8162:
8208:
711:
1264:
to the mediator of the electromagnetic interaction.) As seen from a
Newtonian point of view, if gravitational influence does propagate at a finite speed, then at all points in time a planet is attracted to a point where the Sun was some time before, and not towards the instantaneous position of the Sun. On the assumption of the classical fundamentals, Laplace had shown that if gravity would propagate at a velocity on the order of the speed of light then the solar system would be unstable, and would not exist for a long time. The observation that the solar system is old enough allowed him to put a lower limit on the
2906:
8189:
4452:
8697:
41:
8630:
8606:
8618:
5337:
6034:
7170:
1323:
10103:
724:
5148:
8222:
5856:
8136:
1157:
4845:
7550:
5583:
7821:
7814:
6369:
8161:
1300:(1898) even succeeded in deriving the correct formula for the perihelion shift (which was identical to that formula later used by Einstein). However, because the basic laws of Weber and others were wrong (for example, Weber's law was superseded by Maxwell's theory), those hypotheses were rejected. Another attempt by
6659:
3849:
8279:
gives a thorough treatment of the analogous problem in electromagnetism. Feynman shows that for a moving charge, the non-radiative field is an attraction/repulsion not toward the apparent position of the particle, but toward the extrapolated position assuming that the particle continues in a straight
7242:
is a calculational method that provides a series of ever more accurate solutions to a given problem. The method is iterative; an initial solution for particle motions is used to calculate the gravitational fields; from these derived fields, new particle motions can be calculated, from which even more
1422:
should be curved in order to reproduce the physical laws observed in Nature, particularly gravitation. Gravity is distinct from the fictitious forces centrifugal force and coriolis force in the sense that the curvature of spacetime is regarded as physically real, whereas the fictitious forces are not
1263:
had shown that if the gravitational influence does not act instantaneously, then the motions of the planets themselves will not exactly conserve momentum (and consequently some of the momentum would have to be ascribed to the mediator of the gravitational interaction, analogous to ascribing momentum
1164:
eternally. The presence of other forces (such as the gravitation of other planets), causes this ellipse to rotate gradually. The rate of this rotation (called orbital precession) can be measured very accurately. The rate can also be predicted knowing the magnitudes and directions of the other forces.
831:
of the other. If so, the larger mass may be taken as stationary and the sole contributor to the gravitational field. This is a good approximation for a photon passing a star and for a planet orbiting its sun. The motion of the lighter body (called the "particle" below) can then be determined from the
4113:
3128:
1271:
Laplace's estimate for the speed of gravity is not correct in a field theory which respects the principle of relativity. Since electric and magnetic fields combine, the attraction of a point charge which is moving at a constant velocity is towards the extrapolated instantaneous position, not to the
6876:
5332:
3386:
1385:
as a key concept to guide his way. According to this principle, a uniform gravitational field acts equally on everything within it and, therefore, cannot be detected by a free-falling observer. Conversely, all local gravitational effects should be reproducible in a linearly accelerating reference
7301:
5143:{\displaystyle {\begin{aligned}r_{\mathrm {outer} }&={\frac {a^{2}}{r_{s}}}\left(1+{\sqrt {1-{\frac {3r_{s}^{2}}{a^{2}}}}}\right)\\r_{\mathrm {inner} }&={\frac {a^{2}}{r_{s}}}\left(1-{\sqrt {1-{\frac {3r_{s}^{2}}{a^{2}}}}}\right)={\frac {3a^{2}}{r_{\mathrm {outer} }}},\end{aligned}}}
899:
1177:
is observed for all the planets orbiting the Sun, primarily due to the oblateness of the Sun (it is not perfectly spherical) and the attractions of the other planets to one another. The apsides are the two points of closest and furthest distance of the orbit (the periapsis and apoapsis,
7264:
Einstein's equations can also be solved on a computer using sophisticated numerical methods. Given sufficient computer power, such solutions can be more accurate than post-Newtonian solutions. However, such calculations are demanding because the equations must generally be solved in a
7575:
6190:
1138:. This model can be applied approximately to the Solar System. Since the mass of the Sun is much larger than those of the planets, the force acting on each planet is principally due to the Sun; the gravity of the planets for each other can be neglected to first approximation.
2672:
4827:
i.e., when the two attractive forcesâNewtonian gravity (first term) and the attraction unique to general relativity (third term)âare exactly balanced by the repulsive centrifugal force (second term). There are two radii at which this balancing can occur, denoted here as
1860:
Another illustration would be a world in which the rulers used to measure length were untrustworthy, rulers that changed their length with their position and even their orientation. In the most general case, one must allow for cross-terms when calculating the distance
1223:
was not quite what it should be; the ellipse of its orbit was rotating (precessing) slightly faster than predicted by the traditional theory of
Newtonian gravity, even after all the effects of the other planets had been accounted for. The effect is small (roughly 43
8188:
1369:. It followed that Newton's law of gravitation would have to be replaced with another law, compatible with the principle of relativity, while still obtaining the Newtonian limit for circumstances where relativistic effects are negligible. Such attempts were made by
5851:{\displaystyle \omega _{\varphi }^{2}\approx {\frac {GM}{r_{\mathrm {outer} }^{3}}}=\left({\frac {r_{s}c^{2}}{2r_{\mathrm {outer} }^{3}}}\right)=\left({\frac {r_{s}c^{2}}{2}}\right)\left({\frac {r_{s}^{3}}{8a^{6}}}\right)={\frac {c^{2}r_{s}^{4}}{16a^{6}}}}
8131:{\displaystyle -\left\langle {\frac {dP_{b}}{dt}}\right\rangle ={\frac {192\pi G^{5/3}m_{1}m_{2}(m_{1}+m_{2})^{-1/3}}{5c^{5}\left(1-e^{2}\right)^{7/2}}}\left(1+{\frac {73}{24}}e^{2}+{\frac {37}{96}}e^{4}\right)\left({\frac {P_{b}}{2\pi }}\right)^{-{5/3}}}
2399:
4481:
equals the square root of three, a metastable circular orbit is possible at the radius highlighted with a green circle. At higher angular momenta, there is a significant centrifugal barrier (orange curve) and an unstable inner radius, highlighted in
1232:
per century). Le
Verrier realized the importance of his discovery immediately, and challenged astronomers and physicists alike to account for it. Several classical explanations were proposed, such as interplanetary dust, unobserved oblateness of the
856:, an iterative method in which an initial solution is gradually corrected. More recently, it has become possible to solve Einstein's field equation using a computer instead of mathematical formulae. As the two bodies orbit each other, they will emit
6485:
3652:
2048:
4266:
3864:
2956:
6694:
5197:
3226:
6476:
8150:
The losses in energy and angular momentum increase significantly as the eccentricity approaches one, i.e., as the ellipse of the orbit becomes ever more elongated. The radiation losses also increase significantly with a decreasing size
2534:
in the space-time. In uncurved space-time, far from a source of gravity, these geodesics correspond to straight lines; however, they may deviate from straight lines when the space-time is curved. The equation for the geodesic lines is
7265:
four-dimensional space. Nevertheless, beginning in the late 1990s, it became possible to solve difficult problems such as the merger of two black holes, which is a very difficult version of the Kepler problem in general relativity.
7069:
4379:
1856:
5472:
5362:
in blue and red, respectively. These curves meet at a unique circular orbit (green circle) when the normalized angular momentum equals the square root of three. For comparison, the classical radius predicted from the
1708:
6185:
2538:
112:
7200:
follows a geodesic path through that fixed space-time. This is a reasonable approximation for photons and the orbit of
Mercury, which is roughly 6 million times lighter than the Sun. However, it is inadequate for
5925:
4825:
7160:
2245:
2493:
3640:
5551:
1118:. Given this force law and his equations of motion, Newton was able to show that two point masses attracting each other would each follow perfectly elliptical orbits. The ratio of sizes of these ellipses is
7545:{\displaystyle -\left\langle {\frac {dE}{dt}}\right\rangle ={\frac {32G^{4}m_{1}^{2}m_{2}^{2}(m_{1}+m_{2})}{5c^{5}a^{5}\left(1-e^{2}\right)^{7/2}}}\left(1+{\frac {73}{24}}e^{2}+{\frac {37}{96}}e^{4}\right)}
6972:
2874:); at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The Schwarzschild radius of the Sun is much larger, roughly 2953 meters, but at its surface, the ratio
2508:. Einstein developed those field equations to match the then known laws of Nature; however, they predicted never-before-seen phenomena (such as the bending of light by gravity) that were confirmed later.
851:
If both masses are considered to contribute to the gravitational field, as in binary stars, the Kepler problem can be solved only approximately. The earliest approximation method to be developed was the
8194:
Two neutron stars rotating rapidly around one another gradually lose energy by emitting gravitational radiation. As they lose energy, they orbit each other more quickly and more closely to one another.
1259:
Others argued that Newton's law should be supplemented with a velocity-dependent potential. However, this implied a conflict with
Newtonian celestial dynamics. In his treatise on celestial mechanics,
1197:
predict the apsidal precessions of the planets accurately, it would have to be discarded as a theory of gravitation. Such an anomalous precession was observed in the second half of the 19th century.
2250:
7809:{\displaystyle -\left\langle {\frac {dL_{z}}{dt}}\right\rangle ={\frac {32G^{7/2}m_{1}^{2}m_{2}^{2}{\sqrt {m_{1}+m_{2}}}}{5c^{5}a^{7/2}\left(1-e^{2}\right)^{2}}}\left(1+{\frac {7}{8}}e^{2}\right)}
7243:
accurate estimates of the fields can be computed, and so on. This approach is called "post-Newtonian" because the
Newtonian solution for the particle orbits is often used as the initial solution.
6364:{\displaystyle \omega _{r}^{2}=\left({\frac {c^{2}r_{s}}{2r_{\text{outer}}^{4}}}\right)\left(r_{\text{outer}}-r_{\text{inner}}\right)=\omega _{\varphi }^{2}{\sqrt {1-{\frac {3r_{s}^{2}}{a^{2}}}}}}
5202:
4850:
1867:
4121:
4653:
2711:
for timelike orbits (which are traveled by massive particles), and is usually taken to be equal to it. For lightlike (or null) orbits (which are traveled by massless particles such as the
7223:
for details). In (1+1) dimensions, i.e. a space made of one spatial dimension and one time dimension, the metric for two bodies of equal masses can be solved analytically in terms of the
1527:
6049:
introduces a third force that attracts the particle slightly more strongly than
Newtonian gravity, especially at small radii. This third force causes the particle's elliptical orbit to
3459:, have been introduced. They are constants of the motion and depend on the initial conditions (position and velocity) of the test particle. Hence, the solution of the orbit equation is
2821:
6380:
3197:
1418:, to allow for the curvature of space and time that was necessary for the reconciliation; after eight years of work (1907â1915), he succeeded in discovering the precise way in which
1381:(1910). In 1907 Einstein came to the conclusion that to achieve this a successor to special relativity was needed. From 1907 to 1915, Einstein worked towards a new theory, using his
1104:
1185:
Newton's law of gravitation soon became accepted because it gave very accurate predictions of the motion of all the planets. These calculations were carried out initially by
6977:
4287:
7572:
orbit. The angular brackets on the left-hand side of the equation represent the averaging over a single orbit. Similarly, the average rate of losing angular momentum equals
3457:
3419:
1745:
7298:
due to gravitational radiation from the two bodies of the Kepler problem have been calculated. The rate of losing energy (averaged over a complete orbit) is given by
1406:
and to incorporate the equivalence principle, something had to be sacrificed; that something was the long-held classical assumption that our space obeys the laws of
844:, which is a key piece of evidence supporting the theory of general relativity. They also describe the bending of light in a gravitational field, another prediction
9806:
482:
5571:
5403:
3217:
6654:{\displaystyle \delta \varphi =T(\omega _{\varphi }-\omega _{r})\approx 2\pi \left({\frac {3r_{s}^{2}}{4a^{2}}}\right)={\frac {3\pi m^{2}c^{2}}{2L^{2}}}r_{s}^{2}}
3844:{\displaystyle \left({\frac {dr}{d\tau }}\right)^{2}={\frac {E^{2}}{m^{2}c^{2}}}-c^{2}+{\frac {r_{s}c^{2}}{r}}-{\frac {h^{2}}{r^{2}}}+{\frac {r_{s}h^{2}}{r^{3}}}}
2889:
star is much denser, but even here the ratio at its surface is roughly 250 parts in a million. The ratio only becomes large close to ultra-dense objects such as
5176: = 0. At the outer radius, however, the circular orbits are stable; the third term is less important and the system behaves more like the non-relativistic
4455:
Effective radial potential for various angular momenta. At small radii, the energy drops precipitously, causing the particle to be pulled inexorably inwards to
3148:
2951:
2931:
1560:
6083:
4108:{\displaystyle {\frac {1}{2}}m\left({\frac {dr}{d\tau }}\right)^{2}=\left+{\frac {GMm}{r}}-{\frac {L^{2}}{2\mu r^{2}}}+{\frac {G(M+m)L^{2}}{c^{2}\mu r^{3}}}}
3123:{\displaystyle \left({\frac {dr}{d\tau }}\right)^{2}={\frac {E^{2}}{m^{2}c^{2}}}-\left(1-{\frac {r_{s}}{r}}\right)\left(c^{2}+{\frac {h^{2}}{r^{2}}}\right).}
6871:{\displaystyle \delta \varphi \approx {\frac {3\pi m^{2}c^{2}}{2L^{2}}}\left({\frac {4G^{2}M^{2}}{c^{4}}}\right)={\frac {6\pi G^{2}M^{2}m^{2}}{c^{2}L^{2}}}}
5874:
5327:{\displaystyle {\begin{aligned}r_{\mathrm {outer} }&\approx {\frac {2a^{2}}{r_{s}}}\\r_{\mathrm {inner} }&\approx {\frac {3}{2}}r_{s}\end{aligned}}}
2909:
Comparison between the orbit of a test particle in
Newtonian (left) and Schwarzschild (right) spacetime. Please click for high resolution animated graphics.
4660:
3381:{\displaystyle \left({\frac {dr}{d\varphi }}\right)^{2}={\frac {r^{4}}{b^{2}}}-\left(1-{\frac {r_{s}}{r}}\right)\left({\frac {r^{4}}{a^{2}}}+r^{2}\right),}
934:. Brahe took extraordinarily accurate measurements of the motion of the planets of the Solar System. From these measurements, Kepler was able to formulate
755:
7094:
4281:
2134:
7208:
The metric for the case of two comparable masses cannot be solved in closed form and therefore one has to resort to approximation techniques such as the
2414:
2123:
between two spatial points is not constant, but depends on the motion of the observer. However, there is a measure of separation between two points in
1173:
potential of Newton's gravitational law but differs only slightly, then the ellipse of the orbit gradually rotates (among other possible effects). This
871:, the numerical solution of the two-body problem was achieved after four decades of research in 2005 when three groups devised breakthrough techniques.
3462:
5490:
9213:
8479:; Choi, Dae-Il; Koppitz, Michael; van Meter, James (2006). "Gravitational-Wave Extraction from an Inspiraling Configuration of Merging Black Holes".
1722:
are arbitrary functions of position. It is not hard to imagine such a world; we live on one. The surface of the earth is curved, which is why it is
1272:
apparent position it seems to occupy when looked at. To avoid those problems, between 1870 and 1900 many scientists used the electrodynamic laws of
6894:
134:
8553:
10125:
47:
996:
Kepler published the first two laws in 1609 and the third law in 1619. They supplanted earlier models of the Solar System, such as those of
339:
1178:
respectively); apsidal precession corresponds to the rotation of the line joining the apsides. It also corresponds to the rotation of the
837:
9853:
845:
10088:
692:
9877:
9623:
2753:, which corresponds to the external gravitational field of a stationary, uncharged, non-rotating, spherically symmetric body of mass
1043:. Newton then posed the question: what must the force be that produces the elliptical orbits seen by Kepler? His answer came in his
502:
344:
4505:
1044:
889:
748:
457:
1160:
In the absence of any other forces, a particle orbiting another under the influence of
Newtonian gravity follows the same perfect
8253:
2667:{\displaystyle {\frac {d^{2}x^{\mu }}{dq^{2}}}+\Gamma _{\nu \lambda }^{\mu }{\frac {dx^{\nu }}{dq}}{\frac {dx^{\lambda }}{dq}}=0}
1454:
2771:
1165:
However, the predictions of
Newtonian gravity do not match the observations, as discovered in 1859 from observations of Mercury.
