- Kepler problem in general relativity
The

**Kepler problem in general relativity**involves solving for the motion of two spherical bodies interacting with one another bygravitation , as described by the theory ofgeneral relativity .Typically, and in this article, one body is assumed to have a mass "m" that is negligible compared to the mass "M" of the other body; this is a good approximation for the case of a planet revolving around the Sun, or a photon passing by a star. In such cases, one may assume that only the heavier body contributes to the

curvature ofspace-time and that it is fixed in space. This curved space-time is described by the Schwarzschild solution to thevacuum Einstein equations of general relativity. The motion of the lighter body (called the "particle" below) is described by the space-timegeodesic s of the Schwarzschild solution. It is here assumed that the lighter body is point-like so that tidal forces can be ignored.These geodesic solutions account for the anomalous precession of the planet Mercury, which is a key piece of evidence supporting the theory of general relativity. They also describe the deflection of light in a gravitational field, another prediction famously used as evidence for general relativity.

Orbit decay due to emission of gravitational radiation is not described by the Schwarzschild solution.

**Historical context and intuitive understanding**[

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.]In 1859,

Urbain Le Verrier discovered that the orbitalprecession of theplanet 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. [*cite journal | last = Le Verrier | first = UJJ | authorlink = Urbain Le Verrier | year = 1859 | title = Unknown title | journal = Comptes Revues d'Academie de la Science de Paris | volume = 49 | pages = 379–?*] The effect is small (roughly 43arcsecond s of rotation per century), but well above the measurement error (roughly 0.1arcsecond 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 theSun , an undetected moon of Mercury, or a new planet named Vulcan.Pais 1982] After these explanations were discounted, some physicists were driven to the more radical hypothesis that Newton'sinverse-square law of gravitation was incorrect. For example, some physicists proposed apower law with anexponent that was slightly different from 2.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 (some of the momentum must then 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 instanteneous position of the Sun. On the assumption of the classical fundamentals, Laplace had shown that if gravity would propagate at a velocity in 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 thespeed of gravity that is many orders of magnitude faster than the speed of light.Pais 1982] 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 [*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 . Presumably the same holds in linearized gravity.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 thespeed of gravity is equal to thespeed of light in his theory. And in another attemptPaul 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 hypothesis were rejected. [*Roseveare 1982*] Another attempt byHendrik Lorentz (1900), who already used Maxwell's theory, produced a Perihelion shift which was too low.Around 1904–1905, the works of

Hendrik Lorentz ,Henri Poincaré and finallyAlbert Einstein 'sspecial theory of relativity , exclude the possibility of propagation of any effects faster than thespeed 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 byHenri Poincaré (1905),Hermann Minkowski (1907) andArnold Sommerfeld (1910). [*Walter 2007*] In 1907 Einstein came to the conclusion that to achieve this a successor to special relativity was needed. From 1907 to 1915, Einstein gradually groped towards a new theory, using hisEquivalence 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 afictitious force such as thecentrifugal force or theCoriolis force , which result from being in an accelerated reference frame; all fictitious forces are proportional to theinertial mass , just as gravity is. To effect the reconciliation of gravity andspecial relativity and to incorporate theequivalence principle , something had to be sacrificed; that something was the long-held classical assumption that our space obeys the laws ofEuclidean geometry , e.g., that thePythagorean theorem is true experimentally. Einstein used a more general geometry,Riemannian geometry , to allow for the curvature ofspace andtime that was necessary for the reconciliation; after eight years of work (1907–1915), he succeeded in discovering the precise way in whichspace-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 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.**Geometrical background**In the normal

Euclidean geometry , triangles obey thePythagorean theorem , which states that the square distance "ds"^{2}between two points in space is the sum of the squares of its perpendicular components:$ds^\{2\}\; =\; dx^\{2\}\; +\; dy^\{2\}\; +\; dz^\{2\}\; ,!$

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:$ds^\{2\}\; =\; F(x,\; y,\; z)\; dx^\{2\}\; +\; G(x,\; y,\; z)\; dy^\{2\}\; +\; H(x,\; y,\; z)dz^\{2\}\; ,!$

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

:$ds^\{2\}\; =\; dr^\{2\}\; +\; r^\{2\}\; d\; heta^\{2\}\; +\; r^\{2\}\; sin^\{2\}\; heta\; dvarphi^\{2\}\; ,!$

