4503:
6027:, 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
9374:
712:
4036:
5017:, 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
4498:{\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}}
6092:
11224:
6996:
42:
1584:
6629:
13205:
12054:
725:
8614:
1216:
5038:
12064:
9360:
6991:{\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}}}
5889:). 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
9297:, 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
8784:, 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.
4791:
10359:
1579:{\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}}}
8780:
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
8470:. 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.
8463:(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.
5021:, 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
2558:
1092:) 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
8086:
3969:
5989:
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
4538:
8779:
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
4995:
989:
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
6573:
8787:
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
7146:
2980:
5874:, 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.
8934:
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
7819:
5046:
5511:
9037:
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
7824:
2367:
8339:
9196:
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.
5775:
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
5263:
8186:
5624:
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
9164:
4786:{\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}.}
1034:; 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
6422:
3754:
9292:
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
8905:
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
4833:
7360:
6084:
3230:
8596:), 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
5596:
7700:
113:
9042:. 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
6210:
2673:
3513:
8409:
7450:
7004:
2797:
1799:
1713:
5374:
2774:
7231:
5865:
2163:
6634:
1221:
3325:
2246:
1964:
8894:
8239:
5406:
8090:
8549:
7566:
2080:
1115:
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
1855:
6379:
6095:
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
1634:
3744:
3654:
2553:{\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}}
1075:
9298:
8592:
2352:
1898:
957:
909:
10971:
Gives the relativistic multipole moments for the Ernst vacuums (plus the electromagnetic and gravitational relativistic multipole moments for the charged generalization).
10460:
Bardeen, James M.; Press, William H.; Teukolsky, Saul A. (1972). "Rotating Black Holes: Locally
Nonrotating Frames, Energy Extraction, and Scalar Synchrotron Radiation".
9489:
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,
8988:
6239:
3599:
11949:
8826:
5749:
3115:
3032:
2294:
9344:
8692:
8504:
7480:
6339:
5025:
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
2008:
7504:
7385:
5174:
7669:
7597:
6622:
6071:
5688:
5306:
2032:
9170:
4013:
12908:
7528:
6298:
6047:
483:
6139:
5783:
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
9023:
8858:
8439:
6023:
3547:
3380:
9220:
9056:
8775:
8748:
8720:
8232:
8208:
7693:
7645:
7621:
6411:
6267:
5664:
5605:
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
5117:
5093:
5069:
3709:
3682:
3623:
3571:
3350:
2321:
1192:
1165:
8898:
8081:{\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}}}}
3964:{\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}}}.}
9276:
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
3412:
10404:
4990:{\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 }}.}
2298:. 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.
10123:
756:
5710:). 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
10180:
8728:. 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
6146:
12219:
10927:
Sotiriou, Thomas P.; Apostolatos, Theocharis A. (2004). "Corrections and
Comments on the Multipole Moments of Axisymmetric Electrovacuum Spacetimes".
7237:
9169:
In particular, we see that the
Schwarzschild vacuum has nonzero second order Weyl moment, corresponding to the fact that the "Weyl monopole" is the
8343:
6247:
of the particle. Furthermore, there are two constants of motion given by the time translation and rotation symmetries of Kerr spacetime, the energy
9180:
In weak field general relativity, it is convenient to treat isolated sources using another type of multipole, which generalize the Weyl moments to
8724:
corresponding to these roots. The larger of these roots determines the location of the event horizon, and the smaller determines the location of a
6083:
1086:
into two disconnected patches; using a different coordinate transformation one can then relate the extended external patch to the inner patch (see
135:
9200:
Perez and
Moreschi have given an alternative notion of "monopole solutions" by expanding the standard NP tetrad of the Ernst vacuums in powers of
6577:
Since there are four (independent) constants of motion for degrees of freedom, the equations of motion for a test particle in Kerr spacetime are
3131:
13227:
8815:
6568:{\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).}
5522:
5897:. Rotating black holes in astrophysics are a potential source of large amounts of energy and are used to explain energetic phenomena, such as
5390:
of the metric changes sign from positive to negative, in Kerr metric that happens at a different distance. Again solving a quadratic equation
11305:
48:
3403:
In the KerrâSchild form of the Kerr metric, the determinant of the metric tensor is everywhere equal to negative one, even near the source.
11310:
9173:
solution, not the
Schwarzschild vacuum solution, which arises from the Newtonian potential of a certain finite length uniform density thin
5045:
340:
12955:
10127:
13190:
12029:
11686:
9289:
2605:
693:
7141:{\displaystyle \Theta (\theta )=Q-\cos ^{2}\theta \left(a^{2}\left(\mu ^{2}-E^{2}\right)+{\frac {L_{z}^{2}}{\sin ^{2}\theta }}\right)}
2975:{\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)}
12979:
12725:
10084:
9264:
In this sense, the Kerr vacuums are the simplest stationary axisymmetric asymptotically flat vacuum solutions in general relativity.
3436:
503:
345:
10374:
The LIGO Scientific
Collaboration; the Virgo Collaboration (2007). "The Kerr spacetime: A brief introduction". p. 13, below eq. 52.
5013:
Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is called
11624:
11450:
11350:
10549:
Riazuelo, Alain (December 2020). "Seeing Relativity -- III. Journeying within the Kerr metric toward the negative gravity region".
7398:
3980:
1736:
1650:
1104:
749:
458:
9284:, or the Earth. This works out very nicely for the non-rotating case, where the Schwarzschild vacuum exterior can be matched to a
7814:{\displaystyle {\frac {t}{\tau }}={\sqrt {\frac {\left(a^{2}+r^{2}\right)^{2}-a^{2}\Delta \sin ^{2}\theta }{\Delta \ \Sigma }}}.}
4809:
This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radius
12761:
12715:
10495:
Frolov, Andrei V.; Frolov, Valeri P. (2014). "Rigidly rotating zero-angular-momentum observer surfaces in the Kerr spacetime".
10197:
Bhat, Manjiri; Dhurandhar, Sanjeev; Dadhich, Naresh (1985). "Energetics of the KerrâNewman black hole by the penrose process".
5877:
The region outside the event horizon but inside the surface where the rotational velocity is the speed of light, is called the
5316:
985:), a distinctive prediction of general relativity. The first measurement of this frame dragging effect was done in 2011 by the
371:
205:
3658:. 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
12899:
12100:
11127:
11023:
10885:
10858:
10835:
10809:
10786:
10767:
10743:
10724:
9902:
9875:
3393:
797:
410:
10030:
Balasin, Herbert; Nachbagauer, Herbert (1994). "Distributional energyâmomentum tensor of the KerrâNewman spacetime family".
2696:
12383:
12166:
7152:
5789:
5041:
Location of the horizons, ergospheres and the ring singularity of the Kerr spacetime in Cartesian KerrâSchild coordinates.
2103:
852:, was discovered shortly thereafter in 1965. These four related solutions may be summarized by the following table, where
9870:. Course of Theoretical Physics. Vol. 2 (Revised 4th English ed.). New York: Pergamon Press. pp. 321â330.
1088:
742:
3261:
2186:
1911:
12636:
12178:
12133:
11666:
10153:
9494:
8700:, and, with an appropriate choice of new coordinates, the Kerr solution can be smoothly extended through the values of
122:
8752:
coordinate now behaves like a time coordinate, so it must decrease until the curve passes through the Cauchy horizon.
13007:
12641:
11755:
11033:
Kerr, R. P.; Schild, A. (2009). "Republication of: A new class of vacuum solutions of the Einstein field equations".
9938:
Visser, Matt (2007). "The Kerr spacetime: A brief introduction". p. 15, Eq. 60â61, p. 24, p. 35.
9433:
Do not confuse the relativistic multipole moments computed by Hansen with the Weyl multipole moments discussed below.
8653:
6584:
Using these constants of motion, the trajectory equations for a test particle can be written (using natural units of
6344:
148:
12143:
12123:
11750:
10911:
Perez, Alejandro; Moreschi, Osvaldo M. (2000). "Characterizing exact solutions from asymptotic physical concepts".
10266:
12961:
11635:
9397:
9373:
962:
837:
711:
428:
41:
12674:
12267:
11671:
11467:
10443:
5993:
A surface on which light can orbit a black hole is called a photon sphere. The Kerr solution has infinitely many
5871:
5506:{\displaystyle r_{\rm {E}}^{\pm }={\frac {r_{\text{s}}\pm {\sqrt {r_{\text{s}}^{2}-4a^{2}\cos ^{2}\theta }}}{2}}}
3416:
1205:
1136:
335:
10253:
9348:
Kerr geometry. Physical thin-disk solutions obtained by identifying parts of the Kerr spacetime are also known.
3521:, the remaining mass cannot shrink below the irreducible mass. Therefore, if a black hole rotates with the spin
12777:
12138:
11893:
11574:
11213:
8635:
6305:
11072:"Editorial note to: R. P. Kerr and A. Schild, A new class of vacuum solutions of the Einstein field equations"
990:
bodies. In the case of a rotating black hole, at close enough distances, all objects â even light â
13208:
12950:
12855:
12785:
12239:
11661:
11536:
2582:
2357:
1639:
982:
422:
130:
17:
12918:
8821:
The Kerr solution is also related to various non-vacuum solutions which model black holes. For example, the
468:
12772:
12249:
12067:
11579:
11482:
11270:
10177:
9994:
Debney, G. C.; Kerr, R. P.; Schild, A. (1969). "Solutions of the Einstein and Einstein-Maxwell Equations".
8789:
8631:
6384:
5909:
The Kerr geometry exhibits many noteworthy features: the maximal analytic extension includes a sequence of
1120:
1002:
488:
11589:
12813:
12547:
12532:
11883:
11848:
11838:
8814:. The family of all stationary axially symmetric vacuum solutions to the Einstein field equation are the
8513:
7537:
5986:
of the Kerr spacetime can be simply characterized as a line with circles attached at each integer point.
5692:. Thus, no particle can move in the direction opposite to central mass's rotation within the ergosphere.
3659:
2586:
729:
220:
140:
5049:
Comparison of the shadow (black) and the important surfaces (white) of a black hole. The spin parameter
2044:
13100:
12542:
12495:
12229:
12198:
11569:
11521:
11504:
11203:
10823:
10610:
10145:
9918:
8902:
8466:
In physics, symmetries are typically associated with conserved constants of motion, in accordance with
5979:
1116:
633:
181:
10733:
9586:
Abbot, B.P. (11 February 2016). "Observation of Gravitational Waves from a Binary Black Hole Merger".
8804:
1822:
13130:
12756:
12437:
12093:
12024:
11526:
11440:
11188:
11120:
8907:
8879:
5930:
1607:
1006:
433:
245:
10734:
Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius; Herlt, Eduard (2003).
10219:
9280:
to model the gravitational field around a rotating massive object other than a black hole such as a
6087:
Animation of a test-particle's orbit around a spinning black hole. Left: top view, right: side view.
3722:
3632:
1047:
1001:
The light from distant sources can travel around the event horizon several times (if close enough);
832:
around an uncharged, spherically symmetric, and non-rotating body. The corresponding solution for a
13155:
12705:
12465:
12376:
11701:
9402:
8334:{\displaystyle C=p_{\theta }^{2}+\left(aE\sin {\theta }-{\frac {L_{z}}{\sin {\theta }}}\right)^{2}}
3354:
approaches zero. In this form of solution, units are selected so that the speed of light is unity (
1005:. To a distant viewer, the apparent perpendicular distance between images decreases at a factor of
801:
668:
658:
508:
325:
8558:
6091:
2330:
1876:
1103:
The LIGO experiment that first detected gravitational waves, announced in 2016, also provided the
940:
892:
13237:
12884:
12806:
12426:
12417:
11878:
11414:
11377:
11280:
11275:
11193:
9959:
Boyer, Robert H.; Lindquist, Richard W. (1967). "Maximal Analytic Extension of the Kerr Metric".
9455:(1963). "Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics".
8941:
8811:
8624:
8444:
6217:
5258:{\displaystyle r_{\rm {H}}^{\pm }={\frac {r_{\text{s}}\pm {\sqrt {r_{\text{s}}^{2}-4a^{2}}}}{2}}}
3580:
653:
387:
11102:"... This note is meant to be a guide for those readers who wish to verify all the details ..."
8181:{\displaystyle {\ddot {z}}=-Mz\left({\frac {4r^{2}}{\Sigma }}-1\right){\frac {C}{r\Sigma ^{2}}}}
5997:, lying between an inner one and an outer one. In the nonrotating, Schwarzschild solution, with
5722:
3091:
3079:
3003:
2269:
1011:(about 500). However, fast spinning black holes have less distance between multiplicity images.
13110:
12973:
12879:
12837:
12511:
12475:
12161:
12153:
11908:
11796:
11781:
11640:
11333:
11147:
10214:
9392:
9310:
8822:
8697:
8671:
8483:
7459:
5707:
1987:
1015:
967:
849:
623:
608:
453:
215:
7489:
7370:
13115:
12720:
12659:
12302:
12128:
11786:
11599:
11343:
11198:
9892:
8875:
7654:
7582:
6589:
6056:
5972:
5968:
5938:
5926:
5673:
5273:
5018:
4025:
3420:
2017:
598:
166:
10628:
9568:
9050:
Weyl moments vanish. For the Kerr vacuum solutions, the first few Weyl moments are given by
3988:
3086:
is a constant rotational parameter of the spinning object. It is understood that the vector
840:, was discovered soon afterwards (1916â1918). However, the exact solution for an uncharged,
13232:
13020:
12946:
12790:
12392:
12086:
12057:
11765:
11737:
11620:
11548:
11516:
11360:
11168:
11113:
11083:
11042:
10987:
10946:
10674:
10568:
10514:
10469:
10419:
10301:
10279:
10206:
10049:
10003:
9968:
9813:
9766:
9717:
9660:
9605:
9523:
9464:
9387:
8994:
8863:
8801:
8467:
7513:
6276:
6271:, and the component of the orbital angular momentum parallel to the spin of the black hole
6032:
5772:
5696:
5380:
5134:
3662:
this energy also has a mass-equivalent, which adds to the total massâenergy of the system,
3397:
1901:
1039:
923:
821:
418:
10336:
9159:{\displaystyle a_{0}=M,\qquad a_{1}=0,\qquad a_{2}=M\left({\frac {M^{2}}{3}}-a^{2}\right)}
6124:
8:
13095:
13090:
13080:
12912:
12861:
12679:
12664:
12490:
12421:
12369:
12350:
11888:
11863:
11833:
11791:
11745:
11382:
11248:
11243:
11173:
9412:
9285:
9273:
9245:
9046:
relativistic moments. In the case of solutions symmetric across the equatorial plane the
9002:
8919:
8837:
8418:
7577:-effect, a zero-angular-momentum observer (ZAMO) is corotating with the angular velocity
6108:
6002:
5910:
5026:
3526:
3359:
613:
583:
578:
330:
13085:
11087:
11046:
10991:
10950:
10678:
10665:
BiÄĂĄk, JĆĂ; Ledvinka, TomĂĄĆĄ (1993). "Relativistic disks as sources of the Kerr metric".
10642:
10572:
10518:
10473:
10423:
10373:
10305:
10210:
10098:
Visser, Matt (14 January 2008). "The Kerr spacetime: A brief introduction". p. 12.
10053:
10007:
9972:
9817:
9770:
9721:
9664:
9609:
9527:
9468:
593:
563:
13160:
13060:
12984:
12842:
12823:
12817:
12768:
12710:
12619:
12537:
12455:
12432:
12400:
12296:
11722:
11594:
11499:
11290:
11258:
11223:
11183:
11178:
11058:
11012:
10962:
10936:
10912:
10874:
10798:
10592:
10558:
10530:
10504:
10435:
10375:
10317:
10291:
10232:
10099:
10065:
10039:
9939:
9836:
9801:
9754:
9707:
9629:
9595:
9544:
9519:
9507:
9205:
8760:
8733:
8705:
8217:
8193:
7678:
7630:
7606:
6396:
6252:
6116:
5649:
5102:
5078:
5054:
4021:
in BoyerâLindquist coordinates are shown below in the expression for the square of the
3694:
3667:
3608:
3556:
3335:
2306:
1177:
1150:
825:
805:
663:
548:
473:
320:
225:
210:
171:
33:
10958:
10160:
13170:
13002:
12994:
12595:
12552:
12224:
11868:
11858:
11853:
11553:
11541:
11494:
11435:
11404:
11394:
11328:
11019:
10881:
10854:
10831:
10805:
10782:
10763:
10739:
10720:
10690:
10596:
10584:
10534:
10439:
10321:
10149:
10061:
9919:
Lecture XXVI: Kerr black holes: I. Metric structure and regularity of particle orbits
9898:
9871:
9841:
9782:
9735:
9676:
9633:
9621:
9549:
9490:
9277:
9224:(the radial coordinate in the WeylâPapapetrou chart). According to this formulation:
8597:
5934:
3714:
1035:
844:
black hole, the Kerr metric, remained unsolved until 1963, when it was discovered by
782:
493:
268:
13040:
10966:
10756:; Ansorg, Marcus; Kleinwachter, Andreas; Neugebauer, Gernot; Petroff, David (2008).
10236:
8456:
543:
13075:
13065:
13012:
12989:
12567:
12345:
12310:
12257:
12190:
12019:
11979:
11727:
11676:
11531:
11445:
11238:
11208:
11091:
11050:
10995:
10954:
10682:
10576:
10522:
10477:
10427:
10309:
10224:
10069:
10057:
10011:
9976:
9831:
9821:
9774:
9725:
9668:
9617:
9613:
9539:
9531:
9472:
9365:
9039:
8931:
8923:
8871:
8781:
8460:
8459:
which takes the two-dimensional locus of the singularity to itself. It retains the
7390:
5943:
3979:
Since even a direct check on the Kerr metric involves cumbersome calculations, the
2590:
1170:
1023:
865:
809:
618:
573:
553:
498:
13135:
11062:
9826:
9648:
8922:
Ernst vacuum can be characterized by giving the infinite sequence of relativistic
8455:
The group of isometries of the Kerr metric is the subgroup of the ten-dimensional
638:
13150:
13125:
13050:
13045:
12928:
12889:
12851:
12795:
12669:
12605:
12340:
12039:
11989:
11964:
11760:
11604:
11511:
11477:
11389:
10753:
10249:
10184:
9863:
9294:
8807:
6416:
5898:
5894:
5762:
4822:
3518:
986:
857:
673:
648:
533:
528:
392:
273:
235:
12933:
12320:
10686:
9778:
7355:{\displaystyle R(r)=P(r)^{2}-\Delta \left(\mu ^{2}r^{2}+(L_{z}-aE)^{2}+Q\right)}
443:
13175:
12847:
12831:
12827:
12730:
12700:
12557:
12460:
12330:
12315:
12282:
11954:
11939:
11706:
11645:
11489:
11163:
10999:
10975:
10923:
Characterization of three standard families of vacuum solutions as noted above.
10712:
10526:
10400:
9730:
9695:
9535:
9476:
9379:
8725:
7574:
6388:
5952:
5922:
5022:
5014:
1124:
1093:
978:
716:
683:
678:
366:
230:
11096:
11071:
11054:
10710:
10580:
6419:. It is related to the total angular momentum of the particle and is given by
5644:
is negative, unless the particle is co-rotating around the interior mass
13221:
13165:
13145:
13140:
13055:
12923:
12751:
12695:
12527:
12480:
12014:
11934:
11462:
11457:
10899:
10588:
10431:
9786:
9739:
9680:
9302:
6244:
6112:
5994:
5983:
5918:
5890:
5768:
5130:
4022:
4018:
2594:
1031:
793:
789:
643:
558:
538:
463:
361:
240:
603:
13185:
13105:
13070:
12600:
12562:
12335:
11969:
11843:
11811:
11419:
11338:
10694:
10617:. International Series of Monographs on Physics. Vol. 69. p. 375.
9845:
9625:
9553:
9281:
9028:
5990:
Kerr geometry described above may not be present inside such a black hole.
