose process".
5866:
The region outside the event horizon but inside the surface where the rotational velocity is the speed of light, is called the
5305:
974:), a distinctive prediction of general relativity. The first measurement of this frame dragging effect was done in 2011 by the
360:
194:
3647:. The reason for this is that in order to get a static body to spin, energy needs to be applied to the system. Because of the
12888:
12089:
11116:
11012:
10874:
10847:
10824:
10798:
10775:
10756:
10732:
10713:
9891:
9864:
3382:
786:
399:
10019:
Balasin, Herbert; Nachbagauer, Herbert (1994). "Distributional energyâmomentum tensor of the KerrâNewman spacetime family".
2685:
12372:
12155:
7141:
5778:
5030:
Location of the horizons, ergospheres and the ring singularity of the Kerr spacetime in Cartesian KerrâSchild coordinates.
2092:
841:, was discovered shortly thereafter in 1965. These four related solutions may be summarized by the following table, where
9859:. Course of Theoretical Physics. Vol. 2 (Revised 4th English ed.). New York: Pergamon Press. pp. 321â330.
1077:
731:
3250:
2175:
1900:
12625:
12167:
12122:
11655:
10142:
9483:
8689:, and, with an appropriate choice of new coordinates, the Kerr solution can be smoothly extended through the values of
111:
8741:
coordinate now behaves like a time coordinate, so it must decrease until the curve passes through the Cauchy horizon.
12996:
12630:
11744:
11022:
Kerr, R. P.; Schild, A. (2009). "Republication of: A new class of vacuum solutions of the Einstein field equations".
9927:
Visser, Matt (2007). "The Kerr spacetime: A brief introduction". p. 15, Eq. 60â61, p. 24, p. 35.
9422:
Do not confuse the relativistic multipole moments computed by Hansen with the Weyl multipole moments discussed below.
8642:
6573:
Using these constants of motion, the trajectory equations for a test particle can be written (using natural units of
6333:
137:
12132:
12112:
11739:
10900:
Perez, Alejandro; Moreschi, Osvaldo M. (2000). "Characterizing exact solutions from asymptotic physical concepts".
10255:
12950:
11624:
9386:
9362:
951:
826:
700:
417:
30:
12663:
12256:
11660:
11456:
10432:
5982:
A surface on which light can orbit a black hole is called a photon sphere. The Kerr solution has infinitely many
5860:
5495:{\displaystyle r_{\rm {E}}^{\pm }={\frac {r_{\text{s}}\pm {\sqrt {r_{\text{s}}^{2}-4a^{2}\cos ^{2}\theta }}}{2}}}
3405:
1194:
1125:
324:
10242:
9337:
Kerr geometry. Physical thin-disk solutions obtained by identifying parts of the Kerr spacetime are also known.
3510:, the remaining mass cannot shrink below the irreducible mass. Therefore, if a black hole rotates with the spin
12766:
12127:
11882:
11563:
11202:
8624:
6294:
11061:"Editorial note to: R. P. Kerr and A. Schild, A new class of vacuum solutions of the Einstein field equations"
979:
bodies. In the case of a rotating black hole, at close enough distances, all objects â even light â
13197:
12939:
12844:
12774:
12228:
11650:
11525:
2571:
2346:
1628:
971:
411:
119:
12907:
8810:
The Kerr solution is also related to various non-vacuum solutions which model black holes. For example, the
457:
12761:
12238:
12056:
11568:
11471:
11259:
10166:
9983:
Debney, G. C.; Kerr, R. P.; Schild, A. (1969). "Solutions of the Einstein and Einstein-Maxwell Equations".
8778:
8620:
6373:
5898:
The Kerr geometry exhibits many noteworthy features: the maximal analytic extension includes a sequence of
1109:
991:
477:
11578:
12802:
12536:
12521:
11872:
11837:
11827:
8803:. The family of all stationary axially symmetric vacuum solutions to the Einstein field equation are the
8502:
7526:
5975:
of the Kerr spacetime can be simply characterized as a line with circles attached at each integer point.
5681:. Thus, no particle can move in the direction opposite to central mass's rotation within the ergosphere.
3648:
2575:
718:
209:
129:
5038:
Comparison of the shadow (black) and the important surfaces (white) of a black hole. The spin parameter
2033:
13089:
12531:
12484:
12218:
12187:
11558:
11510:
11493:
11192:
10812:
10599:
10134:
9907:
8891:
8455:
In physics, symmetries are typically associated with conserved constants of motion, in accordance with
5968:
1105:
622:
170:
10722:
9575:
Abbot, B.P. (11 February 2016). "Observation of Gravitational Waves from a Binary Black Hole Merger".
8793:
1811:
13119:
12745:
12426:
12082:
12013:
11515:
11429:
11177:
11109:
8896:
8868:
5919:
1596:
995:
422:
234:
10723:
Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius; Herlt, Eduard (2003).
10208:
9269:
to model the gravitational field around a rotating massive object other than a black hole such as a
6076:
Animation of a test-particle's orbit around a spinning black hole. Left: top view, right: side view.
3711:
3621:
1036:
990:
The light from distant sources can travel around the event horizon several times (if close enough);
821:
around an uncharged, spherically symmetric, and non-rotating body. The corresponding solution for a
13144:
12694:
12454:
12365:
11690:
9391:
8323:{\displaystyle C=p_{\theta }^{2}+\left(aE\sin {\theta }-{\frac {L_{z}}{\sin {\theta }}}\right)^{2}}
3343:
approaches zero. In this form of solution, units are selected so that the speed of light is unity (
994:. To a distant viewer, the apparent perpendicular distance between images decreases at a factor of
790:
657:
647:
497:
314:
8547:
6080:
2319:
1865:
1092:
The LIGO experiment that first detected gravitational waves, announced in 2016, also provided the
929:
881:
13226:
12873:
12795:
12415:
12406:
11867:
11403:
11366:
11269:
11264:
11182:
9948:
Boyer, Robert H.; Lindquist, Richard W. (1967). "Maximal Analytic Extension of the Kerr Metric".
9444:(1963). "Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics".
8930:
8800:
8613:
8433:
6206:
5247:{\displaystyle r_{\rm {H}}^{\pm }={\frac {r_{\text{s}}\pm {\sqrt {r_{\text{s}}^{2}-4a^{2}}}}{2}}}
3569:
642:
376:
11091:"... This note is meant to be a guide for those readers who wish to verify all the details ..."
8170:{\displaystyle {\ddot {z}}=-Mz\left({\frac {4r^{2}}{\Sigma }}-1\right){\frac {C}{r\Sigma ^{2}}}}
5986:, lying between an inner one and an outer one. In the nonrotating, Schwarzschild solution, with
5711:
3080:
3068:
2992:
2258:
1000:(about 500). However, fast spinning black holes have less distance between multiplicity images.
13099:
12962:
12868:
12826:
12500:
12464:
12150:
12142:
11897:
11785:
11770:
11629:
11322:
11136:
10203:
9381:
9299:
8811:
8686:
8660:
8472:
7448:
5696:
1976:
1004:
956:
838:
612:
597:
442:
204:
7478:
7359:
13104:
12709:
12648:
12291:
12117:
11775:
11588:
11332:
11187:
9881:
8864:
7643:
7571:
6578:
6045:
5961:
5957:
5927:
5915:
5662:
5262:
5007:
4014:
3409:
2006:
587:
155:
10617:
9557:
9039:
Weyl moments vanish. For the Kerr vacuum solutions, the first few Weyl moments are given by
3977:
3075:
is a constant rotational parameter of the spinning object. It is understood that the vector
829:, was discovered soon afterwards (1916â1918). However, the exact solution for an uncharged,
13221:
13009:
12935:
12779:
12381:
12075:
12046:
11754:
11726:
11609:
11537:
11505:
11349:
11157:
11102:
11072:
11031:
10976:
10935:
10663:
10557:
10503:
10458:
10408:
10290:
10268:
10195:
10038:
9992:
9957:
9802:
9755:
9706:
9649:
9594:
9512:
9453:
9376:
8983:
8852:
8790:
8456:
7502:
6265:
6260:, and the component of the orbital angular momentum parallel to the spin of the black hole
6021:
5761:
5685:
5369:
5123:
3651:
this energy also has a mass-equivalent, which adds to the total massâenergy of the system,
3386:
1890:
1028:
912:
810:
407:
10325:
9148:{\displaystyle a_{0}=M,\qquad a_{1}=0,\qquad a_{2}=M\left({\frac {M^{2}}{3}}-a^{2}\right)}
6113:
8:
13084:
13079:
13069:
12901:
12850:
12668:
12653:
12479:
12410:
12358:
12339:
11877:
11852:
11822:
11780:
11734:
11371:
11237:
11232:
11162:
9401:
9274:
9262:
9234:
9035:
relativistic moments. In the case of solutions symmetric across the equatorial plane the
8991:
8908:
8826:
8407:
7566:-effect, a zero-angular-momentum observer (ZAMO) is corotating with the angular velocity
6097:
5991:
5899:
5015:
3515:
3348:
602:
572:
567:
319:
13074:
11076:
11035:
10980:
10939:
10667:
10654:
BiÄĂĄk, JĆĂ; Ledvinka, TomĂĄĆĄ (1993). "Relativistic disks as sources of the Kerr metric".
10631:
10561:
10507:
10462:
10412:
10362:
10294:
10199:
10087:
Visser, Matt (14 January 2008). "The Kerr spacetime: A brief introduction". p. 12.
10042:
9996:
9961:
9806:
9759:
9710:
9653:
9598:
9516:
9457:
582:
552:
13149:
13049:
12973:
12831:
12812:
12806:
12757:
12699:
12608:
12526:
12444:
12421:
12389:
12285:
11711:
11583:
11488:
11279:
11247:
11212:
11172:
11167:
11047:
11001:
10951:
10925:
10901:
10863:
10787:
10581:
10547:
10519:
10493:
10424:
10364:
10306:
10280:
10221:
10088:
10054:
10028:
9928:
9825:
9790:
9743:
9696:
9618:
9584:
9533:
9508:
9496:
9194:
8749:
8722:
8694:
8206:
8182:
7667:
7619:
7595:
6385:
6241:
6105:
5638:
5091:
5067:
5043:
4010:
in BoyerâLindquist coordinates are shown below in the expression for the square of the
3683:
3656:
3597:
3545:
3324:
2295:
1166:
1139:
814:
794:
652:
537:
462:
309:
214:
199:
160:
22:
10947:
10149:
13159:
12991:
12983:
12584:
12541:
12213:
11857:
11847:
11842:
11542:
11530:
11483:
11424:
11393:
11383:
11317:
11008:
10870:
10843:
10820:
10794:
10771:
10752:
10728:
10709:
10679:
10585:
10573:
10523:
10428:
10310:
10138:
10050:
9908:
Lecture XXVI: Kerr black holes: I. Metric structure and regularity of particle orbits
9887:
9860:
9830:
9771:
9724:
9665:
9622:
9610:
9538:
9479:
9266:
9213:(the radial coordinate in the WeylâPapapetrou chart). According to this formulation:
8586:
5923:
3703:
1024:
833:
black hole, the Kerr metric, remained unsolved until 1963, when it was discovered by
771:
482:
257:
13029:
10955:
10745:; Ansorg, Marcus; Kleinwachter, Andreas; Neugebauer, Gernot; Petroff, David (2008).
10225:
8445:
532:
13064:
13054:
13001:
12978:
12556:
12334:
12299:
12246:
12179:
12008:
11968:
11716:
11665:
11520:
11434:
11227:
11197:
11080:
11039:
10984:
10943:
10671:
10565:
10511:
10466:
10416:
10298:
10213:
10058:
10046:
10000:
9965:
9820:
9810:
9763:
9714:
9657:
9606:
9602:
9528:
9520:
9461:
9354:
9028:
8920:
8912:
8860:
8770:
8449:
8448:
which takes the two-dimensional locus of the singularity to itself. It retains the
7379:
5932:
3968:
Since even a direct check on the Kerr metric involves cumbersome calculations, the
2579:
1159:
1012:
854:
798:
607:
562:
542:
487:
13124:
11051:
9815:
9637:
8911:
Ernst vacuum can be characterized by giving the infinite sequence of relativistic
8444:
The group of isometries of the Kerr metric is the subgroup of the ten-dimensional
627:
13139:
13114:
13039:
13034:
12917:
12878:
12840:
12784:
12658:
12594:
12329:
12028:
11978:
11953:
11749:
11593:
11500:
11466:
11378:
10742:
10238:
10173:
9852:
9283:
8796:
6405:
5887:
5883:
5751:
4811:
3507:
975:
846:
662:
637:
522:
517:
381:
262:
224:
12922:
12309:
10675:
9767:
7344:{\displaystyle R(r)=P(r)^{2}-\Delta \left(\mu ^{2}r^{2}+(L_{z}-aE)^{2}+Q\right)}
432:
13164:
12836:
12820:
12816:
12719:
12689:
12546:
12449:
12319:
12304:
12271:
11943:
11928:
11695:
11634:
11478:
11152:
10988:
10964:
10912:
Characterization of three standard families of vacuum solutions as noted above.
10701:
10515:
10389:
9719:
9684:
9524:
9465:
9368:
8714:
7563:
6377:
5941:
5911:
5011:
5003:
1113:
1082:
967:
705:
672:
667:
355:
219:
11085:
11060:
11043:
10699:
10569:
6408:. It is related to the total angular momentum of the particle and is given by
5633:
is negative, unless the particle is co-rotating around the interior mass
13210:
13154:
13134:
13129:
13044:
12912:
12740:
12684:
12516:
12469:
12003:
11923:
11451:
11446:
10888:
10577:
10420:
9775:
9728:
9669:
9291:
6233:
6101:
5983:
5972:
5907:
5879:
5757:
5119:
4011:
4007:
2583:
1020:
782:
778:
632:
547:
527:
452:
350:
229:
592:
13174:
13094:
13059:
12589:
12551:
12324:
11958:
11832:
11800:
11408:
11327:
10683:
10606:. International Series of Monographs on Physics. Vol. 69. p. 375.
9834:
9614:
9542:
9270:
9017:
5979:
Kerr geometry described above may not be present inside such a black hole.
3214:{\displaystyle {\frac {x^{2}+y^{2}}{r^{2}+a^{2}}}+{\frac {z^{2}}{r^{2}}}=1}
1186:
1019:
and has a shape similar to a flattened sphere. The inner surface marks the
1003:
Rotating black holes have surfaces where the metric seems to have apparent
577:
557:
8814:
models a (rotating) black hole endowed with an electric charge, while the
8744:
The region beyond the Cauchy horizon has several surprising features. The
12927:
12474:
12251:
12192:
12098:
11998:
11948:
11902:
11805:
11680:
11344:
11252:
10996:
10746:
10344:
10302:
5953:
5615:
5580:{\displaystyle r_{\rm {E}}^{\pm }=1\pm {\sqrt {1-a^{2}\cos ^{2}\theta }}}
5110:, while the left side of the black hole is rotating towards the observer.
3048:
3044:
1190:
467:
10906:
5372:
the event horizon is also the place where the purely temporal component
3408:
can be applied. This implies that the Kerr black hole has the form of a
13169:
12735:
12579:
12574:
12223:
11988:
11918:
11887:
11862:
11815:
11810:
11795:
11461:
11398:
11388:
11289:
11125:
10217:
9848:
8627: in this section. Unsourced material may be challenged and removed.
6567:
6085:
5903:
5603:
4803:
1016:
984:
774:
677:
239:
165:
97:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
10930:
10033:
10004:
9969:
9744:"BĂ€cklund Transformation for the Ernst Equation of General Relativity"
9661:
8467:
The location of the event horizon is determined by the larger root of
2345:) goes to zero, the Kerr metric becomes the orthogonal metric for the
983:
rotate with the black hole; the region where this holds is called the
12566:
12281:
12266:
12162:
12023:
11670:
11573:
11361:
11356:
9396:
9294:
which models a rotating thin disk, approaches in a limiting case the
7810:
In Cartesian KerrâSchild coordinates, the equations for a photon are
7638:
of the test-particle is measured relative to a probe corotating with
5623:
5619:
1131:
818:
767:
248:
10369:
9933:
8602:
13109:
12459:
12261:
11933:
11892:
11790:
10967:(1971). "Axisymmetric Black Hole Has Only Two Degrees of Freedom".
10552:
10470:
10071:
Berman, Marcelo. "Energy of Black Holes and Hawking's Universe" in
9589:
9441:
9182:
8879:
8818:
models a (rotating) hole with infalling electromagnetic radiation.
5706:
is not singular by itself even in Kerr coordinates due to non-zero
1007:; the size and shape of these surfaces depends on the black hole's
834:
617:
427:
267:
10741:
10498:
10285:
10093:
9701:
8655:
Although the Kerr solution appears to be singular at the roots of
7590:
which is defined with respect to the bookkeeper's coordinate time
3592:
in comparison with a corresponding Schwarzschild black hole where
11973:
6194:{\displaystyle -\mu ^{2}=p^{\alpha }g_{\alpha \beta }p^{\beta },}
2657:{\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+fk_{\mu }k_{\nu }\!}
11059:
KrasiĆski, Andrzej; Verdaguer, Enric; Kerr, Roy Patrick (2009).
10808:
for a readable introduction at the advanced undergraduate level.
9558:"Black holes warp the universe into a grotesque hall of mirrors"
9497:"Divergent reflections around the photon sphere of a black hole"
5026:
3404:) is axially symmetric, it can be cast into a form to which the
12789:
11993:
11983:
11963:
11685:
11284:
11274:
5756:
A black hole in general is surrounded by a surface, called the
5133:
of the metric goes to infinity. Solving the quadratic equation
3497:{\displaystyle E_{\rm {rot}}=c^{2}\left(M-M_{\rm {irr}}\right)}
837:. The natural extension to a charged, rotating black hole, the
12067:
11094:
11007:. Chicago: The University of Chicago Press. pp. 312â324.
10706:
The Kerr Spacetime: Rotating Black Holes in General Relativity
8393:{\displaystyle W={\dot {t}}-a\sin ^{2}{\theta }{\dot {\phi }}}
809:
The Kerr metric is a generalization to a rotating body of the
12350:
10394:"Global structure of the Kerr family of gravitational fields"
8871:. (In this case, every Geroch moment but the mass vanishes.)
7434:{\displaystyle {\frac {dx^{\alpha }}{d\lambda }}=p^{\alpha }}
6089:
1783:{\displaystyle y={\sqrt {r^{2}+a^{2}}}\sin \theta \sin \phi }
1697:{\displaystyle x={\sqrt {r^{2}+a^{2}}}\sin \theta \cos \phi }
966:
According to the Kerr metric, a rotating body should exhibit
8878:
of the Kerr geometry, or rather a portion of it, is locally
2253:
A key feature to note in the above metric is the cross-term
12770:
9178:
8916:
5626:. However, this is impossible within the ergosphere, where
1008:
7662:. The gravitational time-dilation between a ZAMO at fixed
10861:
Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975).
10834:
for a very thorough study at the advanced graduate level.