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"

:$ds^\{2\}\; =\; g\_\{xx\}\; dx^\{2\}\; +\; g\_\{xy\}\; dx\; dy\; +\; g\_\{xz\}\; dx\; dz\; +\; cdots\; +\; g\_\{zy\}\; dz\; dy\; +\; g\_\{zz\}\; dz^\{2\}\; ,!$

where the nine functions "g"

_{xx}, "g"_{xy}constitute themetric tensor , which defines the geometry of the space inRiemannian geometry . In the spherical-coordinates example above, there are no cross-terms; the only nonzero metric tensor components are "g"_{rr}= 1, "g"_{θθ}= "r"^{2}and "g"_{φφ}= "r"^{2}sin^{2}θ.In his 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 inspace-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.:$c^\{2\}\; d\; au^\{2\}\; =\; c^\{2\}\; dt^\{2\}\; -\; dx^\{2\}\; -\; dy^\{2\}\; -\; dz^\{2\}\; ,!$

which may be written in spherical coordinates as

:$c^\{2\}\; d\; au^\{2\}\; =\; c^\{2\}\; dt^\{2\}\; -\; dr^\{2\}\; -\; r^\{2\}\; d\; heta^\{2\}\; -\; r^\{2\}\; sin^\{2\}\; heta\; dvarphi^\{2\}\; ,!$

This formula is the natural extension of the

Pythagorean theorem and similarly holds only when there is no curvature in space-time. Ingeneral relativity , however, space and time may have curvature, so this distance formula must be modified to a more general form:$c^\{2\}\; d\; au^\{2\}\; =\; g\_\{mu\; u\}\; dx^\{mu\}\; dx^\{\; u\}\; ,!$

just as we generalized the formula to measure distance on the surface of the Earth. The exact form of the metric "g"

_{μν}depends on the gravitating mass, momentum and energy, as described by theEinstein 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.**chwarzschild metric**One solution to the

Einstein field equations is theSchwarzschild metric , which corresponds to the gravitational field of an uncharged, non-rotating, spherically symmetric body of mass "M". The Schwarzschild solution can be written as:$c^2\; \{d\; au\}^\{2\}\; =\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; c^\{2\}\; dt^\{2\}\; -\; frac\{dr^\{2\{1\; -\; frac\{r\_\{s\{r\; -\; r^\{2\}\; d\; heta^\{2\}\; -\; r^\{2\}\; sin^\{2\}\; heta\; ,\; dvarphi^\{2\}$

where :"τ" is the proper time (time measured by a clock moving with the particle) in seconds, :"c" is the

speed of light in meters per second, :"t" is the time coordinate (measured by a stationary clock at infinity) in seconds, :"r" is the radial coordinate (circumference of a circle centered on the star divided by 2π) in meters, :"θ" is thecolatitude (angle from North) in radians, :"φ" is thelongitude in radians, and :"r"_{s}is theSchwarzschild radius (in meters) of the massive body, which is related to its mass "M" by::$r\_\{s\}\; =\; frac\{2GM\}\{c^\{2$:where "G" is the

gravitational constant .Landau 1975.]The classical Newtonian theory of gravity is recovered in the limit as the ratio "r"

_{s}/"r" goes to zero. In that limit, the metric returns to the form given above forspecial relativity . In practice, this ratio is almost always extremely small. For example, the Schwarzschild radius "r"_{s}of theEarth is roughly 9 mm (frac|3|8inch ), whereas asatellite in ageosynchronous orbit has a radius "r" that is roughly four billion times larger, at 42,164 km (26,200mile s). Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The ratio only becomes large close toblack hole s and other ultra-dense objects such asneutron star s.**Geodesic equation**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 [*Weinberg 1972.*]:$frac\{d^2x^\{mu\{d\; q^2\}\; +\; Gamma^\{mu\}\_\{\; ulambda\}\; frac\{dx^\{\; u\{d\; q\}\; frac\{dx^\{lambda\{dq\}\; =\; 0$