3225:{\displaystyle {\frac {x^{2}+y^{2}}{r^{2}+a^{2}}}+{\frac {z^{2}}{r^{2}}}=1}
1197:
1030:
and has a shape similar to a flattened sphere. The inner surface marks the
1014:
Rotating black holes have surfaces where the metric seems to have apparent
588:
568:
8825:
models a (rotating) black hole endowed with an electric charge, while the
8755:
The region beyond the Cauchy horizon has several surprising features. The
12938:
12485:
12262:
12203:
12109:
12009:
11959:
11913:
11816:
11691:
11355:
11263:
11007:
10757:
10355:
10313:
5964:
5626:
5591:{\displaystyle r_{\rm {E}}^{\pm }=1\pm {\sqrt {1-a^{2}\cos ^{2}\theta }}}
5121:, while the left side of the black hole is rotating towards the observer.
3059:
3055:
1201:
478:
10917:
5383:
the event horizon is also the place where the purely temporal component
3419:
can be applied. This implies that the Kerr black hole has the form of a
13180:
12746:
12590:
12585:
12234:
11999:
11929:
11898:
11873:
11826:
11821:
11806:
11472:
11409:
11399:
11300:
11136:
10228:
9859:
8638: in this section. Unsourced material may be challenged and removed.
6578:
6096:
5914:
5614:
4814:
1027:
995:
785:
688:
250:
176:
108:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
10941:
10044:
10015:
9980:
9755:"BĂ€cklund Transformation for the Ernst Equation of General Relativity"
9672:
8478:
The location of the event horizon is determined by the larger root of
2356:) goes to zero, the Kerr metric becomes the orthogonal metric for the
994:
rotate with the black hole; the region where this holds is called the
12577:
12292:
12277:
12173:
12034:
11681:
11584:
11372:
11367:
9407:
9305:
which models a rotating thin disk, approaches in a limiting case the
7821:
In Cartesian KerrâSchild coordinates, the equations for a photon are
7649:
of the test-particle is measured relative to a probe corotating with
5634:
5630:
1142:
829:
778:
259:
10380:
9944:
8613:
13120:
12470:
12272:
11944:
11903:
11801:
10978:(1971). "Axisymmetric Black Hole Has Only Two Degrees of Freedom".
10563:
10481:
10082:
Berman, Marcelo. "Energy of Black Holes and Hawking's Universe" in
9600:
9452:
9193:
8890:
8829:
models a (rotating) hole with infalling electromagnetic radiation.
5717:
is not singular by itself even in Kerr coordinates due to non-zero
1018:; the size and shape of these surfaces depends on the black hole's
845:
628:
438:
278:
10752:
10509:
10296:
10104:
9712:
8666:
Although the Kerr solution appears to be singular at the roots of
7601:
which is defined with respect to the bookkeeper's coordinate time
3603:
in comparison with a corresponding Schwarzschild black hole where
11984:
6205:{\displaystyle -\mu ^{2}=p^{\alpha }g_{\alpha \beta }p^{\beta },}
2668:{\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+fk_{\mu }k_{\nu }\!}
11070:
KrasiĆski, Andrzej; Verdaguer, Enric; Kerr, Roy Patrick (2009).
10819:
for a readable introduction at the advanced undergraduate level.
9569:"Black holes warp the universe into a grotesque hall of mirrors"
9508:"Divergent reflections around the photon sphere of a black hole"
5037:
3415:) is axially symmetric, it can be cast into a form to which the
12800:
12004:
11994:
11974:
11696:
11295:
11285:
5767:
A black hole in general is surrounded by a surface, called the
5144:
of the metric goes to infinity. Solving the quadratic equation
3508:{\displaystyle E_{\rm {rot}}=c^{2}\left(M-M_{\rm {irr}}\right)}
848:. The natural extension to a charged, rotating black hole, the
12078:
11105:
11018:. Chicago: The University of Chicago Press. pp. 312â324.
10717:
The Kerr Spacetime: Rotating Black Holes in General Relativity
8404:{\displaystyle W={\dot {t}}-a\sin ^{2}{\theta }{\dot {\phi }}}
820:
The Kerr metric is a generalization to a rotating body of the
12361:
10405:"Global structure of the Kerr family of gravitational fields"
8882:. (In this case, every Geroch moment but the mass vanishes.)
7445:{\displaystyle {\frac {dx^{\alpha }}{d\lambda }}=p^{\alpha }}
6100:
1794:{\displaystyle y={\sqrt {r^{2}+a^{2}}}\sin \theta \sin \phi }
1708:{\displaystyle x={\sqrt {r^{2}+a^{2}}}\sin \theta \cos \phi }
977:
According to the Kerr metric, a rotating body should exhibit
8889:
of the Kerr geometry, or rather a portion of it, is locally
2264:
A key feature to note in the above metric is the cross-term
12781:
9189:
8927:
5637:. However, this is impossible within the ergosphere, where
1019:
7673:. The gravitational time-dilation between a ZAMO at fixed
10872:
Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975).
10845:
for a very thorough study at the advanced graduate level.
5937:
can be solved exactly in closed form. In addition to two
5369:{\displaystyle r_{\rm {H}}^{\pm }=1\pm {\sqrt {1-a^{2}}}}
5125:
There are several important surfaces in the Kerr metric (
1089:
Schwarzschild metric § Singularities and black holes
9249:
family (two parameters; not quite asymptotically flat),
3123:
is not the radius, but rather is implicitly defined by
11069:
6391:
showed that there exists a fourth constant of motion,
2769:{\displaystyle f={\frac {2GMr^{3}}{r^{4}+a^{2}z^{2}}}}
812:, which makes exact solutions very difficult to find.
11032:
10459:
10282:(May 2011). "Finally, results from Gravity Probe B".
10196:
9313:
9208:
9059:
9005:
8944:
8901:
model. This is particularly interesting, because the
8840:
8763:
8736:
8708:
8674:
8561:
8516:
8486:
8421:
8346:
8242:
8220:
8196:
8093:
7827:
7703:
7681:
7657:
7633:
7609:
7585:
7540:
7516:
7492:
7462:
7401:
7373:
7240:
7226:{\displaystyle P(r)=E\left(r^{2}+a^{2}\right)-aL_{z}}
7155:
7007:
6632:
6592:
6425:
6399:
6347:
6308:
6279:
6255:
6220:
6149:
6127:
6059:
6035:
6005:
5955:. There is a pair of principal null congruences (one
5860:{\displaystyle (r-M)^{2}=M^{2}-J^{2}\cos ^{2}\theta }
5792:
5756:
5725:
5676:
5652:
5613:; the space between these two surfaces is called the
5525:
5409:
5319:
5276:
5177:
5105:
5081:
5057:
4836:
4541:
4039:
3991:
3757:
3725:
3697:
3670:
3635:
3611:
3583:
3559:
3529:
3439:
3362:
3338:
3264:
3134:
3094:
3006:
2800:
2699:
2608:
2370:
2333:
2309:
2272:
2189:
2106:
2047:
2020:
1990:
1914:
1879:
1825:
1739:
1653:
1610:
1219:
1180:
1153:
1050:
943:
895:
51:
9355:
8795:
7697:
and a stationary observer far away from the mass is
5617:. Within this volume, the purely temporal component
3119:
is directed along the positive z-axis. The quantity
2158:{\displaystyle \Sigma =r^{2}+a^{2}\cos ^{2}\theta ,}
1642:, which are equivalent to the cartesian coordinates
10926:
10020:
Especially see equations (7.10), (7.11) and (7.14).
9802:"A radiating Kerr black hole and Hawking radiation"
8926:, the first two of which can be interpreted as the
5706:is due to the choice of coordinates (i.e., it is a
3517:of a black hole is extracted, for example with the
1038:. A similar situation obtains when considering the
11011:
10873:
10871:
10797:
10640:Roy Kerr (Crafoord Prize Symposium in Astronomy):
10603:
10029:
9338:
9214:
9188:, characterizing respectively the distribution of
9158:
9017:
8982:
8852:
8769:
8742:
8714:
8686:
8586:
8543:
8498:
8433:
8403:
8333:
8226:
8202:
8180:
8080:
7813:
7687:
7663:
7639:
7615:
7591:
7560:
7522:
7498:
7474:
7444:
7379:
7354:
7225:
7140:
6990:
6616:
6567:
6405:
6373:
6333:
6292:
6261:
6233:
6204:
6133:
6065:
6041:
6017:
5859:
5743:
5682:
5658:
5590:
5505:
5368:
5300:
5257:
5111:
5087:
5063:
4989:
4785:
4497:
4007:
3963:
3738:
3713:(the gravitating mass) of the body (including its
3703:
3676:
3648:
3617:
3593:
3565:
3541:
3507:
3374:
3344:
3320:{\displaystyle r\to R={\sqrt {x^{2}+y^{2}+z^{2}}}}
3319:
3224:
3109:
3026:
2974:
2768:
2667:
2552:
2346:
2315:
2288:
2241:{\displaystyle \Delta =r^{2}-r_{\text{s}}r+a^{2}.}
2240:
2157:
2074:
2026:
2002:
1959:{\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}},}
1958:
1892:
1849:
1793:
1707:
1628:
1578:
1186:
1159:
1069:
951:
903:
107:
10139:
3023:
2664:
1042:which also appears to result in a singularity at
13219:
10144:. Cambridge Monographs on Mathematical Physics.
10088:, page 148 (Kreitler ed., Nova Publishers 2006).
9993:
9272:The Kerr geometry is often used as a model of a
8603:
5904:
5137:; this occurs where the purely radial component
1123:(also known as the spinâcoefficient formalism),
9890:
8800:The Kerr geometry is a particular example of a
27:Exact solution for the Einstein field equations
10395:
10393:
10391:
9958:
9858:
6143:of the test particle, defined by the relation
1130:
12377:
12094:
11121:
10910:
10822:
10795:
10736:Exact Solutions of Einstein's Field Equations
10664:
10609:
10351:
10349:
10120:Exact Solutions of Einstein's Field Equations
8473:
3074:is the constant mass of the spinning object,
2589:as follows. These solutions were proposed by
750:
10902:(1968). ed C. de Witt and J. Wheeler (ed.).
10494:
10360:About Time: Einstein's Unfinished Revolution
10126:regarding determinant of metric tensor. See
9897:. Oxford University Press. pp. 55â57 .
6051:variable after completing one period in the
3426:
10851:Colliding Plane Waves in General Relativity
10776:
10453:
10388:
10369:
10367:
9933:
9931:
9929:
9927:
9891:Rezzolla, Luciano; Zanotti, Olindo (2013).
9693:
9649:"Note on the Kerr Spinning-Particle Metric"
9646:
6115:in the Kerr spacetime are governed by four
1003:creating multiple images of the same object
12384:
12370:
12101:
12087:
11687:Magnetospheric eternally collapsing object
11128:
11114:
10880:(Second ed.). New York: McGraw-Hill.
10781:. Wellesley, Massachusetts: A. K. Peters.
10346:
10254:Black Holes: Energetics and Thermodynamics
9647:Newman, E. T.; Janis, A. I. (1965-06-01).
9290:static spherically symmetric perfect fluid
5913:exterior regions, each associated with an
2576:
1141:The Kerr metric describes the geometry of
1127:, or Ellipsoid coordinate transformation.
757:
743:
11095:
10940:
10916:
10848:
10762:. Cambridge: Cambridge University Press.
10738:. Cambridge: Cambridge University Press.
10719:. Cambridge: Cambridge University Press.
10634:
10562:
10551:International Journal of Modern Physics D
10508:
10379:
10295:
10218:
10103:
10043:
9943:
9835:
9825:
9729:
9711:
9599:
9543:
8654:Learn how and when to remove this message
5951:), the Kerr geometry admits a remarkable
2279:
1982:and where for brevity, the length scales
1565:
1558:
945:
897:
828:in 1915, which described the geometry of
10906:. W. A. Benjamin, New York. p. 222.
10868:for the Chandrasekhar/Ferrari CPW model.
10548:
10364:
10259:
9924:
9752:
9483:
9445:
6090:
6082:
5044:
5036:
10898:
10122:(Cambridge University Press 2003). See
9911:
9884:
9505:
9252:the isolated mass monopole source with
9239:the isolated mass monopole source with
9228:the isolated mass monopole source with
6078:
5668:with an angular speed at least of
5129:). The inner surface corresponds to an
3384:). At large distances from the source (
14:
13220:
10974:
10828:The Mathematical Theory of Black Holes
10615:The Mathematical Theory of Black Holes
10399:
10172:
10170:
10168:
10097:
9987:
9937:
9566:
9499:
8212:is analogous to Carter's constant and
3406:
13228:Exact solutions in general relativity
12365:
12082:
11109:
10199:Journal of Astrophysics and Astronomy
10112:
9585:
9288:interior, and indeed to more general
5032:
12063:
11006:
10334:
10278:
10190:
10140:Belinski, V.; Verdaguer, E. (2001).
9799:
9451:
8913:
8636:adding citations to reliable sources
8607:
4827:
4532:
4030:
3125:
2997:
2791:
2690:
2599:
2581:The Kerr metric can be expressed in
2361:
2301:In the non-relativistic limit where
2180:
2097:
2038:
1905:
1816:
1730:
1644:
1210:
836:, spherical, non-rotating body, the
12956:TolmanâOppenheimerâVolkoff equation
12909:FriedmannâLemaĂźtreâRobertsonâWalker
10853:. Oxford: Oxford University Press.
10759:Relativistic Figures of Equilibrium
10711:Wiltshire, David L.; Visser, Matt;
10165:
8544:{\displaystyle r_{\text{s}}/2<a}
7561:{\displaystyle \lambda =\tau /\mu }
3411:As the Kerr metric (along with the
1079:dividing the space above and below
24:
12134:Novikov self-consistency principle
11076:General Relativity and Gravitation
11035:General Relativity and Gravitation
10876:Introduction to General Relativity
10704:
8675:
8487:
8166:
8140:
8066:
8015:
7879:
7801:
7795:
7774:
7658:
7586:
7278:
7008:
6967:
6856:
6835:
6743:
6725:
6690:
6637:
6119:. The first is the invariant mass
5757:Ergosphere and the Penrose process
5677:
5532:
5416:
5326:
5268:which in natural units (that give
5184:
4903:
4837:
4654:
4651:
4476:
4472:
4460:
4433:
4429:
4417:
4414:
4390:
4386:
4369:
4319:
4299:
4295:
4284:
4280:
4272:
4269:
4220:
4216:
4183:
4119:
4081:
4077:
4059:
4055:
3934:
3931:
3928:
3880:
3877:
3874:
3773:
3770:
3767:
3494:
3491:
3488:
3452:
3449:
3446:
3431:If the complete rotational energy
2469:
2423:
2190:
2107:
2075:{\displaystyle a={\frac {J}{Mc}},}
2021:
1997:
1550:
1450:
1373:
1352:
1349:
1311:
1196:. The metric (or equivalently its
68:
25:
13249:
12726:HamiltonâJacobiâEinstein equation
10800:Introducing Einstein's Relativity
10337:"Shadows of rotating black holes"
9694:Newman, Ezra; Adamo, Tim (2014).
9506:Sneppen, Albert (December 2021).
8796:Relation to other exact solutions
5695:As with the event horizon in the
4521:
3392:), these equations reduce to the
1026:. The outer surface encloses the
13204:
13203:
12144:Quantum mechanics of time travel
12124:Chronology protection conjecture
12062:
12053:
12052:
11351:TolmanâOppenheimerâVolkoff limit
11222:
10779:The Geometry of Kerr Black Holes
9753:Harrison, B. Kent (1978-10-30).
9372:
9358:
9267:
8895:ChandrasekharâFerrari CPW vacuum
8612:
7508:, is related to the proper time
5133:similar to that observed in the
4526:We may rewrite the Kerr metric (
3974:
2802:
1850:{\displaystyle z=r\cos \theta ,}
777:describes the geometry of empty
724:
723:
710:
40:
12108:
11468:Innermost stable circular orbit
11135:
10658:
10649:
10621:
10542:
10488:
10328:
10272:
10243:
10133:
10091:
10076:
10023:
9996:Journal of Mathematical Physics
9952:
9852:
9800:Chou, Yu-Ching (January 2020).
9653:Journal of Mathematical Physics
9424:
9099:
9079:
8623:needs additional citations for
6374:{\displaystyle L_{z}=p_{\phi }}
5893:in 1969 and is thus called the
1629:{\displaystyle r,\theta ,\phi }
1107:of a pair of Kerr black holes.
12533:Massâenergy equivalence (E=mc)
12391:
11894:Timeline of black hole physics
10176:Misner, Thorne & Wheeler:
9868:The Classical Theory of Fields
9793:
9746:
9687:
9640:
9618:10.1103/PhysRevLett.116.061102
9579:
9560:
8993:Thus, the special case of the
8971:
8961:
8862:of the Kerr metric yields the
7990:
7975:
7332:
7309:
7266:
7259:
7250:
7244:
7165:
7159:
7017:
7011:
6981:
6975:
6849:
6843:
6734:
6728:
6681:
6675:
5806:
5793:
5699:, the apparent singularity at
3852:
3739:{\displaystyle M_{\text{irr}}}
3649:{\displaystyle M_{\text{irr}}}
3330:when the rotational parameter
3268:
3101:
2848:
2809:
2384:
2376:
1070:{\displaystyle r=r_{\text{s}}}
13:
1:
12240:Parallel universes in fiction
11662:Nonsingular black hole models
10085:Trends in Black Hole Research
10032:Classical and Quantum Gravity
9827:10.1016/j.heliyon.2020.e03336
9567:Sutter, Paul (22 July 2021).
9439:
8604:Kerr black holes as wormholes
8450:
5982:is known. Topologically, the
5917:, stationary limit surfaces,
5905:Features of the Kerr geometry
2358:oblate spheroidal coordinates
1640:oblate spheroidal coordinates
808:; these equations are highly
12250:Philosophy of space and time
9418:
8587:{\displaystyle GM^{2}<Jc}
5003:
4799:
4511:
3551:, its total mass-equivalent
3238:
3040:
2988:
2782:
2681:
2585:, using a particular set of
2347:{\displaystyle r_{\text{s}}}
1893:{\displaystyle r_{\text{s}}}
952:{\displaystyle \mathbb {R} }
904:{\displaystyle \mathbb {R} }
781:around a rotating uncharged
7:
12548:Relativistic Doppler effect
11884:Rossi X-ray Timing Explorer
11849:Hypercompact stellar system
11839:Gamma-ray burst progenitors
10959:10.1088/0264-9381/21/24/003
10830:. Oxford: Clarendon Press.
10804:. Oxford: Clarendon Press.
10687:10.1103/physrevlett.71.1669
10629:Black hole Penrose diagrams
9779:10.1103/PhysRevLett.41.1197
9351:
8983:{\displaystyle M_{n}=M^{n}}
6234:{\displaystyle p^{\alpha }}
5872:BoyerâLindquist coordinates
5126:
4527:
3717:) and its irreducible mass
3594:{\displaystyle {\sqrt {2}}}
3417:BelinskiâZakharov transform
2566:
2254:
2171:
2088:
1972:
1863:
1807:
1721:
1592:
1206:BoyerâLindquist coordinates
1137:BoyerâLindquist coordinates
1131:BoyerâLindquist coordinates
815:
221:Gravitational time dilation
10:
13254:
13019:In computational physics:
12543:Relativity of simultaneity
12230:Many-worlds interpretation
12117:General terms and concepts
11570:Black hole complementarity
11537:Bousso's holographic bound
11522:Quasi-periodic oscillation
11220:
11214:MalamentâHogarth spacetime
11000:10.1103/PhysRevLett.26.331
10646:. (YouTube, Timestamp 26m)
10527:10.1103/PhysRevD.90.124010
10146:Cambridge University Press
10130:regarding generalizations.
10062:10.1088/0264-9381/11/6/010
9894:Relativistic Hydrodynamics
9731:10.4249/scholarpedia.31791
9536:10.1038/s41598-021-93595-w
9477:10.1103/PhysRevLett.11.237
9186:momentum multipole moments
8474:Overextreme Kerr solutions
6103:region around the equator.
