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| {{Expert-subject|General relativity|date=May 2013}}
| | The name of the author is Luther. His working day occupation is a monetary officer but he plans on altering it. Delaware is our beginning location. To play croquet is something that I've carried out for many years.<br><br>My site ... [http://Letuschaton.com/blogs/post/27011 extended auto warranty] |
| {{General_relativity|cTopic=Phenomena}}
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| The '''two-body problem in general relativity''' is to determine the motion and gravitational field of two bodies interacting with one another by [[gravitation]], as described by the field equations of [[general relativity]]. Solving the Kepler problem is essential to calculate the bending of light by gravity and the motion of a [[planet]] orbiting its sun. Solutions are also used to describe the motion of [[binary star]]s around each other, and estimate their gradual loss of energy through [[gravitational radiation]]. It is customary to assume that both bodies are point-like, so that [[tidal force]]s and the specifics of their material composition can be neglected. | |
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| General relativity describes the gravitational field by curved space-time; the [[Einstein equation|field equations governing this curvature]] are [[nonlinear system|nonlinear]] and therefore difficult to solve in a [[closed-form expression|closed form]]. Only one exact solution, the [[Schwarzschild metric|Schwarzschild solution]], has been found for the Kepler problem; this solution pertains when the mass ''M'' of one body is overwhelmingly greater than the mass ''m'' 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 Schwarzschild solution; the motion is a [[geodesic]] ("shortest path between two points") in the curved space-time. Such geodesic solutions account for the [[Tests_of_general_relativity#Perihelion_precession_of_Mercury|anomalous precession]] of the [[Mercury (planet)|planet Mercury]], 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 [[Tests_of_general_relativity#Deflection of light by the Sun|famously used as evidence]] for general relativity.
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| 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 [[post-Newtonian expansion]], 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 [[Gravitational wave|gravitational radiation]]; this causes them to lose energy and angular momentum gradually, as illustrated by the binary pulsar [[PSR B1913+16]].
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| {{TOClimit|limit=3}} <!-- Simplify Table of Contents -->
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| ==Historical context==
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| ===Classical Kepler problem===
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| {{See also|Kepler's laws of planetary motion|Newton's law of universal gravitation|Two-body problem}}
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| [[File:Elliptic orbit.gif|thumb|right|Figure 1. Typical elliptical path of a smaller mass ''m'' orbiting a much larger mass ''M''. The larger mass is also moving on an elliptical orbit, but it is too small to be seen because ''M'' is much greater than ''m''. The ends of the diameter indicate the [[apsis|apsides]], the points of closest and farthest distance. ]]
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| The Kepler problem derives its name from [[Johannes Kepler]], who worked as an assistant to the Danish astronomer [[Tycho Brahe]]. Brahe took extraordinarily accurate measurements of the motion of the planets of the Solar System. From these measurements, Kepler was able to formulate [[Kepler's laws of planetary motion|Kepler's laws]], the first modern description of planetary motion:
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| #The [[orbit]] of every [[planet]] is an [[ellipse]] with the Sun at one of the two [[Focus (geometry)|foci]].
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| #A [[line (geometry)|line]] joining a planet and the Sun sweeps out equal [[area]]s during equal intervals of time.
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| #The [[square (algebra)|square]] of the [[orbital period]] of a planet is directly [[Proportionality (mathematics)|proportional]] to the [[cube (arithmetic)|cube]] of the [[semi-major axis]] of its orbit.
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| 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 [[Ptolemy]] and [[Copernicus]]. Kepler's laws apply only in the limited case of the two-body problem. [[Voltaire]] and [[Émilie du Châtelet]] were the first to call them "Kepler's laws".
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| Nearly a century later, [[Isaac Newton]] had formulated his [[Newton's laws of motion|three laws of motion]]. In particular, Newton's second law states that a force ''F'' applied to a mass ''m'' produces an acceleration ''a'' given by the equation ''F''=''ma''. Newton then posed the question: what must the force be that produces the elliptical orbits seen by Kepler? His answer came in his [[Newton's law of universal gravitation|law of universal gravitation]], which states that the force between a mass ''M'' and another mass ''m'' is given by the formula
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| :<math>
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| F = G \frac{M m}{r^2}
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| </math>,
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| where ''r'' is the distance between the masses and ''G'' is the [[gravitational constant]]. 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 ''m''/''M'', with the larger mass moving on a smaller ellipse. If ''M'' is much larger than ''m'', then the larger mass will appear to be stationary at the focus of the elliptical orbit of the lighter mass ''m''. 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.
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| ===Apsidal precession===
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| {{See also|Apsidal precession|Laplace–Runge–Lenz vector}}
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| [[File:Precessing Kepler orbit 280frames e0.6 smaller.gif|thumb|right|345px|In the absence of any other forces, a particle orbiting another under the influence of Newtonian gravity follows the same perfect [[ellipse]] 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. However, the predictions of Newtonian gravity do not match the observations, as discovered in 1859 from observations of Mercury.]]