9613:
7187:
370:
5172:(where all three forces are in balance), the third force dominates the other two and draws the particle inexorably inwards to
9797:
9226:
9198:
9144:
9114:
9069:
9047:
8990:
8960:
8934:
8912:
8861:
8833:
7196:
is stationary and it alone determines the gravitational field (i.e., the geometry of space-time) and, hence, the lesser mass
1058:
409:
5928:
5151:
9281:
7250:
that captures the GR corrections to the Newtonian potential. Secondly, one should solve the resulting equations of motion.
935:
885:
8779:
8649:
Ohta, T.; Mann, R. B. (1997). "Exact solution for the metric and the motion of two bodies in (1+1)-dimensional gravity".
8551:(1859). "Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planÚte".
8285:
741:
9214:
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies
9155:
7173:
Diagram of the parameter space of compact binaries with the various approximation schemes and their regions of validity.
9534:
3151:
1313:
121:
2079:. In the spherical-coordinates example above, there are no cross-terms; the only nonzero metric tensor components are
9905:
9539:
9092:
9025:
6889:
1179:
1151:
147:
8281:
8985:. San Francisco: W. H. Freeman. pp. Chapter 25 (pp. 636â687), §33.5 (pp. 897â901), and §40.5 (pp. 1110â1116).
9859:
8207:
710:
427:
40:
9572:
2495:
just as we generalized the formula to measure distance on the surface of the Earth. The exact form of the metric
1723:
334:
8401:; Marronetti, P.; Zlochower, Y. (2006). "Accurate Evolutions of Orbiting Black-Hole Binaries without Excision".
1730:
of the earth. Non-Cartesian coordinate systems illustrate this well; for example, in the spherical coordinates (
9675:
8394:
8243:
4269:
2411:, however, space and time may have curvature, so this distance formula must be modified to a more general form
2394:{\displaystyle c^{2}\,d\tau ^{2}=c^{2}\,dt^{2}-dr^{2}-r^{2}\,d\theta ^{2}-r^{2}\sin ^{2}\theta \,d\varphi ^{2}}
19:
10106:
9848:
9753:
9683:
8952:
1427:
explained the anomalous precession of Mercury and predicted an unusual bending of light, which was confirmed
1134:, then the larger mass will appear to be stationary at the focus of the elliptical orbit of the lighter mass
981:
421:
129:
9816:
4439:= 0; this inner radius is a function of the particle's angular momentum per unit mass or, equivalently, the
3157:
1241:. After these explanations were discounted, some physicists were driven to the more radical hypothesis that
467:
9670:
7209:
1398:, which result from being in an accelerated reference frame; all fictitious forces are proportional to the
1362:
487:
9711:
9445:
9430:
2043:{\displaystyle ds^{2}=g_{xx}\,dx^{2}+g_{xy}\,dx\,dy+g_{xz}\,dx\,dz+\cdots +g_{zy}\,dz\,dy+g_{zz}\,dz^{2}}
1554:
1206:
728:
219:
139:
4284:, this inverse-cubic energy causes elliptical orbits to precess gradually by an angle ÎŽÏ per revolution
4261:{\displaystyle V(r)=-{\frac {GMm}{r}}+{\frac {L^{2}}{2\mu r^{2}}}-{\frac {G(M+m)L^{2}}{c^{2}\mu r^{3}}}}
9998:
9440:
9393:
5483:
is the orbital angular speed of the particle, is obtained in non-relativistic mechanics by setting the
5161:
is unstable, because the attractive third force strengthens much faster than the other two forces when
1415:
1020:
836:("shortest path between two points") in the curved space-time. Such geodesic solutions account for the
632:
180:
7212:
or numerical approximations. In passing, we mention one particular exception in lower dimensions (see
1557:. Now imagine a world in which this is not quite true; a world where the distance is instead given by
860:; this causes them to lose energy and angular momentum gradually, as illustrated by the binary pulsar
10028:
9654:
9335:
8848:
8767:
432:
244:
10053:
9603:
9363:
9274:
7239:
7183:
5364:
3424:
2746:
2505:
1424:
1326:
853:
667:
657:
507:
324:
9218:
3391:
9782:
9704:
9324:
9315:
8289:
8258:
7284:
7274:
6471:{\displaystyle \omega _{r}=\omega _{\varphi }\left(1-{\frac {3r_{s}^{2}}{4a^{2}}}+\cdots \right)}
2740:
2720:
2517:
1156:
1009:
819:. No exact solutions of the Kepler problem have been found, but an approximate solution has: the
801:
652:
386:
9190:
9183:
9136:
8825:
4268:
The first two terms are well-known classical energies, the first being the attractive Newtonian
2849:
In practice, this ratio is almost always extremely small. For example, the Schwarzschild radius
2530:
According to Einstein's theory of general relativity, particles of negligible mass travel along
910:. The larger mass is also moving on an elliptical orbit, but it is too small to be seen because
10008:
9871:
9777:
9735:
9409:
9373:
8978:
8719:
Peters PC, Mathews J (1963). "Gravitational Radiation from Point Masses in a Keplerian Orbit".
2828:
1115:
816:
622:
607:
452:
214:
9017:
8771:
7082:
10013:
9618:
9557:
8874:(1931). "Theory of the relativistic trajectories in a gravitational field of Schwarzschild".
8853:
6061:
and Earth. The yellow dot within the orbits represents the center of attraction, such as the
4418:
1382:
597:
165:
9918:
9844:
9688:
9290:
8883:
8793:
8728:
8670:
8498:
8420:
8339:
7557:
7259:
6682:
3852:
2905:
2765:
2750:
1277:
1273:
1186:
820:
417:
8955:. Vol. 2 (revised 4th English ed.). New York: Pergamon Press. pp. 299â309.
5556:
3202:
1304:(1900), who already used Maxwell's theory, produced a perihelion shift which was too low.
1284:
to produce stable orbits and to explain the perihelion shift of Mercury's orbit. In 1890,
8:
9993:
9988:
9978:
9810:
9759:
9577:
9562:
9388:
9319:
9267:
7247:
7091:
Hence, the final formula of angular apsidal precession for a unit complete revolution is
4116:
2525:
2404:
2076:
1444:
1411:
777:
612:
582:
577:
329:
9983:
8887:
8797:
8732:
8674:
8502:
8424:
8343:
7064:{\displaystyle \delta \varphi \approx {\frac {6\pi G(M+m)}{c^{2}A\left(1-e^{2}\right)}}}
4374:{\displaystyle \delta \varphi \approx {\frac {6\pi G(M+m)}{c^{2}A\left(1-e^{2}\right)}}}
592:
562:
10058:
9958:
9882:
9740:
9721:
9715:
9666:
9608:
9517:
9435:
9353:
9330:
9298:
9129:
9102:
9081:
8998:
8818:
8783:
8686:
8660:
8530:
8488:
8452:
8410:
8371:
8329:
8179:
7280:
7224:
6050:
6046:
4277:
3133:
2936:
2916:
2843:
2675:
2408:
2112:
1440:
1407:
1403:
1342:
1317:
1245:
1174:
1147:
857:
785:
662:
547:
472:
319:
224:
209:
170:
32:
1851:{\displaystyle ds^{2}=dr^{2}+r^{2}\,d\theta ^{2}+r^{2}\sin ^{2}\theta \,d\varphi ^{2}}
1451:
between two points in space is the sum of the squares of its perpendicular components
1190:
10068:
9900:
9892:
9493:
9450:
9246:
showing relativistic precession of stars around the Milky Way supermassive black hole
9222:
9194:
9140:
9110:
9088:
9065:
9043:
9021:
9010:
8986:
8970:
8956:
8930:
8922:
8908:
8891:
8857:
8829:
8690:
8548:
8522:
8514:
8444:
8436:
8363:
8355:
5484:
4273:
1391:
1378:
1374:
1212:
985:
868:
808:
492:
267:
9938:
9156:"Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905â1910"
8772:"The Relativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis"
8534:
8375:
4451:
1370:
1354:
542:
9973:
9963:
9910:
9887:
9465:
9057:
8736:
8678:
8506:
8456:
8428:
8347:
8227:
7295:
6374:
6054:
2831:. The classical Newtonian theory of gravity is recovered in the limit as the ratio
1387:
1289:
1281:
1265:
1220:
973:
955:
893:
841:
812:
617:
572:
552:
497:
10033:
8510:
8432:
8351:
8178:(red dots) matches the change due to the reduction in orbital period predicted by
2715:), the proper time is zero and, strictly speaking, cannot be used as the variable
637:
10048:
10023:
9948:
9943:
9826:
9787:
9749:
9693:
9567:
9503:
9208:
9178:
9042:. Translated by G. Field. New York: Dover Publications. pp. 40â41, 166â169.
8871:
8843:
8651:
8317:
7565:
6069:
The orbital precession rate may be derived using this radial effective potential
5340:
The stable and unstable radii are plotted versus the normalized angular momentum
2116:
1358:
1350:
1301:
1238:
989:
962:
927:
672:
647:
532:
527:
391:
272:
234:
9831:
6053:(cyan orbit) in the direction of its rotation; this effect has been measured in
1285:
442:
10073:
9745:
9729:
9725:
9628:
9598:
9455:
9358:
9035:
8476:
8248:
8235:
8213:
6038:
5336:
5177:
1727:
1395:
1366:
1293:
1268:
that turned out to be many orders of magnitude faster than the speed of light.
977:
807:
General relativity describes the gravitational field by curved space-time; the
789:
781:
715:
682:
677:
365:
229:
23:
9243:
1288:
succeeded in doing so by combining the laws of Weber and Riemann, whereby the
792:
is essential to calculate the bending of light by gravity and the motion of a
10119:
10063:
10043:
10038:
9953:
9821:
9649:
9593:
9425:
9378:
9249:
9163:
8895:
8682:
8518:
8440:
8398:
8359:
8175:
8172:
6022:
5467:{\displaystyle r_{\text{outer}}^{3}={\frac {G(M+m)}{\omega _{\varphi }^{2}}}}
2691:
2131:
invariant; in other words, it does not depend on the motion of the observer.
2072:
861:
642:
557:
537:
462:
360:
239:
8852:(5th ed.). Princeton, New Jersey: Princeton University Press. pp.
8740:
7227:. However, the gravitational energy between the two bodies is exchanged via
1228:
of rotation per century), but well above the measurement error (roughly 0.1
602:
10083:
10003:
9968:
9498:
9460:
9005:
8526:
8448:
8367:
7213:
5574:
3220:
2890:
1242:
1016:
587:
567:
6033:
2504:
depends on the gravitating mass, momentum and energy, as described by the
1193:
in the later 19th century. Conversely, if Newton's law of gravitation did
9867:
9836:
9383:
9124:
8788:
7202:
5997:
goes to infinity, implying that there is a circular orbit for photons at
2886:
2705:
1703:{\displaystyle ds^{2}=F(x,y,z)\,dx^{2}+G(x,y,z)\,dy^{2}+H(x,y,z)\,dz^{2}}
1399:
1338:
1297:
931:
797:
477:
447:
8903:
Kopeikin, Sergei; Efroimsky, Michael; Kaplan, George (25 October 2011).
7169:
7073:
Since the closed classical orbit is an ellipse in general, the quantity
6180:{\displaystyle \omega _{r}^{2}={\frac {1}{m}}\left_{r=r_{\text{outer}}}}
1365:, exclude the possibility of propagation of any effects faster than the
1322:
1004:. Kepler's laws apply only in the limited case of the two-body problem.
10078:
9644:
9488:
9483:
8974:
8944:
8168:
7246:
The theory can be divided into two parts: first one finds the two-body
2894:
2857:
2687:
2683:
2124:
1419:
1216:
1001:
687:
249:
175:
107:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
8665:
8493:
8415:
8334:
5920:{\displaystyle r_{\text{outer}}\approx r_{\text{inner}}\approx 3r_{s}}
2407:
and similarly holds only when there is no curvature in space-time. In
1434:
1248:
of gravitation was incorrect. For example, some physicists proposed a
9475:
9064:(revised 2nd ed.). New York: Springer Verlag. pp. 143â149.
8594:
7232:
4820:{\displaystyle F=-{\frac {dV}{dr}}=-{\frac {mc^{2}}{2r^{4}}}\left=0;}
1249:
1229:
1225:
800:
around each other, and estimate their gradual loss of energy through
258:
7155:{\displaystyle \delta \varphi \approx {\frac {6\pi G(M+m)}{c^{2}l}}}
6880:
This may be simplified using the elliptical orbit's semi-major axis
2240:{\displaystyle c^{2}\,d\tau ^{2}=c^{2}\,dt^{2}-dx^{2}-dy^{2}-dz^{2}}
1169:
If the potential energy between the two bodies is not exactly the 1/
796:
orbiting its sun. Solutions are also used to describe the motion of
10018:
9368:
8393:
6482:
of one revolution gives the precession of the orbit per revolution
2842:
goes to zero. In that limit, the metric returns to that defined by
2531:
2521:
1253:
1005:
833:
627:
437:
277:
7192:
In the Schwarzschild solution, it is assumed that the larger mass
2488:{\displaystyle c^{2}\,d\tau ^{2}=g_{\mu \nu }dx^{\mu }\,dx^{\nu }}
1402:, just as gravity is. To effect the reconciliation of gravity and
9162:. Vol. 3. Berlin: Springer. pp. 193â252. Archived from
8747:
7569:
7228:
6042:
3635:{\displaystyle \varphi =\int {\frac {1}{r^{2}}}\left^{-1/2}\,dr.}
1260:
1161:
997:
951:
898:
8929:(4th ed.). New York: Dover Publications. pp. 330â338.
5546:{\displaystyle {\frac {GMm}{r^{2}}}=\mu \omega _{\varphi }^{2}r}
9698:
9012:
Subtle is the Lord: The Science and the Life of Albert Einstein
8280:
line in a constant velocity. This is a notable property of the
7291:
4657:
Circular orbits are possible when the effective force is zero:
2728:
2712:
1431:
his theory was published. These solutions are explained below.
1414:
is true experimentally. Einstein used a more general geometry,
947:
793:
7279:
If there is no incoming gravitational radiation, according to
4115:
which is equivalent to a particle moving in a one-dimensional
2127:â called "proper time" and denoted with the symbol dÏ â that
9259:
8288:. Presumably the same holds in linearized gravity: e.g., see
6967:{\displaystyle {\frac {h^{2}}{G(M+m)}}=A\left(1-e^{2}\right)}
6373:
Taking the square root of both sides and expanding using the
6058:
943:
919:
6080:
will oscillate in a stable manner with an angular frequency
5580:
In our notation, the classical orbital angular speed equals
4276:; however, the third term is an attractive energy unique to
2723:
of timelike orbits, that is, the limit as the particle mass
9679:
6073:. A small radial deviation from a circular orbit of radius
5165:
becomes small; if the particle slips slightly inwards from
2871:
1330:
966:
8969:
5194:(the classical case), these formulae become approximately
2700:, or rather on how it changes with position. The variable
1307:
9109:. Amsterdam: North-Holland Publishing. pp. 289â298.
8572:
8570:
8568:
8474:
6062:
1334:
1234:
1023:. In particular, Newton's second law states that a force
9062:
Essential Relativity: Special, General, and Cosmological
9217:(4th ed.). New York: Dover Publications. pp.