5760:
5744:{\displaystyle dt\ d\phi }
3689:The total mass equivalent
3394:EddingtonâFinkelstein form
3110:{\displaystyle {\vec {a}}}
3027:{\displaystyle k_{0}=1.\!}
2289:{\displaystyle dt\,d\phi }
1145:in the vicinity of a mass
1134:
341:MathissonâPapapetrouâDixon
182:Pseudo-Riemannian manifold
13201:
13033:
12898:
12870:
12856:LenseâThirring precession
12739:
12688:
12650:
12629:
12618:
12576:
12520:
12504:
12446:
12438:Doubly special relativity
12410:
12399:
12291:
12248:
12212:
12189:
12152:
12116:
12048:
11922:
11774:
11736:
11715:
11654:
11613:
11562:
11441:Gravitational singularity
11428:
11321:
11231:
11156:
11143:
11097:10.1007/s10714-009-0856-0
11055:10.1007/s10714-009-0857-z
10849:Griffiths, J. B. (1991).
10777:O'Neill, Barrett (1995).
10581:10.1142/S0218271820501096
10462:The Astrophysical Journal
9398:ReissnerâNordström metric
9339:{\displaystyle GM^{2}=cJ}
9031:" of general relativity.
8908:gravitational plane waves
8880:Schwarzschild coordinates
8823:KerrâNewman electrovacuum
8687:{\displaystyle \Delta =0}
8499:{\displaystyle \Delta =0}
7475:{\displaystyle \mu >0}
6415:, now referred to as the
6334:{\displaystyle E=-p_{t},}
4530:) in the following form:
3575:is higher by a factor of
3427:Mass of rotational energy
3252:becomes the usual radius
3248:Notice that the quantity
2003:{\displaystyle a,\Sigma }
1110:
983:LenseâThirring precession
838:ReissnerâNordström metric
12716:Post-Newtonian formalism
12706:Einstein field equations
12642:Mathematical formulation
12466:Hyperbolic orthogonality
12139:Self-fulfilling prophecy
12025:PSO J030947.49+271757.31
11950:SDSS J150243.09+111557.3
11483:BlandfordâZnajek process
10432:10.1103/PhysRev.174.1559
9260:family (two parameters).
9256:angular momentum is the
9243:angular momentum is the
9232:angular momentum is the
8698:coordinate singularities
7499:{\displaystyle \lambda }
7380:{\displaystyle \lambda }
4821:, where Ω is called the
3058:, making the 4-vector a
2036:have been introduced as
1121:NewmanâPenrose formalism
1105:first direct observation
802:Einstein field equations
346:HamiltonâJacobiâEinstein
326:Einstein field equations
149:Mathematical formulation
12427:Galilean transformation
12418:Principle of relativity
11281:Active galactic nucleus
10980:Physical Review Letters
10796:D'Inverno, Ray (1992).
10667:Physical Review Letters
10557:(16): 2050109â2050202.
9917:Christopher M. Hirata:
9759:Physical Review Letters
9588:Physical Review Letters
9457:Physical Review Letters
9236:family (one parameter),
8812:Einstein field equation
8445:Schwarzschild geodesics
7664:{\displaystyle \Omega }
7592:{\displaystyle \Omega }
6617:{\displaystyle G=M=c=1}
6066:{\displaystyle \theta }
5683:{\displaystyle \Omega }
5301:{\displaystyle G=M=c=1}
3660:massâenergy equivalence
3062:, with respect to both
2577:KerrâSchild coordinates
2027:{\displaystyle \Delta }
12512:Lorentz transformation
12303:closed timelike curves
12154:Time travel in fiction
11909:Tidal disruption event
11879:Supermassive dark star
11797:Black holes in fiction
11782:Outline of black holes
11415:Supermassive dark star
11334:Gravitational collapse
10142:Gravitational Solitons
10118:Stephani, Hans et al.
9340:
9299:NeugebauerâMeinel disk
9216:
9182:mass multipole moments
9160:
9035:Weyl multipole moments
9027:) gives the "monopole
9019:
8984:
8854:
8771:
8744:
8716:
8688:
8588:
8545:
8500:
8435:
8405:
8335:
8228:
8204:
8182:
8082:
7815:
7689:
7665:
7641:
7617:
7593:
7562:
7524:
7500:
7476:
7454:. In particular, when
7446:
7381:
7356:
7227:
7142:
6992:
6618:
6569:
6407:
6385:HamiltonâJacobi theory
6375:
6335:
6294:
6263:
6235:
6206:
6135:
6104:
6088:
6067:
6043:
6019:
5927:closed timelike curves
5861:
5745:
5708:coordinate singularity
5684:
5660:
5592:
5507:
5370:
5302:
5259:
5122:
5113:
5089:
5065:
5042:
4991:
4787:
4499:
4009:
4008:{\displaystyle g^{ik}}
3965:
3740:
3705:
3678:
3650:
3619:
3595:
3567:
3543:
3509:
3376:
3346:
3321:
3226:
3111:
3028:
2976:
2770:
2669:
2554:
2348:
2317:
2290:
2242:
2159:
2076:
2028:
2004:
1960:
1894:
1851:
1795:
1709:
1630:
1602:where the coordinates
1580:
1188:
1161:
1117:NewmanâJanis algorithm
1071:
953:
905:
856:represents the body's
788:with a quasispherical
216:Gravitational redshift
109:
12980:WeylâLewisâPapapetrou
12721:Raychaudhuri equation
12660:Equivalence principle
12351:Traversable wormholes
12129:Closed timelike curve
11787:Black Hole Initiative
11600:Holographic principle
10187:, pages 899, 900, 908
9341:
9217:
9161:
9020:
8985:
8876:spherically symmetric
8855:
8827:KerrâVaidya null dust
8772:
8745:
8717:
8696:, these are actually
8689:
8589:
8546:
8501:
8436:
8406:
8336:
8236:is a useful quantity
8229:
8205:
8183:
8083:
7816:
7690:
7666:
7642:
7625:. The local velocity
7618:
7594:
7563:
7525:
7523:{\displaystyle \tau }
7501:
7484:the affine parameter
7477:
7447:
7382:
7357:
7228:
7143:
6993:
6619:
6570:
6408:
6376:
6336:
6295:
6293:{\displaystyle L_{z}}
6264:
6236:
6207:
6136:
6094:
6086:
6068:
6044:
6042:{\displaystyle \phi }
6020:
5969:algebraically special
5939:Killing vector fields
5931:curvature singularity
5862:
5746:
5685:
5661:
5593:
5516:or in natural units:
5508:
5400:yields the solution:
5371:
5303:
5260:
5168:yields the solution:
5114:
5090:
5066:
5048:
5040:
5019:equivalence principle
4992:
4788:
4500:
4010:
3966:
3741:
3706:
3679:
3651:
3620:
3596:
3568:
3544:
3510:
3421:gravitational soliton
3377:
3347:
3322:
3227:
3112:
3029:
2977:
2771:
2670:
2587:Cartesian coordinates
2555:
2349:
2318:
2291:
2243:
2160:
2077:
2029:
2005:
1961:
1895:
1852:
1796:
1710:
1631:
1581:
1189:
1162:
1072:
954:
906:
504:WeylâLewisâPapapetrou
459:KerrâNewmanâde Sitter
279:EinsteinâRosen bridge
211:Gravitational lensing
167:Equivalence principle
110:
13021:Numerical relativity
12862:pulsar timing arrays
12162:Timelines in fiction
11590:Final parsec problem
11549:Schwarzschild radius
10643:Spinning Black Holes
10314:10.1103/Physics.4.43
10267:Rotating Black Holes
10265:Stijn van Tongeren:
9696:"KerrâNewman metric"
9403:HartleâThorne metric
9388:Schwarzschild metric
9311:
9234:Schwarzschild vacuum
9206:
9057:
9003:
8995:Schwarzschild vacuum
8942:
8899:colliding plane wave
8870:black hole which is
8864:Schwarzschild metric
8838:
8761:
8734:
8706:
8672:
8632:improve this article
8559:
8514:
8484:
8419:
8344:
8240:
8218:
8194:
8091:
7825:
7701:
7679:
7655:
7631:
7607:
7583:
7538:
7514:
7490:
7460:
7399:
7371:
7238:
7153:
7005:
6630:
6590:
6423:
6397:
6345:
6306:
6277:
6253:
6218:
6147:
6134:{\displaystyle \mu }
6125:
6079:Trajectory equations
6057:
6033:
6003:
5929:, and a ring-shaped
5790:
5773:Schwarzschild radius
5771:and situated at the
5723:
5697:Schwarzschild metric
5674:
5650:
5523:
5407:
5381:Schwarzschild metric
5317:
5274:
5175:
5135:Schwarzschild metric
5103:
5079:
5055:
4834:
4539:
4037:
3989:
3755:
3723:
3695:
3668:
3633:
3609:
3581:
3557:
3527:
3437:
3398:Schwarzschild metric
3360:
3336:
3262:
3132:
3092:
3004:
2798:
2697:
2606:
2368:
2331:
2307:
2270:
2187:
2104:
2045:
2018:
1988:
1912:
1902:Schwarzschild radius
1877:
1823:
1737:
1651:
1608:
1217:
1178:
1151:
1048:
1040:Schwarzschild metric
941:
893:
864:represents its spin
822:Schwarzschild metric
434:EinsteinâRosen waves
160:Fundamental concepts
49:
12913:Friedmann equations
12807:HulseâTaylor binary
12769:Gravitational waves
12665:Riemannian geometry
12491:Proper acceleration
12476:Maxwell's equations
12422:Galilean relativity
12199:Grandfather paradox
11889:Superluminal motion
11864:Population III star
11834:Gravitational waves
11792:Black hole starship
11575:Information paradox
11088:2009GReGr..41.2469K
11047:2009GReGr..41.2485K
10992:1971PhRvL..26..331C
10951:2004CQGra..21.5727S
10929:Class. Quantum Grav
10904:Battelle Rencontres
10679:1993PhRvL..71.1669B
10573:2020IJMPD..2950109R
10519:2014PhRvD..90l4010F
10474:1972ApJ...178..347B
10424:1968PhRv..174.1559C
10335:de Vries, Andreas.
10306:2011PhyOJ...4...43W
10211:1985JApA....6...85B
10054:1994CQGra..11.1453B
10008:1969JMP....10.1842D
9973:1967JMP.....8..265B
9818:2020Heliy...603336C
9771:1978PhRvL..41.1197H
9722:2014SchpJ...931791N
9665:1965JMP.....6..915N
9610:2016PhRvL.116f1102A
9528:2021NatSR..1114247S
9469:1963PhRvL..11..237K
9413:Rotating black hole
9286:Schwarzschild fluid
9274:rotating black hole
9171:ChazyâCurzon vacuum
9018:{\displaystyle a=0}
8920:asymptotically flat
8853:{\displaystyle a=0}
8434:{\displaystyle a=0}
8263:
7112:
6446:
6117:constants of motion
6109:equations of motion
6018:{\displaystyle a=0}
5911:asymptotically flat
5633:, its path through
5542:
5462:
5426:
5336:
5230:
5194:
4608:
3944:
3890:
3783:
3542:{\displaystyle a=M}
3407:Soliton coordinates
3375:{\displaystyle c=1}
2325:(or, equivalently,
388:KaluzaâKlein theory
274:Minkowski spacetime
226:Gravitational waves
12962:ReissnerâNordström
12880:BransâDicke theory
12711:Linearized gravity
12538:Length contraction
12456:Frame of reference
12433:Special relativity
12297:general relativity
12220:Alternative future
12213:Parallel timelines
12191:Temporal paradoxes
12167:in science fiction
11723:Optical black hole
11636:ReissnerâNordström
11595:Firewall (physics)
11500:Gravitational lens
11014:General Relativity
10843:See chapters 6--10
10229:10.1007/BF02715080
10183:2017-08-22 at the
9520:Cosmic Dawn Center
9512:Scientific Reports
9393:KerrâNewman metric
9336:
9212:
9156:
9015:
8980:
8897:, an example of a
8850:
8767:
8740:
8712:
8684:
8584:
8541:
8496:
8431:
8401:
8331:
8249:
8224:
8200:
8178:
8078:
7811:
7685:
7661:
7637:
7613:
7589:
7558:
7520:
7496:
7472:
7442:
7377:
7352:
7223:
7138:
7098:
6988:
6986:
6614:
6565:
6432:
6403:
6371:
6331:
6290:
6259:
6231:
6202:
6131:
6105:
6089:
6063:
6039:
6015:
5941:(corresponding to
5857:
5741:
5680:
5656:
5588:
5526:
5503:
5448:
5410:
5366:
5320:
5298:
5255:
5216:
5178:
5123:
5109:
5085:
5061:
5043:
5033:Important surfaces
4987:
4783:
4591:
4495:
4005:
3961:
3922:
3868:
3761:
3736:
3701:
3674:
3646:
3615:
3591:
3563:
3539:
3505:
3372:
3342:
3317:
3222:
3107:
3024:
2972:
2766:
2665:
2583:"KerrâSchild" form
2550:
2391:
2344:
2313:
2286:
2238:
2155:
2072:
2024:
2000:
1956:
1890:
1847:
1791:
1705:
1626:
1576:
1574:
1184:
1157:
1067:
963:ReissnerâNordström
949:
901:
850:KerrâNewman metric
826:Karl Schwarzschild
806:general relativity
717:Physics portal
489:OppenheimerâSnyder
429:ReissnerâNordström
321:Linearized gravity
269:Spacetime diagrams
172:Special relativity
105:
34:General relativity
13215:
13214:
13029:
13028:
13008:OzsvĂĄthâSchĂŒcking
12614:
12613:
12596:Minkowski diagram
12553:Thomas precession
12496:Relativistic mass
12359:
12358:
12311:Alcubierre metric
12225:Alternate history
12076:
12075:
11869:Supermassive star
11859:Naked singularity
11854:Membrane paradigm
11580:Cosmic censorship
11554:Spaghettification
11542:Immirzi parameter
11495:Hawking radiation
11436:Astrophysical jet
11405:Supermassive star
11395:Binary black hole
11329:Stellar evolution
11271:Intermediate-mass
11082:(10): 2469â2484.
11041:(10): 2485â2499.
11025:978-0-226-87032-8
10935:(24): 5727â5733.
10887:978-0-07-000423-8
10860:978-0-19-853209-5
10837:978-0-19-850370-5
10824:Chandrasekhar, S.
10811:978-0-19-859686-8
10788:978-1-56881-019-5
10769:978-0-521-86383-4
10745:978-0-521-46136-8
10726:978-0-521-88512-6
10673:(11): 1669â1672.
10627:Andrew Hamilton:
10611:Chandrasekhar, S.
10497:Physical Review D
10280:Will, Clifford M.
10016:10.1063/1.1664769
10002:(10): 1842â1854.
9981:10.1063/1.1705193
9904:978-0-19-852890-6
9877:978-0-08-018176-9
9765:(18): 1197â1200.
9673:10.1063/1.1704350
9278:exterior solution
9215:{\displaystyle r}
9136:
9040:multipole moments
8924:multipole moments
8914:Multipole moments
8866:, which models a
8832:The special case
8805:axially symmetric
8790:cosmic censorship
8770:{\displaystyle r}
8743:{\displaystyle r}
8715:{\displaystyle r}
8664:
8663:
8656:
8598:naked singularity
8524:
8468:Noether's theorem
8461:time translations
8398:
8362:
8318:
8227:{\displaystyle W}
8203:{\displaystyle C}
8176:
8143:
8103:
8076:
8018:
7961:
7950:
7925:
7907:
7888:
7855:
7837:
7806:
7805:
7800:
7712:
7688:{\displaystyle r}
7640:{\displaystyle v}
7616:{\displaystyle t}
7427:
7131:
6970:
6877:
6838:
6820:
6764:
6737:
6711:
6684:
6658:
6545:
6406:{\displaystyle Q}
6262:{\displaystyle E}
5971:, in fact it has
5935:geodesic equation
5734:
5659:{\displaystyle M}
5586:
5501:
5495:
5455:
5440:
5364:
5310:) simplifies to:
5253:
5247:
5223:
5208:
5112:{\displaystyle M}
5088:{\displaystyle 0}
5073:is animated from
5064:{\displaystyle a}
5011:
5010:
4982:
4949:
4889:
4874:
4807:
4806:
4761:
4622:
4519:
4518:
4483:
4463:
4440:
4420:
4397:
4372:
4362:
4339:
4308:
4291:
4276:
4255:
4227:
4186:
4166:
4123:
4095:
4073:
3956:
3850:
3733:
3715:rotational energy
3704:{\displaystyle M}
3677:{\displaystyle M}
3643:
3618:{\displaystyle M}
3589:
3566:{\displaystyle M}
3345:{\displaystyle a}
3315:
3246:
3245:
3214:
3187:
3104:
3048:
3047:
2996:
2995:
2965:
2952:
2903:
2790:
2789:
2764:
2689:
2688:
2574:
2573:
2451:
2374:
2341:
2316:{\displaystyle M}
2262:
2261:
2216:
2179:
2178:
2096:
2095:
2067:
1980:
1979:
1951:
1922:
1887:
1871:
1870:
1815:
1814:
1771:
1729:
1728:
1685:
1600:
1599:
1553:
1524:
1494:
1453:
1433:
1355:
1314:
1304:
1187:{\displaystyle J}
1160:{\displaystyle M}
1064:
1036:coordinate system
973:
972:
783:axially symmetric
767:
766:
400:
399:
286:
285:
16:(Redirected from
13245:
13207:
13206:
12990:van Stockum dust
12762:Two-body problem
12680:Mach's principle
12627:
12626:
12568:Terrell rotation
12408:
12407:
12386:
12379:
12372:
12363:
12362:
12346:van Stockum dust
12258:Butterfly effect
12103:
12096:
12089:
12080:
12079:
12066:
12065:
12056:
12055:
11728:Sonic black hole
11677:Dark-energy star
11532:Bekenstein bound
11517:Mâsigma relation
11446:Ring singularity
11226:
11130:
11123:
11116:
11107:
11106:
11101:
11099:
11066:
11029:
11017:
11003:
10970:
10944:
10922:
10920:
10907:
10891:
10879:
10864:
10841:
10815:
10803:
10792:
10773:
10754:Meinel, Reinhard
10749:
10730:
10699:
10698:
10662:
10656:
10653:
10647:
10638:
10632:
10625:
10619:
10618:
10607:
10601:
10600:
10566:
10546:
10540:
10538:
10512:
10492:
10486:
10485:
10457:
10451:
10450:
10448:
10442:. Archived from
10418:(5): 1559â1571.
10409:
10397:
10386:
10385:
10383:
10371:
10362:
10353:
10344:
10343:
10341:
10332:
10326:
10325:
10299:
10276:
10270:
10263:
10257:
10247:
10241:
10240:
10222:
10194:
10188:
10174:
10163:
10159:
10137:
10131:
10116:
10110:
10109:
10107:
10095:
10089:
10080:
10074:
10073:
10047:
10038:(6): 1453â1461.