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| If the potential energy between the two bodies is not exactly the 1/''r'' potential of Newton's gravitational law but differs only slightly, then the ellipse of the orbit gradually rotates (among other possible effects). This [[apsidal precession]] 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 for one another. The apsides are the two points of closest and furthest distance of the orbit (the periapsis and apoapsis, respectively); apsidal precession corresponds to the rotation of the line joining the apsides. It also corresponds to the rotation of the [[Laplace–Runge–Lenz vector]], which points along the line of apsides.
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| Newton's law of gravitation soon became accepted because it gave very accurate predictions of the apsidal precessions of all the planets.{{dubious||Mercury's value would have been off by ~10% due to general relativity; the total precessions of other planets are smaller than the error in Mercury's value, so could not have been measured to be non-zero at all if accuracy was so low not to notice Mercury's anomlay|date=January 2013}} These calculations were carried out initially by [[Pierre-Simon Laplace]] in the late 18th century, and refined by [[Félix Tisserand]] in the later 19th century. Conversely, if Newton's law of gravitation did ''not'' 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, and it led to the overthrow of Newtonian model of gravity and the development of [[general relativity]].
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| ===Anomalous precession of Mercury===
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| {{See also|Tests of general relativity}}
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| In 1859, [[Urbain Le Verrier]] discovered that the orbital [[precession]] of the [[planet]] [[Mercury (planet)|Mercury]] 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.<ref>{{cite journal | last = Le Verrier | first = UJJ | authorlink = Urbain Le Verrier | year = 1859 | title = Unknown title | journal = [[Comptes rendus de l'Académie des sciences|Comptes Rendus]] | volume = 49 | pages = 379–?}}</ref> The effect is small (roughly 43 [[arcsecond]]s of rotation per century), but well above the measurement error (roughly 0.1 [[arcsecond]]s 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 [[Sun]], an undetected moon of Mercury, or a new planet named [[Vulcan (planet)|Vulcan]].<ref name="pais_1982"/> After these explanations were discounted, some physicists were driven to the more radical hypothesis that [[Isaac Newton|Newton's]] [[inverse-square law]] of gravitation was incorrect. For example, some physicists proposed a [[power law]] with an [[exponent]] that was slightly different from 2.
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| 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, [[Laplace]] 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 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 allows one to put a lower limit on the [[speed of gravity]] that is many orders of magnitude faster than the speed of light.<ref name="pais_1982">Pais 1982</ref> Laplace's estimate for the velocity of gravity is not correct, because in a field theory which respects the principle of relativity, the attraction of a point charge which is moving at a constant velocity is towards the extrapolated instantaneous position, not to the apparent position it seems to occupy when looked at<ref group=note>Feynman Lectures on Physics vol. II 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 line in a constant velocity. This is a notable property of the [[Liénard-Wiechert Potentials]] which are used in the [[Wheeler-Feynman absorber theory]]. Presumably the same holds in linearized gravity.</ref>
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| To avoid those problems, between 1870 and 1900 many scientists used the electrodynamic laws of [[Wilhelm Eduard Weber]], [[Carl Friedrich Gauss]], [[Bernhard Riemann]] to produce stable orbits and to explain the perihelion shift of Mercury's orbit. In 1890 Lévy succeeded in doing so by combining the laws of Weber and Riemann, whereby the [[speed of gravity]] is equal to the [[speed of light]] in his theory. And in another attempt [[Paul Gerber]] (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.<ref>Roseveare 1982</ref> Another attempt by [[Hendrik Lorentz]] (1900), who already used Maxwell's theory, produced a perihelion shift which was too low.<ref name="pais_1982" /> | |
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| ===Einstein's theory of general relativity===
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| {{See also|Introduction to general relativity|General relativity}}
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| [[File:1919 eclipse negative.jpg|thumb|200px|right|[[Arthur Stanley Eddington|Eddington]]'s 1919 measurements of the bending of [[star]]-light by the [[Sun]]'s [[gravitation|gravity]] led to the acceptance of [[general relativity]] worldwide.]]
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| Around 1904–1905, the works of [[Hendrik Lorentz]], [[Henri Poincaré]] and finally [[Albert Einstein]]'s [[special theory of relativity]], exclude the possibility of propagation of any effects faster than the [[speed of light]]. 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 [[Henri Poincaré]] (1905), [[Hermann Minkowski]] (1907) and [[Arnold Sommerfeld]] (1910).<ref>Walter 2007</ref> 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 [[equivalence principle]] 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 frame, and vice versa. Thus, gravity acts like a [[fictitious force]] such as the [[centrifugal force]] or the [[Coriolis force]], which result from being in an accelerated reference frame; all fictitious forces are proportional to the [[inertial mass]], just as gravity is. To effect the reconciliation of gravity and [[special relativity]] 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 [[Euclidean geometry]], e.g., that the [[Pythagorean theorem]] is true experimentally. Einstein used a more general geometry, [[Pseudo-Riemannian manifold|pseudo-Riemannian geometry]], to allow for the curvature of [[space]] and [[time]] that was necessary for the reconciliation; after eight years of work (1907–1915), he succeeded in discovering the precise way in which [[space-time]] 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 regarded as forces. The very first solutions of [[Einstein field equations|his field equations]] explained the anomalous precession of Mercury and predicted an unusual bending of light, which was confirmed ''after'' his theory was published. These solutions are explained below.