5926:
can be solved exactly in closed form. In addition to two
5358:{\displaystyle r_{\rm {H}}^{\pm }=1\pm {\sqrt {1-a^{2}}}}
5114:
There are several important surfaces in the Kerr metric (
1078:
Schwarzschild metric § Singularities and black holes
9238:
family (two parameters; not quite asymptotically flat),
3112:
is not the radius, but rather is implicitly defined by
11058:
6380:
showed that there exists a fourth constant of motion,
2758:{\displaystyle f={\frac {2GMr^{3}}{r^{4}+a^{2}z^{2}}}}
801:, which makes exact solutions very difficult to find.
11021:
10448:
10271:(May 2011). "Finally, results from Gravity Probe B".
10185:
9302:
9197:
9048:
8994:
8933:
8890:
model. This is particularly interesting, because the
8829:
8752:
8725:
8697:
8663:
8550:
8505:
8475:
8410:
8335:
8231:
8209:
8185:
8082:
7816:
7692:
7670:
7646:
7622:
7598:
7574:
7529:
7505:
7481:
7451:
7390:
7362:
7229:
7215:{\displaystyle P(r)=E\left(r^{2}+a^{2}\right)-aL_{z}}
7144:
6996:
6621:
6581:
6414:
6388:
6336:
6297:
6268:
6244:
6209:
6138:
6116:
6048:
6024:
5994:
5944:. There is a pair of principal null congruences (one
5849:{\displaystyle (r-M)^{2}=M^{2}-J^{2}\cos ^{2}\theta }
5781:
5745:
5714:
5665:
5641:
5602:; the space between these two surfaces is called the
5514:
5398:
5308:
5265:
5166:
5094:
5070:
5046:
4825:
4530:
4028:
3980:
3746:
3714:
3686:
3659:
3624:
3600:
3572:
3548:
3518:
3428:
3351:
3327:
3253:
3123:
3083:
2995:
2789:
2688:
2597:
2359:
2322:
2298:
2261:
2178:
2095:
2036:
2009:
1979:
1903:
1868:
1814:
1728:
1642:
1599:
1208:
1169:
1142:
1039:
932:
884:
40:
9344:
8784:
7686:
and a stationary observer far away from the mass is
5606:. Within this volume, the purely temporal component
3108:
is directed along the positive z-axis. The quantity
2147:{\displaystyle \Sigma =r^{2}+a^{2}\cos ^{2}\theta ,}
1631:, which are equivalent to the cartesian coordinates
10915:
10009:
Especially see equations (7.10), (7.11) and (7.14).
9791:"A radiating Kerr black hole and Hawking radiation"
8915:, the first two of which can be interpreted as the
5695:is due to the choice of coordinates (i.e., it is a
3506:of a black hole is extracted, for example with the
1027:. A similar situation obtains when considering the
11000:
10862:
10860:
10786:
10629:Roy Kerr (Crafoord Prize Symposium in Astronomy):
10592:
10018:
9327:
9203:
9177:, characterizing respectively the distribution of
9147:
9006:
8971:
8841:
8758:
8731:
8703:
8675:
8575:
8532:
8487:
8422:
8392:
8322:
8215:
8191:
8169:
8069:
7802:
7676:
7652:
7628:
7604:
7580:
7549:
7511:
7487:
7463:
7433:
7368:
7343:
7214:
7129:
6979:
6605:
6556:
6394:
6362:
6322:
6281:
6250:
6222:
6193:
6122:
6054:
6030:
6006:
5848:
5732:
5671:
5647:
5579:
5494:
5357:
5289:
5246:
5100:
5076:
5052:
4978:
4774:
4486:
3996:
3952:
3727:
3702:(the gravitating mass) of the body (including its
3692:
3665:
3637:
3606:
3582:
3554:
3530:
3496:
3363:
3333:
3309:{\displaystyle r\to R={\sqrt {x^{2}+y^{2}+z^{2}}}}
3308:
3213:
3098:
3015:
2963:
2757:
2656:
2541:
2335:
2304:
2277:
2230:{\displaystyle \Delta =r^{2}-r_{\text{s}}r+a^{2}.}
2229:
2146:
2063:
2015:
1991:
1948:{\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}},}
1947:
1881:
1838:
1782:
1696:
1617:
1567:
1175:
1148:
1058:
940:
892:
96:
10128:
3012:
2653:
1031:which also appears to result in a singularity at
13208:
10133:. Cambridge Monographs on Mathematical Physics.
10077:, page 148 (Kreitler ed., Nova Publishers 2006).
9982:
9261:The Kerr geometry is often used as a model of a
8592:
5893:
5126:; this occurs where the purely radial component
1112:(also known as the spinâcoefficient formalism),
9879:
8789:The Kerr geometry is a particular example of a
16:Exact solution for the Einstein field equations
10384:
10382:
10380:
9947:
9847:
6132:of the test particle, defined by the relation
1119:
12366:
12083:
11110:
10899:
10811:
10784:
10725:Exact Solutions of Einstein's Field Equations
10653:
10598:
10340:
10338:
10109:Exact Solutions of Einstein's Field Equations
8462:
3063:is the constant mass of the spinning object,
2578:as follows. These solutions were proposed by
739:
10891:(1968). ed C. de Witt and J. Wheeler (ed.).
10483:
10349:About Time: Einstein's Unfinished Revolution
10115:regarding determinant of metric tensor. See
9886:. Oxford University Press. pp. 55â57 .
6040:variable after completing one period in the
3415:
10840:Colliding Plane Waves in General Relativity
10765:
10442:
10377:
10358:
10356:
9922:
9920:
9918:
9916:
9880:Rezzolla, Luciano; Zanotti, Olindo (2013).
9682:
9638:"Note on the Kerr Spinning-Particle Metric"
9635:
6104:in the Kerr spacetime are governed by four
992:creating multiple images of the same object
12373:
12359:
12090:
12076:
11676:Magnetospheric eternally collapsing object
11117:
11103:
10869:(Second ed.). New York: McGraw-Hill.
10770:. Wellesley, Massachusetts: A. K. Peters.
10335:
10243:Black Holes: Energetics and Thermodynamics
9636:Newman, E. T.; Janis, A. I. (1965-06-01).
9279:static spherically symmetric perfect fluid
5902:exterior regions, each associated with an
2565:
1130:The Kerr metric describes the geometry of
1116:, or Ellipsoid coordinate transformation.
746:
732:
11084:
10929:
10905:
10837:
10751:. Cambridge: Cambridge University Press.
10727:. Cambridge: Cambridge University Press.
10708:. Cambridge: Cambridge University Press.
10623:
10551:
10540:International Journal of Modern Physics D
10497:
10368:
10284:
10207:
10092:
10032:
9932:
9824:
9814:
9718:
9700:
9588:
9532:
8643:Learn how and when to remove this message
5940:), the Kerr geometry admits a remarkable
2268:
1971:and where for brevity, the length scales
1554:
1547:
934:
886:
817:in 1915, which described the geometry of
10895:. W. A. Benjamin, New York. p. 222.
10857:for the Chandrasekhar/Ferrari CPW model.
10537:
10353:
10248:
9913:
9741:
9472:
9434:
6079:
6071:
5033:
5025:
10887:
10111:(Cambridge University Press 2003). See
9900:
9873:
9494:
9241:the isolated mass monopole source with
9228:the isolated mass monopole source with
9217:the isolated mass monopole source with
6067:
5657:with an angular speed at least of
5118:). The inner surface corresponds to an
3373:). At large distances from the source (
13209:
10963:
10817:The Mathematical Theory of Black Holes
10604:The Mathematical Theory of Black Holes
10388:
10161:
10159:
10157:
10086:
9976:
9926:
9555:
9488:
8201:is analogous to Carter's constant and
3395:
13217:Exact solutions in general relativity
12354:
12071:
11098:
10188:Journal of Astrophysics and Astronomy
10101:
9574:
9277:interior, and indeed to more general
5021:
12052:
10995:
10323:
10267:
10179:
10129:Belinski, V.; Verdaguer, E. (2001).
9788:
9440:
8902:
8625:adding citations to reliable sources
8596:
4816:
4521:
4019:
3114:
2986:
2780:
2679:
2588:
2570:The Kerr metric can be expressed in
2350:
2290:In the non-relativistic limit where
2169:
2086:
2027:
1894:
1805:
1719:
1633:
1199:
825:, spherical, non-rotating body, the
12945:TolmanâOppenheimerâVolkoff equation
12898:FriedmannâLemaĂźtreâRobertsonâWalker
10842:. Oxford: Oxford University Press.
10748:Relativistic Figures of Equilibrium
10700:Wiltshire, David L.; Visser, Matt;
10154:
8533:{\displaystyle r_{\text{s}}/2<a}
7550:{\displaystyle \lambda =\tau /\mu }
3400:As the Kerr metric (along with the
1068:dividing the space above and below
13:
12123:Novikov self-consistency principle
11065:General Relativity and Gravitation
11024:General Relativity and Gravitation
10865:Introduction to General Relativity
10693:
8664:
8476:
8155:
8129:
8055:
8004:
7868:
7790:
7784:
7763:
7647:
7575:
7267:
6997:
6956:
6845:
6824:
6732:
6714:
6679:
6626:
6108:. The first is the invariant mass
5746:Ergosphere and the Penrose process
5666:
5521:
5405:
5315:
5257:which in natural units (that give
5173:
4892:
4826:
4643:
4640:
4465:
4461:
4449:
4422:
4418:
4406:
4403:
4379:
4375:
4358:
4308:
4288:
4284:
4273:
4269:
4261:
4258:
4209:
4205:
4172:
4108:
4070:
4066:
4048:
4044:
3923:
3920:
3917:
3869:
3866:
3863:
3762:
3759:
3756:
3483:
3480:
3477:
3441:
3438:
3435:
3420:If the complete rotational energy
2458:
2412:
2179:
2096:
2064:{\displaystyle a={\frac {J}{Mc}},}
2010:
1986:
1539:
1439:
1362:
1341:
1338:
1300:
1185:. The metric (or equivalently its
57:
14:
13238:
12715:HamiltonâJacobiâEinstein equation
10789:Introducing Einstein's Relativity
10326:"Shadows of rotating black holes"
9683:Newman, Ezra; Adamo, Tim (2014).
9495:Sneppen, Albert (December 2021).
8785:Relation to other exact solutions
5684:As with the event horizon in the
4510:
3381:), these equations reduce to the
1015:. The outer surface encloses the
13193:
13192:
12133:Quantum mechanics of time travel
12113:Chronology protection conjecture
12051:
12042:
12041:
11340:TolmanâOppenheimerâVolkoff limit
11211:
10768:The Geometry of Kerr Black Holes
9742:Harrison, B. Kent (1978-10-30).
9361:
9347:
9256:
8884:ChandrasekharâFerrari CPW vacuum
8601:
7497:, is related to the proper time
5122:similar to that observed in the
4515:We may rewrite the Kerr metric (
3963:
2791:
1839:{\displaystyle z=r\cos \theta ,}
766:describes the geometry of empty
713:
712:
699:
29:
12097:
11457:Innermost stable circular orbit
11124:
10647:
10638:
10610:
10531:
10477:
10317:
10261:
10232:
10122:
10080:
10065:
10012:
9985:Journal of Mathematical Physics
9941:
9841:
9789:Chou, Yu-Ching (January 2020).
9642:Journal of Mathematical Physics
9413:
9088:
9068:
8612:needs additional citations for
6363:{\displaystyle L_{z}=p_{\phi }}
5882:in 1969 and is thus called the
1618:{\displaystyle r,\theta ,\phi }
1096:of a pair of Kerr black holes.
12522:Massâenergy equivalence (E=mc)
12380:
11883:Timeline of black hole physics
10165:Misner, Thorne & Wheeler:
9857:The Classical Theory of Fields
9782:
9735:
9676:
9629:
9607:10.1103/PhysRevLett.116.061102
9568:
9549:
8982:Thus, the special case of the
8960:
8950:
8851:of the Kerr metric yields the
7979:
7964:
7321:
7298:
7255:
7248:
7239:
7233:
7154:
7148:
7006:
7000:
6970:
6964:
6838:
6832:
6723:
6717:
6670:
6664:
5795:
5782:
5688:, the apparent singularity at
3841:
3728:{\displaystyle M_{\text{irr}}}
3638:{\displaystyle M_{\text{irr}}}
3319:when the rotational parameter
3257:
3090:
2837:
2798:
2373:
2365:
1059:{\displaystyle r=r_{\text{s}}}
1:
12229:Parallel universes in fiction
11651:Nonsingular black hole models
10074:Trends in Black Hole Research
10021:Classical and Quantum Gravity
9816:10.1016/j.heliyon.2020.e03336
9556:Sutter, Paul (22 July 2021).
9428:
8593:Kerr black holes as wormholes
8439:
5971:is known. Topologically, the
5906:, stationary limit surfaces,
5894:Features of the Kerr geometry
2347:oblate spheroidal coordinates
1629:oblate spheroidal coordinates
797:; these equations are highly
12239:Philosophy of space and time
9407:
8576:{\displaystyle GM^{2}<Jc}
4992:
4788:
4500:
3540:, its total mass-equivalent
3227:
3029:
2977:
2771:
2670:
2574:, using a particular set of
2336:{\displaystyle r_{\text{s}}}
1882:{\displaystyle r_{\text{s}}}
941:{\displaystyle \mathbb {R} }
893:{\displaystyle \mathbb {R} }
770:around a rotating uncharged
7:
12537:Relativistic Doppler effect
11873:Rossi X-ray Timing Explorer
11838:Hypercompact stellar system
11828:Gamma-ray burst progenitors
10948:10.1088/0264-9381/21/24/003
10819:. Oxford: Clarendon Press.
10793:. Oxford: Clarendon Press.
10676:10.1103/physrevlett.71.1669
10618:Black hole Penrose diagrams
9768:10.1103/PhysRevLett.41.1197
9340:
8972:{\displaystyle M_{n}=M^{n}}
6223:{\displaystyle p^{\alpha }}
5861:BoyerâLindquist coordinates
5115:
4516:
3706:) and its irreducible mass
3583:{\displaystyle {\sqrt {2}}}
3406:BelinskiâZakharov transform
2555:
2243:
2160:
2077:
1961:
1852:
1796:
1710:
1581:
1195:BoyerâLindquist coordinates
1126:BoyerâLindquist coordinates
1120:BoyerâLindquist coordinates
804:
210:Gravitational time dilation
10:
13243:
13008:In computational physics:
12532:Relativity of simultaneity
12219:Many-worlds interpretation
12106:General terms and concepts
11559:Black hole complementarity
11526:Bousso's holographic bound
11511:Quasi-periodic oscillation
11209:
11203:MalamentâHogarth spacetime
10989:10.1103/PhysRevLett.26.331
10635:. (YouTube, Timestamp 26m)
10516:10.1103/PhysRevD.90.124010
10135:Cambridge University Press
10119:regarding generalizations.
10051:10.1088/0264-9381/11/6/010
9883:Relativistic Hydrodynamics
9720:10.4249/scholarpedia.31791
9525:10.1038/s41598-021-93595-w
9466:10.1103/PhysRevLett.11.237
9175:momentum multipole moments
8463:Overextreme Kerr solutions
6092:region around the equator.
5749:
5733:{\displaystyle dt\ d\phi }
3678:The total mass equivalent
3383:EddingtonâFinkelstein form
3099:{\displaystyle {\vec {a}}}
3016:{\displaystyle k_{0}=1.\!}
2278:{\displaystyle dt\,d\phi }
1134:in the vicinity of a mass
1123:
330:MathissonâPapapetrouâDixon
171:Pseudo-Riemannian manifold
13190:
13022:
12887:
12859:
12845:LenseâThirring precession
12728:
12677:
12639:
12618:
12607:
12565:
12509:
12493:
12435:
12427:Doubly special relativity
12399:
12388:
12280:
12237:
12201:
12178:
12141:
12105:
12037:
11911:
11763:
11725:
11704:
11643:
11602:
11551:
11430:Gravitational singularity
11417:
11310:
11220:
11145:
11132:
11086:10.1007/s10714-009-0856-0
11044:10.1007/s10714-009-0857-z
10838:Griffiths, J. B. (1991).
10766:O'Neill, Barrett (1995).
10570:10.1142/S0218271820501096
10451:The Astrophysical Journal
9387:ReissnerâNordström metric
9328:{\displaystyle GM^{2}=cJ}
9020:" of general relativity.
8897:gravitational plane waves
8869:Schwarzschild coordinates
8812:KerrâNewman electrovacuum
8676:{\displaystyle \Delta =0}
8488:{\displaystyle \Delta =0}
7464:{\displaystyle \mu >0}
6404:, now referred to as the
6323:{\displaystyle E=-p_{t},}
4519:) in the following form:
3564:is higher by a factor of
3416:Mass of rotational energy
3241:becomes the usual radius
3237:Notice that the quantity
1992:{\displaystyle a,\Sigma }
1099:
972:LenseâThirring precession
827:ReissnerâNordström metric
12705:Post-Newtonian formalism
12695:Einstein field equations
12631:Mathematical formulation
12455:Hyperbolic orthogonality
12128:Self-fulfilling prophecy
12014:PSO J030947.49+271757.31
11939:SDSS J150243.09+111557.3
11472:BlandfordâZnajek process
10421:10.1103/PhysRev.174.1559
9249:family (two parameters).
9245:angular momentum is the
9232:angular momentum is the
9221:angular momentum is the
8687:coordinate singularities
7488:{\displaystyle \lambda }
7369:{\displaystyle \lambda }
4810:, where Ω is called the
3047:, making the 4-vector a
2025:have been introduced as
1110:NewmanâPenrose formalism
1094:first direct observation
791:Einstein field equations
335:HamiltonâJacobiâEinstein
315:Einstein field equations
138:Mathematical formulation
12416:Galilean transformation
12407:Principle of relativity
11270:Active galactic nucleus
10969:Physical Review Letters
10785:D'Inverno, Ray (1992).
10656:Physical Review Letters
10546:(16): 2050109â2050202.
9906:Christopher M. Hirata:
9748:Physical Review Letters
9577:Physical Review Letters
9446:Physical Review Letters
9225:family (one parameter),
8801:Einstein field equation
8434:Schwarzschild geodesics
7653:{\displaystyle \Omega }
7581:{\displaystyle \Omega }
6606:{\displaystyle G=M=c=1}
6055:{\displaystyle \theta }
5672:{\displaystyle \Omega }
5290:{\displaystyle G=M=c=1}
3649:massâenergy equivalence
3051:, with respect to both
2566:KerrâSchild coordinates
2016:{\displaystyle \Delta }
12501:Lorentz transformation
12292:closed timelike curves
12143:Time travel in fiction
11898:Tidal disruption event
11868:Supermassive dark star
11786:Black holes in fiction
11771:Outline of black holes
11404:Supermassive dark star
11323:Gravitational collapse
10131:Gravitational Solitons
10107:Stephani, Hans et al.