8902:
8778:. ASP Conference Series. Vol. 328. San Francisco:
8703:
8600:
3223:. This can be converted into an equation for the orbit
2719:. Nevertheless, lightlike orbits can be derived as the
1237:, an undetected moon of Mercury, or a new planet named
1126:, with the larger mass moving on a smaller ellipse. If
8815:
8565:
8389:
8387:
8385:
6021:. The sphere of this radius is sometimes known as the
4446:
3649:
The equation of motion for the particle derived above
3427:
3394:
3160:
930:, who worked as an assistant to the Danish astronomer
9135:. Chicago: The University of Chicago Press. pp.
8320:(2005). "Evolution of Binary Black-Hole Spacetimes".
7824:
7578:
7304:
7097:
6980:
6897:
6697:
6488:
6383:
6193:
6086:
5877:
5871:
from above, the two radii converge to a single value
5586:
5559:
5493:
5406:
5200:
4848:
4663:
4508:
4290:
4124:
3867:
3655:
3465:
3229:
3205:
3136:
2959:
2939:
2919:
2774:
2541:
2417:
2253:
2137:
1870:
1748:
1563:
1457:
1061:
50:
8905:
Relativistic Celestial Mechanics of the Solar System
8470:
8468:
8466:
8203:
7164:
5389:
yields the classical formula for a particle of mass
4424:
The third term is attractive and dominates at small
3644:
2913:
The orbits of a test particle of infinitesimal mass
1541:
represent the infinitesimal differences between the
938:, the first modern description of planetary motion:
902:
Figure 1. Typical elliptical path of a smaller mass
827:
of one body is overwhelmingly greater than the mass
8382:
8167:Experimentally observed changes in the time of the
7205:, in which the masses may be of similar magnitude.
4459:= 0. However, when the normalized angular momentum
4435:at which a particle is drawn inexorably inwards to
1435:
General relativity, special relativity and geometry
9182:
9128:
9080:
9009:
8817:
8582:
8130:
7808:
7544:
7253:
7154:
7063:
6966:
6870:
6653:
6470:
6363:
6179:
6028:
5919:
5850:
5565:
5545:
5466:
5326:
5142:
4819:
4647:
4373:
4260:
4107:
3843:
3634:
3451:
3413:
3380:
3211:
3191:
3142:
3122:
2945:
2925:
2815:
2666:
2487:
2393:
2239:
2042:
1850:
1702:
1521:
1200:
1098:
106:
8463:
7235:which require three-space in which to propagate.
2247:which may be written in spherical coordinates as
1386:frame, and vice versa. Thus, gravity acts like a
10117:
9083:Mercury's perihelion, from Leverrier to Einstein
5367:and Newton's law of gravity is plotted in black.
2900:
1423:regarded as forces. The very first solutions of
8824:. New York: McGraw-Hill Book Company. pp.
8312:
8310:
8308:
7287:, causing the orbits to gradually lose energy.
922:, the points of closest and farthest distance.
8943:
8765:
8753:
4648:{\displaystyle V(r)={\frac {mc^{2}}{2}}\left.}
4272:and the second corresponding to the repulsive
965:joining a planet and the Sun sweeps out equal
9275:
8718:
3851:can be rewritten using the definition of the
2690:. The Christoffel symbol depends only on the
2403:This formula is the natural extension of the
2075:, which defines the geometry of the space in
1047:, which states that the force between a mass
1012:were the first to call them "Kepler's laws".
749:
8876:Japanese Journal of Astronomy and Geophysics
8305:
7283:, two bodies orbiting one another will emit
5487:equal to the Newtonian gravitational force:
7177:
1522:{\displaystyle ds^{2}=dx^{2}+dy^{2}+dz^{2}}
879:
9282:
9268:
9189:. New York: John Wiley and Sons. pp.
8714:
8712:
8547:
7268:
5993:are not possible. For massless particles,
2816:{\displaystyle r_{s}={\frac {2GM}{c^{2}}}}
1182:, which points along the line of apsides.
756:
742:
9207:
9078:
8787:
8664:
8612:
8492:
8414:
8333:
8316:
4490:can be re-written in terms of the length
3622:
2734:
2682:parametrizes the particle's path through
2471:
2428:
2377:
2334:
2291:
2264:
2175:
2148:
2026:
2003:
1996:
1967:
1960:
1937:
1930:
1900:
1834:
1791:
1742:), the Euclidean distance can be written
1686:
1645:
1604:
1189:in the late 18th century, and refined by
926:The Kepler problem derives its name from
832:Schwarzschild solution; the motion is a
776:) is the determination of the motion and
9177:
8870:
8842:
8648:
8636:
7818:The rate of period decrease is given by
7168:
6032:
5335:
4450:
2904:
2757:. It is characterized by a length scale
1447:, which states that the square distance
1321:
1155:
918:. The ends of the diameter indicate the
897:
809:field equations governing this curvature
9056:
8927:The Variational Principles of Mechanics
8921:
8709:
6677:and the definition of the length-scale
4428:values, giving a critical inner radius
3388:where, for brevity, two length-scales,
3192:{\textstyle h=r\times v={L \over \mu }}
1329:'s 1919 measurements of the bending of
1308:Einstein's theory of general relativity
1110:is the distance between the masses and
1099:{\displaystyle F=G{\frac {Mm}{r^{2}}},}
823:. This solution pertains when the mass
10118:
9153:
8816:Adler, R; Bazin M; Schiffer M (1965).
8759:
8624:
7188:Parameterized post-Newtonian formalism
6041:, a particle follows the same perfect
1296:in his theory. And in another attempt
815:and therefore difficult to solve in a
770:two-body problem in general relativity
10126:Exact solutions in general relativity
9263:
9101:
9034:
8774:. In F.A. Rasio; I.H. Stairs (eds.).
8704:Kopeikin, Efroimsky & Kaplan 2011
8601:Kopeikin, Efroimsky & Kaplan 2011
6681:. Substituting the definition of the
2893:(where the ratio is roughly 50%) and
2727:goes to zero while holding its total
1141:
890:Newton's law of universal gravitation
874:
9123:
9016:. Oxford University Press. pp.
9004:
8588:
8576:
8254:Newton's theorem of revolving orbits
7290:The formulae describing the loss of
4417:) coordinates but the change in the
2511:
1256:that was slightly different from 2.
9854:TolmanâOppenheimerâVolkoff equation
9807:FriedmannâLemaĂźtreâRobertsonâWalker
8780:Astronomical Society of the Pacific
8277:Feynman Lectures on Physics vol. II
4447:Circular orbits and their stability
2953:is given by the equation of motion
2885:is roughly 4 parts in a million. A
13:
8820:Introduction to General Relativity
5699:
5696:
5693:
5690:
5687:
5633:
5630:
5627:
5624:
5621:
5287:
5284:
5281:
5278:
5275:
5223:
5220:
5217:
5214:
5211:
5125:
5122:
5119:
5116:
5113:
4990:
4987:
4984:
4981:
4978:
4871:
4868:
4865:
4862:
4859:
3537:
3152:specific relative angular momentum
2768:, which is defined by the formula
2587:
1314:Introduction to general relativity
780:of two bodies as described by the
67:
14:
10137:
9624:HamiltonâJacobiâEinstein equation
9237:
9160:The Genesis of General Relativity
7165:Beyond the Schwarzschild solution
3645:Effective radial potential energy
886:Kepler's laws of planetary motion
10102:
10101:
8220:
8206:
8187:
8160:
5371:Substituting the definitions of
723:
722:
709:
39:
8808:
8642:
8286:WheelerâFeynman absorber theory
7254:Modern computational approaches
6029:Precession of elliptical orbits
5976:. Circular orbits smaller than
4421:of the classical closed orbit.
2856:of the Earth is roughly 9
1553:coordinates of two points in a
1349:Around 1904â1905, the works of
1201:Anomalous precession of Mercury
969:during equal intervals of time.
954:with the Sun at one of the two
9431:Massâenergy equivalence (E=mc)
9289:
9107:Relativity: The General Theory
8949:The Classical Theory of Fields
8541:
8270:
8244:Gravitational two-body problem
7940:
7913:
7411:
7385:
7131:
7119:
7014:
7002:
6926:
6914:
6527:
6501:
5443:
5431:
4518:
4512:
4324:
4312:
4274:"centrifugal" potential energy
4270:gravitational potential energy
4217:
4205:
4134:
4128:
4064:
4052:
3452:{\textstyle b={\frac {Lc}{E}}}
2704:is a constant multiple of the
1683:
1665:
1642:
1624:
1601:
1583:
20:gravitational two-body problem
1:
8953:Course of Theoretical Physics
8511:10.1103/PhysRevLett.96.111102
8433:10.1103/PhysRevLett.96.111101
8352:10.1103/PhysRevLett.95.121101
8298:
8240:Center of mass (relativistic)
6974:to give the precession angle
5150:which are obtained using the
3414:{\textstyle a={\frac {h}{c}}}
2901:Orbits about the central mass
1726:to make a perfectly accurate
774:relativistic two-body problem
9087:. Oxford: University Press.
8182:(blue curve) almost exactly.
7210:post-Newtonian approximation
4443:length-scale defined above.
2113:special theory of relativity
1363:special theory of relativity
1215:discovered that the orbital
1045:law of universal gravitation
906:orbiting a much larger mass
7:
9446:Relativistic Doppler effect
8282:LiĂ©nardâWiechert potentials
8199:
5860:At the other extreme, when
4385:is the semi-major axis and
1555:Cartesian coordinate system
1207:Tests of general relativity
220:Gravitational time dilation
10:
10142:
9917:In computational physics:
9441:Relativity of simultaneity
8754:Landau & Lifshitz 1975
7272:
7257:
7181:
6478:Multiplying by the period
4389:is the eccentricity. Here
2738:
2515:
1416:pseudo-Riemannian geometry
1311:
1204:
1145:
883:
340:MathissonâPapapetrouâDixon
181:Pseudo-Riemannian manifold
17:
10099:
9931:
9796:
9768:
9754:LenseâThirring precession
9637:
9586:
9548:
9527:
9516:
9474:
9418:
9402:
9344:
9336:Doubly special relativity
9308:
9297:
9254:Reflections on Relativity
9185:Gravitation and Cosmology
9079:Roseveare, N. T. (1982).
8907:. John Wiley & Sons.
8849:The Meaning of Relativity
2745:An exact solution to the
2119:showed that the distance
2050:where the nine functions
1341:led to the acceptance of
1180:LaplaceâRungeâLenz vector
1152:LaplaceâRungeâLenz vector
1031:produces an acceleration
846:famously used as evidence
9614:Post-Newtonian formalism
9604:Einstein field equations
9540:Mathematical formulation
9364:Hyperbolic orthogonality
8683:10.1103/PhysRevD.55.4723
8264:
7240:post-Newtonian expansion
7184:Post-Newtonian expansion
7178:Post-Newtonian expansion
6037:In the non-relativistic
5938:is always greater than 3
5393:orbiting a body of mass
5365:centripetal acceleration
4486:The effective potential
2747:Einstein field equations
2506:Einstein field equations
1055:is given by the formula
1015:Nearly a century later,
980:of a planet is directly
880:Classical Kepler problem
854:post-Newtonian expansion
848:for general relativity.
345:HamiltonâJacobiâEinstein
325:Einstein field equations
148:Mathematical formulation
18:Not to be confused with
9325:Galilean transformation
9316:Principle of relativity
8947:; Lifshitz, EM (1975).
8741:10.1103/PhysRev.131.435
8481:Physical Review Letters
8403:Physical Review Letters
8322:Physical Review Letters
8290:Gravitoelectromagnetism
8259:Schwarzschild geodesics
7285:gravitational radiation
7275:Gravitational radiation
7269:Gravitational radiation
6045:(red orbit) eternally.
5400:The following equation
2933:about the central mass
2741:Schwarzschild geodesics
2721:ultrarelativistic limit
2674:where Î represents the
2518:Schwarzschild geodesics
858:gravitational radiation
802:gravitational radiation
9410:Lorentz transformation
8284:which are used in the
8132:
7810:
7546:
7174:
7156:
7065:
6968:
6872:
6655:
6472:
6365:
6181:
6066:
5921:
5852:
5567:
5547:
5468:
5368:
5328:
5144:
4821:
4649:
4483:
4375:
4262:
4109:
3845:
3636:
3453:
3415:
3382:
3213:
3193:
3144:
3124:
2947:
2927:
2910:
2829:gravitational constant
2817:
2735:Schwarzschild solution
2668:
2489:
2395:
2241:
2044:
1852:
1704:
1523:
1346:
1166:
1116:gravitational constant
1100:
1035:given by the equation
923:
821:Schwarzschild solution
215:Gravitational redshift
108:
9878:WeylâLewisâPapapetrou
9619:Raychaudhuri equation
9558:Equivalence principle
9158:. In Renn, J. (ed.).
8133:
7811:
7547:
7172:
7157:
7066:
6969:
6873:
6656:
6473:
6366:
6182:
6036:
5922:
5853:
5568:
5548:
5469:
5339:
5329:
5187:is much greater than
5145:
4822:
4650:
4454:
4419:argument of periapsis
4376:
4280:. As shown below and
4263:
4110:
3846:
3637:
3454:
3416:
3383:
3214:
3194:
3145:
3125:
2948:
2928:
2908:
2818:
2669:
2490:
2396:
2242:
2045:
1853:
1705:
1524:
1443:, triangles obey the
1383:equivalence principle
1325:
1159:
1101:
914:is much greater than
901:
503:WeylâLewisâPapapetrou
458:KerrâNewmanâde Sitter
278:EinsteinâRosen bridge
210:Gravitational lensing
166:Equivalence principle
109:
9919:Numerical relativity
9760:pulsar timing arrays
9040:Theory of Relativity
8776:Binary Radio Pulsars
7822:
7576:
7558:orbital eccentricity
7302:
7260:Numerical relativity
7095:
6978:
6895:
6695:
6683:Schwarzschild radius
6486:
6381:
6191:
6084:
5875:
5584:
5566:{\displaystyle \mu }
5557:
5491:
5404:
5198:
4846:
4661:
4506:
4288:
4122:
3865:
3853:Schwarzschild radius
3653:
3463:
3425:
3392:
3227:
3212:{\displaystyle \mu }
3203:
3158:
3134:
2957:
2937:
2917:
2772:
2766:Schwarzschild radius
2751:Schwarzschild metric
2539:
2415:
2251:
2135:
1868:
1746:
1561:
1455:
1278:Carl Friedrich Gauss
1274:Wilhelm Eduard Weber
1187:Pierre-Simon Laplace
1130:is much larger than
1059:
1021:three laws of motion
838:anomalous precession
433:EinsteinâRosen waves
159:Fundamental concepts
48:
9811:Friedmann equations
9705:HulseâTaylor binary
9667:Gravitational waves
9563:Riemannian geometry
9389:Proper acceleration
9374:Maxwell's equations
9320:Galilean relativity
9154:Walter, S. (2007).
8888:1931AOTok..31...67H
8798:2005ASPC..328...25W
8756:, pp. 356â357.
8733:1963PhRv..131..435P
8675:1997PhRvD..55.4723M
8579:, pp. 253â256.
8503:2006PhRvL..96k1102B
8425:2006PhRvL..96k1101C
8344:2005PhRvL..95l1101P
8147:is orbital period.