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| ==General relativity, special relativity and geometry==
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| In the normal [[Euclidean geometry]], triangles obey the [[Pythagorean theorem]], which states that the square distance ''ds''<sup>2</sup> between two points in space is the sum of the squares of its perpendicular components
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| :<math>
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| ds^{2} = dx^{2} + dy^{2} + dz^{2} \,\!
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| </math>
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| where ''dx'', ''dy'' and ''dz'' represent the infinitesimal differences between the two points along the ''x'', ''y'' and ''z'' axes of a [[Cartesian coordinate system]] (add Figure here). Now imagine a world in which this is not quite true; a world where the distance is instead given by
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| :<math>
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| ds^{2} = F(x, y, z) dx^{2} + G(x, y, z) dy^{2} + H(x, y, z)dz^{2} \,\!
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| </math>
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| where ''F'', ''G'' and ''H'' are arbitrary functions of position. It is not hard to imagine such a world; we live on one. The surface of the world is curved, which is why it's impossible to make a perfectly accurate flat map of the world. Non-Cartesian coordinate systems illustrate this well; for example, in the spherical coordinates (''r'', ''θ'', ''φ''), the Euclidean distance can be written
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| :<math>
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| ds^{2} = dr^{2} + r^{2} d\theta^{2} + r^{2} \sin^{2} \theta d\varphi^{2} \,\!
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| </math>
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| 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 ''ds''
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| :<math>
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| ds^{2} = g_{xx} dx^{2} + g_{xy} dx dy + g_{xz} dx dz + \cdots + g_{zy} dz dy + g_{zz} dz^{2} \,\!
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| </math>
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| where the nine functions ''g''<sub>xx</sub>, ''g''<sub>xy</sub> constitute the [[metric tensor]], which defines the geometry of the space in [[Riemannian geometry]]. In the spherical-coordinates example above, there are no cross-terms; the only nonzero metric tensor components are ''g''<sub>rr</sub> = 1, ''g''<sub>θθ</sub> = ''r''<sup>2</sup> and ''g''<sub>φφ</sub> = ''r''<sup>2</sup> sin<sup>2</sup> θ.
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| In his [[special relativity|special theory of relativity]], [[Albert Einstein]] showed that the distance ''ds'' 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 [[space-time]] — called "proper time" and denoted with the symbol dτ — that ''is'' invariant; in other words, it doesn't depend on the motion of the observer.
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| :<math>
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| c^{2} d\tau^{2} = c^{2} dt^{2} - dx^{2} - dy^{2} - dz^{2} \,\!
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| </math>
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| which may be written in spherical coordinates as
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| :<math>
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| c^{2} d\tau^{2} = c^{2} dt^{2} - dr^{2} - r^{2} d\theta^{2} - r^{2} \sin^{2} \theta d\varphi^{2} \,\!
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| </math>
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| This formula is the natural extension of the [[Pythagorean theorem]] and similarly holds only when there is no curvature in space-time. In [[general relativity]], however, space and time may have curvature, so this distance formula must be modified to a more general form
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| :<math>
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| c^{2} d\tau^{2} = g_{\mu\nu} dx^{\mu} dx^{\nu} \,\!
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| </math>
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| just as we generalized the formula to measure distance on the surface of the Earth. The exact form of the metric ''g''<sub>μν</sub> depends on the gravitating mass, momentum and energy, as described by the [[Einstein field equations]]. 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.
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| ===Geodesic equation===
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| {{See also|Geodesic|Christoffel symbols}}
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| According to Einstein's theory of general relativity, particles of negligible mass travel along [[geodesic]]s 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<ref>Weinberg 1972.</ref>
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| :<math>
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| \frac{d^2x^{\mu}}{d q^2} + \Gamma^{\mu}_{\nu\lambda} \frac{dx^{\nu}}{d q} \frac{dx^{\lambda}}{dq} = 0
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| </math>
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| where Γ represents the [[Christoffel symbol]] and the variable ''q'' parametrizes the particle's path through [[space-time]], its so-called [[world line]]. The Christoffel symbol depends only on the [[metric tensor]] ''g''<sub>μν</sub>, or rather on how it changes with position. The variable ''q'' is a constant multiple of the [[proper time]] ''τ'' 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 [[photon]]), the proper time is zero and, strictly speaking, cannot be used as the variable ''q''. Nevertheless, lightlike orbits can be derived as the [[ultrarelativistic limit]] of timelike orbits, that is, the limit as the particle mass ''m'' goes to zero while holding its total [[energy]] fixed.
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| ==Schwarzschild solution==
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| {{See also|Schwarzschild solution}}
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| An exact solution to the [[Einstein field equations]] is the [[Schwarzschild metric]], which corresponds to the external gravitational field of a stationary, uncharged, non-rotating, spherically symmetric body of mass ''M''. It is characterized by a length scale ''r''<sub>s</sub>, known as the [[Schwarzschild radius]], which is defined by the formula
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| ::<math>
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| r_{s} = \frac{2GM}{c^{2}}
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| </math>
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| where ''G'' is the [[gravitational constant]]. The classical Newtonian theory of gravity is recovered in the limit as the ratio ''r''<sub>s</sub>/''r'' goes to zero. In that limit, the metric returns to that defined by [[special relativity]].