9329:
9288:NeugebauerâMeinel disk
9205:
9171:mass multipole moments
9149:
9024:Weyl multipole moments
9016:) gives the "monopole
9008:
8973:
8843:
8760:
8733:
8705:
8677:
8577:
8534:
8489:
8424:
8394:
8324:
8217:
8193:
8171:
8071:
7804:
7678:
7654:
7630:
7606:
7582:
7551:
7513:
7489:
7465:
7443:. In particular, when
7435:
7370:
7345:
7216:
7131:
6981:
6607:
6558:
6396:
6374:HamiltonâJacobi theory
6364:
6324:
6283:
6252:
6224:
6195:
6124:
6093:
6077:
6056:
6032:
6008:
5916:closed timelike curves
5850:
5734:
5697:coordinate singularity
5673:
5649:
5581:
5496:
5359:
5291:
5248:
5111:
5102:
5078:
5054:
5031:
4980:
4776:
4488:
3998:
3997:{\displaystyle g^{ik}}
3954:
3729:
3694:
3667:
3639:
3608:
3584:
3556:
3532:
3498:
3365:
3335:
3310:
3215:
3100:
3017:
2965:
2759:
2658:
2543:
2337:
2306:
2279:
2231:
2148:
2065:
2017:
1993:
1949:
1883:
1840:
1784:
1698:
1619:
1591:where the coordinates
1569:
1177:
1150:
1106:NewmanâJanis algorithm
1060:
942:
894:
845:represents the body's
777:with a quasispherical
205:Gravitational redshift
98:
12969:WeylâLewisâPapapetrou
12710:Raychaudhuri equation
12649:Equivalence principle
12340:Traversable wormholes
12118:Closed timelike curve
11776:Black Hole Initiative
11589:Holographic principle
10176:, pages 899, 900, 908
9330:
9206:
9150:
9009:
8974:
8865:spherically symmetric
8844:
8816:KerrâVaidya null dust
8761:
8734:
8706:
8685:, these are actually
8678:
8578:
8535:
8490:
8425:
8395:
8325:
8225:is a useful quantity
8218:
8194:
8172:
8072:
7805:
7679:
7655:
7631:
7614:. The local velocity
7607:
7583:
7552:
7514:
7512:{\displaystyle \tau }
7490:
7473:the affine parameter
7466:
7436:
7371:
7346:
7217:
7132:
6982:
6608:
6559:
6397:
6365:
6325:
6284:
6282:{\displaystyle L_{z}}
6253:
6225:
6196:
6125:
6083:
6075:
6057:
6033:
6031:{\displaystyle \phi }
6009:
5958:algebraically special
5928:Killing vector fields
5920:curvature singularity
5851:
5735:
5674:
5650:
5582:
5505:or in natural units:
5497:
5389:yields the solution:
5360:
5292:
5249:
5157:yields the solution:
5103:
5079:
5055:
5037:
5029:
5008:equivalence principle
4981:
4777:
4489:
3999:
3955:
3730:
3695:
3668:
3640:
3609:
3585:
3557:
3533:
3499:
3410:gravitational soliton
3366:
3336:
3311:
3216:
3101:
3018:
2966:
2760:
2659:
2576:Cartesian coordinates
2544:
2338:
2307:
2280:
2232:
2149:
2066:
2018:
1994:
1950:
1884:
1841:
1785:
1699:
1620:
1570:
1178:
1151:
1061:
943:
895:
493:WeylâLewisâPapapetrou
448:KerrâNewmanâde Sitter
268:EinsteinâRosen bridge
200:Gravitational lensing
156:Equivalence principle
99:
13010:Numerical relativity
12851:pulsar timing arrays
12151:Timelines in fiction
11579:Final parsec problem
11538:Schwarzschild radius
10632:Spinning Black Holes
10303:10.1103/Physics.4.43
10256:Rotating Black Holes
10254:Stijn van Tongeren:
9685:"KerrâNewman metric"
9392:HartleâThorne metric
9377:Schwarzschild metric
9300:
9223:Schwarzschild vacuum
9195:
9046:
8992:
8984:Schwarzschild vacuum
8931:
8888:colliding plane wave
8859:black hole which is
8853:Schwarzschild metric
8827:
8750:
8723:
8695:
8661:
8621:improve this article
8548:
8503:
8473:
8408:
8333:
8229:
8207:
8183:
8080:
7814:
7690:
7668:
7644:
7620:
7596:
7572:
7527:
7503:
7479:
7449:
7388:
7360:
7227:
7142:
6994:
6619:
6579:
6412:
6386:
6334:
6295:
6266:
6242:
6207:
6136:
6123:{\displaystyle \mu }
6114:
6068:Trajectory equations
6046:
6022:
5992:
5918:, and a ring-shaped
5779:
5762:Schwarzschild radius
5760:and situated at the
5712:
5686:Schwarzschild metric
5663:
5639:
5512:
5396:
5370:Schwarzschild metric
5306:
5263:
5164:
5124:Schwarzschild metric
5092:
5068:
5044:
4823:
4528:
4026:
3978:
3744:
3712:
3684:
3657:
3622:
3598:
3570:
3546:
3516:
3426:
3387:Schwarzschild metric
3349:
3325:
3251:
3121:
3081:
2993:
2787:
2686:
2595:
2357:
2320:
2296:
2259:
2176:
2093:
2034:
2007:
1977:
1901:
1891:Schwarzschild radius
1866:
1812:
1726:
1640:
1597:
1206:
1167:
1140:
1037:
1029:Schwarzschild metric
930:
882:
853:represents its spin
811:Schwarzschild metric
423:EinsteinâRosen waves
149:Fundamental concepts
38:
12902:Friedmann equations
12796:HulseâTaylor binary
12758:Gravitational waves
12654:Riemannian geometry
12480:Proper acceleration
12465:Maxwell's equations
12411:Galilean relativity
12188:Grandfather paradox
11878:Superluminal motion
11853:Population III star
11823:Gravitational waves
11781:Black hole starship
11564:Information paradox
11077:2009GReGr..41.2469K
11036:2009GReGr..41.2485K
10981:1971PhRvL..26..331C
10940:2004CQGra..21.5727S
10918:Class. Quantum Grav
10893:Battelle Rencontres
10668:1993PhRvL..71.1669B
10562:2020IJMPD..2950109R
10508:2014PhRvD..90l4010F
10463:1972ApJ...178..347B
10413:1968PhRv..174.1559C
10324:de Vries, Andreas.
10295:2011PhyOJ...4...43W
10200:1985JApA....6...85B
10043:1994CQGra..11.1453B
9997:1969JMP....10.1842D
9962:1967JMP.....8..265B
9807:2020Heliy...603336C
9760:1978PhRvL..41.1197H
9711:2014SchpJ...931791N
9654:1965JMP.....6..915N
9599:2016PhRvL.116f1102A
9517:2021NatSR..1114247S
9458:1963PhRvL..11..237K
9402:Rotating black hole
9275:Schwarzschild fluid
9263:rotating black hole
9160:ChazyâCurzon vacuum
9007:{\displaystyle a=0}
8909:asymptotically flat
8842:{\displaystyle a=0}
8423:{\displaystyle a=0}
8252:
7101:
6435:
6106:constants of motion
6098:equations of motion
6007:{\displaystyle a=0}
5900:asymptotically flat
5622:, its path through
5531:
5451:
5415:
5325:
5219:
5183:
4597:
3933:
3879:
3772:
3531:{\displaystyle a=M}
3396:Soliton coordinates
3364:{\displaystyle c=1}
2314:(or, equivalently,
377:KaluzaâKlein theory
263:Minkowski spacetime
215:Gravitational waves
12951:ReissnerâNordström
12869:BransâDicke theory
12700:Linearized gravity
12527:Length contraction
12445:Frame of reference
12422:Special relativity
12286:general relativity
12209:Alternative future
12202:Parallel timelines
12180:Temporal paradoxes
12156:in science fiction
11712:Optical black hole
11625:ReissnerâNordström
11584:Firewall (physics)
11489:Gravitational lens
11003:General Relativity
10832:See chapters 6--10
10218:10.1007/BF02715080
10172:2017-08-22 at the
9509:Cosmic Dawn Center
9501:Scientific Reports
9382:KerrâNewman metric
9325:
9201:
9145:
9004:
8969:
8886:, an example of a
8839:
8756:
8729:
8701:
8673:
8573:
8530:
8485:
8420:
8390:
8320:
8238:
8213:
8189:
8167:
8067:
7800:
7674:
7650:
7626:
7602:
7578:
7547:
7509:
7485:
7461:
7431:
7366:
7341:
7212:
7127:
7087:
6977:
6975:
6603:
6554:
6421:
6392:
6360:
6320:
6279:
6248:
6220:
6191:
6120:
6094:
6078:
6052:
6028:
6004:
5930:(corresponding to
5846:
5730:
5669:
5645:
5577:
5515:
5492:
5437:
5399:
5355:
5309:
5287:
5244:
5205:
5167:
5112:
5098:
5074:
5050:
5032:
5022:Important surfaces
4976:
4772:
4580:
4484:
3994:
3950:
3911:
3857:
3750:
3725:
3690:
3663:
3635:
3604:
3580:
3552:
3528:
3494:
3361:
3331:
3306:
3211:
3096:
3013:
2961:
2755:
2654:
2572:"KerrâSchild" form
2539:
2380:
2333:
2302:
2275:
2227:
2144:
2061:
2013:
1989:
1945:
1879:
1836:
1780:
1694:
1615:
1565:
1563:
1173:
1146:
1056:
952:ReissnerâNordström
938:
890:
839:KerrâNewman metric
815:Karl Schwarzschild
795:general relativity
706:Physics portal
478:OppenheimerâSnyder
418:ReissnerâNordström
310:Linearized gravity
258:Spacetime diagrams
161:Special relativity
94:
23:General relativity
13204:
13203:
13018:
13017:
12997:OzsvĂĄthâSchĂŒcking
12603:
12602:
12585:Minkowski diagram
12542:Thomas precession
12485:Relativistic mass
12348:
12347:
12300:Alcubierre metric
12214:Alternate history
12065:
12064:
11858:Supermassive star
11848:Naked singularity
11843:Membrane paradigm
11569:Cosmic censorship
11543:Spaghettification
11531:Immirzi parameter
11484:Hawking radiation
11425:Astrophysical jet
11394:Supermassive star
11384:Binary black hole
11318:Stellar evolution
11260:Intermediate-mass
11071:(10): 2469â2484.
11030:(10): 2485â2499.
11014:978-0-226-87032-8
10924:(24): 5727â5733.
10876:978-0-07-000423-8
10849:978-0-19-853209-5
10826:978-0-19-850370-5
10813:Chandrasekhar, S.
10800:978-0-19-859686-8
10777:978-1-56881-019-5
10758:978-0-521-86383-4
10734:978-0-521-46136-8
10715:978-0-521-88512-6
10662:(11): 1669â1672.
10616:Andrew Hamilton:
10600:Chandrasekhar, S.
10486:Physical Review D
10269:Will, Clifford M.
10005:10.1063/1.1664769
9991:(10): 1842â1854.
9970:10.1063/1.1705193
9893:978-0-19-852890-6
9866:978-0-08-018176-9
9754:(18): 1197â1200.
9662:10.1063/1.1704350
9267:exterior solution
9204:{\displaystyle r}
9125:
9029:multipole moments
8913:multipole moments
8903:Multipole moments
8855:, which models a
8821:The special case
8794:axially symmetric
8779:cosmic censorship
8759:{\displaystyle r}
8732:{\displaystyle r}
8704:{\displaystyle r}
8653:
8652:
8645:
8587:naked singularity
8513:
8457:Noether's theorem
8450:time translations
8387:
8351:
8307:
8216:{\displaystyle W}
8192:{\displaystyle C}
8165:
8132:
8092:
8065:
8007:
7950:
7939:
7914:
7896:
7877:
7844:
7826:
7795:
7794:
7789:
7701:
7677:{\displaystyle r}
7629:{\displaystyle v}
7605:{\displaystyle t}
7416:
7120:
6959:
6866:
6827:
6809:
6753:
6726:
6700:
6673:
6647:
6534:
6395:{\displaystyle Q}
6251:{\displaystyle E}
5960:, in fact it has
5924:geodesic equation
5723:
5648:{\displaystyle M}
5575:
5490:
5484:
5444:
5429:
5353:
5299:) simplifies to:
5242:
5236:
5212:
5197:
5101:{\displaystyle M}
5077:{\displaystyle 0}
5062:is animated from
5053:{\displaystyle a}
5000:
4999:
4971:
4938:
4878:
4863:
4796:
4795:
4750:
4611:
4508:
4507:
4472:
4452:
4429:
4409:
4386:
4361:
4351:
4328:
4297:
4280:
4265:
4244:
4216:
4175:
4155:
4112:
4084:
4062:
3945:
3839:
3722:
3704:rotational energy
3693:{\displaystyle M}
3666:{\displaystyle M}
3632:
3607:{\displaystyle M}
3578:
3555:{\displaystyle M}
3334:{\displaystyle a}
3304:
3235:
3234:
3203:
3176:
3093:
3037:
3036:
2985:
2984:
2954:
2941:
2892:
2779:
2778:
2753:
2678:
2677:
2563:
2562:
2440:
2363:
2330:
2305:{\displaystyle M}
2251:
2250:
2205:
2168:
2167:
2085:
2084:
2056:
1969:
1968:
1940:
1911:
1876:
1860:
1859:
1804:
1803:
1760:
1718:
1717:
1674:
1589:
1588:
1542:
1513:
1483:
1442:
1422:
1344:
1303:
1293:
1176:{\displaystyle J}
1149:{\displaystyle M}
1053:
1025:coordinate system
962:
961:
772:axially symmetric
756:
755:
389:
388:
275:
274:
13234:
13196:
13195:
12979:van Stockum dust
12751:Two-body problem
12669:Mach's principle
12616:
12615:
12557:Terrell rotation
12397:
12396:
12375:
12368:
12361:
12352:
12351:
12335:van Stockum dust
12247:Butterfly effect
12092:
12085:
12078:
12069:
12068:
12055:
12054:
12045:
12044:
11717:Sonic black hole
11666:Dark-energy star
11521:Bekenstein bound
11506:Mâsigma relation
11435:Ring singularity
11215:
11119:
11112:
11105:
11096:
11095:
11090:
11088:
11055:
11018:
11006:
10992:
10959:
10933:
10911:
10909:
10896:
10880:
10868:
10853:
10830:
10804:
10792:
10781:
10762:
10743:Meinel, Reinhard
10738:
10719:
10688:
10687:
10651:
10645:
10642:
10636:
10627:
10621:
10614:
10608:
10607:
10596:
10590:
10589:
10555:
10535:
10529:
10527:
10501:
10481:
10475:
10474:
10446:
10440:
10439:
10437:
10431:. Archived from
10407:(5): 1559â1571.
10398:
10386:
10375:
10374:
10372:
10360:
10351:
10342:
10333:
10332:
10330:
10321:
10315:
10314:
10288:
10265:
10259:
10252:
10246:
10236:
10230:
10229:
10211:
10183:
10177:
10163:
10152:
10148:
10126:
10120:
10105:
10099:
10098:
10096:
10084:
10078:
10069:
10063:
10062:
10036:
10027:(6): 1453â1461.