7673:
7658:
7384:
7369:
7248:effective potential
6661:where we have used
6650:
6563:
6438:
6377:yields the formula
6345:
6316:
6258:
6208:
6101:
5929:quadratic solutions
5829:
5777:
5709:
5643:
5601:
5539:
5461:
5421:
5154:. The inner radius
5066:
4947:
4117:effective potential
2603:
2526:Christoffel symbols
2405:Pythagorean theorem
2077:Riemannian geometry
1445:Pythagorean theorem
1425:his field equations
1412:Pythagorean theorem
1019:had formulated his
778:gravitational field
387:KaluzaâKlein theory
273:Minkowski spacetime
225:Gravitational waves
9860:ReissnerâNordström
9778:BransâDicke theory
9609:Linearized gravity
9436:Length contraction
9354:Frame of reference
9331:Special relativity
9131:General Relativity
8999:Gravitation (book)
8180:general relativity
8128:
7806:
7659:
7644:
7542:
7370:
7355:
7281:general relativity
7225:Lambert W function
7175:
7152:
7061:
6964:
6868:
6651:
6636:
6549:
6468:
6424:
6361:
6331:
6302:
6244:
6194:
6177:
6087:
6067:
6047:General relativity
5931:above ensure that
5917:
5848:
5815:
5763:
5681:
5615:
5587:
5563:
5543:
5525:
5464:
5447:
5407:
5369:
5324:
5322:
5140:
5138:
5052:
4933:
4817:
4645:
4484:
4397:the change in the
4371:
4278:general relativity
4258:
4105:
3841:
3632:
3449:
3411:
3378:
3209:
3189:
3140:
3120:
2943:
2923:
2911:
2844:special relativity
2813:
2676:Christoffel symbol
2664:
2586:
2485:
2409:general relativity
2391:
2237:
2040:
1848:
1700:
1519:
1441:Euclidean geometry
1408:Euclidean geometry
1404:special relativity
1347:
1343:general relativity
1318:General relativity
1246:inverse-square law
1175:apsidal precession
1167:
1148:Apsidal precession
1142:Apsidal precession
1096:
1027:applied to a mass
1010:Ămilie du ChĂątelet
924:
875:Historical context
869:binary black holes
786:general relativity
716:Physics portal
488:OppenheimerâSnyder
428:ReissnerâNordström
320:Linearized gravity
268:Spacetime diagrams
171:Special relativity
104:
33:General relativity
10113:
10112:
9927:
9926:
9906:OzsvĂĄthâSchĂŒcking
9512:
9511:
9494:Minkowski diagram
9451:Thomas precession
9394:Relativistic mass
9228:978-1-114-28944-4
9200:978-0-471-92567-5
9146:978-0-226-87032-8
9116:978-0-7204-0066-3
9071:978-0-387-10090-6
9049:978-0-486-64152-2
8992:978-0-7167-0344-0
8962:978-0-08-018176-9
8936:978-0-486-65067-8
8914:978-3-527-63457-6
8863:978-0-691-02352-6
8835:978-0-07-000420-7
8103:
8061:
8038:
8017:
7857:
7789:
7768:
7699:
7611:
7525:
7502:
7481:
7330:
7150:
7059:
6930:
6884:and eccentricity
6866:
6795:
6752:
6634:
6580:
6455:
6359:
6357:
6291:
6278:
6260:
6251:
6172:
6152:
6113:
5898:
5885:
5846:
5793:
5751:
5711:
5644:
5517:
5485:centrifugal force
5462:
5414:
5308:
5263:
5152:quadratic formula
5131:
5080:
5078:
5025:
4961:
4959:
4906:
4731:
4691:
4635:
4596:
4569:
4544:
4369:
4256:
4194:
4159:
4103:
4041:
4006:
3967:
3954:
3904:
3876:
3839:
3800:
3773:
3728:
3679:
3593:
3573:
3546:
3513:
3490:
3447:
3409:
3355:
3321:
3290:
3253:
3187:
3143:{\displaystyle h}
3110:
3063:
3032:
2983:
2946:{\displaystyle M}
2926:{\displaystyle m}
2811:
2678:and the variable
2656:
2629:
2581:
2512:Geodesic equation
1410:, e.g., that the
1392:centrifugal force
1379:Arnold Sommerfeld
1375:Hermann Minkowski
1213:Urbain Le Verrier
1091:
1051:and another mass
766:
765:
399:
398:
285:
284:
10133:
10105:
10104:
9888:van Stockum dust
9660:Two-body problem
9578:Mach's principle
9525:
9524:
9466:Terrell rotation
9306:
9305:
9284:
9277:
9270:
9261:
9260:
9232:
9204:
9188:
9174:
9172:
9171:
9150:
9134:
9120:
9098:
9086:
9075:
9053:
9031:
9015:
8996:
8966:
8940:
8918:
8899:
8867:
8839:
8823:
8802:
8801:
8791:
8789:astro-ph/0407149
8766:Weisberg, J.M.;
8763:
8757:
8751:
8745:
8744:
8716:
8707:
8701:
8695:
8694:
8668:
8659:(8): 4723â4747.
8646:
8640:
8634:
8628:
8622:
8616:
8610:
8604:
8598:
8592:
8586:
8580:
8574:
8563:
8562:
8545:
8539:
8538:
8496:
8475:Baker, John G.;
8472:
8461:
8460:
8418:
8391:
8380:
8379:
8337:
8318:Pretorius, Frans
8314:
8293:
8274:
8230:
8228:Astronomy portal
8225:
8224:
8223:
8216:
8211:
8210:
8191:
8164:
8137:
8135:
8134:
8129:
8127:
8126:
8125:
8121:
8108:
8104:
8102:
8094:
8093:
8084:
8077:
8073:
8072:
8071:
8062:
8054:
8049:
8048:
8039:
8031:
8018:
8016:
8015:
8014:
8010:
8001:
7997:
7996:
7995:
7974:
7973:
7960:
7959:
7958:
7954:
7938:
7937:
7925:
7924:
7912:
7911:
7902:
7901:
7892:
7891:
7887:
7867:
7862:
7858:
7856:
7848:
7847:
7846:
7833:
7815:
7813:
7812:
7807:
7805:
7801:
7800:
7799:
7790:
7782:
7769:
7767:
7766:
7765:
7760:
7756:
7755:
7754:
7733:
7732:
7728:
7715:
7714:
7701:
7700:
7698:
7697:
7685:
7684:
7675:
7672:
7667:
7657:
7652:
7643:
7642:
7638:
7621:
7616:
7612:
7610:
7602:
7601:
7600:
7587:
7551:
7549:
7548:
7543:
7541:
7537:
7536:
7535:
7526:
7518:
7513:
7512:
7503:
7495:
7482:
7480:
7479:
7478:
7474:
7465:
7461:
7460:
7459:
7438:
7437:
7428:
7427:
7414:
7410:
7409:
7397:
7396:
7383:
7378:
7368:
7363:
7354:
7353:
7340:
7335:
7331:
7329:
7321:
7313:
7296:angular momentum
7161:
7159:
7158:
7153:
7151:
7149:
7145:
7144:
7134:
7108:
7088:of the ellipse.
7070:
7068:
7067:
7062:
7060:
7058:
7057:
7053:
7052:
7051:
7028:
7027:
7017:
6991:
6973:
6971:
6970:
6965:
6963:
6959:
6958:
6957:
6931:
6929:
6909:
6908:
6899:
6877:
6875:
6874:
6869:
6867:
6865:
6864:
6863:
6854:
6853:
6843:
6842:
6841:
6832:
6831:
6822:
6821:
6805:
6800:
6796:
6794:
6793:
6784:
6783:
6782:
6773:
6772:
6759:
6753:
6751:
6750:
6749:
6736:
6735:
6734:
6725:
6724:
6708:
6676:
6660:
6658:
6657:
6652:
6649:
6644:
6635:
6633:
6632:
6631:
6618:
6617:
6616:
6607:
6606:
6590:
6585:
6581:
6579:
6578:
6577:
6564:
6562:
6557:
6544:
6526:
6525:
6513:
6512:
6477:
6475:
6474:
6469:
6467:
6463:
6456:
6454:
6453:
6452:
6439:
6437:
6432:
6419:
6406:
6405:
6393:
6392:
6375:binomial theorem
6370:
6368:
6367:
6362:
6360:
6358:
6356:
6355:
6346:
6344:
6339:
6326:
6318:
6315:
6310:
6298:
6294:
6293:
6292:
6289:
6280:
6279:
6276:
6265:
6261:
6259:
6257:
6252:
6249:
6239:
6238:
6237:
6228:
6227:
6217:
6207:
6202:
6186:
6184:
6183:
6178:
6176:
6175:
6174:
6173:
6170:
6157:
6153:
6151:
6150:
6149:
6136:
6132:
6131:
6121:
6114:
6106:
6100:
6095:
6013:
6012:
6008:
5985:
5984:
5980:
5961:
5960:
5956:
5926:
5924:
5923:
5918:
5916:
5915:
5900:
5899:
5896:
5887:
5886:
5883:
5857:
5855:
5854:
5849:
5847:
5845:
5844:
5843:
5830:
5828:
5823:
5814:
5813:
5803:
5798:
5794:
5792:
5791:
5790:
5776:
5771:
5762:
5756:
5752:
5747:
5746:
5745:
5736:
5735:
5725:
5716:
5712:
5710:
5708:
5703:
5702:
5676:
5675:
5674:
5665:
5664:
5654:
5645:
5642:
5637:
5636:
5614:
5606:
5600:
5595:
5572:
5570:
5569:
5564:
5552:
5550:
5549:
5544:
5538:
5533:
5518:
5516:
5515:
5506:
5495:
5473:
5471:
5470:
5465:
5463:
5460:
5455:
5446:
5426:
5420:
5415:
5412:
5333:
5331:
5330:
5325:
5323:
5319:
5318:
5309:
5301:
5292:
5291:
5290:
5264:
5262:
5261:
5252:
5251:
5250:
5237:
5228:
5227:
5226:
5149:
5147:
5146:
5141:
5139:
5132:
5130:
5129:
5128:
5106:
5105:
5104:
5091:
5086:
5082:
5081:
5079:
5077:
5076:
5067:
5065:
5060:
5047:
5039:
5026:
5024:
5023:
5014:
5013:
5004:
4995:
4994:
4993:
4967:
4963:
4962:
4960:
4958:
4957:
4948:
4946:
4941:
4928:
4920:
4907:
4905:
4904:
4895:
4894:
4885:
4876:
4875:
4874:
4826:
4824:
4823:
4818:
4807:
4803:
4802:
4801:
4792:
4791:
4773:
4772:
4757:
4756:
4747:
4746:
4732:
4730:
4729:
4728:
4715:
4714:
4713:
4700:
4692:
4690:
4682:
4674:
4654:
4652:
4651:
4646:
4641:
4637:
4636:
4634:
4633:
4624:
4623:
4622:
4613:
4612:
4602:
4597:
4595:
4594:
4585:
4584:
4575:
4570:
4565:
4564:
4555:
4545:
4540:
4539:
4538:
4525:
4401:-coordinate in (
4380:
4378:
4377:
4372:
4370:
4368:
4367:
4363:
4362:
4361:
4338:
4337:
4327:
4301:
4267:
4265:
4264:
4259:
4257:
4255:
4254:
4253:
4241:
4240:
4230:
4229:
4228:
4200:
4195:
4193:
4192:
4191:
4175:
4174:
4165:
4160:
4155:
4144:
4114:
4112:
4111:
4106:
4104:
4102:
4101:
4100:
4088:
4087:
4077:
4076:
4075:
4047:
4042:
4040:
4039:
4038:
4022:
4021:
4012:
4007:
4002:
3991:
3986:
3982:
3981:
3980:
3968:
3960:
3955:
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3952:
3951:
3935:
3934:
3925:
3915:
3914:
3909:
3905:
3903:
3895:
3887:
3877:
3869:
3850:
3848:
3847:
3842:
3840:
3838:
3837:
3828:
3827:
3826:
3817:
3816:
3806:
3801:
3799:
3798:
3789:
3788:
3779:
3774:
3769:
3768:
3767:
3758:
3757:
3747:
3742:
3741:
3729:
3727:
3726:
3725:
3716:
3715:
3705:
3704:
3695:
3690:
3689:
3684:
3680:
3678:
3670:
3662:
3641:
3639:
3638:
3633:
3621:
3620:
3616:
3604:
3600:
3599:
3595:
3594:
3592:
3591:
3579:
3574:
3572:
3571:
3559:
3552:
3548:
3547:
3542:
3541:
3540:
3530:
3514:
3512:
3511:
3499:
3491:
3489:
3488:
3476:
3458:
3456:
3455:
3450:
3448:
3443:
3435:
3420:
3418:
3417:
3412:
3410:
3402:
3387:
3385:
3384:
3379:
3374:
3370:
3369:
3368:
3356:
3354:
3353:
3344:
3343:
3334:
3327:
3323:
3322:
3317:
3316:
3307:
3291:
3289:
3288:
3279:
3278:
3269:
3264:
3263:
3258:
3254:
3252:
3244:
3236:
3218:
3216:
3215:
3210:
3198:
3196:
3195:
3190:
3188:
3180:
3149:
3147:
3146:
3141:
3129:
3127:
3126:
3121:
3116:
3112:
3111:
3109:
3108:
3099:
3098:
3089:
3084:
3083:
3069:
3065:
3064:
3059:
3058:
3049:
3033:
3031:
3030:
3029:
3020:
3019:
3009:
3008:
2999:
2994:
2993:
2988:
2984:
2982:
2974:
2966:
2952:
2950:
2949:
2944:
2932:
2930:
2929:
2924:
2869:
2868:
2864:
2822:
2820:
2819:
2814:
2812:
2810:
2809:
2800:
2789:
2784:
2783:
2686:, its so-called
2673:
2671:
2670:
2665:
2657:
2655:
2647:
2646:
2645:
2632:
2630:
2628:
2620:
2619:
2618:
2605:
2602:
2597:
2582:
2580:
2579:
2578:
2565:
2564:
2563:
2554:
2553:
2543:
2494:
2492:
2491:
2486:
2484:
2483:
2470:
2469:
2457:
2456:
2441:
2440:
2427:
2426:
2400:
2398:
2397:
2392:
2390:
2389:
2370:
2369:
2360:
2359:
2347:
2346:
2333:
2332:
2320:
2319:
2304:
2303:
2290:
2289:
2277:
2276:
2263:
2262:
2246:
2244:
2243:
2238:
2236:
2235:
2220:
2219:
2204:
2203:
2188:
2187:
2174:
2173:
2161:
2160:
2147:
2146:
2049:
2047:
2046:
2041:
2039:
2038:
2025:
2024:
1995:
1994:
1959:
1958:
1929:
1928:
1913:
1912:
1899:
1898:
1883:
1882:
1857:
1855:
1854:
1849:
1847:
1846:
1827:
1826:
1817:
1816:
1804:
1803:
1790:
1789:
1777:
1776:
1761:
1760:
1709:
1707:
1706:
1701:
1699:
1698:
1658:
1657:
1617:
1616:
1576:
1575:
1528:
1526:
1525:
1520:
1518:
1517:
1502:
1501:
1486:
1485:
1470:
1469:
1388:fictitious force
1292:is equal to the
1290:speed of gravity
1282:Bernhard Riemann
1266:speed of gravity
1105:
1103:
1102:
1097:
1092:
1090:
1089:
1080:
1072:
894:Two-body problem
758:
751:
744:
731:
726:
725:
718:
714:
713:
498:van Stockum dust
483:RobertsonâWalker
309:
308:
199:
198:
113:
111:
110:
105:
103:
102:
90:
82:
81:
63:
62:
43:
29:
28:
10141:
10140:
10136:
10135:
10134:
10132:
10131:
10130:
10116:
10115:
10114:
10109:
10095:
9923:
9827:BKL singularity
9817:LemaĂźtreâTolman
9792:
9788:Quantum gravity
9770:
9764:
9750:geodetic effect
9724:(together with
9694:LISA Pathfinder
9633:
9582:
9568:Penrose diagram
9550:
9544:
9519:
9508:
9504:Minkowski space
9470:
9414:
9398:
9346:
9340:
9300:
9293:
9288:
9256:by Kevin Brown.