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| In practice, this ratio is almost always extremely small. For example, the Schwarzschild radius ''r''<sub>s</sub> of the [[Earth]] is roughly 9 [[millimeter|mm]] ({{frac|3|8}} [[inch]]); 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 ''r''<sub>s</sub>/''r'' is roughly 4 parts in a million. A [[white dwarf]] 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 [[neutron star]]s (where the ratio is roughly 50%) and [[black hole]]s.
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| ===Orbits about the central mass===
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| The orbits of a test particle of infinitesimal mass ''m'' about the central mass ''M'' is given by the equation of motion
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| :<math>
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| \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).
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| </math>
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| which can be converted into an equation for the orbit
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| :<math>
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| \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)
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| </math>
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| where, for brevity, two length-scales, ''a'' and ''b'', 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
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| :<math>
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| \varphi = \int \frac{dr}{r^{2} \sqrt{\frac{1}{b^{2}} - \left( 1 - \frac{r_{s}}{r} \right) \left( \frac{1}{a^{2}} + \frac{1}{r^{2}} \right)}}.
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| </math>
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| ===Bending of light by gravity===
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| {{See also|Gravitational lens|Shapiro delay}}
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| [[File:Light deflection.png|thumb|left|200px|Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray)]]
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| The orbit of photons and particles moving close to the speed of light (ultrarelativistic particles) is obtained by taking the limit as the length-scale ''a'' goes to infinity. In this limit, the equation for the orbit becomes
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| :<math>
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| \varphi = \int \frac{dr}{r^{2} \sqrt{\frac{1}{b^{2}} - \left( 1 - \frac{r_{s}}{r} \right) \frac{1}{r^{2}}}}
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| </math>
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| Expanding in powers of ''r''<sub>s</sub>/''r'', the leading order term in this formula gives the approximate angular deflection δ''φ'' for a massless particle coming in from infinity and going back out to infinity:
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| :<math>
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| \delta \varphi \approx \frac{2r_{s}}{b} = \frac{4GM}{c^{2}b}.
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| </math> | |
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| Here, the length-scale ''b'' can be interpreted as the distance of closest approach. Although this formula is approximate, it is accurate for most measurements of [[gravitational lensing]], due to the smallness of the ratio ''r''<sub>s</sub>/''r''. For light grazing the surface of the sun, the approximate angular deflection is roughly 1.75 [[Minute of arc|arcseconds]], roughly one millionth part of a circle.<ref>http://www.mathpages.com/rr/s6-03/6-03.htm - 6.3 Bending Light</ref>
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| ===Effective radial potential energy===
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| The equation of motion for the particle derived above
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| :<math>
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| \left( \frac{dr}{d\tau} \right)^{2} =
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| \frac{E^2}{m^2 c^2} - c^{2} + \frac{ r_{s} c^2}{r} -
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| \frac{h^2}{ r^2 } + \frac{ r_{s} h^2 }{ r^3 }
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| </math>
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| can be rewritten using the definition of the [[Schwarzschild metric|Schwarzschild radius]] ''r''<sub>s</sub> as
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| :<math>
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| \frac{1}{2} m \left( \frac{dr}{d\tau} \right)^{2} =
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| \left[ \frac{E^2}{2 m c^2} - \frac{1}{2} m c^2 \right]
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| + \frac{GMm}{r} - \frac{ L^2 }{ 2 \mu r^2 } + \frac{ G(M+m) L^2 }{c^2 \mu r^3}
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| </math>
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| which is equivalent to a particle moving in a one-dimensional [[effective potential]]
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| :<math>
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| V(r) = -\frac{GMm}{r} + \frac{ L^2 }{ 2 \mu r^2 } - \frac{ G(M+m) L^2 }{ c^2 \mu r^3 }
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| </math>
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| The first two terms are well-known classical energies, the first being the attractive Newtonian gravitational potential energy and the second corresponding to the repulsive [[Centrifugal force|"centrifugal" potential energy]]; however, the third term is an attractive energy unique to [[general relativity]]. As shown below and [[Laplace–Runge–Lenz_vector#Evolution_under_perturbed_potentials|elsewhere]], this inverse-cubic energy causes elliptical orbits to precess gradually by an angle δφ per revolution
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| :<math>
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| \delta \varphi \approx \frac{ 6\pi G(M+m) }{ c^2 A \left( 1- e^{2} \right)}
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| </math>
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| where ''A'' is the semi-major axis and ''e'' is the eccentricity.
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| The third term is attractive and dominates at small ''r'' values, giving a critical inner radius ''r''<sub>inner</sub> at which a particle is drawn inexorably inwards to ''r''=0; this inner radius is a function of the particle's angular momentum per unit mass or, equivalently, the ''a'' length-scale defined above.