10027:
10021:
10019:
9991:
9985:
9984:
9956:
9950:
9949:
9947:
9935:
9922:
9915:
9909:
9908:
9888:
9882:
9881:
9856:
9850:
9849:
9839:
9829:
9797:
9791:
9790:
9750:
9744:
9743:
9733:
9715:
9691:
9685:
9684:
9644:
9638:
9637:
9603:
9583:
9577:
9576:
9564:
9558:
9557:
9547:
9503:
9497:
9487:
9481:
9480:
9449:
9434:
9428:
9382:
9377:
9376:
9368:
9366:Astronomy portal
9363:
9362:
9361:
9347:
9345:
9343:
9342:
9337:
9326:
9325:
9223:
9221:
9219:
9218:
9213:
9165:
9163:
9162:
9157:
9155:
9151:
9150:
9149:
9137:
9132:
9131:
9122:
9109:
9108:
9089:
9088:
9069:
9068:
9026:
9024:
9022:
9021:
9016:
8989:
8987:
8986:
8981:
8979:
8978:
8954:
8953:
8932:angular momentum
8903:global structure
8861:
8859:
8857:
8856:
8851:
8778:
8776:
8774:
8773:
8768:
8751:
8749:
8747:
8746:
8741:
8723:
8721:
8719:
8718:
8713:
8695:
8693:
8691:
8690:
8685:
8659:
8652:
8648:
8645:
8639:
8616:
8608:
8595:
8593:
8591:
8590:
8585:
8574:
8573:
8552:
8550:
8548:
8547:
8542:
8531:
8526:
8525:
8522:
8507:
8505:
8503:
8502:
8497:
8442:
8440:
8438:
8437:
8432:
8410:
8408:
8407:
8402:
8400:
8399:
8391:
8388:
8380:
8379:
8364:
8363:
8355:
8340:
8338:
8337:
8332:
8330:
8329:
8324:
8320:
8319:
8317:
8316:
8304:
8303:
8294:
8289:
8262:
8257:
8235:
8233:
8231:
8230:
8225:
8211:
8209:
8207:
8206:
8201:
8187:
8185:
8184:
8179:
8177:
8175:
8174:
8173:
8157:
8155:
8151:
8144:
8139:
8138:
8137:
8124:
8105:
8104:
8096:
8087:
8085:
8084:
8079:
8077:
8075:
8074:
8073:
8060:
8059:
8058:
8049:
8048:
8032:
8030:
8026:
8019:
8014:
8013:
8012:
7999:
7968:
7964:
7963:
7962:
7954:
7951:
7946:
7932:
7927:
7926:
7918:
7909:
7908:
7900:
7889:
7887:
7886:
7874:
7857:
7856:
7848:
7839:
7838:
7830:
7820:
7818:
7817:
7812:
7807:
7804:
7798:
7793:
7786:
7785:
7773:
7772:
7760:
7759:
7754:
7750:
7749:
7748:
7736:
7735:
7719:
7718:
7713:
7705:
7696:
7694:
7692:
7691:
7686:
7672:
7670:
7668:
7667:
7662:
7648:
7646:
7644:
7643:
7638:
7624:
7622:
7620:
7619:
7614:
7600:
7598:
7596:
7595:
7590:
7569:
7567:
7565:
7564:
7559:
7554:
7531:
7529:
7527:
7526:
7521:
7507:
7505:
7503:
7502:
7497:
7483:
7481:
7479:
7478:
7473:
7453:
7451:
7449:
7448:
7443:
7441:
7440:
7428:
7426:
7418:
7417:
7416:
7403:
7391:affine parameter
7388:
7386:
7384:
7383:
7378:
7361:
7359:
7358:
7353:
7351:
7347:
7340:
7339:
7321:
7320:
7305:
7304:
7295:
7294:
7274:
7273:
7232:
7230:
7229:
7224:
7222:
7221:
7206:
7202:
7201:
7200:
7188:
7187:
7147:
7145:
7144:
7139:
7137:
7133:
7132:
7130:
7123:
7122:
7111:
7106:
7097:
7092:
7088:
7087:
7086:
7074:
7073:
7059:
7058:
7038:
7037:
6997:
6995:
6994:
6989:
6987:
6971:
6966:
6965:
6964:
6952:
6951:
6941:
6936:
6932:
6931:
6930:
6912:
6911:
6878:
6876:
6868:
6860:
6839:
6831:
6826:
6822:
6821:
6819:
6812:
6811:
6801:
6800:
6791:
6765:
6763:
6755:
6747:
6738:
6724:
6712:
6710:
6702:
6694:
6685:
6671:
6659:
6657:
6649:
6641:
6625:
6623:
6621:
6620:
6615:
6574:
6572:
6571:
6566:
6561:
6557:
6556:
6555:
6550:
6546:
6544:
6533:
6532:
6523:
6513:
6509:
6508:
6507:
6495:
6494:
6480:
6479:
6459:
6458:
6445:
6440:
6414:
6412:
6410:
6409:
6404:
6380:
6378:
6377:
6372:
6370:
6369:
6357:
6356:
6340:
6338:
6337:
6332:
6327:
6326:
6301:
6299:
6297:
6296:
6291:
6289:
6288:
6270:
6268:
6266:
6265:
6260:
6242:
6240:
6238:
6237:
6232:
6230:
6229:
6211:
6209:
6208:
6203:
6198:
6197:
6188:
6187:
6175:
6174:
6162:
6161:
6142:
6140:
6138:
6137:
6132:
6074:
6072:
6070:
6069:
6064:
6050:
6048:
6046:
6045:
6040:
6026:
6024:
6022:
6021:
6016:
5980:global structure
5944:time translation
5899:gamma-ray bursts
5866:
5864:
5863:
5858:
5850:
5849:
5840:
5839:
5827:
5826:
5814:
5813:
5752:
5750:
5748:
5747:
5742:
5732:
5691:
5689:
5687:
5686:
5681:
5667:
5665:
5663:
5662:
5657:
5597:
5595:
5594:
5589:
5587:
5579:
5578:
5569:
5568:
5553:
5541:
5536:
5535:
5512:
5510:
5509:
5504:
5502:
5497:
5496:
5488:
5487:
5478:
5477:
5461:
5456:
5453:
5447:
5442:
5441:
5438:
5431:
5425:
5420:
5419:
5399:
5375:
5373:
5372:
5367:
5365:
5363:
5362:
5347:
5335:
5330:
5329:
5309:
5307:
5305:
5304:
5299:
5264:
5262:
5261:
5256:
5254:
5249:
5248:
5246:
5245:
5229:
5224:
5221:
5215:
5210:
5209:
5206:
5199:
5193:
5188:
5187:
5167:
5165:
5163:
5162:
5154:
5151:
5128:
5120:
5118:
5116:
5115:
5110:
5096:
5094:
5092:
5091:
5086:
5072:
5070:
5068:
5067:
5062:
5027:Mach's principle
5005:
4996:
4994:
4993:
4988:
4983:
4981:
4974:
4973:
4964:
4963:
4951:
4950:
4947:
4938:
4934:
4933:
4932:
4920:
4919:
4901:
4891:
4890:
4887:
4880:
4875:
4873:
4872:
4860:
4859:
4847:
4828:
4801:
4792:
4790:
4789:
4784:
4779:
4778:
4773:
4769:
4762:
4760:
4759:
4747:
4746:
4734:
4717:
4716:
4701:
4700:
4688:
4687:
4672:
4671:
4659:
4658:
4657:
4641:
4640:
4628:
4624:
4623:
4621:
4620:
4607:
4602:
4590:
4585:
4584:
4564:
4563:
4551:
4550:
4533:
4529:
4513:
4504:
4502:
4501:
4496:
4494:
4493:
4488:
4484:
4482:
4471:
4464:
4456:
4451:
4450:
4445:
4441:
4439:
4428:
4421:
4413:
4408:
4407:
4402:
4398:
4396:
4385:
4378:
4374:
4373:
4368:
4364:
4363:
4360:
4353:
4340:
4338:
4331:
4330:
4314:
4309:
4307:
4306:
4294:
4292:
4290:
4279:
4277:
4275:
4264:
4257:
4256:
4253:
4243:
4238:
4237:
4232:
4228:
4226:
4215:
4208:
4204:
4197:
4196:
4187:
4182:
4181:
4180:
4168:
4167:
4164:
4157:
4152:
4151:
4139:
4138:
4124:
4122:
4118:
4117:
4104:
4096:
4094:
4093:
4092:
4076:
4074:
4072:
4071:
4070:
4054:
4052:
4051:
4031:
4016:
4014:
4012:
4011:
4006:
4004:
4003:
3970:
3968:
3967:
3962:
3957:
3955:
3954:
3953:
3943:
3938:
3937:
3917:
3916:
3915:
3906:
3905:
3895:
3889:
3884:
3883:
3864:
3863:
3851:
3849:
3848:
3839:
3834:
3833:
3824:
3823:
3811:
3810:
3801:
3796:
3795:
3782:
3777:
3776:
3747:
3745:
3743:
3742:
3737:
3735:
3734:
3731:
3712:
3710:
3708:
3707:
3702:
3685:
3683:
3681:
3680:
3675:
3657:
3655:
3653:
3652:
3647:
3645:
3644:
3641:
3626:
3624:
3622:
3621:
3616:
3602:
3600:
3598:
3597:
3592:
3590:
3585:
3574:
3572:
3570:
3569:
3564:
3550:
3548:
3546:
3545:
3540:
3516:
3514:
3512:
3511:
3506:
3504:
3500:
3499:
3498:
3497:
3470:
3469:
3457:
3456:
3455:
3383:
3381:
3379:
3378:
3373:
3353:
3351:
3349:
3348:
3343:
3326:
3324:
3323:
3318:
3316:
3314:
3313:
3301:
3300:
3288:
3287:
3278:
3240:
3231:
3229:
3228:
3223:
3215:
3213:
3212:
3203:
3202:
3193:
3188:
3186:
3185:
3184:
3172:
3171:
3161:
3160:
3159:
3147:
3146:
3136:
3126:
3118:
3116:
3114:
3113:
3108:
3106:
3105:
3097:
3080:Minkowski tensor
3042:
3033:
3031:
3030:
3025:
3016:
3015:
2998:
2990:
2981:
2979:
2978:
2973:
2971:
2967:
2966:
2958:
2953:
2951:
2950:
2949:
2937:
2936:
2926:
2909:
2904:
2902:
2901:
2900:
2888:
2887:
2877:
2860:
2847:
2846:
2834:
2833:
2821:
2820:
2805:
2792:
2784:
2775:
2773:
2772:
2767:
2765:
2763:
2762:
2761:
2752:
2751:
2739:
2738:
2728:
2727:
2726:
2707:
2691:
2683:
2674:
2672:
2671:
2666:
2663:
2662:
2653:
2652:
2637:
2636:
2621:
2620:
2600:
2568:
2559:
2557:
2556:
2551:
2549:
2548:
2530:
2529:
2520:
2516:
2515:
2514:
2502:
2501:
2484:
2483:
2465:
2464:
2452:
2450:
2449:
2448:
2436:
2435:
2422:
2417:
2416:
2404:
2403:
2390:
2379:
2362:
2355:
2353:
2351:
2350:
2345:
2343:
2342:
2339:
2324:
2322:
2320:
2319:
2314:
2297:
2295:
2293:
2292:
2287:
2256:
2247:
2245:
2244:
2239:
2234:
2233:
2218:
2217:
2214:
2205:
2204:
2181:
2173:
2164:
2162:
2161:
2156:
2145:
2144:
2135:
2134:
2122:
2121:
2098:
2090:
2081:
2079:
2078:
2073:
2068:
2066:
2055:
2039:
2035:
2033:
2031:
2030:
2025:
2011:
2009:
2007:
2006:
2001:
1974:
1965:
1963:
1962:
1957:
1952:
1950:
1949:
1940:
1929:
1924:
1923:
1920:
1906:
1899:
1897:
1896:
1891:
1889:
1888:
1885:
1865:
1856:
1854:
1853:
1848:
1817:
1809:
1800:
1798:
1797:
1792:
1772:
1770:
1769:
1757:
1756:
1747:
1731:
1723:
1714:
1712:
1711:
1706:
1686:
1684:
1683:
1671:
1670:
1661:
1645:
1637:
1635:
1633:
1632:
1627:
1594:
1585:
1583:
1582:
1577:
1575:
1554:
1549:
1542:
1541:
1526:
1525:
1522:
1512:
1507:
1506:
1492:
1485:
1484:
1475:
1471:
1464:
1463:
1454:
1449:
1448:
1447:
1435:
1434:
1431:
1424:
1419:
1418:
1406:
1405:
1388:
1387:
1369:
1368:
1356:
1348:
1343:
1342:
1330:
1329:
1320:
1316:
1315:
1310:
1306:
1305:
1302:
1295:
1273:
1269:
1268:
1256:
1255:
1236:
1235:
1211:
1195:
1193:
1191:
1190:
1185:
1171:angular momentum
1168:
1166:
1164:
1163:
1158:
1078:
1076:
1074:
1073:
1068:
1066:
1065:
1062:
1024:angular momentum
1009:
958:
956:
955:
950:
948:
910:
908:
907:
902:
900:
873:
872:
866:angular momentum
824:, discovered by
759:
752:
745:
732:
727:
726:
719:
715:
714:
499:van Stockum dust
484:RobertsonâWalker
310:
309:
200:
199:
114:
112:
111:
106:
104:
103:
91:
83:
82:
64:
63:
44:
30:
29:
21:
13253:
13252:
13248:
13247:
13246:
13244:
13243:
13242:
13218:
13217:
13216:
13211:
13197:
13025:
12929:BKL singularity
12919:LemaĂźtreâTolman
12894:
12890:Quantum gravity
12872:
12866:
12852:geodetic effect
12826:(together with
12796:LISA Pathfinder
12735:
12684:
12670:Penrose diagram
12652:
12646:
12621:
12610:
12606:Minkowski space
12572:
12516:
12500:
12448:
12442:
12402:
12395:
12390:
12360:
12355:
12341:Tipler cylinder
12300:
12287:
12244:
12208:
12185:
12148:
12112:
12107:
12077:
12072:
12044:
12020:ULAS J1342+0928
11980:SDSS J0849+1114
11965:Phoenix Cluster
11918:
11770:
11732:
11711:
11650:
11609:
11605:No-hair theorem
11558:
11512:Bondi accretion
11478:Penrose process
11424:
11390:Gamma-ray burst
11317:
11227:
11218:
11204:Direct collapse
11152:
11139:
11134:
11026:
10918:gr-qc/0012100v1
10888:
10861:
10838:
10812:
10789:
10770:
10746:
10727:
10715:, eds. (2009).
10713:Scott, Susan M.
10707:
10705:Further reading
10702:
10663:
10659:
10654:
10650:
10639:
10635:
10631:(JILA Colorado)
10626:
10622:
10608:
10604:
10547:
10543:
10493:
10489:
10458:
10454:
10446:
10412:Physical Review
10407:
10401:Carter, Brandon
10398:
10389:
10372:
10365:
10354:
10347:
10339:
10333:
10329:
10277:
10273:
10264:
10260:
10250:Thibault Damour
10248:
10244:
10220:10.1.1.512.1400
10195:
10191:
10185:Wayback Machine
10175:
10166:
10156:
10138:
10134:
10117:
10113:
10096:
10092:
10081:
10077:
10028:
10024:
9992:
9988:
9957:
9953:
9936:
9925:
9916:
9912:
9905:
9889:
9885:
9878:
9864:Lifshitz, E. M.