9240:
9235:
9229:
9201:
9169:
9167:
9147:
9117:
9095:
9072:
9050:
9028:
8993:
8963:
8937:
8915:
8864:
8836:
8811:
8806:
8805:
8764:
8760:
8752:
8748:
8721:Physical Review
8717:
8710:
8702:
8698:
8647:
8643:
8635:
8631:
8623:
8619:
8611:
8607:
8599:
8595:
8587:
8583:
8575:
8566:
8549:Le Verrier, UJJ
8546:
8542:
8477:Centrella, Joan
8473:
8464:
8392:
8383:
8315:
8306:
8301:
8296:
8275:
8271:
8267:
8226:
8221:
8219:
8212:
8205:
8202:
8195:
8192:
8183:
8165:
8146:
8117:
8113:
8109:
8095:
8089:
8085:
8083:
8079:
8078:
8067:
8063:
8053:
8044:
8040:
8030:
8023:
8019:
8006:
8002:
7991:
7987:
7980:
7976:
7975:
7969:
7965:
7961:
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7875:
7868:
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7842:
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7795:
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7359:
7349:
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7300:
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7277:
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7256:
7190:
7180:
7167:
7140:
7136:
7135:
7109:
7107:
7096:
7093:
7092:
7081:) is the semi-
7047:
7043:
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7032:
7023:
7019:
7018:
6992:
6990:
6979:
6976:
6975:
6953:
6949:
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6910:
6904:
6900:
6898:
6896:
6893:
6892:
6888:related by the
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6855:
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6844:
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6813:
6806:
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6758:
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3990:
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3401:
3393:
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3333:
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3259:
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3179:
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3070:
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3025:
3021:
3015:
3011:
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3004:
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2998:
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2938:
2935:
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2915:
2914:
2903:
2880:
2866:
2862:
2861:
2855:
2837:
2805:
2801:
2790:
2788:
2779:
2775:
2773:
2770:
2769:
2764:, known as the
2763:
2743:
2737:
2699:
2648:
2641:
2637:
2633:
2631:
2621:
2614:
2610:
2606:
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2528:
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2422:
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2413:
2412:
2385:
2381:
2365:
2361:
2355:
2351:
2342:
2338:
2328:
2324:
2315:
2311:
2299:
2295:
2285:
2281:
2272:
2268:
2258:
2254:
2252:
2249:
2248:
2231:
2227:
2215:
2211:
2199:
2195:
2183:
2179:
2169:
2165:
2156:
2152:
2142:
2138:
2136:
2133:
2132:
2117:Albert Einstein
2103:
2092:
2085:
2071:constitute the
2070:
2063:
2056:
2034:
2030:
2017:
2013:
1987:
1983:
1951:
1947:
1921:
1917:
1908:
1904:
1891:
1887:
1878:
1874:
1869:
1866:
1865:
1842:
1838:
1822:
1818:
1812:
1808:
1799:
1795:
1785:
1781:
1772:
1768:
1756:
1752:
1747:
1744:
1743:
1694:
1690:
1653:
1649:
1612:
1608:
1571:
1567:
1562:
1559:
1558:
1513:
1509:
1497:
1493:
1481:
1477:
1465:
1461:
1456:
1453:
1452:
1437:
1359:Albert Einstein
1351:Hendrik Lorentz
1320:
1310:
1302:Hendrik Lorentz
1209:
1203:
1191:FĂ©lix Tisserand
1154:
1144:
1085:
1081:
1073:
1071:
1060:
1057:
1056:
990:semi-major axis
928:Johannes Kepler
896:
882:
877:
782:field equations
762:
721:
708:
707:
700:
699:
523:
522:
513:
512:
468:LemaĂźtreâTolman
413:
412:
401:
400:
392:Quantum gravity
379:Advanced theory
306:
305:
304:
287:
286:
235:Geodetic effect
196:
195:
186:
185:
161:
160:
144:
114:
95:
91:
86:
74:
70:
55:
51:
49:
46:
45:
27:
12:
11:
5:
10139:
10129:
10128:
10111:
10110:
10100:
10097:
10096:
10094:
10093:
10086:
10081:
10076:
10071:
10066:
10061:
10056:
10051:
10046:
10041:
10036:
10031:
10026:
10021:
10016:
10014:Choquet-Bruhat
10011:
10006:
10001:
9996:
9991:
9986:
9981:
9976:
9971:
9966:
9961:
9956:
9951:
9946:
9941:
9935:
9933:
9929:
9928:
9925:
9924:
9922:
9921:
9914:
9913:
9908:
9903:
9896:
9895:
9890:
9885:
9880:
9875:
9866:Axisymmetric:
9863:
9862:
9857:
9851:
9840:
9839:
9834:
9829:
9824:
9819:
9814:
9805:Cosmological:
9802:
9800:
9794:
9793:
9791:
9790:
9785:
9780:
9774:
9772:
9766:
9765:
9763:
9762:
9757:
9746:frame-dragging
9743:
9738:
9733:
9730:Einstein rings
9726:Einstein cross
9719:
9708:
9707:
9702:
9696:
9691:
9686:
9673:
9663:
9662:
9657:
9652:
9647:
9641:
9639:
9635:
9634:
9632:
9631:
9629:Ernst equation
9626:
9621:
9616:
9611:
9606:
9601:
9599:BSSN formalism
9596:
9590:
9588:
9584:
9583:
9581:
9580:
9575:
9570:
9565:
9560:
9554:
9552:
9546:
9545:
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9542:
9537:
9531:
9529:
9522:
9514:
9513:
9510:
9509:
9507:
9506:
9501:
9496:
9491:
9486:
9480:
9478:
9472:
9471:
9469:
9468:
9463:
9458:
9456:Ladder paradox
9453:
9448:
9443:
9438:
9433:
9428:
9422:
9420:
9416:
9415:
9413:
9412:
9406:
9404:
9400:
9399:
9397:
9396:
9391:
9386:
9381:
9376:
9371:
9366:
9361:
9359:Speed of light
9356:
9350:
9348:
9342:
9341:
9339:
9338:
9333:
9328:
9322:
9312:
9310:
9303:
9295:
9294:
9287:
9286:
9279:
9272:
9264:
9258:
9257:
9247:
9239:
9238:External links
9236:
9234:
9233:
9227:
9205:
9199:
9175:
9151:
9145:
9121:
9115:
9099:
9093:
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8919:
8913:
8900:
8868:
8862:
8840:
8834:
8812:
8810:
8807:
8804:
8803:
8782:. p. 25.
8758:
8746:
8727:(1): 435â440.
8708:
8696:
8641:
8629:
8617:
8613:Roseveare 1982
8605:
8603:, p. 177.
8593:
8591:, p. 254.
8581:
8564:
8554:Comptes Rendus
8540:
8487:(11): 111102.
8462:
8409:(11): 111101.
8395:Campanelli, M.
8381:
8328:(12): 121101.
8303:
8302:
8300:
8297:
8295:
8294:
8268:
8266:
8263:
8262:
8261:
8256:
8251:
8249:Kepler problem
8246:
8241:
8238:
8236:Binet equation
8232:
8231:
8217:
8214:Physics portal
8201:
8198:
8197:
8196:
8193:
8186:
8184:
8166:
8159:
8155:of the orbit.
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7625:
7619:
7615:
7609:
7606:
7599:
7595:
7591:
7585:
7581:
7566:semimajor axis
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7521:
7516:
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7013:
7010:
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7001:
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6995:
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6648:
6643:
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6622:
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6605:
6601:
6597:
6594:
6588:
6584:
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6572:
6568:
6561:
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6552:
6548:
6542:
6538:
6535:
6532:
6529:
6524:
6520:
6516:
6511:
6507:
6503:
6500:
6497:
6494:
6491:
6466:
6462:
6459:
6451:
6447:
6443:
6436:
6431:
6427:
6423:
6417:
6414:
6410:
6404:
6400:
6396:
6391:
6387:
6354:
6350:
6343:
6338:
6334:
6330:
6324:
6321:
6314:
6309:
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6297:
6287:
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6269:
6264:
6256:
6247:
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6236:
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6222:
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6211:
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6201:
6197:
6168:
6164:
6161:
6156:
6148:
6144:
6140:
6135:
6130:
6126:
6119:
6112:
6109:
6104:
6099:
6094:
6090:
6077:
6039:Kepler problem
6030:
6027:
6018:
6001:
5990:
5973:
5966:
5949:
5942:
5935:
5914:
5910:
5906:
5903:
5894:
5890:
5881:
5868:
5842:
5838:
5834:
5827:
5822:
5818:
5812:
5808:
5801:
5797:
5789:
5785:
5781:
5775:
5770:
5766:
5760:
5755:
5750:
5744:
5740:
5734:
5730:
5723:
5719:
5715:
5707:
5701:
5698:
5695:
5692:
5689:
5684:
5680:
5673:
5669:
5663:
5659:
5652:
5648:
5641:
5635:
5632:
5629:
5626:
5623:
5618:
5613:
5610:
5604:
5599:
5594:
5590:
5562:
5542:
5537:
5532:
5528:
5524:
5521:
5514:
5510:
5505:
5502:
5499:
5478:
5459:
5454:
5450:
5445:
5442:
5439:
5436:
5433:
5430:
5424:
5419:
5410:
5386:
5379:
5359:
5348:
5317:
5313:
5307:
5304:
5299:
5296:
5294:
5289:
5286:
5283:
5280:
5277:
5272:
5268:
5267:
5260:
5256:
5249:
5245:
5241:
5235:
5232:
5230:
5225:
5222:
5219:
5216:
5213:
5208:
5204:
5203:
5191:
5178:Kepler problem
5169:
5158:
5135:
5127:
5124:
5121:
5118:
5115:
5110:
5103:
5099:
5095:
5089:
5085:
5075:
5071:
5064:
5059:
5055:
5051:
5045:
5042:
5037:
5034:
5030:
5022:
5018:
5012:
5008:
5002:
4999:
4997:
4992:
4989:
4986:
4983:
4980:
4975:
4971:
4970:
4966:
4956:
4952:
4945:
4940:
4936:
4932:
4926:
4923:
4918:
4915:
4911:
4903:
4899:
4893:
4889:
4883:
4880:
4878:
4873:
4870:
4867:
4864:
4861:
4856:
4852:
4851:
4839:
4832:
4816:
4813:
4810:
4806:
4800:
4796:
4790:
4786:
4782:
4779:
4776:
4771:
4767:
4763:
4760:
4755:
4751:
4745:
4741:
4736:
4727:
4723:
4719:
4712:
4708:
4704:
4698:
4695:
4689:
4686:
4681:
4678:
4672:
4669:
4666:
4644:
4640:
4632:
4628:
4621:
4617:
4611:
4607:
4600:
4593:
4589:
4583:
4579:
4573:
4568:
4563:
4559:
4553:
4549:
4543:
4537:
4533:
4529:
4523:
4520:
4517:
4514:
4511:
4478:
4467:
4448:
4445:
4432:
4366:
4360:
4356:
4352:
4349:
4345:
4341:
4336:
4332:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4299:
4296:
4293:
4252:
4248:
4244:
4239:
4235:
4227:
4223:
4219:
4216:
4213:
4210:
4207:
4204:
4198:
4190:
4186:
4182:
4179:
4173:
4169:
4163:
4158:
4154:
4151:
4148:
4142:
4139:
4136:
4133:
4130:
4127:
4099:
4095:
4091:
4086:
4082:
4074:
4070:
4066:
4063:
4060:
4057:
4054:
4051:
4045:
4037:
4033:
4029:
4026:
4020:
4016:
4010:
4005:
4001:
3998:
3995:
3989:
3985:
3979:
3975:
3971:
3966:
3963:
3958:
3950:
3946:
3942:
3939:
3933:
3929:
3922:
3918:
3913:
3908:
3902:
3899:
3894:
3891:
3885:
3880:
3875:
3872:
3858:
3836:
3832:
3825:
3821:
3815:
3811:
3804:
3797:
3793:
3787:
3783:
3777:
3772:
3766:
3762:
3756:
3752:
3745:
3740:
3736:
3732:
3724:
3720:
3714:
3710:
3703:
3699:
3693:
3688:
3683:
3677:
3674:
3669:
3666:
3660:
3646:
3643:
3631:
3628:
3625:
3619:
3615:
3611:
3608:
3603:
3598:
3590:
3586:
3582:
3577:
3570:
3566:
3562:
3556:
3551:
3545:
3539:
3534:
3528:
3525:
3521:
3517:
3510:
3506:
3502:
3496:
3487:
3483:
3479:
3474:
3471:
3468:
3446:
3442:
3439:
3433:
3430:
3408:
3405:
3400:
3397:
3377:
3373:
3367:
3363:
3359:
3352:
3348:
3342:
3338:
3331:
3326:
3320:
3315:
3311:
3305:
3302:
3298:
3294:
3287:
3283:
3277:
3273:
3267:
3262:
3257:
3251:
3248:
3243:
3240:
3234:
3208:
3186:
3183:
3178:
3175:
3172:
3169:
3166:
3163:
3139:
3119:
3115:
3107:
3103:
3097:
3093:
3087:
3082:
3078:
3073:
3068:
3062:
3057:
3053:
3047:
3044:
3040:
3036:
3028:
3024:
3018:
3014:
3007:
3003:
2997:
2992:
2987:
2981:
2978:
2973:
2970:
2964:
2942:
2922:
2902:
2899:
2878:
2853:
2835:
2808:
2804:
2799:
2796:
2793:
2787:
2782:
2778:
2761:
2739:Main article:
2736:
2733:
2697:
2663:
2660:
2654:
2651:
2644:
2640:
2636:
2627:
2624:
2617:
2613:
2609:
2601:
2596:
2593:
2589:
2585:
2577:
2573:
2569:
2562:
2558:
2552:
2548:
2513:
2510:
2499:
2482:
2478:
2474:
2468:
2464:
2460:
2455:
2452:
2448:
2444:
2439:
2435:
2431:
2425:
2421:
2388:
2384:
2380:
2376:
2373:
2368:
2364:
2358:
2354:
2350:
2345:
2341:
2337:
2331:
2327:
2323:
2318:
2314:
2310:
2307:
2302:
2298:
2294:
2288:
2284:
2280:
2275:
2271:
2267:
2261:
2257:
2234:
2230:
2226:
2223:
2218:
2214:
2210:
2207:
2202:
2198:
2194:
2191:
2186:
2182:
2178:
2172:
2168:
2164:
2159:
2155:
2151:
2145:
2141:
2101:
2090:
2083:
2068:
2061:
2054:
2037:
2033:
2029:
2023:
2020:
2016:
2012:
2009:
2006:
2002:
1999:
1993:
1990:
1986:
1982:
1979:
1976:
1973:
1970:
1966:
1963:
1957:
1954:
1950:
1946:
1943:
1940:
1936:
1933:
1927:
1924:
1920:
1916:
1911:
1907:
1903:
1897:
1894:
1890:
1886:
1881:
1877:
1873:
1845:
1841:
1837:
1833:
1830:
1825:
1821:
1815:
1811:
1807:
1802:
1798:
1794:
1788:
1784:
1780:
1775:
1771:
1767:
1764:
1759:
1755:
1751:
1697:
1693:
1689:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1656:
1652:
1648:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1615:
1611:
1607:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1574:
1570:
1566:
1516:
1512:
1508:
1505:
1500:
1496:
1492:
1489:
1484:
1480:
1476:
1473:
1468:
1464:
1460:
1439:In the normal
1436:
1433:
1396:Coriolis force
1371:Henri Poincaré
1367:speed of light
1355:Henri Poincaré
1333:-light by the
1309:
1306:
1294:speed of light
1219:of the planet
1202:
1199:
1143:
1140:
1095:
1088:
1084:
1079:
1076:
1070:
1067:
1064:
994:
993:
978:orbital period
970:
959:
881:
878:
876:
873:
842:planet Mercury
790:Kepler problem
788:. Solving the
764:
763:
761:
760:
753:
746:
738:
735:
734:
733:
732:
719:
702:
701:
698:
697:
690:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
635:
630:
625:
620:
615:
610:
605:
600:
595:
590:
585:
580:
575:
570:
565:
560:
555:
550:
545:
540:
535:
530:
524:
520:
519:
518:
515:
514:
511:
510:
505:
500:
495:
490:
485:
480:
475:
470:
465:
460:
455:
450:
445:
440:
435:
430:
425:
414:
408:
407:
406:
403:
402:
397:
396:
395:
394:
389:
381:
380:
376:
375:
374:
373:
371:Post-Newtonian
368:
363:
355:
354:
350:
349:
348:
347:
342:
337:
332:
327:
322:
314:
313:
307:
303:
302:
299:
295:
294:
293:
292:
289:
288:
283:
282:
281:
280:
275:
270:
262:
261:
255:
254:
253:
252:
247:
242:
237:
232:
230:Frame-dragging
227:
222:
217:
212:
207:
205:Kepler problem
197:
193:
192:
191:
188:
187:
184:
183:
178:
173:
168:
162:
158:
157:
156:
153:
152:
151:
150:
145:
143:
142:
137:
132:
126:
124:
116:
115:
101:
98:
94:
89:
85:
80:
77:
73:
69:
66:
61:
58:
54:
44:
36:
35:
24:Kepler problem
9:
6:
4:
3:
2:
10138:
10127:
10124:
10123:
10121:
10108:
10098:
10092:
10091:
10087:
10085:
10082:
10080:
10077:
10075:
10072:
10070:
10067:
10065:
10062:
10060:
10057:
10055:
10052:
10050:
10047:
10045:
10042:
10040:
10037:
10035:
10032:
10030:
10027:
10025:
10022:
10020:
10017:
10015:
10012:
10010:
10007:
10005:
10002:
10000:
9999:Chandrasekhar
9997:
9995:
9992:
9990:
9987:
9985:
9982:
9980:
9977:
9975:
9972:
9970:
9967:
9965:
9962:
9960:
9959:Schwarzschild
9957:
9955:
9952:
9950:
9947:
9945:
9942:
9940:
9937:
9936:
9934:
9930:
9920:
9916:
9915:
9912:
9909:
9907:
9904:
9902:
9898:
9897:
9894:
9891:
9889:
9886:
9884:
9881:
9879:
9876:
9873:
9869:
9865:
9864:
9861:
9858:
9855:
9852:
9850:
9846:
9845:Schwarzschild
9842:
9841:
9838:
9835:
9833:
9830:
9828:
9825:
9823:
9820:
9818:
9815:
9812:
9808:
9804:
9803:
9801:
9799:
9795:
9789:
9786:
9784:
9781:
9779:
9776:
9775:
9773:
9767:
9761:
9758:
9755:
9751:
9747:
9744:
9742:
9741:Shapiro delay
9739:
9737:
9734:
9731:
9727:
9723:
9720:
9717:
9713:
9710:
9709:
9706:
9703:
9700:
9697:
9695:
9692:
9690:
9687:
9685:
9684:collaboration
9681:
9677:
9674:
9672:
9668:
9665:
9664:
9661:
9658:
9656:
9653:
9651:
9650:Event horizon
9648:
9646:
9643:
9642:
9640:
9636:
9630:
9627:
9625:
9622:
9620:
9617:
9615:
9612:
9610:
9607:
9605:
9602:
9600:
9597:
9595:
9594:ADM formalism
9592:
9591:
9589:
9585:
9579:
9576:
9574:
9571:
9569:
9566:
9564:
9561:
9559:
9556:
9555:
9553:
9547:
9541:
9538:
9536:
9533:
9532:
9530:
9526:
9523:
9521:
9515:
9505:
9502:
9500:
9499:Biquaternions
9497:
9495:
9492:
9490:
9487:
9485:
9482:
9481:
9479:
9477:
9473:
9467:
9464:
9462:
9459:
9457:
9454:
9452:
9449:
9447:
9444:
9442:
9439:
9437:
9434:
9432:
9429:
9427:
9426:Time dilation
9424:
9423:
9421:
9417:
9411:
9408:
9407:
9405:
9401:
9395:
9392:
9390:
9387:
9385:
9382:
9380:
9379:Proper length
9377:
9375:
9372:
9370:
9367:
9365:
9362:
9360:
9357:
9355:
9352:
9351:
9349:
9343:
9337:
9334:
9332:
9329:
9326:
9323:
9321:
9317:
9314:
9313:
9311:
9307:
9304:
9302:
9296:
9292:
9285:
9280:
9278:
9273:
9271:
9266:
9265:
9262:
9255:
9251:
9248:
9245:
9242:
9241:
9230:
9224:
9220:
9216:
9215:
9210:
9209:Whittaker, ET
9206:
9202:
9196:
9192:
9187:
9186:
9180:
9176:
9166:on 2009-01-30
9165:
9161:
9157:
9152:
9148:
9142:
9138:
9133:
9132:
9126:
9122:
9118:
9112:
9108:
9104:
9100:
9096:
9094:0-19-858174-2
9090:
9085:
9084:
9077:
9073:
9067:
9063:
9059:
9055:
9051:
9045:
9041:
9037:
9033:
9029:
9027:0-19-520438-7
9023:
9019:
9014:
9013:
9007:
9003:
9000:
8994:
8988:
8984:
8980:
8976:
8972:
8968:
8964:
8958:
8954:
8950:
8946:
8942:
8938:
8932:
8928:
8924:
8920:
8916:
8910:
8906:
8901:
8897:
8893:
8889:
8885:
8881:
8877:
8873:
8869:
8865:
8859:
8855:
8851:
8850:
8845:
8841:
8837:
8831:
8827:
8822:
8821:
8814:
8813:
8799:
8795:
8790:
8785:
8781:
8777:
8773:
8770:(July 2005).
8769:
8762:
8755:
8750:
8742:
8738:
8734:
8730:
8726:
8722:
8715:
8713:
8705:
8700:
8692:
8688:
8684:
8680:
8676:
8672:
8667:
8666:gr-qc/9611008
8662:
8658:
8654:
8653:
8645:
8638:
8637:Weinberg 1972
8633:
8626:
8621:
8614:
8609:
8602:
8597:
8590:
8585:
8578:
8573:
8571:
8569:
8560:
8556:
8555:
8550:
8544:
8536:
8532:
8528:
8524:
8520:
8516:
8512:
8508:
8504:
8500:
8495:
8494:gr-qc/0511103
8490:
8486:
8482:
8478:
8471:
8469:
8467:
8458:
8454:
8450:
8446:
8442:
8438:
8434:
8430:
8426:
8422:
8417:
8416:gr-qc/0511048
8412:
8408:
8404:
8400:
8399:Lousto, C. O.
8396:
8390:
8388:
8386:
8377:
8373:
8369:
8365:
8361:
8357:
8353:
8349:
8345:
8341:
8336:
8335:gr-qc/0507014
8331:
8327:
8323:
8319:
8313:
8311:
8309:
8304:
8291:
8287:
8283:
8278:
8273:
8269:
8260:
8257:
8255:
8252:
8250:
8247:
8245:
8242:
8239:
8237:
8234:
8233:
8229:
8218:
8215:
8209:
8204:
8190:
8185:
8181:
8177:
8174:
8173:binary pulsar
8170:
8163:
8158:
8157:
8156:
8154:
8148:
8145:
8141:
8122:
8118:
8114:
8110:
8105:
8099:
8096:
8090:
8086:
8080:
8074:
8068:
8064:
8058:
8055:
8050:
8045:
8041:
8035:
8032:
8027:
8024:
8020:
8011:
8007:
8003:
7998:
7992:
7988:
7984:
7981:
7977:
7970:
7966:
7962:
7955:
7951:
7947:
7944:
7934:
7930:
7926:
7921:
7917:
7908:
7904:
7898:
7894:
7888:
7884:
7880:
7876:
7872:
7869:
7863:
7859:
7853:
7850:
7843:
7839:
7835:
7829:
7825:
7816:
7802:
7796:
7792:
7786:
7783:
7778:
7775:
7771:
7762:
7757:
7751:
7747:
7743:
7740:
7736:
7729:
7725:
7721:
7717:
7711:
7707:
7703:
7694:
7690:
7686:
7681:
7677:
7669:
7664:
7660:
7654:
7649:
7645:
7639:
7635:
7631:
7627:
7623:
7617:
7613:
7607:
7604:
7597:
7593:
7589:
7583:
7579:
7571:
7567:
7563:
7559:
7555:
7538:
7532:
7528:
7522:
7519:
7514:
7509:
7505:
7499:
7496:
7491:
7488:
7484:
7475:
7471:
7467:
7462:
7456:
7452:
7448:
7445:
7441:
7434:
7430:
7424:
7420:
7416:
7406:
7402:
7398:
7393:
7389:
7380:
7375:
7371:
7365:
7360:
7356:
7350:
7346:
7342:
7336:
7332:
7326:
7323:
7318:
7315:
7309:
7305:
7297:
7293:
7288:
7286:
7282:
7276:
7266:
7261:
7251:
7249:
7244:
7241:
7236:
7234:
7230:
7226:
7222:
7220:
7217: =
7216:
7211:
7206:
7204:
7199:
7195:
7189:
7185:
7171:
7162:
7146:
7141:
7137:
7128:
7125:
7122:
7116:
7113:
7110:
7104:
7101:
7098:
7089:
7087:
7084:
7080:
7076:
7071:
7054:
7048:
7044:
7040:
7037:
7033:
7029:
7024:
7020:
7011:
7008:
7005:
6999:
6996:
6993:
6987:
6984:
6981:
6960:
6954:
6950:
6946:
6943:
6939:
6935:
6932:
6923:
6920:
6917:
6911:
6905:
6901:
6891:
6887:
6883:
6878:
6860:
6856:
6850:
6846:
6838:
6834:
6828:
6824:
6818:
6814:
6810:
6807:
6801:
6797:
6790:
6786:
6779:
6775:
6769:
6765:
6761:
6755:
6746:
6742:
6738:
6731:
6727:
6721:
6717:
6713:
6710:
6704:
6701:
6698:
6687:
6684:
6680:
6672:
6668:
6664:
6646:
6641:
6637:
6628:
6624:
6620:
6613:
6609:
6603:
6599:
6595:
6592:
6586:
6582:
6574:
6570:
6566:
6559:
6554:
6550:
6546:
6540:
6536:
6533:
6530:
6522:
6518:
6514:
6509:
6505:
6498:
6495:
6492:
6489:
6481:
6464:
6460:
6457:
6449:
6445:
6441:
6434:
6429:
6425:
6421:
6415:
6412:
6408:
6402:
6398:
6394:
6389:
6385:
6376:
6371:
6352:
6348:
6341:
6336:
6332:
6328:
6322:
6319:
6312:
6307:
6303:
6299:
6295:
6285:
6281:
6272:
6267:
6262:
6254:
6245:
6241:
6234:
6230:
6224:
6220:
6213:
6209:
6204:
6199:
6195:
6187:which equals
6166:
6162:
6159:
6154:
6146:
6142:
6138:
6133:
6128:
6124:
6117:
6110:
6107:
6102:
6097:
6092:
6088:
6076:
6072:
6064:
6060:
6056:
6052:
6048:
6044:
6040:
6035:
6026:
6024:
6023:photon sphere
6017:
6000:
5996:
5989:
5972:
5965:
5952:lies between
5948:
5941:
5934:
5930:
5912:
5908:
5904:
5901:
5892:
5888:
5879:
5867:
5863:
5858:
5840:
5836:
5832:
5825:
5820:
5816:
5810:
5806:
5799:
5795:
5787:
5783:
5779:
5773:
5768:
5764:
5758:
5753:
5748:
5742:
5738:
5732:
5728:
5721:
5717:
5713:
5705:
5682:
5678:
5671:
5667:
5661:
5657:
5650:
5646:
5639:
5616:
5611:
5608:
5602:
5597:
5592:
5588:
5578:
5576:
5560:
5540:
5535:
5530:
5526:
5522:
5519:
5512:
5508:
5503:
5500:
5497:
5486:
5481:
5477:
5457:
5452:
5448:
5440:
5437:
5434:
5428:
5422:
5417:
5408:
5398:
5396:
5392:
5385:
5378:
5374:
5366:
5358:
5354:
5347:
5343:
5338:
5334:
5315:
5311:
5305:
5302:
5297:
5295:
5270:
5258:
5254:
5247:
5243:
5239:
5233:
5231:
5206:
5190:
5186:
5181:
5179:
5175:
5168:
5164:
5157:
5153:
5133:
5108:
5101:
5097:
5093:
5087:
5083:
5073:
5069:
5062:
5057:
5053:
5049:
5043:
5040:
5035:
5032:
5028:
5020:
5016:
5010:
5006:
5000:
4998:
4973:
4964:
4954:
4950:
4943:
4938:
4934:
4930:
4924:
4921:
4916:
4913:
4909:
4901:
4897:
4891:
4887:
4881:
4879:
4854:
4838:
4831:
4814:
4811:
4808:
4804:
4798:
4794:
4788:
4784:
4780:
4777:
4774:
4769:
4765:
4761:
4758:
4753:
4749:
4743:
4739:
4734:
4725:
4721:
4717:
4710:
4706:
4702:
4696:
4693:
4687:
4684:
4679:
4676:
4670:
4667:
4664:
4655:
4642:
4638:
4630:
4626:
4619:
4615:
4609:
4605:
4598:
4591:
4587:
4581:
4577:
4571:
4566:
4561:
4557:
4551:
4547:
4541:
4535:
4531:
4527:
4521:
4515:
4509:
4501:
4497:
4493:
4489:
4477:
4473:
4466:
4462:
4458:
4453:
4444:
4442:
4438:
4431:
4427:
4422:
4420:
4416:
4412:
4408:
4404:
4400:
4396:
4392:
4388:
4384:
4364:
4358:
4354:
4350:
4347:
4343:
4339:
4334:
4330:
4321:
4318:
4315:
4309:
4306:
4303:
4297:
4294:
4291:
4283:
4279:
4275:
4271:
4250:
4246:
4242:
4237:
4233:
4225:
4221:
4214:
4211:
4208:
4202:
4196:
4188:
4184:
4180:
4177:
4171:
4167:
4161:
4156:
4152:
4149:
4146:
4140:
4137:
4131:
4125:
4118:
4097:
4093:
4089:
4084:
4080:
4072:
4068:
4061:
4058:
4055:
4049:
4043:
4035:
4031:
4027:
4024:
4018:
4014:
4008:
4003:
3999:
3996:
3993:
3987:
3983:
3977:
3973:
3969:
3964:
3961:
3956:
3948:
3944:
3940:
3937:
3931:
3927:
3920:
3916:
3911:
3906:
3900:
3897:
3892:
3889:
3883:
3878:
3873:
3870:
3857:
3854:
3834:
3830:
3823:
3819:
3813:
3809:
3802:
3795:
3791:
3785:
3781:
3775:
3770:
3764:
3760:
3754:
3750:
3743:
3738:
3734:
3730:
3722:
3718:
3712:
3708:
3701:
3697:
3691:
3686:
3681:
3675:
3672:
3667:
3664:
3658:
3642:
3629:
3626:
3623:
3617:
3613:
3609:
3606:
3601:
3596:
3588:
3584:
3580:
3575:
3568:
3564:
3560:
3554:
3549:
3543:
3532:
3526:
3523:
3519:
3515:
3508:
3504:
3500:
3494:
3485:
3481:
3477:
3472:
3469:
3466:
3444:
3440:
3437:
3431:
3428:
3406:
3403:
3398:
3395:
3375:
3371:
3365:
3361:
3357:
3350:
3346:
3340:
3336:
3329:
3324:
3318:
3313:
3309:
3303:
3300:
3296:
3292:
3285:
3281:
3275:
3271:
3265:
3260:
3255:
3249:
3246:
3241:
3238:
3232:
3222:
3206:
3184:
3181:
3176:
3173:
3170:
3167:
3164:
3161:
3153:
3137:
3117:
3113:
3105:
3101:
3095:
3091:
3085:
3080:
3076:
3071:
3066:
3060:
3055:
3051:
3045:
3042:
3038:
3034:
3026:
3022:
3016:
3012:
3005:
3001:
2995:
2990:
2985:
2979:
2976:
2971:
2968:
2962:
2940:
2920:
2907:
2898:
2896:
2892:
2891:neutron stars
2888:
2884:
2877:
2873:
2859:
2852:
2847:
2845:
2841:
2834:
2830:
2826:
2806:
2802:
2797:
2794:
2791:
2785:
2780:
2776:
2767:
2760:
2756:
2752:
2748:
2742:
2732:
2730:
2726:
2722:
2718:
2714:
2710:
2707:
2703:
2696:
2693:
2692:metric tensor
2689:
2685:
2681:
2677:
2661:
2658:
2652:
2649:
2642:
2638:
2634:
2625:
2622:
2615:
2611:
2607:
2599:
2594:
2591:
2583:
2575:
2571:
2567:
2560:
2556:
2550:
2546:
2533:
2527:
2523:
2519:
2509:
2507:
2502:
2498:
2480:
2476:
2472:
2466:
2462:
2458:
2453:
2450:
2446:
2442:
2437:
2433:
2429:
2423:
2419:
2410:
2406:
2401:
2386:
2382:
2378:
2374:
2371:
2366:
2362:
2356:
2352:
2348:
2343:
2339:
2335:
2329:
2325:
2321:
2316:
2312:
2308:
2305:
2300:
2296:
2292:
2286:
2282:
2278:
2273:
2269:
2265:
2259:
2255:
2232:
2228:
2224:
2221:
2216:
2212:
2208:
2205:
2200:
2196:
2192:
2189:
2184:
2180:
2176:
2170:
2166:
2162:
2157:
2153:
2149:
2143:
2139:
2130:
2126:
2122:
2118:
2114:
2109:
2107:
2100:
2096:
2089:
2082:
2078:
2074:
2073:metric tensor
2067:
2060:
2053:
2035:
2031:
2027:
2021:
2018:
2014:
2010:
2007:
2004:
2000:
1997:
1991:
1988:
1984:
1980:
1977:
1974:
1971:
1968:
1964:
1961:
1955:
1952:
1948:
1944:
1941:
1938:
1934:
1931:
1925:
1922:
1918:
1914:
1909:
1905:
1901:
1895:
1892:
1888:
1884:
1879:
1875:
1871:
1864:
1858:
1843:
1839:
1835:
1831:
1828:
1823:
1819:
1813:
1809:
1805:
1800:
1796:
1792:
1786:
1782:
1778:
1773:
1769:
1765:
1762:
1757:
1753:
1749:
1741:
1737:
1733:
1729:
1725:
1721:
1717:
1713:
1695:
1691:
1687:
1680:
1677:
1674:
1671:
1668:
1662:
1659:
1654:
1650:
1646:
1639:
1636:
1633:
1630:
1627:
1621:
1618:
1613:
1609:
1605:
1598:
1595:
1592:
1589:
1586:
1580:
1577:
1572:
1568:
1564:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1514:
1510:
1506:
1503:
1498:
1494:
1490:
1487:
1482:
1478:
1474:
1471:
1466:
1462:
1458:
1450:
1446:
1442:
1432:
1430:
1426:
1421:
1417:
1413:
1409:
1405:
1401:
1400:inertial mass
1397:
1393:
1389:
1384:
1380:
1376:
1372:
1368:
1364:
1360:
1356:
1352:
1344:
1340:
1336:
1332:
1328:
1324:
1319:
1315:
1305:
1303:
1299:
1295:
1291:
1287:
1283:
1279:
1275:
1269:
1267:
1262:
1257:
1255:
1251:
1247:
1244:
1240:
1236:
1231:
1227:
1222:
1218:
1214:
1208:
1198:
1196:
1192:
1188:
1183:
1181:
1176:
1172:
1163:
1158:
1153:
1149:
1139:
1137:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1093:
1086:
1082:
1077:
1074:
1068:
1065:
1062:
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1013:
1011:
1007:
1003:
999:
992:of its orbit.