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| ===Circular orbits and their stability===
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| [[File:Schwarzschild effective potential.svg|thumb|350px|right|Effective radial potential for various angular momenta. At small radii, the energy drops precipitously, causing the particle to be pulled inexorably inwards to ''r''=0. However, when the normalized angular momentum ''a''/''r''<sub>s</sub> = ''L''/''mcr''<sub>s</sub> 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 red.]]
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| The effective potential ''V'' can be re-written in terms of the length ''a'' = h/c.
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| :<math>
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| V(r) = \frac{mc^{2}}{2} \left[ - \frac{r_{s}}{r} + \frac{a^{2}}{r^{2}} - \frac{r_{s} a^{2}}{r^{3}} \right]
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| </math>
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| Circular orbits are possible when the effective force is zero
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| :<math>
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| F = -\frac{dV}{dr} = -\frac{mc^{2}}{2r^{4}} \left[ r_{s} r^{2} - 2a^{2} r + 3r_{s} a^{2} \right] = 0
| |
| </math>
| |
| | |
| i.e., when the two attractive forces — Newtonian gravity (first term) and the attraction unique to general relativity (third term) — are exactly balanced by the repulsive centrifugal force (second term). There are two radii at which this balancing can occur, denoted here as ''r''<sub>inner</sub> and ''r''<sub>outer</sub>
| |
| | |
| :<math>
| |
| r_{\mathrm{outer}} = \frac{a^{2}}{r_{s}} \left( 1 + \sqrt{1 - \frac{3r_{s}^{2}}{a^{2}}} \right)
| |
| </math>
| |
| | |
| :<math>
| |
| 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}}}
| |
| </math>
| |
| | |
| which are obtained using the [[Quadratic_equation#Quadratic_formula|quadratic formula]]. The inner radius ''r''<sub>inner</sub> is unstable, because the attractive third force strengthens much faster than the other two forces when ''r'' becomes small; if the particle slips slightly inwards from ''r''<sub>inner</sub> (where all three forces are in balance), the third force dominates the other two and draws the particle inexorably inwards to ''r''=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 [[Kepler problem]].
| |
| | |
| When ''a'' is much greater than ''r''<sub>s</sub> (the classical case), these formulae become approximately
| |
| | |
| :<math>
| |
| r_{\mathrm{outer}} \approx \frac{2a^{2}}{r_{s}}
| |
| </math>
| |
| | |
| :<math>
| |
| r_{\mathrm{inner}} \approx \frac{3}{2} r_{s}
| |
| </math>
| |
| | |
| [[File:Schwarzschild circular radii.svg|thumb|left|320px|The stable and unstable radii are plotted versus the normalized angular momentum ''a''/''r''<sub>s</sub> = ''L''/''mcr''<sub>s</sub> 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 [[centripetal acceleration]] and Newton's law of gravity is plotted in black.]]
| |
| | |
| Substituting the definitions of ''a'' and ''r''<sub>s</sub> into ''r''<sub>outer</sub> yields the classical formula for a particle of mass ''m'' orbiting a body of mass ''M''.
| |
| | |
| :<math>
| |
| r_{\mathrm{outer}}^{3} = \frac{G(M+m)}{\omega_{\varphi}^{2}}
| |
| </math>
| |
| | |
| where ''ω''<sub>φ</sub> is the orbital angular speed of the particle. This formula is obtained in non-relativistic mechanics by setting the [[centrifugal force]] equal to the Newtonian gravitational force:
| |
| | |
| :<math>
| |
| \frac{GMm}{r^{2}} = \mu \omega_{\varphi}^{2} r
| |
| </math>
| |
| Where <math>\mu</math> is the [[reduced mass]].<br>
| |
| In our notation, the classical orbital angular speed equals
| |
| | |
| :<math>
| |
| \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}}{16 a^{6}}
| |
| </math>
| |
| | |
| At the other extreme, when ''a''<sup>2</sup> approaches 3''r''<sub>s</sub><sup>2</sup> from above, the two radii converge to a single value
| |
| | |
| :<math>
| |
| r_{\mathrm{outer}} \approx r_{\mathrm{inner}} \approx 3 r_{s}
| |
| </math>
| |
| | |
| The [[Quadratic_equation#Quadratic_formula|quadratic solutions]] above ensure that ''r''<sub>outer</sub> is always greater than 3''r''<sub>s</sub>, whereas ''r''<sub>inner</sub> lies between {{frac|3|2}} ''r''<sub>s</sub> and 3''r''<sub>s</sub>. Circular orbits smaller than {{frac|3|2}} ''r''<sub>s</sub> are not possible. For massless particles, ''a'' goes to infinity, implying that there is a circular orbit for photons at ''r''<sub>inner</sub> = {{frac|3|2}} ''r''<sub>s</sub>. The sphere of this radius is sometimes known as the [[photon sphere]].
| |
| | |
| ===Precession of elliptical orbits===
| |
| [[File:Relativistic precession.svg|thumb|right|In the non-relativistic [[Kepler problem]], a particle follows the same perfect [[ellipse]] (red orbit) eternally. [[General relativity]] 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 [[Apsidal precession|precess]] (cyan orbit) in the direction of its rotation; this effect has been measured in [[Mercury (planet)|Mercury]], [[Venus]] and [[Earth]]. The yellow dot within the orbits represents the center of attraction, such as the [[Sun]].]]