where Γ represents the

Christoffel symbol and the variable "q" parametrizes the particle's path throughspace-time , its so-calledworld line . The Christoffel symbol depends only on themetric tensor "g"_{μν}, or rather on how it changes with position. The variable "q" is a constant multiple of theproper time "τ" for timelike orbits (which are traveled by massive particles), and is usually taken to be equal to it. For lightlike orbits (which are traveled by massless particles such as thephoton ), the proper time is zero and, strictly speaking, cannot be used as the variable "q". Nevertheless, lightlike orbits can be derived as theultrarelativistic limit of timelike orbits, that is, the limit as the particle mass "m" goes to zero while holding its totalenergy fixed.We may simplify the problem by using symmetry to eliminate one variable from consideration. Since the Schwarzschild metric is symmetrical about "θ" = π/2, any geodesic that begins moving in that plane will remain in that plane indefinitely (the plane is

totally geodesic ). Therefore, we orient the coordinate system so that the orbit of the particle lies in that plane, and fix the "θ" coordinate to be π/2 so that the metric (of this plane) simplifies to:$c^2\; d\; au^\{2\}\; =\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; c^\{2\}\; dt^\{2\}\; -\; frac\{dr^\{2\{1\; -\; frac\{r\_\{s\{r\; -\; r^\{2\}\; dvarphi^\{2\}.$

Two constants of motion can then be identified (cf. the derivation given below)

:$r^\{2\}\; frac\{dvarphi\}\{d\; au\}\; =\; frac\{L\}\{m\},$

:$left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; frac\{dt\}\{d\; au\}\; =\; frac\{E\}\{mc^\{2.$

Substituting these constants into the definition of the Schwarzschild metric

:$c^\{2\}\; =\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; c^\{2\}\; left(\; frac\{dt\}\{d\; au\}\; ight)^\{2\}\; -\; frac\{1\}\{1\; -\; frac\{r\_\{s\{r\; left(\; frac\{dr\}\{d\; au\}\; ight)^\{2\}\; -\; r^\{2\}\; left(\; frac\{dvarphi\}\{d\; au\}\; ight)^\{2\},$

yields the equation of motion for the particle

:$left(\; frac\{dr\}\{d\; au\}\; ight)^\{2\}\; =\; frac\{E^\{2\{m^\{2\}c^\{2\; -\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; left(\; c^\{2\}\; +\; frac\{L^\{2\{m^\{2\}\; r^\{2\; ight).$

The dependence on proper time can be eliminated using the definition of "L"

:$left(\; frac\{dr\}\{dvarphi\}\; ight)^\{2\}\; =\; left(\; frac\{dr\}\{d\; au\}\; ight)^\{2\}\; left(\; frac\{d\; au\}\{dvarphi\}\; ight)^\{2\}\; =left(\; frac\{dr\}\{d\; au\}\; ight)^\{2\}\; left(\; frac\{m\; r^\{2\{L\}\; ight)^\{2\},$

which yields the equation for the orbit

:$left(\; frac\{dr\}\{dvarphi\}\; ight)^\{2\}\; =\; frac\{r^\{4\{b^\{2\; -\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; left(\; frac\{r^\{4\{a^\{2\; +\; r^\{2\}\; ight)$

where, for brevity, two length-scales, "a" and "b", have been defined by

:$a\; =\; frac\{L\}\{mc\},$

:$b\; =\; frac\{cL\}\{E\}.$

The same equation can also be derived using a Lagrangian approach [

*Whittaker 1937.*] or theHamilton–Jacobi equation Landau and Lifshitz (1975), pp. 306–309.] (see below). The solution of the orbit equation is:$varphi\; =\; int\; frac\{dr\}\{r^\{2\}\; sqrt\{frac\{1\}\{b^\{2\; -\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; left(\; frac\{1\}\{a^\{2\; +\; frac\{1\}\{r^\{2\; ight).$

**Approximate formula for the bending of light**In the limit as the particle mass "m" goes to zero (or, equivalently, as the length-scale "a" goes to infinity), the equation for the orbit becomes

:$varphi\; =\; int\; frac\{dr\}\{r^\{2\}\; sqrt\{frac\{1\}\{b^\{2\; -\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; frac\{1\}\{r^\{2$

Expanding in powers of "r"

_{s}/"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::$delta\; varphi\; approx\; frac\{2r\_\{s\{b\}\; =\; frac\{4GM\}\{c^\{2\}b\}.$

Here, "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"_{s}/"r". For light grazing the surface of the sun, the approximate angular deflection is roughly 1.75 arcseconds.**Relation to classical mechanics and precession of elliptical orbits**The equation of motion for the particle derived above