9857:
9853:
9798:
9794:
9751:
9747:
9692:
9688:
9645:
9641:
9584:
9580:
9573:livescience.com
9565:
9561:
9504:
9500:
9488:
9484:
9450:
9446:
9442:
9437:
9429:
9425:
9421:
9378:
9371:
9364:
9359:
9357:
9354:
9321:
9317:
9312:
9309:
9308:
9306:
9295:Wahlquist fluid
9270:
9246:TaubâNUT vacuum
9207:
9204:
9203:
9201:
9145:
9141:
9127:
9123:
9121:
9120:
9116:
9104:
9100:
9084:
9080:
9064:
9060:
9058:
9055:
9054:
9004:
9001:
9000:
8998:
8974:
8970:
8949:
8945:
8943:
8940:
8939:
8916:
8839:
8836:
8835:
8833:
8808:vacuum solution
8798:
8762:
8759:
8758:
8756:
8735:
8732:
8731:
8729:
8707:
8704:
8703:
8701:
8673:
8670:
8669:
8667:
8660:
8649:
8643:
8640:
8629:
8617:
8606:
8569:
8565:
8560:
8557:
8556:
8554:
8527:
8521:
8517:
8515:
8512:
8511:
8509:
8485:
8482:
8481:
8479:
8476:
8453:
8420:
8417:
8416:
8414:
8390:
8389:
8384:
8375:
8371:
8354:
8353:
8345:
8342:
8341:
8325:
8312:
8305:
8299:
8295:
8293:
8285:
8272:
8268:
8267:
8258:
8253:
8241:
8238:
8237:
8219:
8216:
8215:
8213:
8195:
8192:
8191:
8189:
8169:
8165:
8161:
8156:
8133:
8129:
8125:
8123:
8122:
8118:
8095:
8094:
8092:
8089:
8088:
8069:
8065:
8061:
8054:
8050:
8044:
8040:
8033:
8031:
8008:
8004:
8000:
7998:
7997:
7993:
7953:
7952:
7933:
7931:
7917:
7916:
7899:
7898:
7897:
7893:
7882:
7878:
7873:
7847:
7846:
7829:
7828:
7826:
7823:
7822:
7794:
7781:
7777:
7768:
7764:
7755:
7744:
7740:
7731:
7727:
7726:
7722:
7721:
7720:
7717:
7704:
7702:
7699:
7698:
7680:
7677:
7676:
7674:
7656:
7653:
7652:
7650:
7632:
7629:
7628:
7626:
7608:
7605:
7604:
7602:
7584:
7581:
7580:
7578:
7573:Because of the
7550:
7539:
7536:
7535:
7533:
7515:
7512:
7511:
7509:
7491:
7488:
7487:
7485:
7461:
7458:
7457:
7455:
7436:
7432:
7419:
7412:
7408:
7404:
7402:
7400:
7397:
7396:
7394:
7372:
7369:
7368:
7366:
7335:
7331:
7316:
7312:
7300:
7296:
7290:
7286:
7285:
7281:
7269:
7265:
7239:
7236:
7235:
7217:
7213:
7196:
7192:
7183:
7179:
7178:
7174:
7154:
7151:
7150:
7118:
7114:
7113:
7107:
7102:
7096:
7082:
7078:
7069:
7065:
7064:
7060:
7054:
7050:
7049:
7045:
7033:
7029:
7006:
7003:
7002:
6985:
6984:
6960:
6956:
6947:
6943:
6942:
6940:
6926:
6922:
6907:
6903:
6896:
6892:
6879:
6869:
6861:
6859:
6853:
6852:
6830:
6807:
6803:
6802:
6796:
6792:
6790:
6780:
6776:
6766:
6756:
6748:
6746:
6740:
6739:
6723:
6713:
6703:
6695:
6693:
6687:
6686:
6670:
6660:
6650:
6642:
6640:
6633:
6631:
6628:
6627:
6591:
6588:
6587:
6585:
6551:
6534:
6528:
6524:
6522:
6518:
6517:
6503:
6499:
6490:
6486:
6485:
6481:
6475:
6471:
6470:
6466:
6454:
6450:
6441:
6436:
6424:
6421:
6420:
6417:Carter constant
6398:
6395:
6394:
6392:
6365:
6361:
6352:
6348:
6346:
6343:
6342:
6322:
6318:
6307:
6304:
6303:
6284:
6280:
6278:
6275:
6274:
6272:
6254:
6251:
6250:
6248:
6225:
6221:
6219:
6216:
6215:
6213:
6193:
6189:
6180:
6176:
6170:
6166:
6157:
6153:
6148:
6145:
6144:
6126:
6123:
6122:
6120:
6081:
6058:
6055:
6054:
6052:
6034:
6031:
6030:
6028:
6004:
6001:
6000:
5998:
5923:Cauchy horizons
5907:
5895:Penrose process
5845:
5841:
5835:
5831:
5822:
5818:
5809:
5805:
5791:
5788:
5787:
5765:
5763:Penrose process
5759:
5724:
5721:
5720:
5718:
5716:
5705:
5675:
5672:
5671:
5669:
5651:
5648:
5647:
5645:
5643:
5623:
5574:
5570:
5564:
5560:
5552:
5537:
5531:
5530:
5524:
5521:
5520:
5483:
5479:
5473:
5469:
5457:
5452:
5446:
5437:
5433:
5432:
5430:
5421:
5415:
5414:
5408:
5405:
5404:
5397:
5391:
5389:
5358:
5354:
5346:
5331:
5325:
5324:
5318:
5315:
5314:
5275:
5272:
5271:
5269:
5241:
5237:
5225:
5220:
5214:
5205:
5201:
5200:
5198:
5189:
5183:
5182:
5176:
5173:
5172:
5161:
5155:
5152:
5149:
5148:
5146:
5145:
5143:
5104:
5101:
5100:
5098:
5080:
5077:
5076:
5074:
5056:
5053:
5052:
5050:
5035:
4969:
4965:
4959:
4955:
4946:
4942:
4928:
4924:
4915:
4911:
4910:
4906:
4902:
4886:
4882:
4881:
4879:
4865:
4861:
4852:
4848:
4846:
4835:
4832:
4831:
4823:Killing horizon
4774:
4752:
4748:
4739:
4735:
4733:
4723:
4719:
4718:
4709:
4705:
4696:
4692:
4680:
4676:
4667:
4663:
4650:
4649:
4645:
4636:
4632:
4613:
4609:
4603:
4595:
4589:
4577:
4573:
4572:
4568:
4559:
4555:
4546:
4542:
4540:
4537:
4536:
4524:
4489:
4475:
4470:
4466:
4465:
4455:
4446:
4432:
4427:
4423:
4422:
4412:
4403:
4389:
4384:
4380:
4379:
4359:
4355:
4354:
4352:
4345:
4341:
4326:
4322:
4318:
4313:
4302:
4298:
4293:
4283:
4278:
4265:
4252:
4248:
4244:
4242:
4233:
4219:
4214:
4210:
4209:
4192:
4188:
4176:
4172:
4163:
4159:
4158:
4156:
4147:
4143:
4134:
4130:
4129:
4125:
4113:
4109:
4108:
4103:
4088:
4084:
4080:
4075:
4066:
4062:
4058:
4053:
4044:
4040:
4038:
4035:
4034:
3996:
3992:
3990:
3987:
3986:
3984:
3977:
3949:
3945:
3939:
3927:
3926:
3918:
3911:
3907:
3901:
3897:
3896:
3894:
3885:
3873:
3872:
3859:
3855:
3844:
3840:
3835:
3829:
3825:
3819:
3815:
3806:
3802:
3800:
3791:
3787:
3778:
3766:
3765:
3756:
3753:
3752:
3748:are related by
3730:
3726:
3724:
3721:
3720:
3718:
3696:
3693:
3692:
3690:
3669:
3666:
3665:
3663:
3640:
3636:
3634:
3631:
3630:
3628:
3610:
3607:
3606:
3604:
3584:
3582:
3579:
3578:
3576:
3558:
3555:
3554:
3552:
3528:
3525:
3524:
3522:
3519:Penrose process
3487:
3486:
3482:
3475:
3471:
3465:
3461:
3445:
3444:
3440:
3438:
3435:
3434:
3432:
3429:
3413:KerrâNUT metric
3409:
3361:
3358:
3357:
3355:
3337:
3334:
3333:
3331:
3309:
3305:
3296:
3292:
3283:
3279:
3277:
3263:
3260:
3259:
3208:
3204:
3198:
3194:
3192:
3180:
3176:
3167:
3163:
3162:
3155:
3151:
3142:
3138:
3137:
3135:
3133:
3130:
3129:
3096:
3095:
3093:
3090:
3089:
3087:
3011:
3007:
3005:
3002:
3001:
2957:
2945:
2941:
2932:
2928:
2927:
2910:
2908:
2896:
2892:
2883:
2879:
2878:
2861:
2859:
2858:
2854:
2842:
2838:
2829:
2825:
2816:
2812:
2801:
2799:
2796:
2795:
2757:
2753:
2747:
2743:
2734:
2730:
2729:
2722:
2718:
2708:
2706:
2698:
2695:
2694:
2658:
2654:
2648:
2644:
2629:
2625:
2613:
2609:
2607:
2604:
2603:
2579:
2544:
2540:
2525:
2521:
2510:
2506:
2497:
2493:
2492:
2488:
2479:
2475:
2460:
2456:
2444:
2440:
2431:
2427:
2426:
2421:
2412:
2408:
2399:
2395:
2380:
2375:
2369:
2366:
2365:
2338:
2334:
2332:
2329:
2328:
2326:
2308:
2305:
2304:
2302:
2271:
2268:
2267:
2265:
2229:
2225:
2213:
2209:
2200:
2196:
2188:
2185:
2184:
2140:
2136:
2130:
2126:
2117:
2113:
2105:
2102:
2101:
2059:
2054:
2046:
2043:
2042:
2019:
2016:
2015:
2013:
1989:
1986:
1985:
1983:
1945:
1941:
1930:
1928:
1919:
1915:
1913:
1910:
1909:
1884:
1880:
1878:
1875:
1874:
1824:
1821:
1820:
1765:
1761:
1752:
1748:
1746:
1738:
1735:
1734:
1679:
1675:
1666:
1662:
1660:
1652:
1649:
1648:
1609:
1606:
1605:
1603:
1573:
1572:
1537:
1533:
1521:
1517:
1513:
1511:
1502:
1498:
1480:
1476:
1459:
1455:
1443:
1439:
1430:
1426:
1425:
1423:
1414:
1410:
1401:
1397:
1396:
1392:
1383:
1379:
1364:
1360:
1347:
1338:
1334:
1325:
1321:
1301:
1297:
1296:
1294:
1287:
1283:
1271:
1270:
1264:
1260:
1251:
1247:
1237:
1231:
1227:
1220:
1218:
1215:
1214:
1179:
1176:
1175:
1173:
1152:
1149:
1148:
1146:
1139:
1133:
1113:
1085:
1061:
1057:
1049:
1046:
1045:
1043:
1007:
987:Gravity Probe B
981:(also known as
944:
942:
939:
938:
896:
894:
891:
890:
858:electric charge
818:
763:
722:
709:
708:
701:
700:
524:
523:
514:
513:
469:LemaĂźtreâTolman
414:
413:
402:
401:
393:Quantum gravity
380:Advanced theory
307:
306:
305:
288:
287:
236:Geodetic effect
197:
196:
187:
186:
162:
161:
145:
115:
96:
92:
87:
75:
71:
56:
52:
50:
47:
46:
28:
23:
22:
15:
12:
11:
5:
13251:
13241:
13240:
13238:Metric tensors
13235:
13230:
13213:
13212:
13202:
13199:
13198:
13196:
13195:
13188:
13183:
13178:
13173:
13168:
13163:
13158:
13153:
13148:
13143:
13138:
13133:
13128:
13123:
13118:
13116:Choquet-Bruhat
13113:
13108:
13103:
13098:
13093:
13088:
13083:
13078:
13073:
13068:
13063:
13058:
13053:
13048:
13043:
13037:
13035:
13031:
13030:
13027:
13026:
13024:
13023:
13016:
13015:
13010:
13005:
12998:
12997:
12992:
12987:
12982:
12977:
12968:Axisymmetric:
12965:
12964:
12959:
12953:
12942:
12941:
12936:
12931:
12926:
12921:
12916:
12907:Cosmological:
12904:
12902:
12896:
12895:
12893:
12892:
12887:
12882:
12876:
12874:
12868:
12867:
12865:
12864:
12859:
12848:frame-dragging
12845:
12840:
12835:
12832:Einstein rings
12828:Einstein cross
12821:
12810:
12809:
12804:
12798:
12793:
12788:
12775:
12765:
12764:
12759:
12754:
12749:
12743:
12741:
12737:
12736:
12734:
12733:
12731:Ernst equation
12728:
12723:
12718:
12713:
12708:
12703:
12701:BSSN formalism
12698:
12692:
12690:
12686:
12685:
12683:
12682:
12677:
12672:
12667:
12662:
12656:
12654:
12648:
12647:
12645:
12644:
12639:
12633:
12631:
12624:
12616:
12615:
12612:
12611:
12609:
12608:
12603:
12598:
12593:
12588:
12582:
12580:
12574:
12573:
12571:
12570:
12565:
12560:
12558:Ladder paradox
12555:
12550:
12545:
12540:
12535:
12530:
12524:
12522:
12518:
12517:
12515:
12514:
12508:
12506:
12502:
12501:
12499:
12498:
12493:
12488:
12483:
12478:
12473:
12468:
12463:
12461:Speed of light
12458:
12452:
12450:
12444:
12443:
12441:
12440:
12435:
12430:
12424:
12414:
12412:
12405:
12397:
12396:
12389:
12388:
12381:
12374:
12366:
12357:
12356:
12354:
12353:
12348:
12343:
12338:
12333:
12331:Krasnikov tube
12328:
12323:
12318:
12316:BTZ black hole
12313:
12307:
12305:
12289:
12288:
12286:
12285:
12283:Predestination
12280:
12275:
12270:
12265:
12260:
12254:
12252:
12246:
12245:
12243:
12242:
12237:
12232:
12227:
12222:
12216:
12214:
12210:
12209:
12207:
12206:
12201:
12195:
12193:
12187:
12186:
12184:
12183:
12182:
12181:
12171:
12170:
12169:
12158:
12156:
12150:
12149:
12147:
12146:
12141:
12136:
12131:
12126:
12120:
12118:
12114:
12113:
12106:
12105:
12098:
12091:
12083:
12074:
12073:
12071:
12070:
12060:
12049:
12046:
12045:
12043:
12042:
12040:Swift J1644+57
12037:
12032:
12027:
12022:
12017:
12012:
12007:
12002:
11997:
11992:
11990:MS 0735.6+7421
11987:
11982:
11977:
11972:
11967:
11962:
11957:
11955:Sagittarius A*
11952:
11947:
11942:
11937:
11932:
11926:
11924:
11920:
11919:
11917:
11916:
11911:
11906:
11901:
11896:
11891:
11886:
11881:
11876:
11871:
11866:
11861:
11856:
11851:
11846:
11841:
11836:
11831:
11830:
11829:
11824:
11814:
11809:
11804:
11799:
11794:
11789:
11784:
11778:
11776:
11772:
11771:
11769:
11768:
11763:
11758:
11753:
11748:
11742:
11740:
11734:
11733:
11731:
11730:
11725:
11719:
11717:
11713:
11712:
11710:
11709:
11704:
11699:
11694:
11689:
11684:
11679:
11674:
11669:
11664:
11658:
11656:
11652:
11651:
11649:
11648:
11643:
11638:
11633:
11628:
11617:
11615:
11611:
11610:
11608:
11607:
11602:
11597:
11592:
11587:
11582:
11577:
11572:
11566:
11564:
11560:
11559:
11557:
11556:
11551:
11546:
11545:
11544:
11534:
11529:
11527:Thermodynamics
11524:
11519:
11514:
11509:
11508:
11507:
11497:
11492:
11490:Accretion disk
11487:
11486:
11485:
11480:
11470:
11465:
11460:
11455:
11454:
11453:
11448:
11438:
11432:
11430:
11426:
11425:
11423:
11422:
11417:
11412:
11407:
11402:
11397:
11392:
11387:
11386:
11385:
11380:
11375:
11365:
11364:
11363:
11353:
11348:
11347:
11346:
11336:
11331:
11325:
11323:
11319:
11318:
11316:
11315:
11314:
11313:
11308:
11303:
11298:
11293:
11288:
11283:
11273:
11268:
11267:
11266:
11256:
11255:
11254:
11251:
11246:
11235:
11233:
11229:
11228:
11221:
11219:
11217:
11216:
11211:
11206:
11201:
11196:
11191:
11186:
11181:
11176:
11171:
11166:
11164:BTZ black hole
11160:
11158:
11154:
11153:
11151:
11150:
11144:
11141:
11140:
11133:
11132:
11125:
11118:
11110:
11104:
11103:
11067:
11030:
11024:
11004:
10986:(6): 331â333.
10972:
10924:
10908:
10896:
10886:
10869:
10866:See chapter 13
10859:
10846:
10836:
10820:
10817:See chapter 19
10810:
10793:
10787:
10774:
10768:
10750:
10744:
10731:
10725:
10706:
10703:
10701:
10700:
10657:
10648:
10633:
10620:
10602:
10541:
10503:(12): 124010.
10487:
10482:10.1086/151796
10452:
10449:on 2020-02-25.
10387:
10363:
10345:
10327:
10271:
10258:
10242:
10189:
10164:
10155:978-0521805865
10154:
10132:
10111:
10090:
10075:
10022:
9986:
9967:(2): 265â281.
9951:
9923:
9910:
9903:
9883:
9876:
9851:
9792:
9745:
9686:
9659:(6): 915â917.
9639:
9578:
9559:
9498:
9495:978-0226519517
9482:
9463:(5): 237â238.
9443:
9441:
9438:
9436:
9435:
9422:
9420:
9417:
9416:
9415:
9410:
9405:
9400:
9395:
9390:
9384:
9383:
9380:Physics portal
9369:
9353:
9350:
9335:
9332:
9329:
9324:
9320:
9316:
9269:
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9262:
9261:
9250:
9237:
9211:
9167:
9166:
9154:
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9144:
9140:
9135:
9130:
9126:
9119:
9115:
9112:
9107:
9103:
9098:
9095:
9092:
9087:
9083:
9078:
9075:
9072:
9067:
9063:
9014:
9011:
9008:
8991:
8990:
8977:
8973:
8969:
8966:
8963:
8960:
8957:
8952:
8948:
8915:
8912:
8849:
8846:
8843:
8797:
8794:
8766:
8739:
8726:Cauchy horizon
8711:
8683:
8680:
8677:
8662:
8661:
8620:
8618:
8611:
8605:
8602:
8583:
8580:
8577:
8572:
8568:
8564:
8540:
8537:
8534:
8530:
8520:
8495:
8492:
8489:
8475:
8472:
8457:Poincaré group
8452:
8449:
8447:are restored.
8430:
8427:
8424:
8397:
8394:
8387:
8383:
8378:
8374:
8370:
8367:
8361:
8358:
8352:
8349:
8328:
8323:
8315:
8311:
8308:
8302:
8298:
8292:
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8256:
8252:
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8245:
8223:
8199:
8172:
8168:
8164:
8160:
8154:
8150:
8147:
8142:
8136:
8132:
8128:
8121:
8117:
8114:
8111:
8108:
8102:
8099:
8072:
8068:
8064:
8057:
8053:
8047:
8043:
8039:
8036:
8029:
8025:
8022:
8017:
8011:
8007:
8003:
7996:
7992:
7989:
7986:
7983:
7980:
7977:
7974:
7971:
7967:
7960:
7957:
7949:
7945:
7942:
7939:
7936:
7930:
7924:
7921:
7915:
7912:
7906:
7903:
7896:
7892:
7885:
7881:
7877:
7872:
7869:
7866:
7863:
7860:
7854:
7851:
7845:
7842:
7836:
7833:
7810:
7803:
7797:
7792:
7789:
7784:
7780:
7776:
7771:
7767:
7763:
7758:
7753:
7747:
7743:
7739:
7734:
7730:
7725:
7716:
7711:
7708:
7684:
7660:
7636:
7612:
7588:
7575:frame-dragging
7557:
7553:
7549:
7546:
7543:
7519:
7495:
7471:
7468:
7465:
7439:
7435:
7431:
7425:
7422:
7415:
7411:
7407:
7376:
7363:
7362:
7350:
7346:
7343:
7338:
7334:
7330:
7327:
7324:
7319:
7315:
7311:
7308:
7303:
7299:
7293:
7289:
7284:
7280:
7277:
7272:
7268:
7264:
7261:
7258:
7255:
7252:
7249:
7246:
7243:
7233:
7220:
7216:
7212:
7209:
7205:
7199:
7195:
7191:
7186:
7182:
7177:
7173:
7170:
7167:
7164:
7161:
7158:
7148:
7136:
7129:
7126:
7121:
7117:
7110:
7105:
7101:
7095:
7091:
7085:
7081:
7077:
7072:
7068:
7063:
7057:
7053:
7048:
7044:
7041:
7036:
7032:
7028:
7025:
7022:
7019:
7016:
7013:
7010:
6983:
6980:
6977:
6974:
6969:
6963:
6959:
6955:
6950:
6946:
6939:
6935:
6929:
6925:
6921:
6918:
6915:
6910:
6906:
6902:
6899:
6895:
6891:
6888:
6885:
6882:
6880:
6875:
6872:
6867:
6864:
6858:
6855:
6854:
6851:
6848:
6845:
6842:
6837:
6834:
6829:
6825:
6818:
6815:
6810:
6806:
6799:
6795:
6789:
6786:
6783:
6779:
6775:
6772:
6769:
6767:
6762:
6759:
6754:
6751:
6745:
6742:
6741:
6736:
6733:
6730:
6727:
6722:
6719:
6716:
6714:
6709:
6706:
6701:
6698:
6692:
6689:
6688:
6683:
6680:
6677:
6674:
6669:
6666:
6663:
6661:
6656:
6653:
6648:
6645:
6639:
6636:
6635:
6613:
6610:
6607:
6604:
6601:
6598:
6595:
6564:
6560:
6554:
6549:
6543:
6540:
6537:
6531:
6527:
6521:
6516:
6512:
6506:
6502:
6498:
6493:
6489:
6484:
6478:
6474:
6469:
6465:
6462:
6457:
6453:
6449:
6444:
6439:
6435:
6431:
6428:
6402:
6389:Brandon Carter
6368:
6364:
6360:
6355:
6351:
6330:
6325:
6321:
6317:
6314:
6311:
6287:
6283:
6258:
6228:
6224:
6201:
6196:
6192:
6186:
6183:
6179:
6173:
6169:
6165:
6160:
6156:
6152:
6130:
6113:test particles
6080:
6077:
6062:
6038:
6014:
6011:
6008:
5995:photon spheres
5953:Killing tensor
5919:event horizons
5906:
5903:
5868:
5867:
5856:
5853:
5848:
5844:
5838:
5834:
5830:
5825:
5821:
5817:
5812:
5808:
5804:
5801:
5798:
5795:
5761:Main article:
5758:
5755:
5740:
5737:
5731:
5728:
5714:
5703:
5679:
5655:
5641:
5621:
5601:Due to the cos
5599:
5598:
5585:
5582:
5577:
5573:
5567:
5563:
5559:
5556:
5551:
5548:
5545:
5540:
5534:
5529:
5514:
5513:
5500:
5494:
5491:
5486:
5482:
5476:
5472:
5468:
5465:
5460:
5451:
5445:
5436:
5429:
5424:
5418:
5413:
5395:
5387:
5377:
5376:
5361:
5357:
5353:
5350:
5345:
5342:
5339:
5334:
5328:
5323:
5297:
5294:
5291:
5288:
5285:
5282:
5279:
5266:
5265:
5252:
5244:
5240:
5236:
5233:
5228:
5219:
5213:
5204:
5197:
5192:
5186:
5181:
5159:
5141:
5108:
5084:
5060:
5034:
5031:
5023:planetary gear
5015:frame-dragging
5009:
5008:
4999:
4997:
4986:
4980:
4977:
4972:
4968:
4962:
4958:
4954:
4945:
4941:
4937:
4931:
4927:
4923:
4918:
4914:
4909:
4905:
4900:
4897:
4894:
4885:
4878:
4871:
4868:
4864:
4858:
4855:
4851:
4845:
4842:
4839:
4805:
4804:
4795:
4793:
4782:
4777:
4772:
4768:
4765:
4758:
4755:
4751:
4745:
4742:
4738:
4732:
4729:
4726:
4722:
4715:
4712:
4708:
4704:
4699:
4695:
4691:
4686:
4683:
4679:
4675:
4670:
4666:
4662:
4656:
4653:
4648:
4644:
4639:
4635:
4631:
4627:
4619:
4616:
4612:
4606:
4601:
4598:
4594:
4588:
4583:
4580:
4576:
4571:
4567:
4562:
4558:
4554:
4549:
4545:
4523:
4522:Frame dragging
4520:
4517:
4516:
4507:
4505:
4492:
4487:
4481:
4478:
4474:
4469:
4462:
4459:
4454:
4449:
4444:
4438:
4435:
4431:
4426:
4419:
4416:
4411:
4406:
4401:
4395:
4392:
4388:
4383:
4377:
4371:
4367:
4358:
4351:
4348:
4344:
4337:
4334:
4329:
4325:
4321:
4317:
4312:
4305:
4301:
4297:
4289:
4286:
4282:
4274:
4271:
4268:
4263:
4260:
4251:
4247:
4241:
4236:
4231:
4225:
4222:
4218:
4213:
4207:
4203:
4200:
4195:
4191:
4185:
4179:
4175:
4171:
4162:
4155:
4150:
4146:
4142:
4137:
4133:
4128:
4121:
4116:
4112:
4107:
4102:
4099:
4091:
4087:
4083:
4079:
4069:
4065:
4061:
4057:
4050:
4047:
4043:
4002:
3999:
3995:
3976:
3973:
3972:
3971:
3960:
3952:
3948:
3942:
3936:
3933:
3930:
3925:
3921:
3914:
3910:
3904:
3900:
3893:
3888:
3882:
3879:
3876:
3871:
3867:
3862:
3858:
3854:
3847:
3843:
3838:
3832:
3828:
3822:
3818:
3814:
3809:
3805:
3799:
3794:
3790:
3786:
3781:
3775:
3772:
3769:
3764:
3760:
3729:
3700:
3673:
3639:
3614:
3588:
3562:
3538:
3535:
3532:
3503:
3496:
3493:
3490:
3485:
3481:
3478:
3474:
3468:
3464:
3460:
3454:
3451:
3448:
3443:
3428:
3425:
3408:
3405:
3371:
3368:
3365:
3341:
3328:
3327:
3312:
3308:
3304:
3299:
3295:
3291:
3286:
3282:
3276:
3273:
3270:
3267:
3244:
3243:
3234:
3232:
3221:
3218:
3211:
3207:
3201:
3197:
3191:
3183:
3179:
3175:
3170:
3166:
3158:
3154:
3150:
3145:
3141:
3103:
3100:
3046:
3045:
3036:
3034:
3022:
3019:
3014:
3010:
2994:
2993:
2984:
2982:
2970:
2964:
2961:
2956:
2948:
2944:
2940:
2935:
2931:
2925:
2922:
2919:
2916:
2913:
2907:
2899:
2895:
2891:
2886:
2882:
2876:
2873:
2870:
2867:
2864:
2857:
2853:
2850:
2845:
2841:
2837:
2832:
2828:
2824:
2819:
2815:
2811:
2808:
2804:
2788:
2787:
2778:
2776:
2760:
2756:
2750:
2746:
2742:
2737:
2733:
2725:
2721:
2717:
2714:
2711:
2705:
2702:
2687:
2686:
2677:
2675:
2661:
2657:
2651:
2647:
2643:
2640:
2635:
2632:
2628:
2624:
2619:
2616:
2612:
2578:
2575:
2572:
2571:
2562:
2560:
2547:
2543:
2539:
2536:
2533:
2528:
2524:
2519:
2513:
2509:
2505:
2500:
2496:
2491:
2487:
2482:
2478:
2474:
2471:
2468:
2463:
2459:
2455:
2447:
2443:
2439:
2434:
2430:
2425:
2420:
2415:
2411:
2407:
2402:
2398:
2394:
2389:
2386:
2383:
2378:
2373:
2337:
2312:
2285:
2282:
2278:
2275:
2260:
2259:
2250:
2248:
2237:
2232:
2228:
2224:
2221:
2212:
2208:
2203:
2199:
2195:
2192:
2177:
2176:
2167:
2165:
2154:
2151:
2148:
2143:
2139:
2133:
2129:
2125:
2120:
2116:
2112:
2109:
2094:
2093:
2084:
2082:
2071:
2065:
2062:
2058:
2053:
2050:
2023:
1999:
1996:
1993:
1978:
1977:
1968:
1966:
1955:
1948:
1944:
1939:
1936:
1933:
1927:
1918:
1883:
1869:
1868:
1859:
1857:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1813:
1812:
1803:
1801:
1790:
1787:
1784:
1781:
1778:
1775:
1768:
1764:
1760:
1755:
1751:
1745:
1742:
1727:
1726:
1717:
1715:
1704:
1701:
1698:
1695:
1692:
1689:
1682:
1678:
1674:
1669:
1665:
1659:
1656:
1625:
1622:
1619:
1616:
1613:
1598:
1597:
1588:
1586:
1571:
1568:
1564:
1561:
1557:
1552:
1548:
1545:
1540:
1536:
1532:
1529:
1520:
1516:
1510:
1505:
1501:
1497:
1491:
1488:
1483:
1479:
1474:
1470:
1467:
1462:
1458:
1452:
1446:
1442:
1438:
1429:
1422:
1417:
1413:
1409:
1404:
1400:
1395:
1391:
1386:
1382:
1378:
1375:
1372:
1367:
1363:
1359:
1354:
1351:
1346:
1341:
1337:
1333:
1328:
1324:
1319:
1313:
1309:
1300:
1293:
1290:
1286:
1282:
1279:
1276:
1274:
1272:
1267:
1263:
1259:
1254:
1250:
1246:
1243:
1240:
1238:
1234:
1230:
1226:
1223:
1222:
1183:
1169:rotating with
1156:
1135:Main article:
1132:
1129:
1125:Ernst equation
1112:
1109:
1094:invariant mass
1083:
1060:
1056:
1053:
979:frame-dragging
975:
974:
971:
970:
965:
960:
947:
930:
929:
926:
921:
913:
912:
899:
883:
878:Non-rotating (
876:
817:
814:
798:exact solution
765:
764:
762:
761:
754:
747:
739:
736:
735:
734:
733:
720:
703:
702:
699:
698:
691:
686:
681:
676:
671:
666:
661:
656:
651:
646:
641:
636:
631:
626:
621:
616:
611:
606:
601:
596:
591:
586:
581:
576:
571:
566:
561:
556:
551:
546:
541:
536:
531:
525:
521:
520:
519:
516:
515:
512:
511:
506:
501:
496:
491:
486:
481:
476:
471:
466:
461:
456:
451:
446:
441:
436:
431:
426:
415:
409:
408:
407:
404:
403:
398:
397:
396:
395:
390:
382:
381:
377:
376:
375:
374:
372:Post-Newtonian
369:
364:
356:
355:
351:
350:
349:
348:
343:
338:
333:
328:
323:
315:
314:
308:
304:
303:
300:
296:
295:
294:
293:
290:
289:
284:
283:
282:
281:
276:
271:
263:
262:
256:
255:
254:
253:
248:
243:
238:
233:
231:Frame-dragging
228:
223:
218:
213:
208:
206:Kepler problem
198:
194:
193:
192:
189:
188:
185:
184:
179:
174:
169:
163:
159:
158:
157:
154:
153:
152:
151:
146:
144:
143:
138:
133:
127:
125:
117:
116:
102:
99:
95:
90:
86:
81:
78:
74:
70:
67:
62:
59:
55:
45:
37:
36:
26:
9:
6:
4:
3:
2:
13250:
13239:
13236:
13234:
13231:
13229:
13226:
13225:
13223:
13210:
13200:
13194:
13193:
13189:
13187:
13184:
13182:
13179:
13177:
13174:
13172:
13169:
13167:
13164:
13162:
13159:
13157:
13154:
13152:
13149:
13147:
13144:
13142:
13139:
13137:
13134:
13132:
13129:
13127:
13124:
13122:
13119:
13117:
13114:
13112:
13109:
13107:
13104:
13102:
13101:Chandrasekhar
13099:
13097:
13094:
13092:
13089:
13087:
13084:
13082:
13079:
13077:
13074:
13072:
13069:
13067:
13064:
13062:
13061:Schwarzschild
13059:
13057:
13054:
13052:
13049:
13047:
13044:
13042:
13039:
13038:
13036:
13032:
13022:
13018:
13017:
13014:
13011:
13009:
13006:
13004:
13000:
12999:
12996:
12993:
12991:
12988:
12986:
12983:
12981:
12978:
12975:
12971:
12967:
12966:
12963:
12960:
12957:
12954:
12952:
12948:
12947:Schwarzschild
12944:
12943:
12940:
12937:
12935:
12932:
12930:
12927:
12925:
12922:
12920:
12917:
12914:
12910:
12906:
12905:
12903:
12901:
12897:
12891:
12888:
12886:
12883:
12881:
12878:
12877:
12875:
12869:
12863:
12860:
12857:
12853:
12849:
12846:
12844:
12843:Shapiro delay
12841:
12839:
12836:
12833:
12829:
12825:
12822:
12819:
12815:
12812:
12811:
12808:
12805:
12802:
12799:
12797:
12794:
12792:
12789:
12787:
12786:collaboration
12783:
12779:
12776:
12774:
12770:
12767:
12766:
12763:
12760:
12758:
12755:
12753:
12752:Event horizon
12750:
12748:
12745:
12744:
12742:
12738:
12732:
12729:
12727:
12724:
12722:
12719:
12717:
12714:
12712:
12709:
12707:
12704:
12702:
12699:
12697:
12696:ADM formalism
12694:
12693:
12691:
12687:
12681:
12678:
12676:
12673:
12671:
12668:
12666:
12663:
12661:
12658:
12657:
12655:
12649:
12643:
12640:
12638:
12635:
12634:
12632:
12628:
12625:
12623:
12617:
12607:
12604:
12602:
12601:Biquaternions
12599:
12597:
12594:
12592:
12589:
12587:
12584:
12583:
12581:
12579:
12575:
12569:
12566:
12564:
12561:
12559:
12556:
12554:
12551:
12549:
12546:
12544:
12541:
12539:
12536:
12534:
12531:
12529:
12528:Time dilation
12526:
12525:
12523:
12519:
12513:
12510:
12509:
12507:
12503:
12497:
12494:
12492:
12489:
12487:
12484:
12482:
12481:Proper length
12479:
12477:
12474:
12472:
12469:
12467:
12464:
12462:
12459:
12457:
12454:
12453:
12451:
12445:
12439:
12436:
12434:
12431:
12428:
12425:
12423:
12419:
12416:
12415:
12413:
12409:
12406:
12404:
12398:
12394:
12387:
12382:
12380:
12375:
12373:
12368:
12367:
12364:
12352:
12349:
12347:
12344:
12342:
12339:
12337:
12334:
12332:
12329:
12327:
12324:
12322:
12319:
12317:
12314:
12312:
12309:
12308:
12306:
12304:
12298:
12294:
12290:
12284:
12281:
12279:
12276:
12274:
12271:
12269:
12266:
12264:
12261:
12259:
12256:
12255:
12253:
12251:
12247:
12241:
12238:
12236:
12233:
12231:
12228:
12226:
12223:
12221:
12218:
12217:
12215:
12211:
12205:
12202:
12200:
12197:
12196:
12194:
12192:
12188:
12180:
12177:
12176:
12175:
12172:
12168:
12165:
12164:
12163:
12160:
12159:
12157:
12155:
12151:
12145:
12142:
12140:
12137:
12135:
12132:
12130:
12127:
12125:
12122:
12121:
12119:
12115:
12111:
12104:
12099:
12097:
12092:
12090:
12085:
12084:
12081:
12069:
12061:
12059:
12051:
12050:
12047:
12041:
12038:
12036:
12033:
12031:
12028:
12026:
12023:
12021:
12018:
12016:
12015:Markarian 501
12013:
12011:
12008:
12006:
12003:
12001:
11998:
11996:
11993:
11991:
11988:
11986:
11983:
11981:
11978:
11976:
11973:
11971:
11968:
11966:
11963:
11961:
11958:
11956:
11953:
11951:
11948:
11946:
11943:
11941:
11940:XTE J1118+480
11938:
11936:
11935:XTE J1650-500
11933:
11931:
11928:
11927:
11925:
11921:
11915:
11912:
11910:
11907:
11905:
11902:
11900:
11897:
11895:
11892:
11890:
11887:
11885:
11882:
11880:
11877:
11875:
11872:
11870:
11867:
11865:
11862:
11860:
11857:
11855:
11852:
11850:
11847:
11845:
11842:
11840:
11837:
11835:
11832:
11828:
11825:
11823:
11820:
11819:
11818:
11815:
11813:
11810:
11808:
11805:
11803:
11800:
11798:
11795:
11793:
11790:
11788:
11785:
11783:
11780:
11779:
11777:
11773:
11767:
11764:
11762:
11759:
11757:
11754:
11752:
11749:
11747:
11744:
11743:
11741:
11739:
11735:
11729:
11726:
11724:
11721:
11720:
11718:
11714:
11708:
11705:
11703:
11700:
11698:
11695:
11693:
11690:
11688:
11685:
11683:
11680:
11678:
11675:
11673:
11670:
11668:
11665:
11663:
11660:
11659:
11657:
11653:
11647:
11644:
11642:
11639:
11637:
11634:
11632:
11629:
11626:
11622:
11621:Schwarzschild
11619:
11618:
11616:
11612:
11606:
11603:
11601:
11598:
11596:
11593:
11591:
11588:
11586:
11583:
11581:
11578:
11576:
11573:
11571:
11568:
11567:
11565:
11561:
11555:
11552:
11550:
11547:
11543:
11540:
11539:
11538:
11535:
11533:
11530:
11528:
11525:
11523:
11520:
11518:
11515:
11513:
11510:
11506:
11503:
11502:
11501:
11498:
11496:
11493:
11491:
11488:
11484:
11481:
11479:
11476:
11475:
11474:
11471:
11469:
11466:
11464:
11463:Photon sphere
11461:
11459:
11458:Event horizon
11456:
11452:
11449:
11447:
11444:
11443:
11442:
11439:
11437:
11434:
11433:
11431:
11427:
11421:
11418:
11416:
11413:
11411:
11408:
11406:
11403:
11401:
11398:
11396:
11393:
11391:
11388:
11384:
11383:Related links
11381:
11379:
11376:
11374:
11371:
11370:
11369:
11366:
11362:
11361:Related links
11359:
11358:
11357:
11354:
11352:
11349:
11345:
11344:Related links
11342:
11341:
11340:
11337:
11335:
11332:
11330:
11327:
11326:
11324:
11320:
11312:
11309:
11307:
11304:
11302:
11299:
11297:
11294:
11292:
11289:
11287:
11284:
11282:
11279:
11278:
11277:
11274:
11272:
11269:
11265:
11262:
11261:
11260:
11257:
11252:
11250:
11247:
11245:
11242:
11241:
11240:
11237:
11236:
11234:
11230:
11225:
11215:
11212:
11210:
11207:
11205:
11202:
11200:
11197:
11195:
11192:
11190:
11187:
11185:
11182:
11180:
11177:
11175:
11172:
11170:
11169:Schwarzschild
11167:
11165:
11162:
11161:
11159:
11155:
11149:
11146:
11145:
11142:
11138:
11131:
11126:
11124:
11119:
11117:
11112:
11111:
11108:
11098:
11093:
11089:
11085:
11081:
11077:
11073:
11068:
11064:
11060:
11056:
11052:
11048:
11044:
11040:
11036:
11031:
11027:
11021:
11016:
11015:
11009:
11005:
11001:
10997:
10993:
10989:
10985:
10981:
10977:
10973:
10968:
10964:
10960:
10956:
10952:
10948:
10943:
10942:gr-qc/0407064
10938:
10934:
10930:
10925:
10919:
10914:
10909:
10905:
10901:
10897:
10894:
10893:See chapter 7
10889:
10883:
10878:
10877:
10870:
10867:
10862:
10856:
10852:
10847:
10844:
10839:
10833:
10829:
10825:
10821:
10818:
10813:
10807:
10802:
10801:
10794:
10790:
10784:
10780:
10775:
10771:
10765:
10761:
10760:
10755:
10751:
10747:
10741:
10737:
10732:
10728:
10722:
10718:
10714:
10709:
10708:
10696:
10692:
10688:
10684:
10680:
10676:
10672:
10668:
10661:
10652:
10645:
10644:
10637:
10630:
10624:
10616:
10612:
10606:
10598:
10594:
10590:
10586:
10582:
10578:
10574:
10570:
10565:
10560:
10556:
10552:
10545:
10536:
10532:
10528:
10524:
10520:
10516:
10511:
10506:
10502:
10498:
10491:
10483:
10479:
10475:
10471:
10467:
10463:
10456:
10445:
10441:
10437:
10433:
10429:
10425:
10421:
10417:
10413:
10406:
10402:
10396:
10394:
10392:
10382:
10377:
10370:
10368:
10361:
10357:
10352:
10350:
10338:
10331:
10323:
10319:
10315:
10311:
10307:
10303:
10298:
10293:
10289:
10285:
10281:
10275:
10268:
10262:
10255:
10251:
10246:
10238:
10234:
10230:
10226:
10221:
10216:
10212:
10208:
10205:(2): 85â100.
10204:
10200:
10193:
10186:
10182:
10179:
10173:
10171:
10169:
10162:
10157:
10151:
10147:
10143:
10136:
10129:
10125:
10121:
10115:
10106:
10101:
10094:
10087:
10086:
10079:
10071:
10067:
10063:
10059:
10055:
10051:
10046:
10045:gr-qc/9312028
10041:
10037:
10033:
10026:
10017:
10013:
10009:
10005:
10001:
9997:
9990:
9982:
9978:
9974:
9970:
9966:
9962:
9961:J. Math. Phys
9955:
9946:
9941:
9934:
9932:
9930:
9928:
9921:, p. 1, Eq. 1
9920:
9914:
9906:
9900:
9896:
9895:
9887:
9879:
9873:
9869:
9865:
9861:
9860:Landau, L. D.
9855:
9847:
9843:
9838:
9833:
9828:
9823:
9819:
9815:
9812:(1): e03336.
9811:
9807:
9803:
9796:
9788:
9784:
9780:
9776:
9772:
9768:
9764:
9760:
9756:
9749:
9741:
9737:
9732:
9727:
9723:
9719:
9714:
9709:
9706:(10): 31791.
9705:
9701:
9697:
9690:
9682:
9678:
9674:
9670:
9666:
9662:
9658:
9654:
9650:
9643:
9635:
9631:
9627:
9623:
9619:
9615:
9611:
9607:
9602:
9597:
9594:(6): 061102.