| |
| | |
| The orbital precession rate may be derived using this radial effective potential ''V''. A small radial deviation from a circular orbit of radius ''r''<sub>outer</sub> will oscillate in a stable manner with an angular frequency
| |
| | |
| :<math>
| |
| \omega_{r}^{2} = \frac{1}{m} \left[ \frac{d^{2}V}{dr^{2}} \right]_{r=r_{\mathrm{outer}}}
| |
| </math>
| |
| | |
| which equals
| |
| | |
| :<math>
| |
| \omega_{r}^{2} = \left( \frac{c^{2} r_{s}}{2 r_{\mathrm{outer}}^{4}} \right) \left( r_{\mathrm{outer}} - r_{\mathrm{inner}} \right) =
| |
| \omega_{\varphi}^{2} \sqrt{1 - \frac{3r_{s}^{2}}{a^{2}}}
| |
| </math>
| |
| | |
| Taking the square root of both sides and expanding using the [[binomial theorem]] yields the formula
| |
| | |
| :<math>
| |
| \omega_{r} = \omega_{\varphi} \left( 1 - \frac{3r_{s}^{2}}{4a^{2}} + \cdots \right)
| |
| </math>
| |
| | |
| Multiplying by the period ''T'' of one revolution gives the precession of the orbit per revolution
| |
| | |
| :<math>
| |
| \delta \varphi = T \left( \omega_{\varphi} - \omega_{r} \right) \approx 2\pi \left( \frac{3r_{s}^{2}}{4a^{2}} \right) =
| |
| \frac{3\pi m^{2} c^{2}}{2L^{2}} r_{s}^{2}
| |
| </math>
| |
| | |
| where we have used ''ω<sub>φ</sub>T'' = 2''п'' and the definition of the length-scale ''a''. Substituting the definition of the [[Schwarzschild metric|Schwarzschild radius]] ''r''<sub>s</sub> gives
| |
| | |
| :<math>
| |
| \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}}
| |
| </math>
| |
| | |
| This may be simplified using the elliptical orbit's semiaxis ''A'' and eccentricity ''e'' related by the [[Laplace–Runge–Lenz vector|formula]]
| |
| | |
| :<math>
| |
| \frac{ h^2 }{ G(M+m) } = A \left( 1 - e^2 \right)
| |
| </math>
| |
| | |
| to give the precession angle
| |
| | |
| :<math>
| |
| \delta \varphi \approx \frac{6\pi G(M+m)}{c^2 A \left( 1 - e^{2} \right)}
| |
| </math>
| |
| | |
| ==Corrections to the Schwarzschild solution==
| |
| ===Post-Newtonian expansion===
| |
| | |
| {{See also|Post-Newtonian expansion|Parameterized post-Newtonian formalism}}
| |
| | |
| In the Schwarzschild solution, it is assumed that the larger mass ''M'' is stationary and it alone determines the gravitational field (i.e., the geometry of space-time) and, hence, the lesser mass ''m'' 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 [[binary star]]s, in which the masses may be of similar magnitude.
| |
| | |
| 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 [[post-Newtonian approximation]] or numerical approximations. In passing, we mention one particular exception in lower dimensions (see [[R=T model]] 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 [[Lambert W function]].<ref>{{cite journal |first=T. |last=Ohta |first2=R. B. |last2=Mann |year=1997 |title=Exact solution for the metric and the motion of two bodies in (1+1)-dimensional gravity |journal=[[Physical Review|Phys. Rev. D]] |volume=55 |issue=8 |pages=4723–4747 |doi=10.1103/PhysRevD.55.4723 |arxiv = gr-qc/9611008 |bibcode = 1997PhRvD..55.4723M }}</ref> However, the gravitational energy between the two bodies is exchanged via [[dilaton]]s rather than [[gravitons]] which require three-space in which to propagate.
| |
| | |
| The [[post-Newtonian expansion]] 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 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.
| |
| | |
| When this method is applied to the two-body problem without restriction on their masses, the result is remarkably simple. To the lowest order, the relative motion of the two particles is equivalent to the motion of an infinitesimal particle in the field of their combined masses. In other words, the Schwarzschild solution can be applied, provided that the ''M'' + ''m'' is used in place of ''M'' in the formulae for the Schwarzschild radius ''r''<sub>''s''</sub> and the precession angle per revolution δφ.
| |
| | |
| ===Modern computational approaches===
| |
| | |
| {{See also|Numerical relativity}}
| |
| | |
| 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 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.
| |
| | |
| [[File:Psr1913+16-weisberg.png|thumb|left|200px|Experimentally observed decreases of the [[orbital period]] of the [[binary pulsar]] [[PSR B1913+16]] (blue dots) match the predictions of [[general relativity]] (black curve) almost exactly.]]