991:
987:
983:
979:
975:
971:
968:
964:
960:
957:
953:
949:
945:
941:
940:
939:
937:
936:Kepler's laws
933:
929:
921:
917:
913:
909:
905:
900:
895:
891:
887:
872:
870:
865:
863:
859:
855:
849:
847:
843:
839:
835:
830:
826:
822:
818:
814:
810:
805:
803:
799:
795:
791:
787:
783:
779:
775:
771:
759:
754:
752:
747:
745:
740:
739:
737:
736:
730:
720:
717:
712:
706:
705:
704:
703:
696:
695:
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
633:Chandrasekhar
631:
629:
626:
624:
621:
619:
616:
614:
611:
609:
606:
604:
601:
599:
596:
594:
591:
589:
586:
584:
581:
579:
576:
574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
549:
548:Schwarzschild
546:
544:
541:
539:
536:
534:
531:
529:
526:
525:
517:
516:
509:
508:HartleâThorne
506:
504:
501:
499:
496:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
464:
461:
459:
456:
454:
451:
449:
446:
444:
441:
439:
436:
434:
431:
429:
426:
423:
419:
418:Schwarzschild
416:
415:
411:
405:
404:
393:
390:
388:
385:
384:
383:
382:
378:
377:
372:
369:
367:
364:
362:
359:
358:
357:
356:
352:
351:
346:
343:
341:
338:
336:
333:
331:
328:
326:
323:
321:
318:
317:
316:
315:
311:
310:
300:
297:
296:
291:
290:
279:
276:
274:
271:
269:
266:
265:
264:
263:
260:
257:
256:
251:
248:
246:
243:
241:
240:Event horizon
238:
236:
233:
231:
228:
226:
223:
221:
218:
216:
213:
211:
208:
206:
203:
202:
201:
200:
190:
189:
182:
179:
177:
174:
172:
169:
167:
164:
163:
155:
154:
149:
146:
141:
138:
136:
133:
131:
128:
127:
125:
123:
120:
119:
118:
117:
99:
96:
92:
87:
83:
78:
75:
71:
64:
59:
56:
52:
42:
38:
37:
34:
31:
30:
25:
21:
16:
10089:
9783:KaluzaâKlein
9659:
9535:Introduction
9461:Twin paradox
9253:
9212:
9184:
9168:. Retrieved
9164:the original
9159:
9130:
9106:
9082:
9061:
9039:
9011:
8982:
8948:
8926:
8904:
8879:
8875:
8847:
8819:
8809:Bibliography
8775:
8768:Taylor, J.H.
8761:
8749:
8724:
8720:
8699:
8656:
8652:Phys. Rev. D
8650:
8644:
8632:
8620:
8608:
8596:
8584:
8558:
8552:
8543:
8484:
8480:
8406:
8402:
8325:
8321:
8276:
8272:
8176:PSR B1913+16
8152:
8149:
8143:
8139:
7817:
7561:
7553:
7289:
7278:
7263:
7245:
7237:
7231:rather than
7218:
7214:
7207:
7203:binary stars
7197:
7193:
7191:
7090:
7085:
7083:latus rectum
7078:
7074:
7072:
6885:
6881:
6879:
6685:
6678:
6670:
6666:
6662:
6479:
6372:
6074:
6070:
6068:
6015:
5998:
5994:
5987:
5970:
5963:
5946:
5939:
5932:
5865:
5864:approaches 3
5861:
5859:
5579:
5575:reduced mass
5479:
5475:
5399:
5394:
5390:
5383:
5376:
5372:
5370:
5356:
5352:
5345:
5341:
5188:
5184:
5182:
5173:
5166:
5162:
5155:
4836:
4829:
4656:
4499:
4495:
4491:
4487:
4485:
4475:
4471:
4464:
4460:
4456:
4440:
4436:
4429:
4425:
4423:
4414:
4410:
4406:
4402:
4398:
4394:
4390:
4386:
4382:
3855:
3648:
3221:reduced mass
2912:
2882:
2875:
2850:
2848:
2839:
2832:
2824:
2758:
2754:
2744:
2724:
2716:
2708:
2701:
2694:
2679:
2529:
2500:
2496:
2402:
2128:
2120:
2110:
2105:
2098:
2094:
2087:
2080:
2065:
2058:
2051:
1862:
1859:
1739:
1735:
1731:
1719:
1715:
1711:
1550:
1546:
1542:
1538:
1534:
1530:
1448:
1438:
1428:
1390:such as the
1357:and finally
1348:
1286:Maurice LĂ©vy
1270:
1258:
1210:
1194:
1184:
1170:
1168:
1135:
1131:
1127:
1123:
1119:
1111:
1107:
1052:
1048:
1040:
1036:
1032:
1028:
1024:
1017:Isaac Newton
1014:
995:
982:proportional
925:
915:
911:
907:
903:
866:
862:PSR B1913+16
850:
828:
824:
806:
798:binary stars
773:
769:
767:
693:
653:Raychaudhuri
204:
122:Introduction
15:
9872:KerrâNewman
9843:Spherical:
9712:Other tests
9655:Singularity
9587:Formulation
9549:Fundamental
9403:Formulation
9384:Proper time
9345:Fundamental
9179:Weinberg, S
8983:Gravitation
8979:Wheeler, JA
8872:Hagihara, Y
8844:Einstein, A
8706:, p. .
8639:, p. .
8627:, p. .
8625:Walter 2007
8615:, p. .
2895:black holes
2887:white dwarf
2706:proper time
1377:(1907) and
1298:Paul Gerber
932:Tycho Brahe
817:closed form
668:van Stockum
598:Oppenheimer
453:KerrâNewman
245:Singularity
10024:Zel'dovich
9932:Scientists
9911:Alcubierre
9718:of Mercury
9716:precession
9645:Black hole
9528:Background
9520:relativity
9489:World line
9484:Light cone
9309:Background
9301:relativity
9291:Relativity
9170:2008-07-26
9058:Rindler, W
8971:Misner, CW
8945:Landau, LD
8923:Lanczos, C
8882:: 67â176.
8561:: 379â383.
8299:References
8169:periastron
7570:elliptical
7273:See also:
7258:See also:
7182:See also:
5945:, whereas
2688:world line
2684:space-time
2516:See also:
2125:space-time
1724:impossible
1420:space-time
1345:worldwide.
1312:See also:
1230:arcseconds
1226:arcseconds
1217:precession
1205:See also:
1146:See also:
1002:Copernicus
884:See also:
521:Scientists
353:Formalisms
301:Formalisms
250:Black hole
176:World line
9994:Robertson
9979:Friedmann
9974:Eddington
9964:de Sitter
9798:Solutions
9676:detectors
9671:astronomy
9638:Phenomena
9573:Geodesics
9476:Spacetime
9419:Phenomena
9244:Animation
9103:Synge, JL
8975:Thorne, K
8896:0368-346X
8691:119083668
8589:Pais 1982
8577:Pais 1982
8519:0031-9007
8441:0031-9007
8360:0031-9007
8111:−
8100:π
7985:−
7945:−
7873:π
7826:−
7744:−
7580:−
7449:−
7306:−
7233:gravitons
7114:π
7105:≈
7102:φ
7099:δ
7041:−
6997:π
6988:≈
6985:φ
6982:δ
6947:−
6811:π
6714:π
6705:≈
6702:φ
6699:δ
6596:π
6537:π
6531:≈
6519:ω
6515:−
6510:φ
6506:ω
6493:φ
6490:δ
6461:⋯
6416:−
6403:φ
6399:ω
6386:ω
6323:−
6308:φ
6304:ω
6282:−
6196:ω
6089:ω
5902:≈
5889:≈
5603:≈
5593:φ
5589:ω
5561:μ
5531:φ
5527:ω
5523:μ
5453:φ
5449:ω
5298:≈
5234:≈
5044:−
5036:−
4925:−
4759:−
4697:−
4671:−
4599:−
4552:−
4351:−
4307:π
4298:≈
4295:φ
4292:δ
4282:elsewhere
4243:μ
4197:−
4181:μ
4141:−
4090:μ
4028:μ
4009:−
3957:−
3901:τ
3776:−
3731:−
3676:τ
3607:−
3527:−
3516:−
3473:∫
3467:φ
3304:−
3293:−
3250:φ
3207:μ
3185:μ
3171:×
3046:−
3035:−
2980:τ
2643:λ
2616:ν
2600:μ
2595:λ
2592:ν
2588:Γ
2561:μ
2532:geodesics
2481:ν
2467:μ
2454:ν
2451:μ
2434:τ
2383:φ
2375:θ
2372:
2349:−
2340:θ
2322:−
2306:−
2270:τ
2222:−
2206:−
2190:−
2154:τ
1978:⋯
1840:φ
1832:θ
1829:
1797:θ
1327:Eddington
1250:power law
1211:In 1859,
946:of every
813:nonlinear
613:Robertson
578:Friedmann
573:Eddington
563:Nordström
553:de Sitter
410:Solutions
335:Geodesics
330:Friedmann
312:Equations
298:Equations
259:Spacetime
194:Phenomena
100:ν
97:μ
88:κ
79:ν
76:μ
68:Λ
60:ν
57:μ
10120:Category
10107:Category
9984:LemaĂźtre
9949:Einstein
9939:Poincaré
9899:Others:
9883:TaubâNUT
9849:interior
9771:theories
9769:Advanced
9736:redshift
9551:concepts
9369:Rapidity
9347:concepts
9211:(1937).
9181:(1972).
9127:(1984).
9125:Wald, RM
9105:(1960).
9060:(1977).
9038:(1958).
9036:Pauli, W
9008:(1982).
9006:Pais, A.
8981:(1973).
8925:(1986).
8846:(1956).
8535:23409406
8527:16605809
8449:16605808
8376:24225193
8368:16197061
8200:See also
7860:⟩
7830:⟨
7614:⟩
7584:⟨
7333:⟩
7310:⟨
7229:dilatons
2522:Geodesic
1728:flat map
1373:(1905),
1254:exponent
1252:with an
1243:Newton's
1006:Voltaire
834:geodesic
729:Category
593:LemaĂźtre
558:Reissner
543:Poincaré
528:Einstein
473:TaubâNUT
438:Wormhole
422:interior
135:Timeline
10049:Hawking
10044:Penrose
10029:Novikov
10009:Wheeler
9954:Hilbert
9944:Lorentz
9901:pp-wave
9722:lensing
9518:General
9299:Special
9250:Excerpt
9191:185â201
9018:253â256
8884:Bibcode
8794:Bibcode
8729:Bibcode
8671:Bibcode
8499:Bibcode
8457:5954627
8421:Bibcode
8340:Bibcode
8171:of the
7568:of the
7564:is the
7556:is the
6890:formula
6055:Mercury
6051:precess
6043:ellipse
6009:⁄
5981:⁄
5957:⁄
5573:is the
3219:is the
3150:is the
2865:⁄
2827:is the
2749:is the
2731:fixed.
2111:In his
2108:sin Ξ.
2064:, ...,
1394:or the
1339:gravity
1261:Laplace
1221:Mercury
1162:ellipse
1114:is the
998:Ptolemy
988:of the
984:to the
976:of the
952:ellipse
920:apsides
840:of the
648:Hawking
643:Penrose
618:Bardeen
608:Wheeler
538:Hilbert
533:Lorentz
493:pp-wave
130:History
10090:others
10079:Thorne
10069:Misner
10054:Taylor
10039:Geroch
10034:Ehlers
10004:Zwicky
9822:Kasner
9225:
9221:â393.
9197:
9143:
9139:â146.
9113:
9091:
9068:
9046:
9024:
8989:
8959:
8933:
8911:
8894:
8860:
8832:
8828:â193.
8689:
8533:
8525:
8517:
8455:
8447:
8439:
8374:
8366:
8358:
8138:where
7552:where
7292:energy
6691:gives
6014:
5986:
5962:
5553:where
5474:where
4381:where
3130:where
2870:
2823:where
2729:energy
2713:photon
2524:, and
1710:where
1529:where
1239:Vulcan
1106:where
974:square
950:is an
948:planet
892:, and
794:planet
727:
694:others
688:Thorne
678:Newman
658:Taylor
638:Ehlers
623:Walker
588:Zwicky
463:Kasner
10084:Weiss
10064:Bondi
10059:Hulse
9989:Milne
9893:discs
9837:Milne
9832:Gödel
9689:Virgo
9252:from
8997:(See
8856:â97.