9593:
9589:
9582:
9574:
9570:
9563:
9555:
9551:
9546:
9541:
9537:
9533:
9529:
9525:
9521:
9517:
9513:
9509:
9502:
9496:
9492:
9486:
9478:
9474:
9470:
9466:
9462:
9458:
9454:
9448:
9444:
9432:
9427:
9423:
9414:
9411:
9409:
9406:
9404:
9401:
9399:
9396:
9394:
9391:
9389:
9386:
9385:
9381:
9375:
9370:
9367:
9356:
9349:
9333:
9330:
9327:
9322:
9318:
9314:
9304:
9303:dust solution
9300:
9296:
9291:
9287:
9283:
9279:
9275:
9268:Open problems
9265:
9259:
9255:
9251:
9248:
9247:
9242:
9238:
9235:
9231:
9227:
9226:
9225:
9209:
9198:
9195:
9191:
9187:
9183:
9178:
9176:
9172:
9152:
9146:
9142:
9138:
9133:
9128:
9124:
9117:
9113:
9110:
9105:
9101:
9096:
9093:
9090:
9085:
9081:
9076:
9073:
9070:
9065:
9061:
9053:
9052:
9051:
9049:
9045:
9041:
9036:
9032:
9030:
9012:
9009:
9006:
8996:
8975:
8967:
8964:
8958:
8955:
8950:
8946:
8938:
8937:
8936:
8933:
8929:
8925:
8921:
8911:
8909:
8904:
8900:
8896:
8892:
8888:
8883:
8881:
8877:
8873:
8869:
8865:
8847:
8844:
8841:
8830:
8828:
8824:
8819:
8817:
8816:Ernst vacuums
8813:
8809:
8806:
8803:
8793:
8791:
8785:
8783:
8764:
8753:
8737:
8727:
8709:
8699:
8681:
8678:
8658:
8655:
8647:
8644:February 2011
8637:
8633:
8627:
8626:
8621:This section
8619:
8615:
8610:
8609:
8601:
8599:
8581:
8578:
8575:
8570:
8566:
8562:
8538:
8535:
8532:
8528:
8518:
8493:
8490:
8471:
8469:
8464:
8462:
8458:
8448:
8446:
8428:
8425:
8422:
8411:
8395:
8392:
8385:
8381:
8376:
8372:
8368:
8365:
8359:
8356:
8350:
8347:
8326:
8321:
8313:
8309:
8306:
8300:
8296:
8290:
8286:
8282:
8279:
8276:
8273:
8269:
8264:
8259:
8254:
8250:
8246:
8243:
8221:
8197:
8170:
8162:
8158:
8152:
8148:
8145:
8134:
8130:
8126:
8119:
8115:
8112:
8109:
8106:
8100:
8097:
8070:
8062:
8055:
8051:
8045:
8041:
8037:
8034:
8027:
8023:
8020:
8009:
8005:
8001:
7994:
7987:
7984:
7981:
7978:
7972:
7969:
7965:
7958:
7955:
7947:
7943:
7940:
7937:
7934:
7928:
7922:
7919:
7913:
7910:
7904:
7901:
7894:
7890:
7883:
7875:
7870:
7867:
7864:
7861:
7858:
7852:
7849:
7843:
7840:
7834:
7831:
7808:
7790:
7787:
7782:
7778:
7769:
7765:
7761:
7756:
7751:
7745:
7741:
7737:
7732:
7728:
7723:
7714:
7709:
7706:
7682:
7634:
7610:
7576:
7571:
7555:
7551:
7547:
7544:
7541:
7517:
7493:
7469:
7466:
7463:
7437:
7433:
7429:
7423:
7420:
7413:
7409:
7405:
7392:
7374:
7348:
7344:
7341:
7336:
7328:
7325:
7322:
7317:
7313:
7306:
7301:
7297:
7291:
7287:
7282:
7275:
7270:
7262:
7256:
7253:
7247:
7241:
7234:
7218:
7214:
7210:
7207:
7203:
7197:
7193:
7189:
7184:
7180:
7175:
7171:
7168:
7162:
7156:
7149:
7134:
7127:
7124:
7119:
7115:
7108:
7103:
7099:
7093:
7089:
7083:
7079:
7075:
7070:
7066:
7061:
7055:
7051:
7046:
7042:
7039:
7034:
7030:
7026:
7023:
7020:
7014:
7001:
7000:
6999:
6978:
6972:
6961:
6957:
6953:
6948:
6944:
6937:
6933:
6927:
6923:
6919:
6916:
6913:
6908:
6904:
6900:
6897:
6893:
6889:
6886:
6883:
6881:
6873:
6870:
6865:
6862:
6846:
6840:
6832:
6827:
6823:
6816:
6813:
6808:
6804:
6797:
6793:
6787:
6784:
6781:
6777:
6773:
6770:
6768:
6760:
6757:
6752:
6749:
6731:
6720:
6717:
6715:
6707:
6704:
6699:
6696:
6678:
6672:
6667:
6664:
6662:
6654:
6651:
6646:
6643:
6611:
6608:
6605:
6602:
6599:
6596:
6593:
6582:
6580:
6575:
6562:
6558:
6552:
6547:
6541:
6538:
6535:
6529:
6525:
6519:
6514:
6510:
6504:
6500:
6496:
6491:
6487:
6482:
6476:
6472:
6467:
6463:
6460:
6455:
6451:
6447:
6442:
6437:
6433:
6429:
6426:
6418:
6400:
6390:
6386:
6381:
6366:
6362:
6358:
6353:
6349:
6328:
6323:
6319:
6315:
6312:
6309:
6285:
6281:
6256:
6246:
6245:four-momentum
6226:
6222:
6199:
6194:
6190:
6184:
6181:
6177:
6171:
6167:
6163:
6158:
6154:
6150:
6128:
6118:
6114:
6110:
6102:
6098:
6093:
6085:
6076:
6060:
6036:
6012:
6009:
6006:
5996:
5991:
5987:
5985:
5984:homotopy type
5981:
5977:
5974:
5970:
5966:
5962:
5958:
5954:
5950:
5946:
5945:
5940:
5936:
5932:
5928:
5924:
5920:
5916:
5912:
5902:
5900:
5896:
5892:
5891:Roger Penrose
5888:
5884:
5880:
5875:
5873:
5854:
5851:
5846:
5842:
5836:
5832:
5828:
5823:
5819:
5815:
5810:
5802:
5799:
5796:
5786:
5785:
5784:
5781:
5779:
5774:
5770:
5769:event horizon
5764:
5754:
5738:
5735:
5729:
5726:
5713:
5709:
5702:
5698:
5693:
5653:
5640:
5636:
5632:
5628:
5620:
5616:
5612:
5608:
5604:
5583:
5580:
5575:
5571:
5565:
5561:
5557:
5554:
5549:
5546:
5543:
5538:
5527:
5519:
5518:
5517:
5498:
5492:
5489:
5484:
5480:
5474:
5470:
5466:
5463:
5458:
5449:
5443:
5434:
5427:
5422:
5411:
5403:
5402:
5401:
5394:
5386:
5382:
5379:While in the
5359:
5355:
5351:
5348:
5343:
5340:
5337:
5332:
5321:
5313:
5312:
5311:
5295:
5292:
5289:
5286:
5283:
5280:
5277:
5250:
5242:
5238:
5234:
5231:
5226:
5217:
5211:
5202:
5195:
5190:
5179:
5171:
5170:
5169:
5158:
5140:
5136:
5132:
5131:event horizon
5106:
5082:
5058:
5047:
5039:
5030:
5028:
5024:
5020:
5016:
5007:
5000:
4998:
4984:
4978:
4975:
4970:
4966:
4960:
4956:
4952:
4943:
4939:
4935:
4929:
4925:
4921:
4916:
4912:
4907:
4898:
4895:
4892:
4883:
4876:
4869:
4866:
4862:
4856:
4853:
4849:
4843:
4840:
4830:
4829:
4826:
4824:
4820:
4816:
4812:
4803:
4796:
4794:
4780:
4775:
4770:
4766:
4763:
4756:
4753:
4749:
4743:
4740:
4736:
4730:
4727:
4724:
4720:
4713:
4710:
4706:
4702:
4697:
4693:
4689:
4684:
4681:
4677:
4673:
4668:
4664:
4660:
4646:
4642:
4637:
4633:
4629:
4625:
4617:
4614:
4610:
4604:
4599:
4596:
4592:
4586:
4581:
4578:
4574:
4569:
4565:
4560:
4556:
4552:
4547:
4543:
4535:
4534:
4531:
4515:
4508:
4506:
4490:
4485:
4479:
4467:
4457:
4452:
4447:
4442:
4436:
4424:
4409:
4404:
4399:
4393:
4381:
4375:
4365:
4356:
4349:
4346:
4342:
4335:
4332:
4327:
4323:
4315:
4310:
4303:
4287:
4266:
4261:
4258:
4249:
4245:
4239:
4234:
4229:
4223:
4211:
4205:
4201:
4198:
4193:
4189:
4177:
4173:
4169:
4160:
4153:
4148:
4144:
4140:
4135:
4131:
4126:
4114:
4110:
4105:
4100:
4097:
4089:
4085:
4067:
4063:
4048:
4045:
4041:
4033:
4032:
4029:
4027:
4024:
4023:four-gradient
4020:
4019:metric tensor
4000:
3997:
3993:
3982:
3981:contravariant
3975:Wave operator
3958:
3950:
3946:
3940:
3923:
3919:
3912:
3908:
3902:
3898:
3891:
3886:
3869:
3865:
3860:
3856:
3845:
3841:
3836:
3830:
3826:
3820:
3816:
3812:
3807:
3803:
3797:
3792:
3788:
3784:
3779:
3762:
3758:
3751:
3750:
3749:
3727:
3716:
3698:
3687:
3671:
3661:
3637:
3612:
3586:
3560:
3536:
3533:
3530:
3520:
3501:
3483:
3479:
3476:
3472:
3466:
3462:
3458:
3441:
3424:
3422:
3418:
3414:
3404:
3401:
3399:
3395:
3391:
3387:
3369:
3366:
3363:
3339:
3310:
3306:
3302:
3297:
3293:
3289:
3284:
3280:
3274:
3271:
3265:
3258:
3257:
3256:
3255:
3251:
3242:
3235:
3233:
3219:
3216:
3209:
3205:
3199:
3195:
3189:
3181:
3177:
3173:
3168:
3164:
3156:
3152:
3148:
3143:
3139:
3128:
3127:
3124:
3122:
3098:
3085:
3081:
3077:
3073:
3069:
3065:
3061:
3057:
3056:unit 3-vector
3053:
3044:
3037:
3035:
3020:
3017:
3012:
3008:
3000:
2999:
2992:
2985:
2983:
2968:
2962:
2959:
2954:
2946:
2942:
2938:
2933:
2929:
2923:
2920:
2917:
2914:
2911:
2905:
2897:
2893:
2889:
2884:
2880:
2874:
2871:
2868:
2865:
2862:
2855:
2851:
2843:
2839:
2835:
2830:
2826:
2822:
2817:
2813:
2806:
2794:
2793:
2786:
2779:
2777:
2758:
2754:
2748:
2744:
2740:
2735:
2731:
2723:
2719:
2715:
2712:
2709:
2703:
2700:
2693:
2692:
2685:
2678:
2676:
2659:
2655:
2649:
2645:
2641:
2638:
2633:
2630:
2626:
2622:
2617:
2614:
2610:
2602:
2601:
2598:
2596:
2592:
2588:
2584:
2570:
2563:
2561:
2545:
2541:
2537:
2534:
2531:
2526:
2522:
2517:
2511:
2507:
2503:
2498:
2494:
2489:
2485:
2480:
2476:
2472:
2466:
2461:
2457:
2453:
2445:
2441:
2437:
2432:
2428:
2418:
2413:
2409:
2405:
2400:
2396:
2392:
2387:
2381:
2371:
2364:
2363:
2360:
2359:
2335:
2310:
2299:
2283:
2280:
2276:
2273:
2258:
2251:
2249:
2235:
2230:
2226:
2222:
2219:
2210:
2206:
2201:
2197:
2193:
2183:
2182:
2175:
2168:
2166:
2152:
2149:
2146:
2141:
2137:
2131:
2127:
2123:
2118:
2114:
2110:
2100:
2099:
2092:
2085:
2083:
2069:
2063:
2060:
2056:
2051:
2048:
2041:
2040:
2037:
1994:
1991:
1976:
1969:
1967:
1953:
1946:
1942:
1937:
1934:
1931:
1925:
1916:
1908:
1907:
1904:
1903:
1881:
1867:
1860:
1858:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1819:
1818:
1811:
1804:
1802:
1788:
1785:
1782:
1779:
1776:
1773:
1766:
1762:
1758:
1753:
1749:
1743:
1740:
1733:
1732:
1725:
1718:
1716:
1702:
1699:
1696:
1693:
1690:
1687:
1680:
1676:
1672:
1667:
1663:
1657:
1654:
1647:
1646:
1643:
1641:
1638:are standard
1623:
1620:
1617:
1614:
1611:
1596:
1589:
1587:
1569:
1566:
1562:
1559:
1555:
1546:
1543:
1538:
1534:
1530:
1527:
1518:
1514:
1508:
1503:
1499:
1495:
1489:
1486:
1481:
1477:
1472:
1468:
1465:
1460:
1456:
1444:
1440:
1436:
1427:
1420:
1415:
1411:
1407:
1402:
1398:
1393:
1389:
1384:
1380:
1376:
1370:
1365:
1361:
1357:
1344:
1339:
1335:
1331:
1326:
1322:
1317:
1307:
1298:
1291:
1288:
1284:
1280:
1277:
1275:
1265:
1261:
1257:
1252:
1248:
1244:
1241:
1239:
1232:
1228:
1224:
1213:
1212:
1209:
1207:
1203:
1199:
1181:
1172:
1154:
1144:
1138:
1128:
1126:
1122:
1118:
1108:
1106:
1101:
1099:
1096:energy,
1095:
1091:
1090:
1082:
1058:
1054:
1051:
1041:
1037:
1033:
1032:event horizon
1029:
1025:
1021:
1017:
1016:singularities
1012:
1010:
1004:
999:
997:
993:
988:
984:
980:
969:
966:
964:
961:
936:
932:
931:
927:
925:
924:Schwarzschild
922:
919:
915:
914:
888:
884:
881:
877:
875:
874:
871:
870:
869:
867:
863:
859:
855:
851:
847:
843:
839:
835:
831:
827:
823:
813:
811:
807:
803:
799:
795:
791:
790:event horizon
787:
784:
780:
776:
775:Kerr geometry
772:
760:
755:
753:
748:
746:
741:
740:
738:
737:
731:
721:
718:
713:
707:
706:
705:
704:
697:
696:
692:
690:
687:
685:
682:
680:
677:
675:
672:
670:
667:
665:
662:
660:
657:
655:
652:
650:
647:
645:
642:
640:
637:
635:
634:Chandrasekhar
632:
630:
627:
625:
622:
620:
617:
615:
612:
610:
607:
605:
602:
600:
597:
595:
592:
590:
587:
585:
582:
580:
577:
575:
572:
570:
567:
565:
562:
560:
557:
555:
552:
550:
549:Schwarzschild
547:
545:
542:
540:
537:
535:
532:
530:
527:
526:
518:
517:
510:
509:HartleâThorne
507:
505:
502:
500:
497:
495:
492:
490:
487:
485:
482:
480:
477:
475:
472:
470:
467:
465:
462:
460:
457:
455:
452:
450:
447:
445:
442:
440:
437:
435:
432:
430:
427:
424:
420:
419:Schwarzschild
417:
416:
412:
406:
405:
394:
391:
389:
386:
385:
384:
383:
379:
378:
373:
370:
368:
365:
363:
360:
359:
358:
357:
353:
352:
347:
344:
342:
339:
337:
334:
332:
329:
327:
324:
322:
319:
318:
317:
316:
312:
311:
301:
298:
297:
292:
291:
280:
277:
275:
272:
270:
267:
266:
265:
264:
261:
258:
257:
252:
249:
247:
244:
242:
241:Event horizon
239:
237:
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
203:
202:
201:
191:
190:
183:
180:
178:
175:
173:
170:
168:
165:
164:
156:
155:
150:
147:
142:
139:
137:
134:
132:
129:
128:
126:
124:
121:
120:
119:
118:
100:
97:
93:
88:
84:
79:
76:
72:
65:
60:
57:
53:
43:
39:
38:
35:
32:
31:
19:
18:Kerr solution
13191:
12969:
12885:KaluzaâKlein
12637:Introduction
12563:Twin paradox
12336:Misner space
12325:
12321:Gödel metric
12301:can contain
11970:PKS 1302-102
11844:Gravity well
11812:Compact star
11766:Microquasars
11751:Most massive
11655:Alternatives
11630:
11420:X-ray binary
11339:Neutron star
11276:Supermassive
11253:Hawking star
11194:Supermassive
11079:
11075:
11038:
11034:
11013:
10983:
10979:
10932:
10928:
10903:
10892:
10875:
10865:
10850:
10842:
10827:
10816:
10799:
10778:
10758:
10735:
10716:
10670:
10666:
10660:
10655:Penrose 1968
10651:
10641:
10636:
10623:
10614:
10605:
10554:
10550:
10544:
10500:
10496:
10490:
10465:
10461:
10455:
10444:the original
10415:
10411:
10342:. p. 8.
10330:
10287:
10283:
10274:
10261:
10245:
10202:
10198:
10192:
10141:
10135:
10119:
10114:
10093:
10083:
10078:
10035:
10031:
10025:
9999:
9995:
9989:
9964:
9960:
9954:
9913:
9893:
9886:
9867:
9854:
9809:
9805:
9795:
9762:
9758:
9748:
9703:
9700:Scholarpedia
9699:
9689:
9656:
9652:
9642:
9591:
9587:
9581:
9572:
9562:
9515:
9511:
9501:
9485:
9460:
9456:
9453:Kerr, Roy P.
9447:
9430:
9426:
9282:neutron star
9271:
9263:
9257:
9253:
9244:
9240:
9233:
9229:
9199:
9185:
9181:
9179:
9174:
9168:
9047:
9043:
9034:
9033:
9029:point source
8992:
8917:
8886:
8884:
8867:
8831:
8820:
8799:
8786:
8754:
8665:
8650:
8641:
8630:Please help
8625:verification
8622:
8477:
8465:
8454:
8412:
7572:
7364:
6583:
6576:
6382:
6106:
5992:
5988:
5975:
5960:
5956:
5948:
5942:
5908:
5886:
5882:
5881:(from Greek
5878:
5876:
5869:
5782:
5778:static limit
5777:
5766:
5711:
5700:
5694:
5638:
5618:
5610:
5609:equals 0 or
5606:
5602:
5600:
5515:
5392:
5384:
5378:
5267:
5156:
5138:
5124:
5012:
5001:
4818:
4810:
4808:
4797:
4525:
4509:
3978:
3688:
3627:is equal to
3430:
3410:
3402:
3389:
3385:
3329:
3253:
3249:
3247:
3236:
3120:
3083:
3075:
3071:
3067:
3063:
3051:
3050:Notice that
3049:
3038:
2986:
2780:
2679:
2580:
2564:
2300:
2263:
2252:
2169:
2086:
1981:
1970:
1872:
1861:
1805:
1719:
1601:
1590:
1198:line element
1140:
1114:
1102:
1097:
1087:
1080:
1013:
1000:
991:
976:
934:
917:
886:
879:
861:
853:
841:
833:
819:
774:
770:
768:
694:
654:Raychaudhuri
448:
123:Introduction
13233:Black holes
12974:KerrâNewman
12945:Spherical:
12814:Other tests
12757:Singularity
12689:Formulation
12651:Fundamental
12505:Formulation
12486:Proper time
12447:Fundamental
12326:Kerr metric
12263:Determinism
12204:Causal loop
12110:Time travel
11960:Centaurus A
11914:Planet Nine
11817:Exotic star
11746:Black holes
11692:Planck star
11641:KerrâNewman
11356:White dwarf
11306:Radio-Quiet
11264:Microquasar
11137:Black holes
11008:Wald, R. M.
10900:Penrose, R.
10381:0706.0622v3
10356:Paul Davies
10178:Gravitation
9945:0706.0622v3
9301:, an exact
9258:Kerr vacuum
8868:nonrotating
6097:ergodically
5973:Petrov type
5965:Weyl tensor
5949:axisymmetry
5627:proper time
3983:components
3060:null vector
1202:proper time
968:KerrâNewman
916:Uncharged (
792:. The Kerr
771:Kerr metric
669:van Stockum
599:Oppenheimer
454:KerrâNewman
246:Singularity
13222:Categories
13126:Zel'dovich
13034:Scientists
13013:Alcubierre
12820:of Mercury
12818:precession
12747:Black hole
12630:Background
12622:relativity
12591:World line
12586:Light cone
12411:Background
12403:relativity
12393:Relativity
12293:Spacetimes
12268:Eternalism
12235:Multiverse
12010:Q0906+6930
12000:Hercules A
11930:Cygnus X-1
11899:White hole
11874:Quasi-star
11827:Preon star
11822:Quark star
11807:Big Bounce
11667:Black star
11625:Derivation
11473:Ergosphere
11429:Properties
11410:Quasi-star
11400:Quark star
11311:Radio-Loud
11199:Primordial
11189:Kugelblitz
10976:Carter, B.
10564:2008.04384
9601:1602.03837
9440:References
9044:even order
8802:stationary
8451:Symmetries
8413:If we set
7393:such that
6579:integrable
6075:variable.