10016:
10010:
10008:
9980:
9974:
9973:
9945:
9939:
9938:
9936:
9924:
9911:
9904:
9898:
9897:
9877:
9871:
9870:
9845:
9839:
9838:
9828:
9818:
9786:
9780:
9779:
9739:
9733:
9732:
9722:
9704:
9680:
9674:
9673:
9633:
9627:
9626:
9592:
9572:
9566:
9565:
9553:
9547:
9546:
9536:
9492:
9486:
9476:
9470:
9469:
9438:
9423:
9417:
9371:
9366:
9365:
9357:
9355:Astronomy portal
9352:
9351:
9350:
9336:
9334:
9332:
9331:
9326:
9315:
9314:
9212:
9210:
9208:
9207:
9202:
9154:
9152:
9151:
9146:
9144:
9140:
9139:
9138:
9126:
9121:
9120:
9111:
9098:
9097:
9078:
9077:
9058:
9057:
9015:
9013:
9011:
9010:
9005:
8978:
8976:
8975:
8970:
8968:
8967:
8943:
8942:
8921:angular momentum
8892:global structure
8850:
8848:
8846:
8845:
8840:
8767:
8765:
8763:
8762:
8757:
8740:
8738:
8736:
8735:
8730:
8712:
8710:
8708:
8707:
8702:
8684:
8682:
8680:
8679:
8674:
8648:
8641:
8637:
8634:
8628:
8605:
8597:
8584:
8582:
8580:
8579:
8574:
8563:
8562:
8541:
8539:
8537:
8536:
8531:
8520:
8515:
8514:
8511:
8496:
8494:
8492:
8491:
8486:
8431:
8429:
8427:
8426:
8421:
8399:
8397:
8396:
8391:
8389:
8388:
8380:
8377:
8369:
8368:
8353:
8352:
8344:
8329:
8327:
8326:
8321:
8319:
8318:
8313:
8309:
8308:
8306:
8305:
8293:
8292:
8283:
8278:
8251:
8246:
8224:
8222:
8220:
8219:
8214:
8200:
8198:
8196:
8195:
8190:
8176:
8174:
8173:
8168:
8166:
8164:
8163:
8162:
8146:
8144:
8140:
8133:
8128:
8127:
8126:
8113:
8094:
8093:
8085:
8076:
8074:
8073:
8068:
8066:
8064:
8063:
8062:
8049:
8048:
8047:
8038:
8037:
8021:
8019:
8015:
8008:
8003:
8002:
8001:
7988:
7957:
7953:
7952:
7951:
7943:
7940:
7935:
7921:
7916:
7915:
7907:
7898:
7897:
7889:
7878:
7876:
7875:
7863:
7846:
7845:
7837:
7828:
7827:
7819:
7809:
7807:
7806:
7801:
7796:
7793:
7787:
7782:
7775:
7774:
7762:
7761:
7749:
7748:
7743:
7739:
7738:
7737:
7725:
7724:
7708:
7707:
7702:
7694:
7685:
7683:
7681:
7680:
7675:
7661:
7659:
7657:
7656:
7651:
7637:
7635:
7633:
7632:
7627:
7613:
7611:
7609:
7608:
7603:
7589:
7587:
7585:
7584:
7579:
7558:
7556:
7554:
7553:
7548:
7543:
7520:
7518:
7516:
7515:
7510:
7496:
7494:
7492:
7491:
7486:
7472:
7470:
7468:
7467:
7462:
7442:
7440:
7438:
7437:
7432:
7430:
7429:
7417:
7415:
7407:
7406:
7405:
7392:
7380:affine parameter
7377:
7375:
7373:
7372:
7367:
7350:
7348:
7347:
7342:
7340:
7336:
7329:
7328:
7310:
7309:
7294:
7293:
7284:
7283:
7263:
7262:
7221:
7219:
7218:
7213:
7211:
7210:
7195:
7191:
7190:
7189:
7177:
7176:
7136:
7134:
7133:
7128:
7126:
7122:
7121:
7119:
7112:
7111:
7100:
7095:
7086:
7081:
7077:
7076:
7075:
7063:
7062:
7048:
7047:
7027:
7026:
6986:
6984:
6983:
6978:
6976:
6960:
6955:
6954:
6953:
6941:
6940:
6930:
6925:
6921:
6920:
6919:
6901:
6900:
6867:
6865:
6857:
6849:
6828:
6820:
6815:
6811:
6810:
6808:
6801:
6800:
6790:
6789:
6780:
6754:
6752:
6744:
6736:
6727:
6713:
6701:
6699:
6691:
6683:
6674:
6660:
6648:
6646:
6638:
6630:
6614:
6612:
6610:
6609:
6604:
6563:
6561:
6560:
6555:
6550:
6546:
6545:
6544:
6539:
6535:
6533:
6522:
6521:
6512:
6502:
6498:
6497:
6496:
6484:
6483:
6469:
6468:
6448:
6447:
6434:
6429:
6403:
6401:
6399:
6398:
6393:
6369:
6367:
6366:
6361:
6359:
6358:
6346:
6345:
6329:
6327:
6326:
6321:
6316:
6315:
6290:
6288:
6286:
6285:
6280:
6278:
6277:
6259:
6257:
6255:
6254:
6249:
6231:
6229:
6227:
6226:
6221:
6219:
6218:
6200:
6198:
6197:
6192:
6187:
6186:
6177:
6176:
6164:
6163:
6151:
6150:
6131:
6129:
6127:
6126:
6121:
6063:
6061:
6059:
6058:
6053:
6039:
6037:
6035:
6034:
6029:
6015:
6013:
6011:
6010:
6005:
5969:global structure
5933:time translation
5888:gamma-ray bursts
5855:
5853:
5852:
5847:
5839:
5838:
5829:
5828:
5816:
5815:
5803:
5802:
5741:
5739:
5737:
5736:
5731:
5721:
5680:
5678:
5676:
5675:
5670:
5656:
5654:
5652:
5651:
5646:
5586:
5584:
5583:
5578:
5576:
5568:
5567:
5558:
5557:
5542:
5530:
5525:
5524:
5501:
5499:
5498:
5493:
5491:
5486:
5485:
5477:
5476:
5467:
5466:
5450:
5445:
5442:
5436:
5431:
5430:
5427:
5420:
5414:
5409:
5408:
5388:
5364:
5362:
5361:
5356:
5354:
5352:
5351:
5336:
5324:
5319:
5318:
5298:
5296:
5294:
5293:
5288:
5253:
5251:
5250:
5245:
5243:
5238:
5237:
5235:
5234:
5218:
5213:
5210:
5204:
5199:
5198:
5195:
5188:
5182:
5177:
5176:
5156:
5154:
5152:
5151:
5143:
5140:
5117:
5109:
5107:
5105:
5104:
5099:
5085:
5083:
5081:
5080:
5075:
5061:
5059:
5057:
5056:
5051:
5016:Mach's principle
4994:
4985:
4983:
4982:
4977:
4972:
4970:
4963:
4962:
4953:
4952:
4940:
4939:
4936:
4927:
4923:
4922:
4921:
4909:
4908:
4890:
4880:
4879:
4876:
4869:
4864:
4862:
4861:
4849:
4848:
4836:
4817:
4790:
4781:
4779:
4778:
4773:
4768:
4767:
4762:
4758:
4751:
4749:
4748:
4736:
4735:
4723:
4706:
4705:
4690:
4689:
4677:
4676:
4661:
4660:
4648:
4647:
4646:
4630:
4629:
4617:
4613:
4612:
4610:
4609:
4596:
4591:
4579:
4574:
4573:
4553:
4552:
4540:
4539:
4522:
4518:
4502:
4493:
4491:
4490:
4485:
4483:
4482:
4477:
4473:
4471:
4460:
4453:
4445:
4440:
4439:
4434:
4430:
4428:
4417:
4410:
4402:
4397:
4396:
4391:
4387:
4385:
4374:
4367:
4363:
4362:
4357:
4353:
4352:
4349:
4342:
4329:
4327:
4320:
4319:
4303:
4298:
4296:
4295:
4283:
4281:
4279:
4268:
4266:
4264:
4253:
4246:
4245:
4242:
4232:
4227:
4226:
4221:
4217:
4215:
4204:
4197:
4193:
4186:
4185:
4176:
4171:
4170:
4169:
4157:
4156:
4153:
4146:
4141:
4140:
4128:
4127:
4113:
4111:
4107:
4106:
4093:
4085:
4083:
4082:
4081:
4065:
4063:
4061:
4060:
4059:
4043:
4041:
4040:
4020:
4005:
4003:
4001:
4000:
3995:
3993:
3992:
3959:
3957:
3956:
3951:
3946:
3944:
3943:
3942:
3932:
3927:
3926:
3906:
3905:
3904:
3895:
3894:
3884:
3878:
3873:
3872:
3853:
3852:
3840:
3838:
3837:
3828:
3823:
3822:
3813:
3812:
3800:
3799:
3790:
3785:
3784:
3771:
3766:
3765:
3736:
3734:
3732:
3731:
3726:
3724:
3723:
3720:
3701:
3699:
3697:
3696:
3691:
3674:
3672:
3670:
3669:
3664:
3646:
3644:
3642:
3641:
3636:
3634:
3633:
3630:
3615:
3613:
3611:
3610:
3605:
3591:
3589:
3587:
3586:
3581:
3579:
3574:
3563:
3561:
3559:
3558:
3553:
3539:
3537:
3535:
3534:
3529:
3505:
3503:
3501:
3500:
3495:
3493:
3489:
3488:
3487:
3486:
3459:
3458:
3446:
3445:
3444:
3372:
3370:
3368:
3367:
3362:
3342:
3340:
3338:
3337:
3332:
3315:
3313:
3312:
3307:
3305:
3303:
3302:
3290:
3289:
3277:
3276:
3267:
3229:
3220:
3218:
3217:
3212:
3204:
3202:
3201:
3192:
3191:
3182:
3177:
3175:
3174:
3173:
3161:
3160:
3150:
3149:
3148:
3136:
3135:
3125:
3115:
3107:
3105:
3103:
3102:
3097:
3095:
3094:
3086:
3069:Minkowski tensor
3031:
3022:
3020:
3019:
3014:
3005:
3004:
2987:
2979:
2970:
2968:
2967:
2962:
2960:
2956:
2955:
2947:
2942:
2940:
2939:
2938:
2926:
2925:
2915:
2898:
2893:
2891:
2890:
2889:
2877:
2876:
2866:
2849:
2836:
2835:
2823:
2822:
2810:
2809:
2794:
2781:
2773:
2764:
2762:
2761:
2756:
2754:
2752:
2751:
2750:
2741:
2740:
2728:
2727:
2717:
2716:
2715:
2696:
2680:
2672:
2663:
2661:
2660:
2655:
2652:
2651:
2642:
2641:
2626:
2625:
2610:
2609:
2589:
2557:
2548:
2546:
2545:
2540:
2538:
2537:
2519:
2518:
2509:
2505:
2504:
2503:
2491:
2490:
2473:
2472:
2454:
2453:
2441:
2439:
2438:
2437:
2425:
2424:
2411:
2406:
2405:
2393:
2392:
2379:
2368:
2351:
2344:
2342:
2340:
2339:
2334:
2332:
2331:
2328:
2313:
2311:
2309:
2308:
2303:
2286:
2284:
2282:
2281:
2276:
2245:
2236:
2234:
2233:
2228:
2223:
2222:
2207:
2206:
2203:
2194:
2193:
2170:
2162:
2153:
2151:
2150:
2145:
2134:
2133:
2124:
2123:
2111:
2110:
2087:
2079:
2070:
2068:
2067:
2062:
2057:
2055:
2044:
2028:
2024:
2022:
2020:
2019:
2014:
2000:
1998:
1996:
1995:
1990:
1963:
1954:
1952:
1951:
1946:
1941:
1939:
1938:
1929:
1918:
1913:
1912:
1909:
1895:
1888:
1886:
1885:
1880:
1878:
1877:
1874:
1854:
1845:
1843:
1842:
1837:
1806:
1798:
1789:
1787:
1786:
1781:
1761:
1759:
1758:
1746:
1745:
1736:
1720:
1712:
1703:
1701:
1700:
1695:
1675:
1673:
1672:
1660:
1659:
1650:
1634:
1626:
1624:
1622:
1621:
1616:
1583:
1574:
1572:
1571:
1566:
1564:
1543:
1538:
1531:
1530:
1515:
1514:
1511:
1501:
1496:
1495:
1481:
1474:
1473:
1464:
1460:
1453:
1452:
1443:
1438:
1437:
1436:
1424:
1423:
1420:
1413:
1408:
1407:
1395:
1394:
1377:
1376:
1358:
1357:
1345:
1337:
1332:
1331:
1319:
1318:
1309:
1305:
1304:
1299:
1295:
1294:
1291:
1284:
1262:
1258:
1257:
1245:
1244:
1225:
1224:
1200:
1184:
1182:
1180:
1179:
1174:
1160:angular momentum
1157:
1155:
1153:
1152:
1147:
1067:
1065:
1063:
1062:
1057:
1055:
1054:
1051:
1013:angular momentum
998:
947:
945:
944:
939:
937:
899:
897:
896:
891:
889:
862:
861:
855:angular momentum
813:, discovered by
748:
741:
734:
721:
716:
715:
708:
704:
703:
488:van Stockum dust
473:RobertsonâWalker
299:
298:
189:
188:
103:
101:
100:
95:
93:
92:
80:
72:
71:
53:
52:
33:
19:
18:
13242:
13241:
13237:
13236:
13235:
13233:
13232:
13231:
13207:
13206:
13205:
13200:
13186:
13014:
12918:BKL singularity
12908:LemaĂźtreâTolman
12883:
12879:Quantum gravity
12861:
12855:
12841:geodetic effect
12815:(together with
12785:LISA Pathfinder
12724:
12673:
12659:Penrose diagram
12641:
12635:
12610:
12599:
12595:Minkowski space
12561:
12505:
12489:
12437:
12431:
12391:
12384:
12379:
12349:
12344:
12330:Tipler cylinder
12289:
12276:
12233:
12197:
12174:
12137:
12101:
12096:
12066:
12061:
12033:
12009:ULAS J1342+0928
11969:SDSS J0849+1114
11954:Phoenix Cluster
11907:
11759:
11721:
11700:
11639:
11598:
11594:No-hair theorem
11547:
11501:Bondi accretion
11467:Penrose process
11413:
11379:Gamma-ray burst
11306:
11216:
11207:
11193:Direct collapse
11141:
11128:
11123:
11015:
10907:gr-qc/0012100v1
10877:
10850:
10827:
10801:
10778:
10759:
10735:
10716:
10704:, eds. (2009).
10702:Scott, Susan M.
10696:
10694:Further reading
10691:
10652:
10648:
10643:
10639:
10628:
10624:
10620:(JILA Colorado)
10615:
10611:
10597:
10593:
10536:
10532:
10482:
10478:
10447:
10443:
10435:
10401:Physical Review
10396:
10390:Carter, Brandon
10387:
10378:
10361:
10354:
10343:
10336:
10328:
10322:
10318:
10266:
10262:
10253:
10249:
10239:Thibault Damour
10237:
10233:
10209:10.1.1.512.1400
10184:
10180:
10174:Wayback Machine
10164:
10155:
10145:
10127:
10123:
10106:
10102:
10085:
10081:
10070:
10066:
10017:
10013:
9981:
9977:
9946:
9942:
9925:
9914:
9905:
9901:
9894:
9878:
9874:
9867:
9853:Lifshitz, E. M.
9846:
9842:
9787:
9783:
9740:
9736:
9681:
9677:
9634:
9630:
9573:
9569:
9562:livescience.com
9554:
9550:
9493:
9489:
9477:
9473:
9439:
9435:
9431:
9426:
9418:
9414:
9410:
9367:
9360:
9353:
9348:
9346:
9343:
9310:
9306:
9301:
9298:
9297:
9295:
9284:Wahlquist fluid
9259:
9235:TaubâNUT vacuum
9196:
9193:
9192:
9190:
9134:
9130:
9116:
9112:
9110:
9109:
9105:
9093:
9089:
9073:
9069:
9053:
9049:
9047:
9044:
9043:
8993:
8990:
8989:
8987:
8963:
8959:
8938:
8934:
8932:
8929:
8928:
8905:
8828:
8825:
8824:
8822:
8797:vacuum solution
8787:
8751:
8748:
8747:
8745:
8724:
8721:
8720:
8718:
8696:
8693:
8692:
8690:
8662:
8659:
8658:
8656:
8649:
8638:
8632:
8629:
8618:
8606:
8595:
8558:
8554:
8549:
8546:
8545:
8543:
8516:
8510:
8506:
8504:
8501:
8500:
8498:
8474:
8471:
8470:
8468:
8465:
8442:
8409:
8406:
8405:
8403:
8379:
8378:
8373:
8364:
8360:
8343:
8342:
8334:
8331:
8330:
8314:
8301:
8294:
8288:
8284:
8282:
8274:
8261:
8257:
8256:
8247:
8242:
8230:
8227:
8226:
8208:
8205:
8204:
8202:
8184:
8181:
8180:
8178:
8158:
8154:
8150:
8145:
8122:
8118:
8114:
8112:
8111:
8107:
8084:
8083:
8081:
8078:
8077:
8058:
8054:
8050:
8043:
8039:
8033:
8029:
8022:
8020:
7997:
7993:
7989:
7987:
7986:
7982:
7942:
7941:
7922:
7920:
7906:
7905:
7888:
7887:
7886:
7882:
7871:
7867:
7862:
7836:
7835:
7818:
7817:
7815:
7812:
7811:
7783:
7770:
7766:
7757:
7753:
7744:
7733:
7729:
7720:
7716:
7715:
7711:
7710:
7709:
7706:
7693:
7691:
7688:
7687:
7669:
7666:
7665:
7663:
7645:
7642:
7641:
7639:
7621:
7618:
7617:
7615:
7597:
7594:
7593:
7591:
7573:
7570:
7569:
7567:
7562:Because of the
7539:
7528:
7525:
7524:
7522:
7504:
7501:
7500:
7498:
7480:
7477:
7476:
7474:
7450:
7447:
7446:
7444:
7425:
7421:
7408:
7401:
7397:
7393:
7391:
7389:
7386:
7385:
7383:
7361:
7358:
7357:
7355:
7324:
7320:
7305:
7301:
7289:
7285:
7279:
7275:
7274:
7270:
7258:
7254:
7228:
7225:
7224:
7206:
7202:
7185:
7181:
7172:
7168:
7167:
7163:
7143:
7140:
7139:
7107:
7103:
7102:
7096:
7091:
7085:
7071:
7067:
7058:
7054:
7053:
7049:
7043:
7039:
7038:
7034:
7022:
7018:
6995:
6992:
6991:
6974:
6973:
6949:
6945:
6936:
6932:
6931:
6929:
6915:
6911:
6896:
6892:
6885:
6881:
6868:
6858:
6850:
6848:
6842:
6841:
6819:
6796:
6792:
6791:
6785:
6781:
6779:
6769:
6765:
6755:
6745:
6737:
6735:
6729:
6728:
6712:
6702:
6692:
6684:
6682:
6676:
6675:
6659:
6649:
6639:
6631:
6629:
6622:
6620:
6617:
6616:
6580:
6577:
6576:
6574:
6540:
6523:
6517:
6513:
6511:
6507:
6506:
6492:
6488:
6479:
6475:
6474:
6470:
6464:
6460:
6459:
6455:
6443:
6439:
6430:
6425:
6413:
6410:
6409:
6406:Carter constant
6387:
6384:
6383:
6381:
6354:
6350:
6341:
6337:
6335:
6332:
6331:
6311:
6307:
6296:
6293:
6292:
6273:
6269:
6267:
6264:
6263:
6261:
6243:
6240:
6239:
6237:
6214:
6210:
6208:
6205:
6204:
6202:
6182:
6178:
6169:
6165:
6159:
6155:
6146:
6142:
6137:
6134:
6133:
6115:
6112:
6111:
6109:
6070:
6047:
6044:
6043:
6041:
6023:
6020:
6019:
6017:
5993:
5990:
5989:
5987:
5912:Cauchy horizons
5896:
5884:Penrose process
5834:
5830:
5824:
5820:
5811:
5807:
5798:
5794:
5780:
5777:
5776:
5754:
5752:Penrose process
5748:
5713:
5710:
5709:
5707:
5705:
5694:
5664:
5661:
5660:
5658:
5640:
5637:
5636:
5634:
5632:
5612:
5563:
5559:
5553:
5549:
5541:
5526:
5520:
5519:
5513:
5510:
5509:
5472:
5468:
5462:
5458:
5446:
5441:
5435:
5426:
5422:
5421:
5419:
5410:
5404:
5403:
5397:
5394:
5393:
5386:
5380:
5378:
5347:
5343:
5335:
5320:
5314:
5313:
5307:
5304:
5303:
5264:
5261:
5260:
5258:
5230:
5226:
5214:
5209:
5203:
5194:
5190:
5189:
5187:
5178:
5172:
5171:
5165:
5162:
5161:
5150:
5144:
5141:
5138:
5137:
5135:
5134:
5132:
5093:
5090:
5089:
5087:
5069:
5066:
5065:
5063:
5045:
5042:
5041:
5039:
5024:
4958:
4954:
4948:
4944:
4935:
4931:
4917:
4913:
4904:
4900:
4899:
4895:
4891:
4875:
4871:
4870:
4868:
4854:
4850:
4841:
4837:
4835:
4824:
4821:
4820:
4812:Killing horizon
4763:
4741:
4737:
4728:
4724:
4722:
4712:
4708:
4707:
4698:
4694:
4685:
4681:
4669:
4665:
4656:
4652:
4639:
4638:
4634:
4625:
4621:
4602:
4598:
4592:
4584:
4578:
4566:
4562:
4561:
4557:
4548:
4544:
4535:
4531:
4529:
4526:
4525:
4513:
4478:
4464:
4459:
4455:
4454:
4444:
4435:
4421:
4416:
4412:
4411:
4401:
4392:
4378:
4373:
4369:
4368:
4348:
4344:
4343:
4341:
4334:
4330:
4315:
4311:
4307:
4302:
4291:
4287:
4282:
4272:
4267:
4254:
4241:
4237:
4233:
4231:
4222:
4208:
4203:
4199:
4198:
4181:
4177:
4165:
4161:
4152:
4148:
4147:
4145:
4136:
4132:
4123:
4119:
4118:
4114:
4102:
4098:
4097:
4092:
4077:
4073:
4069:
4064:
4055:
4051:
4047:
4042:
4033:
4029:
4027:
4024:
4023:
3985:
3981:
3979:
3976:
3975:
3973:
3966:
3938:
3934:
3928:
3916:
3915:
3907:
3900:
3896:
3890:
3886:
3885:
3883:
3874:
3862:
3861:
3848:
3844:
3833:
3829:
3824:
3818:
3814:
3808:
3804:
3795:
3791:
3789:
3780:
3776:
3767:
3755:
3754:
3745:
3742:
3741:
3737:are related by
3719:
3715:
3713:
3710:
3709:
3707:
3685:
3682:
3681:
3679:
3658:
3655:
3654:
3652:
3629:
3625:
3623:
3620:
3619:
3617:
3599:
3596:
3595:
3593:
3573:
3571:
3568:
3567:
3565:
3547:
3544:
3543:
3541:
3517:
3514:
3513:
3511:
3508:Penrose process
3476:
3475:
3471:
3464:
3460:
3454:
3450:
3434:
3433:
3429:
3427:
3424:
3423:
3421:
3418:
3402:KerrâNUT metric
3398:
3350:
3347:
3346:
3344:
3326:
3323:
3322:
3320:
3298:
3294:
3285:
3281:
3272:
3268:
3266:
3252:
3249:
3248:
3197:
3193:
3187:
3183:
3181:
3169:
3165:
3156:
3152:
3151:
3144:
3140:
3131:
3127:
3126:
3124:
3122:
3119:
3118:
3085:
3084:
3082:
3079:
3078:
3076:
3000:
2996:
2994:
2991:
2990:
2946:
2934:
2930:
2921:
2917:
2916:
2899:
2897:
2885:
2881:
2872:
2868:
2867:
2850:
2848:
2847:
2843:
2831:
2827:
2818:
2814:
2805:
2801:
2790:
2788:
2785:
2784:
2746:
2742:
2736:
2732:
2723:
2719:
2718:
2711:
2707:
2697:
2695:
2687:
2684:
2683:
2647:
2643:
2637:
2633:
2618:
2614:
2602:
2598:
2596:
2593:
2592:
2568:
2533:
2529:
2514:
2510:
2499:
2495:
2486:
2482:
2481:
2477:
2468:
2464:
2449:
2445:
2433:
2429:
2420:
2416:
2415:
2410:
2401:
2397:
2388:
2384:
2369:
2364:
2358:
2355:
2354:
2327:
2323:
2321:
2318:
2317:
2315:
2297:
2294:
2293:
2291:
2260:
2257:
2256:
2254:
2218:
2214:
2202:
2198:
2189:
2185:
2177:
2174:
2173:
2129:
2125:
2119:
2115:
2106:
2102:
2094:
2091:
2090:
2048:
2043:
2035:
2032:
2031:
2008:
2005:
2004:
2002:
1978:
1975:
1974:
1972:
1934:
1930:
1919:
1917:
1908:
1904:
1902:
1899:
1898:
1873:
1869:
1867:
1864:
1863:
1813:
1810:
1809:
1754:
1750:
1741:
1737:
1735:
1727:
1724:
1723:
1668:
1664:
1655:
1651:
1649:
1641:
1638:
1637:
1598:
1595:
1594:
1592:
1562:
1561:
1526:
1522:
1510:
1506:
1502:
1500:
1491:
1487:
1469:
1465:
1448:
1444:
1432:
1428:
1419:
1415:
1414:
1412:
1403:
1399:
1390:
1386:
1385:
1381:
1372:
1368:
1353:
1349:
1336:
1327:
1323:
1314:
1310:
1290:
1286:
1285:
1283:
1276:
1272:
1260:
1259:
1253:
1249:
1240:
1236:
1226:
1220:
1216:
1209:
1207:
1204:
1203:
1168:
1165:
1164:
1162:
1141:
1138:
1137:
1135:
1128:
1122:
1102:
1074:
1050:
1046:
1038:
1035:
1034:
1032:
996:
976:Gravity Probe B
970:(also known as
933:
931:
928:
927:
885:
883:
880:
879:
847:electric charge
807:
752:
711:
698:
697:
690:
689:
513:
512:
503:
502:
458:LemaĂźtreâTolman
403:
402:
391:
390:
382:Quantum gravity
369:Advanced theory
296:
295:
294:
277:
276:
225:Geodetic effect
186:
185:
176:
175:
151:
150:
134:
104:
85:
81:
76:
64:
60:
45:
41:
39:
36:
35:
17:
12:
11:
5:
13240:
13230:
13229:
13227:Metric tensors
13224:
13219:
13202:
13201:
13191:
13188:
13187:
13185:
13184:
13177:
13172:
13167:
13162:
13157:
13152:
13147:
13142:
13137:
13132:
13127:
13122:
13117:
13112:
13107:
13105:Choquet-Bruhat
13102:
13097:
13092:
13087:
13082:
13077:
13072:
13067:
13062:
13057:
13052:
13047:
13042:
13037:
13032:
13026:
13024:
13020:
13019:
13016:
13015:
13013:
13012:
13005:
13004:
12999:
12994:
12987:
12986:
12981:
12976:
12971:
12966:
12957:Axisymmetric:
12954:
12953:
12948:
12942:
12931:
12930:
12925:
12920:
12915:
12910:
12905:
12896:Cosmological:
12893:
12891:
12885:
12884:
12882:
12881:
12876:
12871:
12865:
12863:
12857:
12856:
12854:
12853:
12848:
12837:frame-dragging
12834:
12829:
12824:
12821:Einstein rings
12817:Einstein cross
12810:
12799:
12798:
12793:
12787:
12782:
12777:
12764:
12754:
12753:
12748:
12743:
12738:
12732:
12730:
12726:
12725:
12723:
12722:
12720:Ernst equation
12717:
12712:
12707:
12702:
12697:
12692:
12690:BSSN formalism
12687:
12681:
12679:
12675:
12674:
12672:
12671:
12666:
12661:
12656:
12651:
12645:
12643:
12637:
12636:
12634:
12633:
12628:
12622:
12620:
12613:
12605:
12604:
12601:
12600:
12598:
12597:
12592:
12587:
12582:
12577:
12571:
12569:
12563:
12562:
12560:
12559:
12554:
12549:
12547:Ladder paradox
12544:
12539:
12534:
12529:
12524:
12519:
12513:
12511:
12507:
12506:
12504:
12503:
12497:
12495:
12491:
12490:
12488:
12487:
12482:
12477:
12472:
12467:
12462:
12457:
12452:
12450:Speed of light
12447:
12441:
12439:
12433:
12432:
12430:
12429:
12424:
12419:
12413:
12403:
12401:
12394:
12386:
12385:
12378:
12377:
12370:
12363:
12355:
12346:
12345:
12343:
12342:
12337:
12332:
12327:
12322:
12320:Krasnikov tube
12317:
12312:
12307:
12305:BTZ black hole
12302:
12296:
12294:
12278:
12277:
12275:
12274:
12272:Predestination
12269:
12264:
12259:
12254:
12249:
12243:
12241:
12235:
12234:
12232:
12231:
12226:
12221:
12216:
12211:
12205:
12203:
12199:
12198:
12196:
12195:
12190:
12184:
12182:
12176:
12175:
12173:
12172:
12171:
12170:
12160:
12159:
12158:
12147:
12145:
12139:
12138:
12136:
12135:
12130:
12125:
12120:
12115:
12109:
12107:
12103:
12102:
12095:
12094:
12087:
12080:
12072:
12063:
12062:
12060:
12059:
12049:
12038:
12035:
12034:
12032:
12031:
12029:Swift J1644+57
12026:
12021:
12016:
12011:
12006:
12001:
11996:
11991:
11986:
11981:
11979:MS 0735.6+7421
11976:
11971:
11966:
11961:
11956:
11951:
11946:
11944:Sagittarius A*
11941:
11936:
11931:
11926:
11921:
11915:
11913:
11909:
11908:
11906:
11905:
11900:
11895:
11890:
11885:
11880:
11875:
11870:
11865:
11860:
11855:
11850:
11845:
11840:
11835:
11830:
11825:
11820:
11819:
11818:
11813:
11803:
11798:
11793:
11788:
11783:
11778:
11773:
11767:
11765:
11761:
11760:
11758:
11757:
11752:
11747:
11742:
11737:
11731:
11729:
11723:
11722:
11720:
11719:
11714:
11708:
11706:
11702:
11701:
11699:
11698:
11693:
11688:
11683:
11678:
11673:
11668:
11663:
11658:
11653:
11647:
11645:
11641:
11640:
11638:
11637:
11632:
11627:
11622:
11617:
11606:
11604:
11600:
11599:
11597:
11596:
11591:
11586:
11581:
11576:
11571:
11566:
11561:
11555:
11553:
11549:
11548:
11546:
11545:
11540:
11535:
11534:
11533:
11523:
11518:
11516:Thermodynamics
11513:
11508:
11503:
11498:
11497:
11496:
11486:
11481:
11479:Accretion disk
11476:
11475:
11474:
11469:
11459:
11454:
11449:
11444:
11443:
11442:
11437:
11427:
11421:
11419:
11415:
11414:
11412:
11411:
11406:
11401:
11396:
11391:
11386:
11381:
11376:
11375:
11374:
11369:
11364:
11354:
11353:
11352:
11342:
11337:
11336:
11335:
11325:
11320:
11314:
11312:
11308:
11307:
11305:
11304:
11303:
11302:
11297:
11292:
11287:
11282:
11277:
11272:
11262:
11257:
11256:
11255:
11245:
11244:
11243:
11240:
11235:
11224:
11222:
11218:
11217:
11210:
11208:
11206:
11205:
11200:
11195:
11190:
11185:
11180:
11175:
11170:
11165:
11160:
11155:
11153:BTZ black hole
11149:
11147:
11143:
11142:
11140:
11139:
11133:
11130:
11129:
11122:
11121:
11114:
11107:
11099:
11093:
11092:
11056:
11019:
11013:
10993:
10975:(6): 331â333.
10961:
10913:
10897:
10885:
10875:
10858:
10855:See chapter 13
10848:
10835:
10825:
10809:
10806:See chapter 19
10799:
10782:
10776:
10763:
10757:
10739:
10733:
10720:
10714:
10695:
10692:
10690:
10689:
10646:
10637:
10622:
10609:
10591:
10530:
10492:(12): 124010.
10476:
10471:10.1086/151796
10441:
10438:on 2020-02-25.
10376:
10352:
10334:
10316:
10260:
10247:
10231:
10178:
10153:
10144:978-0521805865
10143:
10121:
10100:
10079:
10064:
10011:
9975:
9956:(2): 265â281.
9940:
9912:
9899:
9892:
9872:
9865:
9840:
9781:
9734:
9675:
9648:(6): 915â917.
9628:
9567:
9548:
9487:
9484:978-0226519517
9471:
9452:(5): 237â238.
9432:
9430:
9427:
9425:
9424:
9411:
9409:
9406:
9405:
9404:
9399:
9394:
9389:
9384:
9379:
9373:
9372:
9369:Physics portal
9358:
9342:
9339:
9324:
9321:
9318:
9313:
9309:
9305:
9258:
9255:
9251:
9250:
9239:
9226:
9200:
9156:
9155:
9143:
9137:
9133:
9129:
9124:
9119:
9115:
9108:
9104:
9101:
9096:
9092:
9087:
9084:
9081:
9076:
9072:
9067:
9064:
9061:
9056:
9052:
9003:
9000:
8997:
8980:
8979:
8966:
8962:
8958:
8955:
8952:
8949:
8946:
8941:
8937:
8904:
8901:
8838:
8835:
8832:
8786:
8783:
8755:
8728:
8715:Cauchy horizon
8700:
8672:
8669:
8666:
8651:
8650:
8609:
8607:
8600:
8594:
8591:
8572:
8569:
8566:
8561:
8557:
8553:
8529:
8526:
8523:
8519:
8509:
8484:
8481:
8478:
8464:
8461:
8446:Poincaré group
8441:
8438:
8436:are restored.
8419:
8416:
8413:
8386:
8383:
8376:
8372:
8367:
8363:
8359:
8356:
8350:
8347:
8341:
8338:
8317:
8312:
8304:
8300:
8297:
8291:
8287:
8281:
8277:
8273:
8270:
8267:
8264:
8260:
8255:
8250:
8245:
8241:
8237:
8234:
8212:
8188:
8161:
8157:
8153:
8149:
8143:
8139:
8136:
8131:
8125:
8121:
8117:
8110:
8106:
8103:
8100:
8097:
8091:
8088:
8061:
8057:
8053:
8046:
8042:
8036:
8032:
8028:
8025:
8018:
8014:
8011:
8006:
8000:
7996:
7992:
7985:
7981:
7978:
7975:
7972:
7969:
7966:
7963:
7960:
7956:
7949:
7946:
7938:
7934:
7931:
7928:
7925:
7919:
7913:
7910:
7904:
7901:
7895:
7892:
7885:
7881:
7874:
7870:
7866:
7861:
7858:
7855:
7852:
7849:
7843:
7840:
7834:
7831:
7825:
7822:
7799:
7792:
7786:
7781:
7778:
7773:
7769:
7765:
7760:
7756:
7752:
7747:
7742:
7736:
7732:
7728:
7723:
7719:
7714:
7705:
7700:
7697:
7673:
7649:
7625:
7601:
7577:
7564:frame-dragging
7546:
7542:
7538:
7535:
7532:
7508:
7484:
7460:
7457:
7454:
7428:
7424:
7420:
7414:
7411:
7404:
7400:
7396:
7365:
7352:
7351:
7339:
7335:
7332:
7327:
7323:
7319:
7316:
7313:
7308:
7304:
7300:
7297:
7292:
7288:
7282:
7278:
7273:
7269:
7266:
7261:
7257:
7253:
7250:
7247:
7244:
7241:
7238:
7235:
7232:
7222:
7209:
7205:
7201:
7198:
7194:
7188:
7184:
7180:
7175:
7171:
7166:
7162:
7159:
7156:
7153:
7150:
7147:
7137:
7125:
7118:
7115:
7110:
7106:
7099:
7094:
7090:
7084:
7080:
7074:
7070:
7066:
7061:
7057:
7052:
7046:
7042:
7037:
7033:
7030:
7025:
7021:
7017:
7014:
7011:
7008:
7005:
7002:
6999:
6972:
6969:
6966:
6963:
6958:
6952:
6948:
6944:
6939:
6935:
6928:
6924:
6918:
6914:
6910:
6907:
6904:
6899:
6895:
6891:
6888:
6884:
6880:
6877:
6874:
6871:
6869:
6864:
6861:
6856:
6853:
6847:
6844:
6843:
6840:
6837:
6834:
6831:
6826:
6823:
6818:
6814:
6807:
6804:
6799:
6795:
6788:
6784:
6778:
6775:
6772:
6768:
6764:
6761:
6758:
6756:
6751:
6748:
6743:
6740:
6734:
6731:
6730:
6725:
6722:
6719:
6716:
6711:
6708:
6705:
6703:
6698:
6695:
6690:
6687:
6681:
6678:
6677:
6672:
6669:
6666:
6663:
6658:
6655:
6652:
6650:
6645:
6642:
6637:
6634:
6628:
6625:
6624:
6602:
6599:
6596:
6593:
6590:
6587:
6584:
6553:
6549:
6543:
6538:
6532:
6529:
6526:
6520:
6516:
6510:
6505:
6501:
6495:
6491:
6487:
6482:
6478:
6473:
6467:
6463:
6458:
6454:
6451:
6446:
6442:
6438:
6433:
6428:
6424:
6420:
6417:
6391:
6378:Brandon Carter
6357:
6353:
6349:
6344:
6340:
6319:
6314:
6310:
6306:
6303:
6300:
6276:
6272:
6247:
6217:
6213:
6190:
6185:
6181:
6175:
6172:
6168:
6162:
6158:
6154:
6149:
6145:
6141:
6119:
6102:test particles
6069:
6066:
6051:
6027:
6003:
6000:
5997:
5984:photon spheres
5942:Killing tensor
5908:event horizons
5895:
5892:
5857:
5856:
5845:
5842:
5837:
5833:
5827:
5823:
5819:
5814:
5810:
5806:
5801:
5797:
5793:
5790:
5787:
5784:
5750:Main article:
5747:
5744:
5729:
5726:
5720:
5717:
5703:
5692:
5668:
5644:
5630:
5610:
5590:Due to the cos
5588:
5587:
5574:
5571:
5566:
5562:
5556:
5552:
5548:
5545:
5540:
5537:
5534:
5529:
5523:
5518:
5503:
5502:
5489:
5483:
5480:
5475:
5471:
5465:
5461:
5457:
5454:
5449:
5440:
5434:
5425:
5418:
5413:
5407:
5402:
5384:
5376:
5366:
5365:
5350:
5346:
5342:
5339:
5334:
5331:
5328:
5323:
5317:
5312:
5286:
5283:
5280:
5277:
5274:
5271:
5268:
5255:
5254:
5241:
5233:
5229:
5225:
5222:
5217:
5208:
5202:
5193:
5186:
5181:
5175:
5170:
5148:
5130:
5097:
5073:
5049:
5023:
5020:
5012:planetary gear
5004:frame-dragging
4998:
4997:
4988:
4986:
4975:
4969:
4966:
4961:
4957:
4951:
4947:
4943:
4934:
4930:
4926:
4920:
4916:
4912:
4907:
4903:
4898:
4894:
4889:
4886:
4883:
4874:
4867:
4860:
4857:
4853:
4847:
4844:
4840:
4834:
4831:
4828:
4794:
4793:
4784:
4782:
4771:
4766:
4761:
4757:
4754:
4747:
4744:
4740:
4734:
4731:
4727:
4721:
4718:
4715:
4711:
4704:
4701:
4697:
4693:
4688:
4684:
4680:
4675:
4672:
4668:
4664:
4659:
4655:
4651:
4645:
4642:
4637:
4633:
4628:
4624:
4620:
4616:
4608:
4605:
4601:
4595:
4590:
4587:
4583:
4577:
4572:
4569:
4565:
4560:
4556:
4551:
4547:
4543:
4538:
4534:
4512:
4511:Frame dragging
4509:
4506:
4505:
4496:
4494:
4481:
4476:
4470:
4467:
4463:
4458:
4451:
4448:
4443:
4438:
4433:
4427:
4424:
4420:
4415:
4408:
4405:
4400:
4395:
4390:
4384:
4381:
4377:
4372:
4366:
4360:
4356:
4347:
4340:
4337:
4333:
4326:
4323:
4318:
4314:
4310:
4306:
4301:
4294:
4290:
4286:
4278:
4275:
4271:
4263:
4260:
4257:
4252:
4249:
4240:
4236:
4230:
4225:
4220:
4214:
4211:
4207:
4202:
4196:
4192:
4189:
4184:
4180:
4174:
4168:
4164:
4160:
4151:
4144:
4139:
4135:
4131:
4126:
4122:
4117:
4110:
4105:
4101:
4096:
4091:
4088:
4080:
4076:
4072:
4068:
4058:
4054:
4050:
4046:
4039:
4036:
4032:
3991:
3988:
3984:
3965:
3962:
3961:
3960:
3949:
3941:
3937:
3931:
3925:
3922:
3919:
3914:
3910:
3903:
3899:
3893:
3889:
3882:
3877:
3871:
3868:
3865:
3860:
3856:
3851:
3847:
3843:
3836:
3832:
3827:
3821:
3817:
3811:
3807:
3803:
3798:
3794:
3788:
3783:
3779:
3775:
3770:
3764:
3761:
3758:
3753:
3749:
3718:
3689:
3662:
3628:
3603:
3577:
3551:
3527:
3524:
3521:
3492:
3485:
3482:
3479:
3474:
3470:
3467:
3463:
3457:
3453:
3449:
3443:
3440:
3437:
3432:
3417:
3414:
3397:
3394:
3360:
3357:
3354:
3330:
3317:
3316:
3301:
3297:
3293:
3288:
3284:
3280:
3275:
3271:
3265:
3262:
3259:
3256:
3233:
3232:
3223:
3221:
3210:
3207:
3200:
3196:
3190:
3186:
3180:
3172:
3168:
3164:
3159:
3155:
3147:
3143:
3139:
3134:
3130:
3092:
3089:
3035:
3034:
3025:
3023:
3011:
3008:
3003:
2999:
2983:
2982:
2973:
2971:
2959:
2953:
2950:
2945:
2937:
2933:
2929:
2924:
2920:
2914:
2911:
2908:
2905:
2902:
2896:
2888:
2884:
2880:
2875:
2871:
2865:
2862:
2859:
2856:
2853:
2846:
2842:
2839:
2834:
2830:
2826:
2821:
2817:
2813:
2808:
2804:
2800:
2797:
2793:
2777:
2776:
2767:
2765:
2749:
2745:
2739:
2735:
2731:
2726:
2722:
2714:
2710:
2706:
2703:
2700:
2694:
2691:
2676:
2675:
2666:
2664:
2650:
2646:
2640:
2636:
2632:
2629:
2624:
2621:
2617:
2613:
2608:
2605:
2601:
2567:
2564:
2561:
2560:
2551:
2549:
2536:
2532:
2528:
2525:
2522:
2517:
2513:
2508:
2502:
2498:
2494:
2489:
2485:
2480:
2476:
2471:
2467:
2463:
2460:
2457:
2452:
2448:
2444:
2436:
2432:
2428:
2423:
2419:
2414:
2409:
2404:
2400:
2396:
2391:
2387:
2383:
2378:
2375:
2372:
2367:
2362:
2326:
2301:
2274:
2271:
2267:
2264:
2249:
2248:
2239:
2237:
2226:
2221:
2217:
2213:
2210:
2201:
2197:
2192:
2188:
2184:
2181:
2166:
2165:
2156:
2154:
2143:
2140:
2137:
2132:
2128:
2122:
2118:
2114:
2109:
2105:
2101:
2098:
2083:
2082:
2073:
2071:
2060:
2054:
2051:
2047:
2042:
2039:
2012:
1988:
1985:
1982:
1967:
1966:
1957:
1955:
1944:
1937:
1933:
1928:
1925:
1922:
1916:
1907:
1872:
1858:
1857:
1848:
1846:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1802:
1801:
1792:
1790:
1779:
1776:
1773:
1770:
1767:
1764:
1757:
1753:
1749:
1744:
1740:
1734:
1731:
1716:
1715:
1706:
1704:
1693:
1690:
1687:
1684:
1681:
1678:
1671:
1667:
1663:
1658:
1654:
1648:
1645:
1614:
1611:
1608:
1605:
1602:
1587:
1586:
1577:
1575:
1560:
1557:
1553:
1550:
1546:
1541:
1537:
1534:
1529:
1525:
1521:
1518:
1509:
1505:
1499:
1494:
1490:
1486:
1480:
1477:
1472:
1468:
1463:
1459:
1456:
1451:
1447:
1441:
1435:
1431:
1427:
1418:
1411:
1406:
1402:
1398:
1393:
1389:
1384:
1380:
1375:
1371:
1367:
1364:
1361:
1356:
1352:
1348:
1343:
1340:
1335:
1330:
1326:
1322:
1317:
1313:
1308:
1302:
1298:
1289:
1282:
1279:
1275:
1271:
1268:
1265:
1263:
1261:
1256:
1252:
1248:
1243:
1239:
1235:
1232:
1229:
1227:
1223:
1219:
1215:
1212:
1211:
1172:
1158:rotating with
1145:
1124:Main article:
1121:
1118:
1114:Ernst equation
1101:
1098:
1083:invariant mass
1072:
1049:
1045:
1042:
968:frame-dragging
964:
963:
960:
959:
954:
949:
936:
919:
918:
915:
910:
902:
901:
888:
872:
867:Non-rotating (
865:
806:
803:
787:exact solution
754:
753:
751:
750:
743:
736:
728:
725:
724:
723:
722:
709:
692:
691:
688:
687:
680:
675:
670:
665:
660:
655:
650:
645:
640:
635:
630:
625:
620:
615:
610:
605:
600:
595:
590:
585:
580:
575:
570:
565:
560:
555:
550:
545:
540:
535:
530:
525:
520:
514:
510:
509:
508:
505:
504:
501:
500:
495:
490:
485:
480:
475:
470:
465:
460:
455:
450:
445:
440:
435:
430:
425:
420:
415:
404:
398:
397:
396:
393:
392:
387:
386:
385:
384:
379:
371:
370:
366:
365:
364:
363:
361:Post-Newtonian
358:
353:
345:
344:
340:
339:
338:
337:
332:
327:
322:
317:
312:
304:
303:
297:
293:
292:
289:
285:
284:
283:
282:
279:
278:
273:
272:
271:
270:
265:
260:
252:
251:
245:
244:
243:
242:
237:
232:
227:
222:
220:Frame-dragging
217:
212:
207:
202:
197:
195:Kepler problem
187:
183:
182:
181:
178:
177:
174:
173:
168:
163:
158:
152:
148:
147:
146:
143:
142:
141:
140:
135:
133:
132:
127:
122:
116:
114:
106:
105:
91:
88:
84:
79:
75:
70:
67:
63:
59:
56:
51:
48:
44:
34:
26:
25:
15:
9:
6:
4:
3:
2:
13239:
13228:
13225:
13223:
13220:
13218:
13215:
13214:
13212:
13199:
13189:
13183:
13182:
13178:
13176:
13173:
13171:
13168:
13166:
13163:
13161:
13158:
13156:
13153:
13151:
13148:
13146:
13143:
13141:
13138:
13136:
13133:
13131:
13128:
13126:
13123:
13121:
13118:
13116:
13113:
13111:
13108:
13106:
13103:
13101:
13098:
13096:
13093:
13091:
13090:Chandrasekhar
13088:
13086:
13083:
13081:
13078:
13076:
13073:
13071:
13068:
13066:
13063:
13061:
13058:
13056:
13053:
13051:
13050:Schwarzschild
13048:
13046:
13043:
13041:
13038:
13036:
13033:
13031:
13028:
13027:
13025:
13021:
13011:
13007:
13006:
13003:
13000:
12998:
12995:
12993:
12989:
12988:
12985:
12982:
12980:
12977:
12975:
12972:
12970:
12967:
12964:
12960:
12956:
12955:
12952:
12949:
12946:
12943:
12941:
12937:
12936:Schwarzschild
12933:
12932:
12929:
12926:
12924:
12921:
12919:
12916:
12914:
12911:
12909:
12906:
12903:
12899:
12895:
12894:
12892:
12890:
12886:
12880:
12877:
12875:
12872:
12870:
12867:
12866:
12864:
12858:
12852:
12849:
12846:
12842:
12838:
12835:
12833:
12832:Shapiro delay
12830:
12828:
12825:
12822:
12818:
12814:
12811:
12808:
12804:
12801:
12800:
12797:
12794:
12791:
12788:
12786:
12783:
12781:
12778:
12776:
12775:collaboration
12772:
12768:
12765:
12763:
12759:
12756:
12755:
12752:
12749:
12747:
12744:
12742:
12741:Event horizon
12739:
12737:
12734:
12733:
12731:
12727:
12721:
12718:
12716:
12713:
12711:
12708:
12706:
12703:
12701:
12698:
12696:
12693:
12691:
12688:
12686:
12685:ADM formalism
12683:
12682:
12680:
12676:
12670:
12667:
12665:
12662:
12660:
12657:
12655:
12652:
12650:
12647:
12646:
12644:
12638:
12632:
12629:
12627:
12624:
12623:
12621:
12617:
12614:
12612:
12606:
12596:
12593:
12591:
12590:Biquaternions
12588:
12586:
12583:
12581:
12578:
12576:
12573:
12572:
12570:
12568:
12564:
12558:
12555:
12553:
12550:
12548:
12545:
12543:
12540:
12538:
12535:
12533:
12530:
12528:
12525:
12523:
12520:
12518:
12517:Time dilation
12515:
12514:
12512:
12508:
12502:
12499:
12498:
12496:
12492:
12486:
12483:
12481:
12478:
12476:
12473:
12471:
12470:Proper length
12468:
12466:
12463:
12461:
12458:
12456:
12453:
12451:
12448:
12446:
12443:
12442:
12440:
12434:
12428:
12425:
12423:
12420:
12417:
12414:
12412:
12408:
12405:
12404:
12402:
12398:
12395:
12393:
12387:
12383:
12376:
12371:
12369:
12364:
12362:
12357:
12356:
12353:
12341:
12338:
12336:
12333:
12331:
12328:
12326:
12323:
12321:
12318:
12316:
12313:
12311:
12308:
12306:
12303:
12301:
12298:
12297:
12295:
12293:
12287:
12283:
12279:
12273:
12270:
12268:
12265:
12263:
12260:
12258:
12255:
12253:
12250:
12248:
12245:
12244:
12242:
12240:
12236:
12230:
12227:
12225:
12222:
12220:
12217:
12215:
12212:
12210:
12207:
12206:
12204:
12200:
12194:
12191:
12189:
12186:
12185:
12183:
12181:
12177:
12169:
12166:
12165:
12164:
12161:
12157:
12154:
12153:
12152:
12149:
12148:
12146:
12144:
12140:
12134:
12131:
12129:
12126:
12124:
12121:
12119:
12116:
12114:
12111:
12110:
12108:
12104:
12100:
12093:
12088:
12086:
12081:
12079:
12074:
12073:
12070:
12058:
12050:
12048:
12040:
12039:
12036:
12030:
12027:
12025:
12022:
12020:
12017:
12015:
12012:
12010:
12007:
12005:
12004:Markarian 501
12002:
12000:
11997:
11995:
11992:
11990:
11987:
11985:
11982:
11980:
11977:
11975:
11972:
11970:
11967:
11965:
11962:
11960:
11957:
11955:
11952:
11950:
11947:
11945:
11942:
11940:
11937:
11935:
11932:
11930:
11929:XTE J1118+480
11927:
11925:
11924:XTE J1650-500
11922:
11920:
11917:
11916:
11914:
11910:
11904:
11901:
11899:
11896:
11894:
11891:
11889:
11886:
11884:
11881:
11879:
11876:
11874:
11871:
11869:
11866:
11864:
11861:
11859:
11856:
11854:
11851:
11849:
11846:
11844:
11841:
11839:
11836:
11834:
11831:
11829:
11826:
11824:
11821:
11817:
11814:
11812:
11809:
11808:
11807:
11804:
11802:
11799:
11797:
11794:
11792:
11789:
11787:
11784:
11782:
11779:
11777:
11774:
11772:
11769:
11768:
11766:
11762:
11756:
11753:
11751:
11748:
11746:
11743:
11741:
11738:
11736:
11733:
11732:
11730:
11728:
11724:
11718:
11715:
11713:
11710:
11709:
11707:
11703:
11697:
11694:
11692:
11689:
11687:
11684:
11682:
11679:
11677:
11674:
11672:
11669:
11667:
11664:
11662:
11659:
11657:
11654:
11652:
11649:
11648:
11646:
11642:
11636:
11633:
11631:
11628:
11626:
11623:
11621:
11618:
11615:
11611:
11610:Schwarzschild
11608:
11607:
11605:
11601:
11595:
11592:
11590:
11587:
11585:
11582:
11580:
11577:
11575:
11572:
11570:
11567:
11565:
11562:
11560:
11557:
11556:
11554:
11550:
11544:
11541:
11539:
11536:
11532:
11529:
11528:
11527:
11524:
11522:
11519:
11517:
11514:
11512:
11509:
11507:
11504:
11502:
11499:
11495:
11492:
11491:
11490:
11487:
11485:
11482:
11480:
11477:
11473:
11470:
11468:
11465:
11464:
11463:
11460:
11458:
11455:
11453:
11452:Photon sphere
11450:
11448:
11447:Event horizon
11445:
11441:
11438:
11436:
11433:
11432:
11431:
11428:
11426:
11423:
11422:
11420:
11416:
11410:
11407:
11405:
11402:
11400:
11397:
11395:
11392:
11390:
11387:
11385:
11382:
11380:
11377:
11373:
11372:Related links
11370:
11368:
11365:
11363:
11360:
11359:
11358:
11355:
11351:
11350:Related links
11348:
11347:
11346:
11343:
11341:
11338:
11334:
11333:Related links
11331:
11330:
11329:
11326:
11324:
11321:
11319:
11316:
11315:
11313:
11309:
11301:
11298:
11296:
11293:
11291:
11288:
11286:
11283:
11281:
11278:
11276:
11273:
11271:
11268:
11267:
11266:
11263:
11261:
11258:
11254:
11251:
11250:
11249:
11246:
11241:
11239:
11236:
11234:
11231:
11230:
11229:
11226:
11225:
11223:
11219:
11214:
11204:
11201:
11199:
11196:
11194:
11191:
11189:
11186:
11184:
11181:
11179:
11176:
11174:
11171:
11169:
11166:
11164:
11161:
11159:
11158:Schwarzschild
11156:
11154:
11151:
11150:
11148:
11144:
11138:
11135:
11134:
11131:
11127:
11120:
11115:
11113:
11108:
11106:
11101:
11100:
11097:
11087:
11082:
11078:
11074:
11070:
11066:
11062:
11057:
11053:
11049:
11045:
11041:
11037:
11033:
11029:
11025:
11020:
11016:
11010:
11005:
11004:
10998:
10994:
10990:
10986:
10982:
10978:
10974:
10970:
10966:
10962:
10957:
10953:
10949:
10945:
10941:
10937:
10932:
10931:gr-qc/0407064
10927:
10923:
10919:
10914:
10908:
10903:
10898:
10894:
10890:
10886:
10883:
10882:See chapter 7
10878:
10872:
10867:
10866:
10859:
10856:
10851:
10845:
10841:
10836:
10833:
10828:
10822:
10818:
10814:
10810:
10807:
10802:
10796:
10791:
10790:
10783:
10779:
10773:
10769:
10764:
10760:
10754:
10750:
10749:
10744:
10740:
10736:
10730:
10726:
10721:
10717:
10711:
10707:
10703:
10698:
10697:
10685:
10681:
10677:
10673:
10669:
10665:
10661:
10657:
10650:
10641:
10634:
10633:
10626:
10619:
10613:
10605:
10601:
10595:
10587:
10583:
10579:
10575:
10571:
10567:
10563:
10559:
10554:
10549:
10545:
10541:
10534:
10525:
10521:
10517:
10513:
10509:
10505:
10500:
10495:
10491:
10487:
10480:
10472:
10468:
10464:
10460:
10456:
10452:
10445:
10434:
10430:
10426:
10422:
10418:
10414:
10410:
10406:
10402:
10395:
10391:
10385:
10383:
10381:
10371:
10366:
10359:
10357:
10350:
10346:
10341:
10339:
10327:
10320:
10312:
10308:
10304:
10300:
10296:
10292:
10287:
10282:
10278:
10274:
10270:
10264:
10257:
10251:
10244:
10240:
10235:
10227:
10223:
10219:
10215:
10210:
10205:
10201:
10197:
10194:(2): 85â100.
10193:
10189:
10182:
10175:
10171:
10168:
10162:
10160:
10158:
10151:
10146:
10140:
10136:
10132:
10125:
10118:
10114:
10110:
10104:
10095:
10090:
10083:
10076:
10075:
10068:
10060:
10056:
10052:
10048:
10044:
10040:
10035:
10034:gr-qc/9312028
10030:
10026:
10022:
10015:
10006:
10002:
9998:
9994:
9990:
9986:
9979:
9971:
9967:
9963:
9959:
9955:
9951:
9950:J. Math. Phys
9944:
9935:
9930:
9923:
9921:
9919:
9917:
9910:, p. 1, Eq. 1
9909:
9903:
9895:
9889:
9885:
9884:
9876:
9868:
9862:
9858:
9854:
9850:
9849:Landau, L. D.
9844:
9836:
9832:
9827:
9822:
9817:
9812:
9808:
9804:
9801:(1): e03336.
9800:
9796:
9792:
9785:
9777:
9773:
9769:
9765:
9761:
9757:
9753:
9749:
9745:
9738:
9730:
9726:
9721:
9716:
9712:
9708:
9703:
9698:
9695:(10): 31791.
9694:
9690:
9686:
9679:
9671:
9667:
9663:
9659:
9655:
9651:
9647:
9643:
9639:
9632:
9624:
9620:
9616:
9612:
9608:
9604:
9600:
9596:
9591:
9586:
9583:(6): 061102.