8784:arXiv
8687:S2CID
8661:arXiv
8531:S2CID
8489:arXiv
8453:S2CID
8411:arXiv
8372:S2CID
8330:arXiv
8265:Notes
7221:model
7077:(1 â
6290:inner
6277:outer
6250:outer
6171:outer
6078:outer
6059:Venus
6002:inner
5969:and 3
5950:inner
5936:outer
5897:inner
5884:outer
5413:outer
5387:outer
5382:into
5183:When
5170:inner
5159:inner
4840:outer
4833:inner
4433:inner
2086:= 1,
1429:after
967:areas
944:orbit
663:Hulse
603:Gödel
583:Milne
478:Milne
443:Gödel
140:Tests
10019:Kerr
9969:Weyl
9868:Kerr
9728:and
9682:and
9680:LIGO
9223:ISBN
9195:ISBN
9141:ISBN
9111:ISBN
9089:ISBN
9066:ISBN
9044:ISBN
9022:ISBN
8987:ISBN
8957:ISBN
8931:ISBN
8909:ISBN
8892:ISSN
8858:ISBN
8830:ISBN
8523:PMID
8515:ISSN
8445:PMID
8437:ISSN
8364:PMID
8356:ISSN
7560:and
7294:and
7238:The
7186:and
5927:The
5375:and
4835:and
4482:red.
3421:and
3199:and
2872:inch
2097:and
1718:and
1549:and
1537:and
1331:star
1316:and
1150:and
1008:and
1000:and
986:cube
972:The
963:line
956:foci
942:The
867:For
811:are
772:(or
768:The
673:Taub
628:Kerr
568:Weyl
448:Kerr
366:BSSN
10074:Yau
9699:GEO
9219:389
9137:136
8826:177
8737:doi
8725:131
8679:doi
8507:doi
8429:doi
8348:doi
7870:192
6673:= 2
6063:Sun
5357:mcr
4476:mcr
4395:not
4393:is
3861:as
2363:sin
1820:sin
1361:'s
1337:'s
1335:Sun
1235:Sun
1195:not
784:of
683:Yau
361:ADM
22:or
10122::
9748:/
9714::
9669::
9193:.
9020:.
9001:.)
8977:;
8973:;
8951:.
8890:.
8878:.
8854:92
8792:.
8735:.
8723:.
8711:^
8685:.
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8669:.
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8655:.
8567:^
8559:49
8557:.
8529:.
8521:.
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8485:96
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8465:^
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8435:.
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8384:^
8370:.
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8059:96
8056:37
8036:24
8033:73
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7497:73
7343:32
6057:,
6025:.
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5833:16
5577:.
5397:.
5351:=
5180:.
4842::
4502::
4494:=
4470:=
4413:,
4409:,
4405:,
4391:ÎŽÏ
3154:,
2897:.
2858:mm
2846:.
2698:ΌΜ
2520:,
2501:ΌΜ
2129:is
2121:ds
2115:,
2104:=
2102:ÏÏ
2093:=
2091:ΞΞ
2084:rr
2069:zz
2062:xy
2057:,
2055:xx
1863:ds
1738:,
1734:,
1714:,
1545:,
1539:dz
1535:dy
1533:,
1531:dx
1449:ds
1353:,
1280:,
1276:,
1041:ma
961:A
888:,
864:.
804:.
9874:)
9870:(
9856:)
9847:(
9813:)
9809:(
9756:)
9752:(
9732:)
9701:)
9678:(
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9318:(
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9276:t
9269:v
9231:.
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9119:.
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9074:.
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8880:8
8866:.
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8786::
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8731::
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8332::
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8153:a
8144:b
8140:P
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8119:/
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8106:)
8097:2
8091:b
8087:P
8081:(
8075:)
8069:4
8065:e
8051:+
8046:2
8042:e
8028:+
8025:1
8021:(
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8004:7
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7993:2
7989:e
7982:1
7978:(
7971:5
7967:c
7963:5
7956:3
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7948:1
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7935:2
7931:m
7927:+
7922:1
7918:m
7914:(
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7905:m
7899:1
7895:m
7889:3
7885:/
7881:5
7877:G
7864:=
7854:t
7851:d
7844:b
7840:P
7836:d
7803:)
7797:2
7793:e
7787:8
7784:7
7779:+
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7763:2
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7726:/
7722:7
7718:a
7712:5
7708:c
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7695:2
7691:m
7687:+
7682:1
7678:m
7670:2
7665:2
7661:m
7655:2
7650:1
7646:m
7640:2
7636:/
7632:7
7628:G
7618:=
7608:t
7605:d
7598:z
7594:L
7590:d
7562:a
7554:e
7539:)
7533:4
7529:e
7515:+
7510:2
7506:e
7492:+
7489:1
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7468:7
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7457:2
7453:e
7446:1
7442:(
7435:5
7431:a
7425:5
7421:c
7417:5
7412:)
7407:2
7403:m
7399:+
7394:1
7390:m
7386:(
7381:2
7376:2
7372:m
7366:2
7361:1
7357:m
7351:4
7347:G
7337:=
7327:t
7324:d
7319:E
7316:d
7219:T
7215:R
7198:m
7194:M
7147:l
7142:2
7138:c
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7129:m
7126:+
7123:M
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7111:6
7086:l
7079:e
7075:A
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7049:2
7045:e
7038:1
7034:(
7030:A
7025:2
7021:c
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7009:+
7006:M
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7000:G
6994:6
6961:)
6955:2
6951:e
6944:1
6940:(
6936:A
6933:=
6927:)
6924:m
6921:+
6918:M
6915:(
6912:G
6906:2
6902:h
6886:e
6882:A
6861:2
6857:L
6851:2
6847:c
6839:2
6835:m
6829:2
6825:M
6819:2
6815:G
6808:6
6802:=
6798:)
6791:4
6787:c
6780:2
6776:M
6770:2
6766:G
6762:4
6756:(
6747:2
6743:L
6739:2
6732:2
6728:c
6722:2
6718:m
6711:3
6689:s
6686:r
6679:a
6675:Ï
6671:T
6667:Ï
6663:Ï
6647:2
6642:s
6638:r
6629:2
6625:L
6621:2
6614:2
6610:c
6604:2
6600:m
6593:3
6587:=
6583:)
6575:2
6571:a
6567:4
6560:2
6555:s
6551:r
6547:3
6541:(
6534:2
6528:)
6523:r
6502:(
6499:T
6496:=
6480:T
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6458:+
6450:2
6446:a
6442:4
6435:2
6430:s
6426:r
6422:3
6413:1
6409:(
6395:=
6390:r
6353:2
6349:a
6342:2
6337:s
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6329:3
6320:1
6313:2
6300:=
6296:)
6286:r
6273:r
6268:(
6263:)
6255:4
6246:r
6242:2
6235:s
6231:r
6225:2
6221:c
6214:(
6210:=
6205:2
6200:r
6167:r
6163:=
6160:r
6155:]
6147:2
6143:r
6139:d
6134:V
6129:2
6125:d
6118:[
6111:m
6108:1
6103:=
6098:2
6093:r
6075:r
6071:V
6065:.
6019:s
6016:r
6011:2
6007:3
5999:r
5995:a
5991:s
5988:r
5983:2
5979:3
5974:s
5971:r
5967:s
5964:r
5959:2
5955:3
5947:r
5943:s
5940:r
5933:r
5913:s
5909:r
5905:3
5893:r
5880:r
5869:s
5866:r
5862:a
5841:6
5837:a
5826:4
5821:s
5817:r
5811:2
5807:c
5800:=
5796:)
5788:6
5784:a
5780:8
5774:3
5769:s
5765:r
5759:(
5754:)
5749:2
5743:2
5739:c
5733:s
5729:r
5722:(
5718:=
5714:)
5706:3
5700:r
5697:e
5694:t
5691:u
5688:o
5683:r
5679:2
5672:2
5668:c
5662:s
5658:r
5651:(
5647:=
5640:3
5634:r
5631:e
5628:t
5625:u
5622:o
5617:r
5612:M
5609:G
5598:2
5541:r
5536:2
5520:=
5513:2
5509:r
5504:m
5501:M
5498:G
5480:Ï
5476:Ï
5458:2
5444:)
5441:m
5438:+
5435:M
5432:(
5429:G
5423:=
5418:3
5409:r
5395:M
5391:m
5384:r
5380:s
5377:r
5373:a
5360:s
5355:/
5353:L
5349:s
5346:r
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5316:s
5312:r
5306:2
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5288:r
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5282:n
5279:n
5276:i
5271:r
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5224:r
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5189:r
5185:a
5174:r
5167:r
5163:r
5156:r
5134:,
5126:r
5123:e
5120:t
5117:u
5114:o
5109:r
5102:2
5098:a
5094:3
5088:=
5084:)
5074:2
5070:a
5063:2
5058:s
5054:r
5050:3
5041:1
5033:1
5029:(
5021:s
5017:r
5011:2
5007:a
5001:=
4991:r
4988:e
4985:n
4982:n
4979:i
4974:r
4965:)
4955:2
4951:a
4944:2
4939:s
4935:r
4931:3
4922:1
4917:+
4914:1
4910:(
4902:s
4898:r
4892:2
4888:a
4882:=
4872:r
4869:e
4866:t
4863:u
4860:o
4855:r
4837:r
4830:r
4815:;
4812:0
4809:=
4805:]
4799:2
4795:a
4789:s
4785:r
4781:3
4778:+
4775:r
4770:2
4766:a
4762:2
4754:2
4750:r
4744:s
4740:r
4735:[
4726:4
4722:r
4718:2
4711:2
4707:c
4703:m
4694:=
4688:r
4685:d
4680:V
4677:d
4668:=
4665:F
4643:.
4639:]
4631:3
4627:r
4620:2
4616:a
4610:s
4606:r
4592:2
4588:r
4582:2
4578:a
4572:+
4567:r
4562:s
4558:r
4548:[
4542:2
4536:2
4532:c
4528:m
4522:=
4519:)
4516:r
4513:(
4510:V
4500:c
4498:/
4496:h
4492:a
4488:V
4479:s
4474:/
4472:L
4468:s
4465:r
4463:/
4461:a
4457:r
4441:a
4437:r
4430:r
4426:r
4415:Ï
4411:Ξ
4407:r
4403:t
4399:Ï
4387:e
4383:A
4365:)
4359:2
4355:e
4348:1
4344:(
4340:A
4335:2
4331:c
4325:)
4322:m
4319:+
4316:M
4313:(
4310:G
4304:6
4251:3
4247:r
4238:2
4234:c
4226:2
4222:L
4218:)
4215:m
4212:+
4209:M
4206:(
4203:G
4189:2
4185:r
4178:2
4172:2
4168:L
4162:+
4157:r
4153:m
4150:M
4147:G
4138:=
4135:)
4132:r
4129:(
4126:V
4098:3
4094:r
4085:2
4081:c
4073:2
4069:L
4065:)
4062:m
4059:+
4056:M
4053:(
4050:G
4044:+
4036:2
4032:r
4025:2
4019:2
4015:L
4004:r
4000:m
3997:M
3994:G
3988:+
3984:]
3978:2
3974:c
3970:m
3965:2
3962:1
3949:2
3945:c
3941:m
3938:2
3932:2
3928:E
3921:[
3917:=
3912:2
3907:)
3898:d
3893:r
3890:d
3884:(
3879:m
3874:2
3871:1
3859:s
3856:r
3835:3
3831:r
3824:2
3820:h
3814:s
3810:r
3803:+
3796:2
3792:r
3786:2
3782:h
3771:r
3765:2
3761:c
3755:s
3751:r
3744:+
3739:2
3735:c
3723:2
3719:c
3713:2
3709:m
3702:2
3698:E
3692:=
3687:2
3682:)
3673:d
3668:r
3665:d
3659:(
3630:.
3627:r
3624:d
3618:2
3614:/
3610:1
3602:]
3597:)
3589:2
3585:r
3581:1
3576:+
3569:2
3565:a
3561:1
3555:(
3550:)
3544:r
3538:s
3533:r
3524:1
3520:(
3509:2
3505:b
3501:1
3495:[
3486:2
3482:r
3478:1
3470:=
3445:E
3441:c
3438:L
3432:=
3429:b
3407:c
3404:h
3399:=
3396:a
3376:,
3372:)
3366:2
3362:r
3358:+
3351:2
3347:a
3341:4
3337:r
3330:(
3325:)
3319:r
3314:s
3310:r
3301:1
3297:(
3286:2
3282:b
3276:4
3272:r
3266:=
3261:2
3256:)
3247:d
3242:r
3239:d
3233:(
3182:L
3177:=
3174:v
3168:r
3165:=
3162:h
3138:h
3118:.
3114:)
3106:2
3102:r
3096:2
3092:h
3086:+
3081:2
3077:c
3072:(
3067:)
3061:r
3056:s
3052:r
3043:1
3039:(
3027:2
3023:c
3017:2
3013:m
3006:2
3002:E
2996:=
2991:2
2986:)
2977:d
2972:r
2969:d
2963:(
2941:M
2921:m
2883:r
2881:/
2879:s
2876:r
2867:8
2863:3
2860:(
2854:s
2851:r
2840:r
2838:/
2836:s
2833:r
2825:G
2807:2
2803:c
2798:M
2795:G
2792:2
2786:=
2781:s
2777:r
2762:s
2759:r
2755:M
2725:m
2717:q
2709:Ï
2702:q
2695:g
2680:q
2662:0
2659:=
2653:q
2650:d
2639:x
2635:d
2626:q
2623:d
2612:x
2608:d
2584:+
2576:2
2572:q
2568:d
2557:x
2551:2
2547:d
2497:g
2477:x
2473:d
2463:x
2459:d
2447:g
2443:=
2438:2
2430:d
2424:2
2420:c
2387:2
2379:d
2367:2
2357:2
2353:r
2344:2
2336:d
2330:2
2326:r
2317:2
2313:r
2309:d
2301:2
2297:t
2293:d
2287:2
2283:c
2279:=
2274:2
2266:d
2260:2
2256:c
2233:2
2229:z
2225:d
2217:2
2213:y
2209:d
2201:2
2197:x
2193:d
2185:2
2181:t
2177:d
2171:2
2167:c
2163:=
2158:2
2150:d
2144:2
2140:c
2106:r
2099:g
2095:r
2088:g
2081:g
2066:g
2059:g
2052:g
2036:2
2032:z
2028:d
2022:z
2019:z
2015:g
2011:+
2008:y
2005:d
2001:z
1998:d
1992:y
1989:z
1985:g
1981:+
1975:+
1972:z
1969:d
1965:x
1962:d
1956:z
1953:x
1949:g
1945:+
1942:y
1939:d
1935:x
1932:d
1926:y
1923:x
1919:g
1915:+
1910:2
1906:x
1902:d
1896:x
1893:x
1889:g
1885:=
1880:2
1876:s
1872:d
1844:2
1836:d
1824:2
1814:2
1810:r
1806:+
1801:2
1793:d
1787:2
1783:r
1779:+
1774:2
1770:r
1766:d
1763:=
1758:2
1754:s
1750:d
1740:Ï
1736:Ξ
1732:r
1720:H
1716:G
1712:F
1696:2
1692:z
1688:d
1684:)
1681:z
1678:,
1675:y
1672:,
1669:x
1666:(
1663:H
1660:+
1655:2
1651:y
1647:d
1643:)
1640:z
1637:,
1634:y
1631:,
1628:x
1625:(
1622:G
1619:+
1614:2
1610:x
1606:d
1602:)
1599:z
1596:,
1593:y
1590:,
1587:x
1584:(
1581:F
1578:=
1573:2
1569:s
1565:d
1551:z
1547:y
1543:x
1515:2
1511:z
1507:d
1504:+
1499:2
1495:y
1491:d
1488:+
1483:2
1479:x
1475:d
1472:=
1467:2
1463:s
1459:d
1171:r
1136:m
1132:m
1128:M
1124:M
1122:/
1120:m
1112:G
1108:r
1094:,
1087:2
1083:r
1078:m
1075:M
1069:G
1066:=
1063:F
1053:m
1049:M
1039:=
1037:F
1033:a
1029:m
1025:F
958:.
916:m
912:M
908:M
904:m
829:m
825:M
757:e
750:t
743:v
424:)
420:(
93:T
84:=
72:g
65:+
53:G
26:.
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