:$left(\; frac\{dr\}\{d\; au\}\; ight)^\{2\}\; =\; frac\{E^\{2\{m^\{2\}c^\{2\; -\; c^\{2\}\; +\; frac\{r\_\{s\}c^\{2\{r\}\; -\; frac\{L^\{2\{m^\{2\}\; r^\{2\; +\; frac\{r\_\{s\}\; L^\{2\{m^\{2\}\; r^\{3$

can be re-written using the definition of the Schwarzschild radius "r"

_{s}as:$frac\{1\}\{2\}\; m\; left(\; frac\{dr\}\{d\; au\}\; ight)^\{2\}\; =\; left\; [\; frac\{E^\{2\{2mc^\{2\; -\; frac\{1\}\{2\}\; mc^\{2\}\; ight]\; +\; frac\{GMm\}\{r\}\; -\; frac\{L^\{2\{2m\; r^\{2\; +\; frac\{GM\; L^\{2\{c^\{2\}\; m\; r^\{3$

which is equivalent to a particle moving in a one-dimensional effective potential

:$V(r)\; =\; -frac\{GMm\}\{r\}\; +\; frac\{L^\{2\{2m\; r^\{2\; -\; frac\{GM\; L^\{2\{c^\{2\}\; m\; r^\{3$

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" potential energy; however, the third term is an attractive energy unique to

general relativity . As shown below and elsewhere, this inverse-cubic energy causes elliptical orbits to precess gradually by an angle δφ per revolution:$delta\; varphi\; approx\; frac\{6pi\; G\; M\}\{c^\{2\}\; A\; left(\; 1-\; e^\{2\}\; ight)\}$

where "A" is the semi-major axis and "e" is the eccentricity.

The third term is attractive and dominates at small "r" values, giving a critical inner radius "r"

_{inner}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.**Circular orbits and their stability**The effective potential "V" can be re-written in terms of the lengths "a" and "b"

:$V(r)\; =\; frac\{mc^\{2\{2\}\; left\; [\; -\; frac\{r\_\{s\{r\}\; +\; frac\{a^\{2\{r^\{2\; -\; frac\{r\_\{s\}\; a^\{2\{r^\{3\; ight]$

Circular orbits are possible when the effective force is zero

:$F\; =\; -frac\{dV\}\{dr\}\; =\; -frac\{mc^\{2\{2r^\{4\; left\; [\; r\_\{s\}\; r^\{2\}\; -\; 2a^\{2\}\; r\; +\; 3r\_\{s\}\; a^\{2\}\; ight]\; =\; 0$

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"

_{inner}and "r"_{outer}:$r\_\{mathrm\{outer\; =\; frac\{a^\{2\{r\_\{s\; left(\; 1\; +\; sqrt\{1\; -\; frac\{3r\_\{s\}^\{2\{a^\{2\}\; ight)$

:$r\_\{mathrm\{inner\; =\; frac\{a^\{2\{r\_\{s\; left(\; 1\; -\; sqrt\{1\; -\; frac\{3r\_\{s\}^\{2\{a^\{2\}\; ight)\; =\; frac\{3a^\{2\{r\_\{mathrm\{outer\}$

which are obtained using the quadratic formula. The inner radius "r"

_{inner}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"_{inner}(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-relativisticKepler problem .When "a" is much greater than "r"

_{s}(the classical case), these formulae become approximately:$r\_\{mathrm\{outer\; approx\; frac\{2a^\{2\{r\_\{s$

:$r\_\{mathrm\{inner\; approx\; frac\{3\}\{2\}\; r\_\{s\}$

Substituting the definitions of "a" and "r"

_{s}into "r"_{outer}yields the classical formula for a particle orbiting a mass "M" in a circle:$r\_\{mathrm\{outer^\{3\}\; approx\; frac\{GM\}\{omega\_\{varphi\}^\{2$

where "ω"

_{φ}is the orbital angular speed of the particle. This formula is obtained in non-relativistic mechanics by setting thecentrifugal force equal to the Newtonian gravitational force:$momega\_\{varphi\}^\{2\}\; r\; =\; frac\{GMm\}\{r^\{2$