9582:
9578:
9571:
9563:
9559:
9552:
9544:
9540:
9535:
9530:
9526:
9522:
9518:
9514:
9510:
9506:
9502:
9498:
9491:
9485:
9481:
9475:
9467:
9463:
9459:
9455:
9451:
9447:
9443:
9437:
9433:
9421:
9416:
9412:
9403:
9400:
9398:
9395:
9393:
9390:
9388:
9385:
9383:
9380:
9378:
9375:
9374:
9370:
9364:
9359:
9356:
9345:
9338:
9322:
9319:
9316:
9311:
9307:
9303:
9293:
9292:dust solution
9289:
9285:
9280:
9276:
9272:
9268:
9264:
9257:Open problems
9254:
9248:
9244:
9240:
9237:
9236:
9231:
9227:
9224:
9220:
9216:
9215:
9214:
9198:
9187:
9184:
9180:
9176:
9172:
9167:
9165:
9161:
9141:
9135:
9131:
9127:
9122:
9117:
9113:
9106:
9102:
9099:
9094:
9090:
9085:
9082:
9079:
9074:
9070:
9065:
9062:
9059:
9054:
9050:
9042:
9041:
9040:
9038:
9034:
9030:
9025:
9021:
9019:
9001:
8998:
8995:
8985:
8964:
8956:
8953:
8947:
8944:
8939:
8935:
8927:
8926:
8925:
8922:
8918:
8914:
8910:
8900:
8898:
8893:
8889:
8885:
8881:
8877:
8872:
8870:
8866:
8862:
8858:
8854:
8836:
8833:
8830:
8819:
8817:
8813:
8808:
8806:
8805:Ernst vacuums
8802:
8798:
8795:
8792:
8782:
8780:
8774:
8772:
8753:
8742:
8726:
8716:
8698:
8688:
8670:
8667:
8647:
8644:
8636:
8633:February 2011
8626:
8622:
8616:
8615:
8610:This section
8608:
8604:
8599:
8598:
8590:
8588:
8570:
8567:
8564:
8559:
8555:
8551:
8527:
8524:
8521:
8517:
8507:
8482:
8479:
8460:
8458:
8453:
8451:
8447:
8437:
8435:
8417:
8414:
8411:
8400:
8384:
8381:
8374:
8370:
8365:
8361:
8357:
8354:
8348:
8345:
8339:
8336:
8315:
8310:
8302:
8298:
8295:
8289:
8285:
8279:
8275:
8271:
8268:
8265:
8262:
8258:
8253:
8248:
8243:
8239:
8235:
8232:
8210:
8186:
8159:
8151:
8147:
8141:
8137:
8134:
8123:
8119:
8115:
8108:
8104:
8101:
8098:
8095:
8089:
8086:
8059:
8051:
8044:
8040:
8034:
8030:
8026:
8023:
8016:
8012:
8009:
7998:
7994:
7990:
7983:
7976:
7973:
7970:
7967:
7961:
7958:
7954:
7947:
7944:
7936:
7932:
7929:
7926:
7923:
7917:
7911:
7908:
7902:
7899:
7893:
7890:
7883:
7879:
7872:
7864:
7859:
7856:
7853:
7850:
7847:
7841:
7838:
7832:
7829:
7823:
7820:
7797:
7779:
7776:
7771:
7767:
7758:
7754:
7750:
7745:
7740:
7734:
7730:
7726:
7721:
7717:
7712:
7703:
7698:
7695:
7671:
7623:
7599:
7565:
7560:
7544:
7540:
7536:
7533:
7530:
7506:
7482:
7458:
7455:
7452:
7426:
7422:
7418:
7412:
7409:
7402:
7398:
7394:
7381:
7363:
7337:
7333:
7330:
7325:
7317:
7314:
7311:
7306:
7302:
7295:
7290:
7286:
7280:
7276:
7271:
7264:
7259:
7251:
7245:
7242:
7236:
7230:
7223:
7207:
7203:
7199:
7196:
7192:
7186:
7182:
7178:
7173:
7169:
7164:
7160:
7157:
7151:
7145:
7138:
7123:
7116:
7113:
7108:
7104:
7097:
7092:
7088:
7082:
7078:
7072:
7068:
7064:
7059:
7055:
7050:
7044:
7040:
7035:
7031:
7028:
7023:
7019:
7015:
7012:
7009:
7003:
6990:
6989:
6988:
6967:
6961:
6950:
6946:
6942:
6937:
6933:
6926:
6922:
6916:
6912:
6908:
6905:
6902:
6897:
6893:
6889:
6886:
6882:
6878:
6875:
6872:
6870:
6862:
6859:
6854:
6851:
6835:
6829:
6821:
6816:
6812:
6805:
6802:
6797:
6793:
6786:
6782:
6776:
6773:
6770:
6766:
6762:
6759:
6757:
6749:
6746:
6741:
6738:
6720:
6709:
6706:
6704:
6696:
6693:
6688:
6685:
6667:
6661:
6656:
6653:
6651:
6643:
6640:
6635:
6632:
6600:
6597:
6594:
6591:
6588:
6585:
6582:
6571:
6569:
6564:
6551:
6547:
6541:
6536:
6530:
6527:
6524:
6518:
6514:
6508:
6503:
6499:
6493:
6489:
6485:
6480:
6476:
6471:
6465:
6461:
6456:
6452:
6449:
6444:
6440:
6436:
6431:
6426:
6422:
6418:
6415:
6407:
6389:
6379:
6375:
6370:
6355:
6351:
6347:
6342:
6338:
6317:
6312:
6308:
6304:
6301:
6298:
6274:
6270:
6245:
6235:
6234:four-momentum
6215:
6211:
6188:
6183:
6179:
6173:
6170:
6166:
6160:
6156:
6152:
6147:
6143:
6139:
6117:
6107:
6103:
6099:
6091:
6087:
6082:
6074:
6065:
6049:
6025:
6001:
5998:
5995:
5985:
5980:
5976:
5974:
5973:homotopy type
5970:
5966:
5963:
5959:
5955:
5951:
5947:
5943:
5939:
5935:
5934:
5929:
5925:
5921:
5917:
5913:
5909:
5905:
5901:
5891:
5889:
5885:
5881:
5880:Roger Penrose
5877:
5873:
5869:
5864:
5862:
5843:
5840:
5835:
5831:
5825:
5821:
5817:
5812:
5808:
5804:
5799:
5791:
5788:
5785:
5775:
5774:
5773:
5770:
5768:
5763:
5759:
5758:event horizon
5753:
5743:
5727:
5724:
5718:
5715:
5702:
5698:
5691:
5687:
5682:
5642:
5629:
5625:
5621:
5617:
5609:
5605:
5601:
5597:
5593:
5572:
5569:
5564:
5560:
5554:
5550:
5546:
5543:
5538:
5535:
5532:
5527:
5516:
5508:
5507:
5506:
5487:
5481:
5478:
5473:
5469:
5463:
5459:
5455:
5452:
5447:
5438:
5432:
5423:
5416:
5411:
5400:
5392:
5391:
5390:
5383:
5375:
5371:
5368:While in the
5348:
5344:
5340:
5337:
5332:
5329:
5326:
5321:
5310:
5302:
5301:
5300:
5284:
5281:
5278:
5275:
5272:
5269:
5266:
5239:
5231:
5227:
5223:
5220:
5215:
5206:
5200:
5191:
5184:
5179:
5168:
5160:
5159:
5158:
5147:
5129:
5125:
5121:
5120:event horizon
5095:
5071:
5047:
5036:
5028:
5019:
5017:
5013:
5009:
5005:
4996:
4989:
4987:
4973:
4967:
4964:
4959:
4955:
4949:
4945:
4941:
4932:
4928:
4924:
4918:
4914:
4910:
4905:
4901:
4896:
4887:
4884:
4881:
4872:
4865:
4858:
4855:
4851:
4845:
4842:
4838:
4832:
4829:
4819:
4818:
4815:
4813:
4809:
4805:
4801:
4792:
4785:
4783:
4769:
4764:
4759:
4755:
4752:
4745:
4742:
4738:
4732:
4729:
4725:
4719:
4716:
4713:
4709:
4702:
4699:
4695:
4691:
4686:
4682:
4678:
4673:
4670:
4666:
4662:
4657:
4653:
4649:
4635:
4631:
4626:
4622:
4618:
4614:
4606:
4603:
4599:
4593:
4588:
4585:
4581:
4575:
4570:
4567:
4563:
4558:
4554:
4549:
4545:
4541:
4536:
4532:
4524:
4523:
4520:
4504:
4497:
4495:
4479:
4474:
4468:
4456:
4446:
4441:
4436:
4431:
4425:
4413:
4398:
4393:
4388:
4382:
4370:
4364:
4354:
4345:
4338:
4335:
4331:
4324:
4321:
4316:
4312:
4304:
4299:
4292:
4276:
4255:
4250:
4247:
4238:
4234:
4228:
4223:
4218:
4212:
4200:
4194:
4190:
4187:
4182:
4178:
4166:
4162:
4158:
4149:
4142:
4137:
4133:
4129:
4124:
4120:
4115:
4103:
4099:
4094:
4089:
4086:
4078:
4074:
4056:
4052:
4037:
4034:
4030:
4022:
4021:
4018:
4016:
4013:
4012:four-gradient
4009:
4008:metric tensor
3989:
3986:
3982:
3971:
3970:contravariant
3964:Wave operator
3947:
3939:
3935:
3929:
3912:
3908:
3901:
3897:
3891:
3887:
3880:
3875:
3858:
3854:
3849:
3845:
3834:
3830:
3825:
3819:
3815:
3809:
3805:
3801:
3796:
3792:
3786:
3781:
3777:
3773:
3768:
3751:
3747:
3740:
3739:
3738:
3716:
3705:
3687:
3676:
3660:
3650:
3626:
3601:
3575:
3549:
3525:
3522:
3519:
3509:
3490:
3472:
3468:
3465:
3461:
3455:
3451:
3447:
3430:
3413:
3411:
3407:
3403:
3393:
3390:
3388:
3384:
3380:
3376:
3358:
3355:
3352:
3328:
3299:
3295:
3291:
3286:
3282:
3278:
3273:
3269:
3263:
3260:
3254:
3247:
3246:
3245:
3244:
3240:
3231:
3224:
3222:
3208:
3205:
3198:
3194:
3188:
3184:
3178:
3170:
3166:
3162:
3157:
3153:
3145:
3141:
3137:
3132:
3128:
3117:
3116:
3113:
3111:
3087:
3074:
3070:
3066:
3062:
3058:
3054:
3050:
3046:
3045:unit 3-vector
3042:
3033:
3026:
3024:
3009:
3006:
3001:
2997:
2989:
2988:
2981:
2974:
2972:
2957:
2951:
2948:
2943:
2935:
2931:
2927:
2922:
2918:
2912:
2909:
2906:
2903:
2900:
2894:
2886:
2882:
2878:
2873:
2869:
2863:
2860:
2857:
2854:
2851:
2844:
2840:
2832:
2828:
2824:
2819:
2815:
2811:
2806:
2802:
2795:
2783:
2782:
2775:
2768:
2766:
2747:
2743:
2737:
2733:
2729:
2724:
2720:
2712:
2708:
2704:
2701:
2698:
2692:
2689:
2682:
2681:
2674:
2667:
2665:
2648:
2644:
2638:
2634:
2630:
2627:
2622:
2619:
2615:
2611:
2606:
2603:
2599:
2591:
2590:
2587:
2585:
2581:
2577:
2573:
2559:
2552:
2550:
2534:
2530:
2526:
2523:
2520:
2515:
2511:
2506:
2500:
2496:
2492:
2487:
2483:
2478:
2474:
2469:
2465:
2461:
2455:
2450:
2446:
2442:
2434:
2430:
2426:
2421:
2417:
2407:
2402:
2398:
2394:
2389:
2385:
2381:
2376:
2370:
2360:
2353:
2352:
2349:
2348:
2324:
2299:
2288:
2272:
2269:
2265:
2262:
2247:
2240:
2238:
2224:
2219:
2215:
2211:
2208:
2199:
2195:
2190:
2186:
2182:
2172:
2171:
2164:
2157:
2155:
2141:
2138:
2135:
2130:
2126:
2120:
2116:
2112:
2107:
2103:
2099:
2089:
2088:
2081:
2074:
2072:
2058:
2052:
2049:
2045:
2040:
2037:
2030:
2029:
2026:
1983:
1980:
1965:
1958:
1956:
1942:
1935:
1931:
1926:
1923:
1920:
1914:
1905:
1897:
1896:
1893:
1892:
1870:
1856:
1849:
1847:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1808:
1807:
1800:
1793:
1791:
1777:
1774:
1771:
1768:
1765:
1762:
1755:
1751:
1747:
1742:
1738:
1732:
1729:
1722:
1721:
1714:
1707:
1705:
1691:
1688:
1685:
1682:
1679:
1676:
1669:
1665:
1661:
1656:
1652:
1646:
1643:
1636:
1635:
1632:
1630:
1627:are standard
1612:
1609:
1606:
1603:
1600:
1585:
1578:
1576:
1558:
1555:
1551:
1548:
1544:
1535:
1532:
1527:
1523:
1519:
1516:
1507:
1503:
1497:
1492:
1488:
1484:
1478:
1475:
1470:
1466:
1461:
1457:
1454:
1449:
1445:
1433:
1429:
1425:
1416:
1409:
1404:
1400:
1396:
1391:
1387:
1382:
1378:
1373:
1369:
1365:
1359:
1354:
1350:
1346:
1333:
1328:
1324:
1320:
1315:
1311:
1306:
1296:
1287:
1280:
1277:
1273:
1269:
1266:
1264:
1254:
1250:
1246:
1241:
1237:
1233:
1230:
1228:
1221:
1217:
1213:
1202:
1201:
1198:
1196:
1192:
1188:
1170:
1161:
1143:
1133:
1127:
1117:
1115:
1111:
1107:
1097:
1095:
1090:
1088:
1085:energy,
1084:
1080:
1079:
1071:
1047:
1043:
1040:
1030:
1026:
1022:
1021:event horizon
1018:
1014:
1010:
1006:
1005:singularities
1001:
999:
993:
988:
986:
982:
977:
973:
969:
958:
955:
953:
950:
925:
921:
920:
916:
914:
913:Schwarzschild
911:
908:
904:
903:
877:
873:
870:
866:
864:
863:
860:
859:
858:
856:
852:
848:
844:
840:
836:
832:
828:
824:
820:
816:
812:
802:
800:
796:
792:
788:
784:
780:
779:event horizon
776:
773:
769:
765:
764:Kerr geometry
761:
749:
744:
742:
737:
735:
730:
729:
727:
726:
720:
710:
707:
702:
696:
695:
694:
693:
686:
685:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
629:
626:
624:
623:Chandrasekhar
621:
619:
616:
614:
611:
609:
606:
604:
601:
599:
596:
594:
591:
589:
586:
584:
581:
579:
576:
574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
539:
538:Schwarzschild
536:
534:
531:
529:
526:
524:
521:
519:
516:
515:
507:
506:
499:
498:HartleâThorne
496:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
464:
461:
459:
456:
454:
451:
449:
446:
444:
441:
439:
436:
434:
431:
429:
426:
424:
421:
419:
416:
413:
409:
408:Schwarzschild
406:
405:
401:
395:
394:
383:
380:
378:
375:
374:
373:
372:
368:
367:
362:
359:
357:
354:
352:
349:
348:
347:
346:
342:
341:
336:
333:
331:
328:
326:
323:
321:
318:
316:
313:
311:
308:
307:
306:
305:
301:
300:
290:
287:
286:
281:
280:
269:
266:
264:
261:
259:
256:
255:
254:
253:
250:
247:
246:
241:
238:
236:
233:
231:
230:Event horizon
228:
226:
223:
221:
218:
216:
213:
211:
208:
206:
203:
201:
198:
196:
193:
192:
191:
190:
180:
179:
172:
169:
167:
164:
162:
159:
157:
154:
153:
145:
144:
139:
136:
131:
128:
126:
123:
121:
118:
117:
115:
113:
110:
109:
108:
107:
89:
86:
82:
77:
73:
68:
65:
61:
54:
49:
46:
42:
32:
28:
27:
24:
21:
20:
13180:
12958:
12874:KaluzaâKlein
12626:Introduction
12552:Twin paradox
12325:Misner space
12314:
12310:Gödel metric
12290:can contain
11959:PKS 1302-102
11833:Gravity well
11801:Compact star
11755:Microquasars
11740:Most massive
11644:Alternatives
11619:
11409:X-ray binary
11328:Neutron star
11265:Supermassive
11242:Hawking star
11183:Supermassive
11068:
11064:
11027:
11023:
11002:
10972:
10968:
10921:
10917:
10892:
10881:
10864:
10854:
10839:
10831:
10816:
10805:
10788:
10767:
10747:
10724:
10705:
10659:
10655:
10649:
10644:Penrose 1968
10640:
10630:
10625:
10612:
10603:
10594:
10543:
10539:
10533:
10489:
10485:
10479:
10454:
10450:
10444:
10433:the original
10404:
10400:
10331:. p. 8.
10319:
10276:
10272:
10263:
10250:
10234:
10191:
10187:
10181:
10130:
10124:
10108:
10103:
10082:
10072:
10067:
10024:
10020:
10014:
9988:
9984:
9978:
9953:
9949:
9943:
9902:
9882:
9875:
9856:
9843:
9798:
9794:
9784:
9751:
9747:
9737:
9692:
9689:Scholarpedia
9688:
9678:
9645:
9641:
9631:
9580:
9576:
9570:
9561:
9551:
9504:
9500:
9490:
9474:
9449:
9445:
9442:Kerr, Roy P.
9436:
9419:
9415:
9271:neutron star
9260:
9252:
9246:
9242:
9233:
9229:
9222:
9218:
9188:
9174:
9170:
9168:
9163:
9157:
9036:
9032:
9023:
9022:
9018:point source
8981:
8906:
8875:
8873:
8856:
8820:
8809:
8788:
8775:
8743:
8654:
8639:
8630:
8619:Please help
8614:verification
8611:
8466:
8454:
8443:
8401:
7561:
7353:
6572:
6565:
6371:
6095:
5981:
5977:
5964:
5949:
5945:
5937:
5931:
5897:
5875:
5871:
5870:(from Greek
5867:
5865:
5858:
5771:
5767:static limit
5766:
5755:
5700:
5689:
5683:
5627:
5607:
5599:
5598:equals 0 or
5595:
5591:
5589:
5504:
5381:
5373:
5367:
5256:
5145:
5127:
5113:
5001:
4990:
4807:
4799:
4797:
4786:
4514:
4498:
3967:
3677:
3616:is equal to
3419:
3399:
3391:
3378:
3374:
3318:
3242:
3238:
3236:
3225:
3109:
3072:
3064:
3060:
3056:
3052:
3040:
3039:Notice that
3038:
3027:
2975:
2769:
2668:
2569:
2553:
2289:
2252:
2241:
2158:
2075:
1970:
1959:
1861:
1850:
1794:
1708:
1590:
1579:
1187:line element
1129:
1103:
1091:
1086:
1076:
1069:
1002:
989:
980:
965:
923:
906:
875:
868:
850:
842:
830:
822:
808:
763:
759:
757:
683:
643:Raychaudhuri
437:
112:Introduction
13222:Black holes
12963:KerrâNewman
12934:Spherical:
12803:Other tests
12746:Singularity
12678:Formulation
12640:Fundamental
12494:Formulation
12475:Proper time
12436:Fundamental
12315:Kerr metric
12252:Determinism
12193:Causal loop
12099:Time travel
11949:Centaurus A
11903:Planet Nine
11806:Exotic star
11735:Black holes
11681:Planck star
11630:KerrâNewman
11345:White dwarf
11295:Radio-Quiet
11253:Microquasar
11126:Black holes
10997:Wald, R. M.
10889:Penrose, R.
10370:0706.0622v3
10345:Paul Davies
10167:Gravitation
9934:0706.0622v3
9290:, an exact
9247:Kerr vacuum
8857:nonrotating
6086:ergodically
5962:Petrov type
5954:Weyl tensor
5938:axisymmetry
5616:proper time
3972:components
3049:null vector
1191:proper time
957:KerrâNewman
905:Uncharged (
781:. The Kerr
760:Kerr metric
658:van Stockum
588:Oppenheimer
443:KerrâNewman
235:Singularity
13211:Categories
13115:Zel'dovich
13023:Scientists
13002:Alcubierre
12809:of Mercury
12807:precession
12736:Black hole
12619:Background
12611:relativity
12580:World line
12575:Light cone
12400:Background
12392:relativity
12382:Relativity
12282:Spacetimes
12257:Eternalism
12224:Multiverse
11999:Q0906+6930
11989:Hercules A
11919:Cygnus X-1
11888:White hole
11863:Quasi-star
11816:Preon star
11811:Quark star
11796:Big Bounce
11656:Black star
11614:Derivation
11462:Ergosphere
11418:Properties
11399:Quasi-star
11389:Quark star
11300:Radio-Loud
11188:Primordial
11178:Kugelblitz
10965:Carter, B.
10553:2008.04384
9590:1602.03837
9429:References
9033:even order
8791:stationary
8440:Symmetries
8402:If we set
7382:such that
6568:integrable
6064:variable.
5904:ergosphere
5868:ergosphere
5618:along its
5604:ergosphere
4804:colatitude
1017:ergosphere
985:ergosphere
874:Rotating (
799:non-linear
775:black hole
511:Scientists
343:Formalisms
291:Formalisms
240:Black hole
166:World line
13085:Robertson
13070:Friedmann
13065:Eddington
13055:de Sitter
12889:Solutions
12767:detectors
12762:astronomy
12729:Phenomena
12664:Geodesics
12567:Spacetime
12510:Phenomena
12267:Free will
12163:Time loop
12024:AT2018hyz
11671:Gravastar
11661:Dark star
11494:Microlens
11367:Hypernova
11362:Micronova
11357:Supernova
11311:Formation
10586:221095833
10578:0218-2718
10524:118417747
10499:1408.6316
10429:123261579
10311:119237335
10286:1106.1198
10258:, page 42
10245:, page 11
10204:CiteSeerX
10094:0706.0622
9776:0031-9007
9729:1941-6016
9702:1410.6626
9670:0022-2488
9623:124959784
9511:: 14247.