5915:ergosphere
5879:ergosphere
5629:along its
5615:ergosphere
4815:colatitude
1028:ergosphere
996:ergosphere
885:Rotating (
810:non-linear
786:black hole
522:Scientists
354:Formalisms
302:Formalisms
251:Black hole
177:World line
13096:Robertson
13081:Friedmann
13076:Eddington
13066:de Sitter
12900:Solutions
12778:detectors
12773:astronomy
12740:Phenomena
12675:Geodesics
12578:Spacetime
12521:Phenomena
12278:Free will
12174:Time loop
12035:AT2018hyz
11682:Gravastar
11672:Dark star
11505:Microlens
11378:Hypernova
11373:Micronova
11368:Supernova
11322:Formation
10597:221095833
10589:0218-2718
10535:118417747
10510:1408.6316
10440:123261579
10322:119237335
10297:1106.1198
10269:, page 42
10256:, page 11
10215:CiteSeerX
10105:0706.0622
9787:0031-9007
9740:1941-6016
9713:1410.6626
9681:0022-2488
9634:124959784
9522:: 14247.
9419:Footnotes
9408:Spin-flip
9139:−
9048:odd order
8891:isometric
8878:, in the
8676:Δ
8488:Δ
8396:˙
8393:ϕ
8386:θ
8382:
8366:−
8360:˙
8314:θ
8310:
8291:−
8287:θ
8283:
8255:θ
8167:Σ
8146:−
8141:Σ
8110:−
8101:¨
8067:Σ
8038:−
8021:−
8016:Σ
7970:−
7959:˙
7929:−
7923:˙
7905:˙
7880:Σ
7853:¨
7835:¨
7802:Σ
7796:Δ
7791:θ
7788:
7775:Δ
7762:−
7710:τ
7659:Ω
7587:Ω
7556:μ
7548:τ
7542:λ
7518:τ
7494:λ
7464:μ
7438:α
7424:λ
7414:α
7375:λ
7323:−
7288:μ
7279:Δ
7276:−
7208:−
7128:θ
7125:
7076:−
7067:μ
7043:θ
7040:
7027:−
7015:θ
7009:Θ
6968:Δ
6920:−
6917:θ
6914:
6887:−
6874:λ
6857:Σ
6836:Δ
6817:θ
6814:
6788:−
6774:−
6761:λ
6753:ϕ
6744:Σ
6732:θ
6726:Θ
6721:±
6708:λ
6700:θ
6691:Σ
6668:±
6655:λ
6638:Σ
6542:θ
6539:
6497:−
6488:μ
6464:θ
6461:
6438:θ
6367:ϕ
6316:−
6227:α
6195:β
6185:β
6182:α
6172:α
6155:μ
6151:−
6129:μ
6101:toruslike
6061:θ
6037:ϕ
5855:θ
5852:
5829:−
5800:−
5739:ϕ
5678:Ω
5635:spacetime
5631:worldline
5584:θ
5581:
5558:−
5550:±
5539:±
5493:θ
5490:
5464:−
5444:±
5423:±
5352:−
5344:±
5333:±
5232:−
5212:±
5191:±
4979:θ
4976:
4904:Σ
4870:ϕ
4867:ϕ
4857:ϕ
4844:−
4838:Ω
4757:ϕ
4754:ϕ
4744:ϕ
4728:ϕ
4714:ϕ
4711:ϕ
4694:θ
4685:θ
4682:θ
4618:ϕ
4615:ϕ
4600:ϕ
4587:−
4557:τ
4480:θ
4477:∂
4473:∂
4461:Σ
4434:∂
4430:∂
4418:Σ
4415:Δ
4394:ϕ
4391:∂
4387:∂
4370:Σ
4350:−
4336:θ
4333:
4320:Δ
4300:∂
4296:∂
4288:ϕ
4285:∂
4281:∂
4273:Δ
4270:Σ
4240:−
4221:∂
4217:∂
4202:θ
4199:
4184:Σ
4120:Δ
4101:−
4090:ν
4082:∂
4078:∂
4068:μ
4060:∂
4056:∂
4049:ν
4046:μ
3853:⟹
3813:−
3480:−
3269:→
3102:→
2918:−
2660:ν
2650:μ
2634:ν
2631:μ
2627:η
2618:ν
2615:μ
2597:in 1965.
2542:ϕ
2535:θ
2532:
2477:θ
2470:Σ
2424:Σ
2393:−
2385:→
2377:⟶
2284:ϕ
2207:−
2191:Δ
2150:θ
2147:
2108:Σ
2022:Δ
1998:Σ
1842:θ
1839:
1789:ϕ
1786:
1780:θ
1777:
1703:ϕ
1700:
1694:θ
1691:
1624:ϕ
1618:θ
1570:ϕ
1551:Σ
1547:θ
1544:
1509:−
1500:ϕ
1490:θ
1487:
1469:θ
1466:
1451:Σ
1381:θ
1374:Σ
1353:Δ
1350:Σ
1312:Σ
1292:−
1281:−
1262:τ
1245:−
1143:spacetime
933:Charged (
830:spacetime
779:spacetime
614:Robertson
579:Friedmann
574:Eddington
564:Nordström
554:de Sitter
411:Solutions
336:Geodesics
331:Friedmann
313:Equations
299:Equations
260:Spacetime
195:Phenomena
101:ν
98:μ
89:κ
80:ν
77:μ
69:Λ
61:ν
58:μ
13209:Category
13086:LemaĂźtre
13051:Einstein
13041:Poincaré
13001:Others:
12985:TaubâNUT
12951:interior
12873:theories
12871:Advanced
12838:redshift
12653:concepts
12471:Rapidity
12449:concepts
12273:Fatalism
12058:Category
11945:A0620-00
11904:Wormhole
11802:Big Bang
11702:Fuzzball
11585:ER = EPR
11451:Theorems
11249:Electron
11244:Extremal
11174:Rotating
11010:(1984).
10967:16858122
10826:(1992).
10695:10054468
10613:(1983).
10403:(1968).
10237:53513572
10181:Archived
10128:page 325
10124:page 485
9866:(1975).
9846:32051884
9626:26918975
9554:34244573
9431:Warning:
9352:See also
9194:momentum
8887:interior
7532:through
5961:outgoing
5959:and one
5885:meaning
4813:and the
4026:operator
846:Roy Kerr
842:rotating
816:Overview
730:Category
594:LemaĂźtre
559:Reissner
544:Poincaré
529:Einstein
474:TaubâNUT
439:Wormhole
423:interior
136:Timeline
13151:Hawking
13146:Penrose
13131:Novikov
13111:Wheeler
13056:Hilbert
13046:Lorentz
13003:pp-wave
12824:lensing
12620:General
12401:Special
12179:in film
12068:Commons
12030:P172+18
11985:TON 618
11923:Notable
11775:Related
11761:Quasars
11756:Nearest
11716:Analogs
11646:Hayward
11614:Metrics
11259:Stellar
11184:Virtual
11179:Charged
11148:Outline
11084:Bibcode
11043:Bibcode
10988:Bibcode
10947:Bibcode
10675:Bibcode
10569:Bibcode
10515:Bibcode
10470:Bibcode
10468:: 347.
10420:Bibcode
10302:Bibcode
10284:Physics
10207:Bibcode
10070:6041750
10050:Bibcode
10004:Bibcode
9969:Bibcode
9837:7002888
9814:Bibcode
9806:Heliyon
9767:Bibcode
9718:Bibcode
9661:Bibcode
9606:Bibcode
9545:8270963
9524:Bibcode
9465:Bibcode
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9025:
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8893:to the
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8810:to the
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1900:is the
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834:charged
800:of the
649:Hawking
644:Penrose
619:Bardeen
609:Wheeler
539:Hilbert
534:Lorentz
494:pp-wave
131:History
13192:others
13181:Thorne
13171:Misner
13156:Taylor
13141:Geroch
13136:Ehlers
13106:Zwicky
12924:Kasner
12005:3C 273
11995:NeVe 1
11975:OJ 287
11697:Q star
11563:Issues
11296:Blazar
11286:Quasar
11063:361088
11061:
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8872:static
8553:(i.e.
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7365:where
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6212:where
5978:. The
5933:. The
5753:term.
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2595:Schild
1873:where
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1111:Metric
796:is an
794:metric
728:
695:others
689:Thorne
679:Newman
659:Taylor
639:Ehlers
624:Walker
589:Zwicky
464:Kasner
13186:Weiss
13166:Bondi
13161:Hulse
13091:Milne
12995:discs
12939:Milne
12934:Gödel
12791:Virgo
11738:Lists
11239:Micro
11209:Rogue
11157:Types
11059:S2CID
10963:S2CID
10937:arXiv
10913:arXiv
10593:S2CID
10559:arXiv
10531:S2CID
10505:arXiv
10447:(PDF)
10436:S2CID
10408:(PDF)
10376:arXiv
10340:(PDF)
10318:S2CID
10292:arXiv
10233:S2CID
10100:arXiv
10066:S2CID
10040:arXiv
9940:arXiv
9708:arXiv
9630:S2CID
9596:arXiv
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8918:Each
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664:Hulse
604:Gödel
584:Milne
479:Milne
444:Gödel
141:Tests
13121:Kerr
13071:Weyl
12970:Kerr
12830:and
12784:and
12782:LIGO
12299:that
11707:Geon
11631:Kerr
11232:Size
11020:ISBN
10882:ISBN
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10832:ISBN
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10783:ISBN
10764:ISBN
10740:ISBN
10721:ISBN
10691:PMID
10585:ISSN
10150:ISBN
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9783:ISSN
9736:ISSN
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9622:PMID
9550:PMID
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9230:zero
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9184:and
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8928:mass
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6341:and
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1200:for
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1020:mass
992:must
860:and
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629:Kerr
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367:BSSN
13176:Yau
12801:GEO
12295:in
11301:OVV
11291:LQG
11092:doi
11051:doi
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10577:doi
10523:doi
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10466:178
10428:doi
10416:174
10310:doi
10225:doi
10161:PDF
10058:doi
10012:doi
9977:doi
9832:PMC
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9669:doi
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8135:2
8131:r
8127:4
8120:(
8116:z
8113:M
8107:=
8098:z
8071:2
8063:r
8056:2
8052:W
8046:2
8042:a
8035:C
8028:)
8024:1
8010:2
8006:r
8002:4
7995:(
7991:)
7988:y
7985:i
7982:+
7979:x
7976:(
7973:M
7966:]
7956:r
7948:r
7944:y
7941:i
7938:+
7935:x
7920:y
7914:i
7911:+
7902:x
7895:[
7891:W
7884:2
7876:r
7871:a
7868:M
7865:i
7862:4
7859:=
7850:y
7844:i
7841:+
7832:x
7809:.
7783:2
7770:2
7766:a
7757:2
7752:)
7746:2
7742:r
7738:+
7733:2
7729:a
7724:(
7715:=
7707:t
7683:r
7635:v
7611:t
7552:/
7545:=
7470:0
7434:p
7430:=
7421:d
7410:x
7406:d
7349:)
7345:Q
7342:+
7337:2
7333:)
7329:E
7326:a
7318:z
7314:L
7310:(
7307:+
7302:2
7298:r
7292:2
7283:(
7271:2
7267:)
7263:r
7260:(
7257:P
7254:=
7251:)
7248:r
7245:(
7242:R
7219:z
7215:L
7211:a
7204:)
7198:2
7194:a
7190:+
7185:2
7181:r
7176:(
7172:E
7169:=
7166:)
7163:r
7160:(
7157:P
7135:)
7120:2
7109:2
7104:z
7100:L
7094:+
7090:)
7084:2
7080:E
7071:2
7062:(
7056:2
7052:a
7047:(
7035:2
7024:Q
7021:=
7018:)
7012:(
6982:)
6979:r
6976:(
6973:P
6962:2
6958:a
6954:+
6949:2
6945:r
6938:+
6934:)
6928:z
6924:L
6909:2
6901:E
6898:a
6894:(
6890:a
6884:=
6871:d
6866:t
6863:d
6850:)
6847:r
6844:(
6841:P
6833:a
6828:+
6824:)
6809:2
6798:z
6794:L
6785:E
6782:a
6778:(
6771:=
6758:d
6750:d
6735:)
6729:(
6718:=
6705:d
6697:d
6682:)
6679:r
6676:(
6673:R
6665:=
6652:d
6647:r
6644:d
6612:1
6609:=
6606:c
6603:=
6600:M
6597:=
6594:G
6563:.
6559:)
6553:2
6548:)
6530:z
6526:L
6520:(
6515:+
6511:)
6505:2
6501:E
6492:2
6483:(
6477:2
6473:a
6468:(
6456:2
6448:+
6443:2
6434:p
6430:=
6427:Q
6401:Q
6363:p
6359:=
6354:z
6350:L
6329:,
6324:t
6320:p
6313:=
6310:E
6286:z
6282:L
6257:E
6223:p
6200:,
6191:p
6178:g
6168:p
6164:=
6159:2
6013:0
6010:=
6007:a
5976:D
5847:2
5837:2
5833:J
5824:2
5820:M
5816:=
5811:2
5807:)
5803:M
5797:r
5794:(
5736:d
5730:t
5727:d
5715:E
5712:r
5704:H
5701:r
5654:M
5639:g
5619:g
5611:Ï
5607:Ξ
5603:Ξ
5576:2
5566:2
5562:a
5555:1
5547:1
5544:=
5533:E
5528:r
5499:2
5485:2
5475:2
5471:a
5467:4
5459:2
5454:s
5450:r
5439:s
5435:r
5428:=
5417:E
5412:r
5393:g
5385:g
5360:2
5356:a
5349:1
5341:1
5338:=
5327:H
5322:r
5296:1
5293:=
5290:c
5287:=
5284:M
5281:=
5278:G
5251:2
5243:2
5239:a
5235:4
5227:2
5222:s
5218:r
5207:s
5203:r
5196:=
5185:H
5180:r
5157:g
5153:/
5150:1
5139:g
5127:1
5107:M
5083:0
5059:a
5006:)
5002:(
4985:.
4971:2
4961:2
4957:a
4953:r
4948:s
4944:r
4940:+
4936:)
4930:2
4926:a
4922:+
4917:2
4913:r
4908:(
4899:c
4896:a
4893:r
4888:s
4884:r
4877:=
4863:g
4854:t
4850:g
4841:=
4819:Ξ
4811:r
4802:)
4798:(
4781:.
4776:2
4771:)
4767:t
4764:d
4750:g
4741:t
4737:g
4731:+
4725:d
4721:(
4707:g
4703:+
4698:2
4690:d
4678:g
4674:+
4669:2
4665:r
4661:d
4655:r
4652:r
4647:g
4643:+
4638:2
4634:t
4630:d
4626:)
4611:g
4605:2
4597:t
4593:g
4582:t
4579:t
4575:g
4570:(
4566:=
4561:2
4553:d
4548:2
4544:c
4528:1
4514:)
4510:(
4491:2
4486:)
4468:(
4458:1
4453:+
4448:2
4443:)
4437:r
4425:(
4410:+
4405:2
4400:)
4382:(
4376:)
4366:r
4361:s
4357:r
4347:1
4343:(
4328:2
4316:1
4311:+
4304:t
4267:c
4262:a
4259:r
4254:s
4250:r
4246:2
4235:2
4230:)
4224:t
4212:(
4206:)
4194:2
4178:2
4174:a
4170:r
4165:s
4161:r
4154:+
4149:2
4145:a
4141:+
4136:2
4132:r
4127:(
4115:2
4111:c
4106:1
4098:=
4086:x
4064:x
4042:g
4001:k
3998:i
3994:g
3959:.
3951:2
3947:G
3941:2
3935:r
3932:r
3929:i
3924:M
3920:4
3913:2
3909:c
3903:2
3899:J
3892:+
3887:2
3881:r
3878:r
3875:i
3870:M
3866:=
3861:2
3857:M
3846:2
3842:G
3837:/
3831:2
3827:c
3821:2
3817:J
3808:4
3804:M
3798:+
3793:2
3789:M
3785:=
3780:2
3774:r
3771:r
3768:i
3763:M
3759:2
3728:M
3699:M
3672:M
3638:M
3613:M
3587:2
3561:M
3537:M
3534:=
3531:a
3502:)
3495:r
3492:r
3489:i
3484:M
3477:M
3473:(
3467:2
3463:c
3459:=
3453:t
3450:o
3447:r
3442:E
3390:a
3386:R
3370:1
3367:=
3364:c
3340:a
3311:2
3307:z
3303:+
3298:2
3294:y
3290:+
3285:2
3281:x
3275:=
3272:R
3266:r
3254:R
3250:r
3241:)
3237:(
3220:1
3217:=
3210:2
3206:r
3200:2
3196:z
3190:+
3182:2
3178:a
3174:+
3169:2
3165:r
3157:2
3153:y
3149:+
3144:2
3140:x
3121:r
3099:a
3084:a
3076:η
3072:M
3068:η
3064:g
3052:k
3043:)
3039:(
3018:=
3013:0
3009:k
2991:)
2987:(
2969:)
2963:r
2960:z
2955:,
2947:2
2943:a
2939:+
2934:2
2930:r
2924:x
2921:a
2915:y
2912:r
2906:,
2898:2
2894:a
2890:+
2885:2
2881:r
2875:y
2872:a
2869:+
2866:x
2863:r
2856:(
2852:=
2849:)
2844:z
2840:k
2836:,
2831:y
2827:k
2823:,
2818:x
2814:k
2810:(
2807:=
2803:k
2785:)
2781:(
2759:2
2755:z
2749:2
2745:a
2741:+
2736:4
2732:r
2724:3
2720:r
2716:M
2713:G
2710:2
2704:=
2701:f
2684:)
2680:(
2656:k
2646:k
2642:f
2639:+
2623:=
2611:g
2569:)
2567:9
2565:(
2546:2
2538:d
2527:2
2518:)
2512:2
2508:a
2504:+
2499:2
2495:r
2490:(
2486:+
2481:2
2473:d
2467:+
2462:2
2458:r
2454:d
2446:2
2442:a
2438:+
2433:2
2429:r
2419:+
2414:2
2410:t
2406:d
2401:2
2397:c
2388:0
2382:M
2372:g
2340:s
2336:r
2311:M
2281:d
2277:t
2274:d
2257:)
2255:8
2253:(
2236:.
2231:2
2227:a
2223:+
2220:r
2215:s
2211:r
2202:2
2198:r
2194:=
2174:)
2172:7
2170:(
2153:,
2142:2
2132:2
2128:a
2124:+
2119:2
2115:r
2111:=
2091:)
2089:6
2087:(
2070:,
2064:c
2061:M
2057:J
2052:=
2049:a
1995:,
1992:a
1975:)
1973:5
1971:(
1954:,
1947:2
1943:c
1938:M
1935:G
1932:2
1926:=
1921:s
1917:r
1886:s
1882:r
1866:)
1864:4
1862:(
1845:,
1833:r
1830:=
1827:z
1810:)
1808:3
1806:(
1767:2
1763:a
1759:+
1754:2
1750:r
1744:=
1741:y
1724:)
1722:2
1720:(
1681:2
1677:a
1673:+
1668:2
1664:r
1658:=
1655:x
1621:,
1615:,
1612:r
1595:)
1593:1
1591:(
1567:d
1563:t
1560:d
1556:c
1539:2
1531:a
1528:r
1523:s
1519:r
1515:2
1504:2
1496:d
1482:2
1473:)
1461:2
1445:2
1441:a
1437:r
1432:s
1428:r
1421:+
1416:2
1412:a
1408:+
1403:2
1399:r
1394:(
1390:+
1385:2
1377:d
1371:+
1366:2
1362:r
1358:d
1345:+
1340:2
1336:t
1332:d
1327:2
1323:c
1318:)
1308:r
1303:s
1299:r
1289:1
1285:(
1278:=
1266:2
1258:d
1253:2
1249:c
1242:=
1233:2
1229:s
1225:d
1182:J
1155:M
1084:s
1081:r
1063:s
1059:r
1055:=
1052:r
1008:e
946:R
935:Q
918:Q
898:R
887:J
880:J
862:J
854:Q
758:e
751:t
744:v
425:)
421:(
94:T
85:=
73:g
66:+
54:G
20:)
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