In our notation, the classical orbital angular speed equals

:$omega\_\{varphi\}^\{2\}\; approx\; frac\{GM\}\{r\_\{mathrm\{outer^\{3\; =\; left(\; frac\{r\_\{s\}\; c^\{2\{2r\_\{mathrm\{outer^\{3\; ight)\; =\; left(\; frac\{r\_\{s\}\; c^\{2\{2\}\; ight)\; left(\; frac\{r\_\{s\}^\{3\{8a^\{6\; ight)\; =\; frac\{c^\{2\}\; r\_\{s\}^\{4\{16\; a^\{6$

At the other extreme, when "a"

^{2}approaches 3"r"_{s}^{2}from above, the two radii converge to a single value:$r\_\{mathrm\{outer\; approx\; r\_\{mathrm\{inner\; approx\; 3\; r\_\{s\}$

The quadratic solutions above ensure that "r"

_{outer}is always greater than 3"r"_{s}, whereas "r"_{inner}lies between frac|3|2 "r"_{s}and 3"r"_{s}. Circular orbits smaller than frac|3|2 "r"_{s}are not possible. For massless particles, "a" goes to infinity, implying that there is a circular orbit for photons at "r"_{inner}= frac|3|2 "r"_{s}. The sphere of this radius is sometimes known as thephoton sphere .**Precession of elliptical orbits**The orbital precession rate may be derived using this radial effective potential "V". A small radial deviation from a circular orbit of radius "r"

_{outer}will oscillate stably with an angular frequency:$omega\_\{r\}^\{2\}\; =\; frac\{1\}\{m\}\; left\; [\; frac\{d^\{2\}V\}\{dr^\{2\; ight]\; \_\{r=r\_\{mathrm\{outer\}$

which equals

:$omega\_\{r\}^\{2\}\; =\; left(\; frac\{c^\{2\}\; r\_\{s\{2\; r\_\{mathrm\{outer^\{4\; ight)\; left(\; r\_\{mathrm\{outer\; -\; r\_\{mathrm\{inner\; ight)\; =\; omega\_\{varphi\}^\{2\}\; sqrt\{1\; -\; frac\{3r\_\{s\}^\{2\{a^\{2\}$

Taking the square root of both sides and expanding using the

binomial theorem yields the formula:$omega\_\{r\}\; =\; omega\_\{varphi\}\; left(\; 1\; -\; frac\{3r\_\{s\}^\{2\{4a^\{2\; +\; cdots\; ight)$

Multiplying by the period "T" of one revolution gives the precession of the orbit per revolution

:$delta\; varphi\; =\; T\; left(\; omega\_\{varphi\}\; -\; omega\_\{r\}\; ight)\; approx\; 2pi\; left(\; frac\{3r\_\{s\}^\{2\{4a^\{2\; ight)\; =\; frac\{3pi\; m^\{2\}\; c^\{2\{2L^\{2\; r\_\{s\}^\{2\}$

where we have used "ω

_{φ}T" = 2"п" and the definition of the length-scale "a". Substituting the definition of the Schwarzschild radius "r"_{s}gives:$delta\; varphi\; approx\; frac\{3pi\; m^\{2\}\; c^\{2\{2L^\{2\; left(\; frac\{4G^\{2\}\; M^\{2\{c^\{4\; ight)\; =\; frac\{6pi\; G^\{2\}\; M^\{2\}\; m^\{2\{c^\{2\}\; L^\{2$

This may be simplified using the elliptical orbit's semiaxis "A" and eccentricity "e" related by the formula

:$frac\{L^\{2\{GMm^\{2\; =\; A\; left(\; 1\; -\; e^\{2\}\; ight)$

to give the most common form of the precession angle

:$delta\; varphi\; approx\; frac\{6pi\; G\; M\}\{c^\{2\}\; A\; left(\; 1\; -\; e^\{2\}\; ight)\}$

**Orbital solution using elliptic functions**The equation for the orbit

:$left(\; frac\{dr\}\{dvarphi\}\; ight)^\{2\}\; =\; frac\{r^\{4\{b^\{2\; -\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; left(\; frac\{r^\{4\{a^\{2\; +\; r^\{2\}\; ight)$

can be simplified by introducing a dimensionless variable

:$zeta\; =\; frac\{r\_\{s\{4r\}\; -\; frac\{1\}\{12\}$

so that it reduces to

:$left(\; frac\{dzeta\}\{dvarphi\}\; ight)^\{2\}\; =\; 4\; zeta^\{3\}\; -\; g\_\{2\}\; zeta\; -\; g\_\{3\},$

where the constant, dimensionless coefficients "g"