| |
| | |
| ===Gravitational radiation===
| |
| | |
| {{See also|Gravitational radiation|PSR B1913+16}}
| |
| | |
| If there is no incoming gravitational radiation, according to [[general relativity]], two bodies revolving about one another will emit [[gravitational radiation]], causing the orbits to gradually lose energy. This has been observed indirectly in a [[binary star]] system known as [[PSR B1913+16]], for which [[Russell Alan Hulse]] and [[Joseph Hooton Taylor, Jr.]] were awarded the 1993 [[Nobel Prize in Physics]]. The two [[neutron star]]s of this system are extremely close and rotate about one another very quickly, completing a revolution in roughly 465 minutes. Their orbit is highly elliptical, with an [[orbital eccentricity|eccentricity]] of 0.62 (62%). According to general relativity, the short [[orbital period]] and high eccentricity should make the system an excellent emitter of gravitational radiation, thereby losing energy and decreasing the orbital period still further. The observed decrease in the orbital period over thirty years matches the predictions of general relativity within even the most precise measurements. General relativity predicts that, in another 300 million years, these two stars will spiral into one another.
| |
| | |
| [[File:Wavy.gif|thumb|300px|right|Two neutron stars rotating rapidly around one another gradually lose energy by emitting gravitational radiation. As they lose energy, they revolve about each other more quickly and more closely to one another.]]
| |
| | |
| The formulae describing the loss of [[energy]] and [[angular momentum]] due to gravitational radiation from the two bodies of the Kepler problem have been calculated.<ref name="Mathews">{{cite journal | author = Peters PC, Mathews J | year = 1963 | title = Gravitational Radiation from Point Masses in a Keplerian Orbit | journal = Physical Review | volume = 131 | pages = 435–440 | doi = 10.1103/PhysRev.131.435|bibcode = 1963PhRv..131..435P }}</ref> The rate of losing energy (averaged over a complete orbit) is given by<ref>Landau and Lifshitz, p. 356–357.</ref>
| |
| | |
| :<math>
| |
| -\Bigl\langle \frac{dE}{dt} \Bigr\rangle =
| |
| \frac{32G^{4}m_{1}^{2}m_{2}^{2}\left(m_{1} + m_{2}\right)}{5c^{5} a^{5} \left( 1 - e^{2} \right)^{7/2}}
| |
| \left( 1 + \frac{73}{24} e^{2} + \frac{37}{96} e^{4} \right)
| |
| </math>
| |
| | |
| where ''e'' is the [[orbital eccentricity]] and ''a'' is the [[semi-major axis|semimajor axis]] of the [[ellipse|elliptical]] 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
| |
| | |
| :<math>
| |
| -\Bigl\langle \frac{dL_{z}}{dt} \Bigr\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)
| |
| </math>
| |
| | |
| The rate of period decrease is given by<ref name="wt2005">{{cite conference |url=http://aspbooks.org/custom/publications/paper/328-0025.html |title=The Relativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis |last1=Weisberg |first1=J.M. |last2=Taylor |first2=J.H. |authorlink2=Joseph Hooton Taylor, Jr. |date=July 2005 |publisher=[[Astronomical Society of the Pacific]] |location=[[San Francisco]] |booktitle=Binary Radio Pulsars |pages=25 |location=[[Aspen, Colorado]], [[USA]] |editor=F.A. Rasio and I.H. Stairs (eds.) |booktitle=ASP Conference Series |volume=328 |arxiv=astro-ph/0407149 |bibcode= 2005ASPC..328...25W }}</ref><ref name="Mathews"/>
| |
| | |
| :<math>
| |
| -\Bigl\langle \frac{dP_{b}}{dt} \Bigr\rangle =
| |
| \frac{192G^{5/3}m_{1}m_{2}\left(m_{1} + m_{2}\right)^{-1/3}}{5c^{5} \left( 1 - e^{2} \right)^{7/2}}
| |
| \left( 1 + \frac{73}{24} e^{2} + \frac{37}{96} e^{4} \right) (\frac{P_{b}}{2 \pi})^{-5/3}
| |
| </math>
| |
| | |
| where P<sub>b</sub> is orbital period.
| |
| | |
| 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 ''a'' of the orbit.