9408:Footnotes
9397:Spin-flip
9128:−
9037:odd order
8880:isometric
8867:, in the
8665:Δ
8477:Δ
8385:˙
8382:ϕ
8375:θ
8371:
8355:−
8349:˙
8303:θ
8299:
8280:−
8276:θ
8272:
8244:θ
8156:Σ
8135:−
8130:Σ
8099:−
8090:¨
8056:Σ
8027:−
8010:−
8005:Σ
7959:−
7948:˙
7918:−
7912:˙
7894:˙
7869:Σ
7842:¨
7824:¨
7791:Σ
7785:Δ
7780:θ
7777:
7764:Δ
7751:−
7699:τ
7648:Ω
7576:Ω
7545:μ
7537:τ
7531:λ
7507:τ
7483:λ
7453:μ
7427:α
7413:λ
7403:α
7364:λ
7312:−
7277:μ
7268:Δ
7265:−
7197:−
7117:θ
7114:
7065:−
7056:μ
7032:θ
7029:
7016:−
7004:θ
6998:Θ
6957:Δ
6909:−
6906:θ
6903:
6876:−
6863:λ
6846:Σ
6825:Δ
6806:θ
6803:
6777:−
6763:−
6750:λ
6742:ϕ
6733:Σ
6721:θ
6715:Θ
6710:±
6697:λ
6689:θ
6680:Σ
6657:±
6644:λ
6627:Σ
6531:θ
6528:
6486:−
6477:μ
6453:θ
6450:
6427:θ
6356:ϕ
6305:−
6216:α
6184:β
6174:β
6171:α
6161:α
6144:μ
6140:−
6118:μ
6090:toruslike
6050:θ
6026:ϕ
5844:θ
5841:
5818:−
5789:−
5728:ϕ
5667:Ω
5624:spacetime
5620:worldline
5573:θ
5570:
5547:−
5539:±
5528:±
5482:θ
5479:
5453:−
5433:±
5412:±
5341:−
5333:±
5322:±
5221:−
5201:±
5180:±
4968:θ
4965:
4893:Σ
4859:ϕ
4856:ϕ
4846:ϕ
4833:−
4827:Ω
4746:ϕ
4743:ϕ
4733:ϕ
4717:ϕ
4703:ϕ
4700:ϕ
4683:θ
4674:θ
4671:θ
4607:ϕ
4604:ϕ
4589:ϕ
4576:−
4546:τ
4469:θ
4466:∂
4462:∂
4450:Σ
4423:∂
4419:∂
4407:Σ
4404:Δ
4383:ϕ
4380:∂
4376:∂
4359:Σ
4339:−
4325:θ
4322:
4309:Δ
4289:∂
4285:∂
4277:ϕ
4274:∂
4270:∂
4262:Δ
4259:Σ
4229:−
4210:∂
4206:∂
4191:θ
4188:
4173:Σ
4109:Δ
4090:−
4079:ν
4071:∂
4067:∂
4057:μ
4049:∂
4045:∂
4038:ν
4035:μ
3842:⟹
3802:−
3469:−
3258:→
3091:→
2907:−
2649:ν
2639:μ
2623:ν
2620:μ
2616:η
2607:ν
2604:μ
2586:in 1965.
2531:ϕ
2524:θ
2521:
2466:θ
2459:Σ
2413:Σ
2382:−
2374:→
2366:⟶
2273:ϕ
2196:−
2180:Δ
2139:θ
2136:
2097:Σ
2011:Δ
1987:Σ
1831:θ
1828:
1778:ϕ
1775:
1769:θ
1766:
1692:ϕ
1689:
1683:θ
1680:
1613:ϕ
1607:θ
1559:ϕ
1540:Σ
1536:θ
1533:
1498:−
1489:ϕ
1479:θ
1476:
1458:θ
1455:
1440:Σ
1370:θ
1363:Σ
1342:Δ
1339:Σ
1301:Σ
1281:−
1270:−
1251:τ
1234:−
1132:spacetime
922:Charged (
819:spacetime
768:spacetime
603:Robertson
568:Friedmann
563:Eddington
553:Nordström
543:de Sitter
400:Solutions
325:Geodesics
320:Friedmann
302:Equations
288:Equations
249:Spacetime
184:Phenomena
90:ν
87:μ
78:κ
69:ν
66:μ
58:Λ
50:ν
47:μ
13198:Category
13075:LemaĂźtre
13040:Einstein
13030:Poincaré
12990:Others:
12974:TaubâNUT
12940:interior
12862:theories
12860:Advanced
12827:redshift
12642:concepts
12460:Rapidity
12438:concepts
12262:Fatalism
12047:Category
11934:A0620-00
11893:Wormhole
11791:Big Bang
11691:Fuzzball
11574:ER = EPR
11440:Theorems
11238:Electron
11233:Extremal
11163:Rotating
10999:(1984).
10956:16858122
10815:(1992).
10684:10054468
10602:(1983).
10392:(1968).
10226:53513572
10170:Archived
10117:page 325
10113:page 485
9855:(1975).
9835:32051884
9615:26918975
9543:34244573
9420:Warning:
9341:See also
9183:momentum
8876:interior
7521:through
5950:outgoing
5948:and one
5874:meaning
4802:and the
4015:operator
835:Roy Kerr
831:rotating
805:Overview
719:Category
583:LemaĂźtre
548:Reissner
533:Poincaré
518:Einstein
463:TaubâNUT
428:Wormhole
412:interior
125:Timeline
13140:Hawking
13135:Penrose
13120:Novikov
13100:Wheeler
13045:Hilbert
13035:Lorentz
12992:pp-wave
12813:lensing
12609:General
12390:Special
12168:in film
12057:Commons
12019:P172+18
11974:TON 618
11912:Notable
11764:Related
11750:Quasars
11745:Nearest
11705:Analogs
11635:Hayward
11603:Metrics
11248:Stellar
11173:Virtual
11168:Charged
11137:Outline
11073:Bibcode
11032:Bibcode
10977:Bibcode
10936:Bibcode
10664:Bibcode
10558:Bibcode
10504:Bibcode
10459:Bibcode
10457:: 347.
10409:Bibcode
10291:Bibcode
10273:Physics
10196:Bibcode
10059:6041750
10039:Bibcode
9993:Bibcode
9958:Bibcode
9826:7002888
9803:Bibcode
9795:Heliyon
9756:Bibcode
9707:Bibcode
9650:Bibcode
9595:Bibcode
9534:8270963
9513:Bibcode
9454:Bibcode
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8882:to the
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1889:is the
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823:charged
789:of the
638:Hawking
633:Penrose
608:Bardeen
598:Wheeler
528:Hilbert
523:Lorentz
483:pp-wave
120:History
13181:others
13170:Thorne
13160:Misner
13145:Taylor
13130:Geroch
13125:Ehlers
13095:Zwicky
12913:Kasner
11994:3C 273
11984:NeVe 1
11964:OJ 287
11686:Q star
11552:Issues
11285:Blazar
11275:Quasar
11052:361088
11050:
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9230:radial
8861:static
8542:(i.e.
8432:, the
8177:where
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7354:where
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6201:where
5967:. The
5922:. The
5742:term.
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3071:, and
2584:Schild
1862:where
1482:
1100:Metric
785:is an
783:metric
717:
684:others
678:Thorne
668:Newman
648:Taylor
628:Ehlers
613:Walker
578:Zwicky
453:Kasner
13175:Weiss
13155:Bondi
13150:Hulse
13080:Milne
12984:discs
12928:Milne
12923:Gödel
12780:Virgo
11727:Lists
11228:Micro
11198:Rogue
11146:Types
11048:S2CID
10952:S2CID
10926:arXiv
10902:arXiv
10582:S2CID
10548:arXiv
10520:S2CID
10494:arXiv
10436:(PDF)
10425:S2CID
10397:(PDF)
10365:arXiv
10329:(PDF)
10307:S2CID
10281:arXiv
10222:S2CID
10089:arXiv
10055:S2CID
10029:arXiv
9929:arXiv
9697:arXiv
9619:S2CID
9585:arXiv
9507:(1).
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8907:Each
6987:with
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653:Hulse
593:Gödel
573:Milne
468:Milne
433:Gödel
130:Tests
13110:Kerr
13060:Weyl
12959:Kerr
12819:and
12773:and
12771:LIGO
12288:that
11696:Geon
11620:Kerr
11221:Size
11009:ISBN
10871:ISBN
10844:ISBN
10821:ISBN
10795:ISBN
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10753:ISBN
10729:ISBN
10710:ISBN
10680:PMID
10574:ISSN
10139:ISBN
9888:ISBN
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9831:PMID
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9725:ISSN
9666:ISSN
9611:PMID
9539:PMID
9480:ISBN
9219:zero
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8874:The
8863:and
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6330:and
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6096:The
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3055:and
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2001:and
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1011:and
1009:mass
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849:and
758:The
663:Taub
618:Kerr
558:Weyl
438:Kerr
356:BSSN
13165:Yau
12790:GEO
12284:in
11290:OVV
11280:LQG
11081:doi
11040:doi
10985:doi
10944:doi
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10566:doi
10512:doi
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10417:doi
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10299:doi
10214:doi
10150:PDF
10047:doi
10001:doi
9966:doi
9821:PMC
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6894:sin
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5956:is
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5470:cos
5387:= 0
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4956:sin
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8052:r
8045:2
8041:W
8035:2
8031:a
8024:C
8017:)
8013:1
7999:2
7995:r
7991:4
7984:(
7980:)
7977:y
7974:i
7971:+
7968:x
7965:(
7962:M
7955:]
7945:r
7937:r
7933:y
7930:i
7927:+
7924:x
7909:y
7903:i
7900:+
7891:x
7884:[
7880:W
7873:2
7865:r
7860:a
7857:M
7854:i
7851:4
7848:=
7839:y
7833:i
7830:+
7821:x
7798:.
7772:2
7759:2
7755:a
7746:2
7741:)
7735:2
7731:r
7727:+
7722:2
7718:a
7713:(
7704:=
7696:t
7672:r
7624:v
7600:t
7541:/
7534:=
7459:0
7423:p
7419:=
7410:d
7399:x
7395:d
7338:)
7334:Q
7331:+
7326:2
7322:)
7318:E
7315:a
7307:z
7303:L
7299:(
7296:+
7291:2
7287:r
7281:2
7272:(
7260:2
7256:)
7252:r
7249:(
7246:P
7243:=
7240:)
7237:r
7234:(
7231:R
7208:z
7204:L
7200:a
7193:)
7187:2
7183:a
7179:+
7174:2
7170:r
7165:(
7161:E
7158:=
7155:)
7152:r
7149:(
7146:P
7124:)
7109:2
7098:2
7093:z
7089:L
7083:+
7079:)
7073:2
7069:E
7060:2
7051:(
7045:2
7041:a
7036:(
7024:2
7013:Q
7010:=
7007:)
7001:(
6971:)
6968:r
6965:(
6962:P
6951:2
6947:a
6943:+
6938:2
6934:r
6927:+
6923:)
6917:z
6913:L
6898:2
6890:E
6887:a
6883:(
6879:a
6873:=
6860:d
6855:t
6852:d
6839:)
6836:r
6833:(
6830:P
6822:a
6817:+
6813:)
6798:2
6787:z
6783:L
6774:E
6771:a
6767:(
6760:=
6747:d
6739:d
6724:)
6718:(
6707:=
6694:d
6686:d
6671:)
6668:r
6665:(
6662:R
6654:=
6641:d
6636:r
6633:d
6601:1
6598:=
6595:c
6592:=
6589:M
6586:=
6583:G
6552:.
6548:)
6542:2
6537:)
6519:z
6515:L
6509:(
6504:+
6500:)
6494:2
6490:E
6481:2
6472:(
6466:2
6462:a
6457:(
6445:2
6437:+
6432:2
6423:p
6419:=
6416:Q
6390:Q
6352:p
6348:=
6343:z
6339:L
6318:,
6313:t
6309:p
6302:=
6299:E
6275:z
6271:L
6246:E
6212:p
6189:,
6180:p
6167:g
6157:p
6153:=
6148:2
6002:0
5999:=
5996:a
5965:D
5836:2
5826:2
5822:J
5813:2
5809:M
5805:=
5800:2
5796:)
5792:M
5786:r
5783:(
5725:d
5719:t
5716:d
5704:E
5701:r
5693:H
5690:r
5643:M
5628:g
5608:g
5600:Ï
5596:Ξ
5592:Ξ
5565:2
5555:2
5551:a
5544:1
5536:1
5533:=
5522:E
5517:r
5488:2
5474:2
5464:2
5460:a
5456:4
5448:2
5443:s
5439:r
5428:s
5424:r
5417:=
5406:E
5401:r
5382:g
5374:g
5349:2
5345:a
5338:1
5330:1
5327:=
5316:H
5311:r
5285:1
5282:=
5279:c
5276:=
5273:M
5270:=
5267:G
5240:2
5232:2
5228:a
5224:4
5216:2
5211:s
5207:r
5196:s
5192:r
5185:=
5174:H
5169:r
5146:g
5142:/
5139:1
5128:g
5116:1
5096:M
5072:0
5048:a
4995:)
4991:(
4974:.
4960:2
4950:2
4946:a
4942:r
4937:s
4933:r
4929:+
4925:)
4919:2
4915:a
4911:+
4906:2
4902:r
4897:(
4888:c
4885:a
4882:r
4877:s
4873:r
4866:=
4852:g
4843:t
4839:g
4830:=
4808:Ξ
4800:r
4791:)
4787:(
4770:.
4765:2
4760:)
4756:t
4753:d
4739:g
4730:t
4726:g
4720:+
4714:d
4710:(
4696:g
4692:+
4687:2
4679:d
4667:g
4663:+
4658:2
4654:r
4650:d
4644:r
4641:r
4636:g
4632:+
4627:2
4623:t
4619:d
4615:)
4600:g
4594:2
4586:t
4582:g
4571:t
4568:t
4564:g
4559:(
4555:=
4550:2
4542:d
4537:2
4533:c
4517:1
4503:)
4499:(
4480:2
4475:)
4457:(
4447:1
4442:+
4437:2
4432:)
4426:r
4414:(
4399:+
4394:2
4389:)
4371:(
4365:)
4355:r
4350:s
4346:r
4336:1
4332:(
4317:2
4305:1
4300:+
4293:t
4256:c
4251:a
4248:r
4243:s
4239:r
4235:2
4224:2
4219:)
4213:t
4201:(
4195:)
4183:2
4167:2
4163:a
4159:r
4154:s
4150:r
4143:+
4138:2
4134:a
4130:+
4125:2
4121:r
4116:(
4104:2
4100:c
4095:1
4087:=
4075:x
4053:x
4031:g
3990:k
3987:i
3983:g
3948:.
3940:2
3936:G
3930:2
3924:r
3921:r
3918:i
3913:M
3909:4
3902:2
3898:c
3892:2
3888:J
3881:+
3876:2
3870:r
3867:r
3864:i
3859:M
3855:=
3850:2
3846:M
3835:2
3831:G
3826:/
3820:2
3816:c
3810:2
3806:J
3797:4
3793:M
3787:+
3782:2
3778:M
3774:=
3769:2
3763:r
3760:r
3757:i
3752:M
3748:2
3717:M
3688:M
3661:M
3627:M
3602:M
3576:2
3550:M
3526:M
3523:=
3520:a
3491:)
3484:r
3481:r
3478:i
3473:M
3466:M
3462:(
3456:2
3452:c
3448:=
3442:t
3439:o
3436:r
3431:E
3379:a
3375:R
3359:1
3356:=
3353:c
3329:a
3300:2
3296:z
3292:+
3287:2
3283:y
3279:+
3274:2
3270:x
3264:=
3261:R
3255:r
3243:R
3239:r
3230:)
3226:(
3209:1
3206:=
3199:2
3195:r
3189:2
3185:z
3179:+
3171:2
3167:a
3163:+
3158:2
3154:r
3146:2
3142:y
3138:+
3133:2
3129:x
3110:r
3088:a
3073:a
3065:η
3061:M
3057:η
3053:g
3041:k
3032:)
3028:(
3007:=
3002:0
2998:k
2980:)
2976:(
2958:)
2952:r
2949:z
2944:,
2936:2
2932:a
2928:+
2923:2
2919:r
2913:x
2910:a
2904:y
2901:r
2895:,
2887:2
2883:a
2879:+
2874:2
2870:r
2864:y
2861:a
2858:+
2855:x
2852:r
2845:(
2841:=
2838:)
2833:z
2829:k
2825:,
2820:y
2816:k
2812:,
2807:x
2803:k
2799:(
2796:=
2792:k
2774:)
2770:(
2748:2
2744:z
2738:2
2734:a
2730:+
2725:4
2721:r
2713:3
2709:r
2705:M
2702:G
2699:2
2693:=
2690:f
2673:)
2669:(
2645:k
2635:k
2631:f
2628:+
2612:=
2600:g
2558:)
2556:9
2554:(
2535:2
2527:d
2516:2
2507:)
2501:2
2497:a
2493:+
2488:2
2484:r
2479:(
2475:+
2470:2
2462:d
2456:+
2451:2
2447:r
2443:d
2435:2
2431:a
2427:+
2422:2
2418:r
2408:+
2403:2
2399:t
2395:d
2390:2
2386:c
2377:0
2371:M
2361:g
2329:s
2325:r
2300:M
2270:d
2266:t
2263:d
2246:)
2244:8
2242:(
2225:.
2220:2
2216:a
2212:+
2209:r
2204:s
2200:r
2191:2
2187:r
2183:=
2163:)
2161:7
2159:(
2142:,
2131:2
2121:2
2117:a
2113:+
2108:2
2104:r
2100:=
2080:)
2078:6
2076:(
2059:,
2053:c
2050:M
2046:J
2041:=
2038:a
1984:,
1981:a
1964:)
1962:5
1960:(
1943:,
1936:2
1932:c
1927:M
1924:G
1921:2
1915:=
1910:s
1906:r
1875:s
1871:r
1855:)
1853:4
1851:(
1834:,
1822:r
1819:=
1816:z
1799:)
1797:3
1795:(
1756:2
1752:a
1748:+
1743:2
1739:r
1733:=
1730:y
1713:)
1711:2
1709:(
1670:2
1666:a
1662:+
1657:2
1653:r
1647:=
1644:x
1610:,
1604:,
1601:r
1584:)
1582:1
1580:(
1556:d
1552:t
1549:d
1545:c
1528:2
1520:a
1517:r
1512:s
1508:r
1504:2
1493:2
1485:d
1471:2
1462:)
1450:2
1434:2
1430:a
1426:r
1421:s
1417:r
1410:+
1405:2
1401:a
1397:+
1392:2
1388:r
1383:(
1379:+
1374:2
1366:d
1360:+
1355:2
1351:r
1347:d
1334:+
1329:2
1325:t
1321:d
1316:2
1312:c
1307:)
1297:r
1292:s
1288:r
1278:1
1274:(
1267:=
1255:2
1247:d
1242:2
1238:c
1231:=
1222:2
1218:s
1214:d
1171:J
1144:M
1073:s
1070:r
1052:s
1048:r
1044:=
1041:r
997:e
935:R
924:Q
907:Q
887:R
876:J
869:J
851:J
843:Q
747:e
740:t
733:v
414:)
410:(
83:T
74:=
62:g
55:+
43:G
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