| |
| | |
| ==See also==
| |
| | |
| * [[Newton's theorem of revolving orbits]]
| |
| * [[Binet equation]]
| |
| * [[Kepler problem]]
| |
| * [[Schwarzschild geodesics]]
| |
| * [[Center of mass (relativistic)]]
| |
| | |
| ==Notes==
| |
| | |
| <references group=note />
| |
| | |
| ==References==
| |
| | |
| <references />
| |
| | |
| ==Bibliography==
| |
| | |
| * {{cite book | last = Adler | first = R | coauthors = Bazin M, and Schiffer M | year = 1965 | title = Introduction to General Relativity | publisher = McGraw-Hill Book Company | location = New York | isbn = 978-0-07-000420-7 | pages = 177–193}}<!--{{LCCN|64|0|16476}}--><!-- Note: 2nd edition, 1975: 978-0-07-000423-8 -->
| |
| | |
| * {{cite book | last = Einstein | first = A | authorlink = Albert Einstein | year = 1956 | title = The Meaning of Relativity | edition = 5th | publisher = Princeton University Press | location = Princeton, NJ | pages = 92–97 | isbn = 978-0-691-02352-6 }}<!-- This is paperback; hardcover ISBN 978-0-691-08007-9. There is a November 1, 2004 reissue in paperback, ISBN 978-0-691-12027-0 -->
| |
| | |
| * {{cite journal | last = Hagihara | first = Y | authorlink = Yusuke Hagihara | year = 1931 | title = Theory of the relativistic trajectories in a gravitational field of Schwarzschild | journal = Japanese Journal of Astronomy and Geophysics | volume = 8 | pages = 67–176 | issn = 0368-346X }}
| |
| | |
| * {{cite book | last = Lanczos | first = C | authorlink = Cornelius Lanczos | title = The Variational Principles of Mechanics | edition = 4th | publisher = Dover Publications | location = New York | isbn = 978-0-486-65067-8 | pages = 330–338 | year = 1986}}
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| | |
| * {{cite book | last = Landau | first = LD | authorlink = Lev Landau | coauthors = Lifshitz, EM | year = 1975 | title = The Classical Theory of Fields |volume=Vol. 2 |series=[[Course of Theoretical Physics]] | edition = revised 4th English | publisher = Pergamon Press | location = New York | isbn = 978-0-08-018176-9 |pages = 299–309}}
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| | |
| * {{cite book | last = Misner | first = CW | authorlink = Charles W. Misner | coauthors = [[Kip Thorne|Thorne, K]], and [[John Archibald Wheeler|Wheeler, JA]] | title = Gravitation | location = San Francisco | publisher = W. H. Freeman | year = 1973 | isbn = 978-0-7167-0344-0 | pages = Chapter 25 (pp. 636–687), §33.5 (pp. 897–901), and §40.5 (pp. 1110–1116) }} (See [[Gravitation (book)]].)
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| | |
| * {{cite book | last = Pais | first = A. | authorlink = Abraham Pais | year = 1982 | title = Subtle is the Lord: The Science and the Life of Albert Einstein | publisher = Oxford University Press | isbn = 0-19-520438-7 | pages = 253–256}}
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| | |
| * {{cite book | last = Pauli | first = W | authorlink = Wolfgang Pauli | year = 1958 | title = Theory of Relativity | others = Translated by G. Field | publisher = Dover Publications | location = New York | isbn = 978-0-486-64152-2 | pages = 40–41, 166–169}}
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| | |
| * {{cite book | last = Rindler | first = W | authorlink = Wolfgang Rindler | year = 1977 | title = Essential Relativity: Special, General, and Cosmological | edition = revised 2nd | publisher = Springer Verlag | location = New York | isbn = 978-0-387-10090-6 | pages = 143–149 }}
| |
| | |
| *{{Cite book
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| |author=Roseveare, N. T
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| |year=1982
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| |title=Mercury's perihelion, from Leverrier to Einstein
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| |location=Oxford
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| |publisher=University Press
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| |isbn=0-19-858174-2}}
| |
| | |
| * {{cite book | last = Synge | first = JL | authorlink = John Lighton Synge | year = 1960 | title = Relativity: The General Theory | publisher = North-Holland Publishing | location = Amsterdam | isbn = 978-0-7204-0066-3 | pages = 289–298}}
| |
| | |
| * {{cite book | last = Wald | first = RM | authorlink = Robert Wald | year = 1984 | title = General Relativity | publisher = The University of Chicago Press | location = Chicago | isbn = 978-0-226-87032-8| pages = 136–146}}
| |
| | |
| *{{cite book
| |
| |author=Walter, S.
| |
| |year=2007
| |
| |editor=Renn, J.
| |
| |chapter= Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910
| |
| |title=The Genesis of General Relativity
| |
| |pages=193–252
| |
| |volume=3
| |
| |location=Berlin
| |
| |publisher=Springer
| |
| |chapterurl=http://www.univ-nancy2.fr/DepPhilo/walter/}}
| |
| | |
| * {{cite book | last = Weinberg | first = S | authorlink = Steven Weinberg | year = 1972 | title = Gravitation and Cosmology | publisher = John Wiley and Sons | location = New York | isbn = 978-0-471-92567-5 | pages = 185–201}}
| |
| | |
| * {{cite book | last = Whittaker | first = ET | authorlink = E. T. Whittaker | year = 1937 | title = A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies | edition = 4th | publisher = Dover Publications | location = New York | pages = 389–393 | isbn = 978-1-114-28944-4 | url = http://www.archive.org/details/treatisanalytdyn00whitrich}}<!-- The scanned version is an earlier edition. -->
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| | |
| ==External links==
| |
| * [http://www.youtube.com/watch?v=uVlcIb-rClI Animation] showing relativistic precession of stars around the Milky Way supermassive black hole
| |
| * [http://www.mathpages.com/rr/s6-02/6-02.htm Excerpt] from ''Reflections on Relativity'' by Kevin Brown.
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| {{Relativity}}
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| [[Category:Exact solutions in general relativity]]
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| {{Link GA|zh}}
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