_{2}and "g"_{3}are defined by:$egin\{align\}g\_\{2\}\; =\; frac\{1\}\{12\}\; -\; frac\{r\_\{s\}^\{2\{4\; a^\{2,\backslash \backslash g\_\{3\}\; =\; frac\{1\}\{216\}\; +\; frac\{r\_\{s\}^\{2\{24\; a^\{2\; -\; frac\{r\_\{s\}^\{2\{16\; b^\{2.end\{align\}$

The solution of this orbital equation is given by

:$varphi\; -\; varphi\_\{0\}\; =\; int\; frac\{dzeta\}\{sqrt\{4zeta^\{3\}\; -\; g\_\{2\}\; zeta\; -\; g\_\{3\}.$

It follows that, up to a phase-shift, $zeta\; =\; wp(varphi\; -\; varphi\_\{0\})$, where $wp$ is Weierstrass's elliptic function with parameters "g"

_{2}and "g"_{3}, and φ_{0}is a constant of integration (possibly complex).**Qualitative character of possible orbits**The orbits are described by the equation of motion

:$left(\; frac\{dzeta\}\{dvarphi\}\; ight)^\{2\}\; =\; 4\; zeta^\{3\}\; -\; g\_\{2\}\; zeta\; -\; g\_\{3\},$

If the discriminant $Delta\; =\; g\_\{2\}^\{3\}\; -\; 27\; g\_\{3\}^\{2\}$ is greater than zero, the

cubic equation :$G(zeta)\; =\; 4\; zeta^\{3\}\; -\; g\_\{2\}\; zeta\; -\; g\_\{3\}\; =\; 0,$has three distinct real roots, "e"_{1}, "e"_{2}, and "e"_{3}, which may be listed in decreasing order:$e\_\{1\}\; >\; e\_\{2\}\; >\; e\_\{3\}.$

In such cases, the solution $zeta\; =\; wp(varphi\; -\; varphi\_\{0\})$ is an elliptic function with two half-periods, one completely real

:$omega\_\{1\}\; =\; int\_\{e\_\{1^\{infty\}\; frac\{dz\}\{sqrt\{4z^\{3\}\; -\; g\_\{2\}z\; -\; g\_\{3\}$

and the other completely imaginary

:$omega\_\{3\}\; =\; i\; int\_\{-e\_\{3^\{infty\}\; frac\{dz\}\{sqrt\{4z^\{3\}\; -\; g\_\{2\}z\; -\; g\_\{3\}$ The remaining root defines a complex half-period ω

_{2}= -ω_{1}- ω_{3}. These three half-periods are related to the corresponding roots by the equation $wp(omega\_\{i\})\; =\; e\_\{i\}$, where "i" can equal 1, 2, or 3. Therefore, if φ_{0}is set equal to any of these half-periods, the derivative of ζ is zero, indicating a periapse or an apoapse, the points of closest or furthest approach, respectively.:$frac\{dzeta\}\{dphi\}\; =\; 0\; mathrm\{when\}\; zeta\; =\; wp(-omega\_\{i\})\; =\; e\_\{i\}$

since

:$left(\; frac\{dzeta\}\{dvarphi\}\; ight)^\{2\}\; =\; G(zeta)\; =\; 4\; zeta^\{3\}\; -\; g\_\{2\}\; zeta\; -\; g\_\{3\}\; =\; 4\; left(\; zeta\; -\; e\_\{1\}\; ight)\; left(\; zeta\; -\; e\_\{2\}\; ight)\; left(\; zeta\; -\; e\_\{3\}\; ight),$

The roots "e"

_{i}are thecritical value s of "ζ" with respect to "φ", i.e., the values at which the derivative is zero.The qualitative character of the orbits depends on the choice of φ

_{0}. Solutions with φ_{0}equal to ω_{2}correspond to oscillatory orbits that vary between ζ="e"_{2}and ζ="e"_{3}, or else diverge to infinity (ζ=-1/12). By contrast, solutions with φ_{0}equal to ω_{1}(or any other real number) correspond to orbits that decay to zero radius, since (to be areal number ) ζ cannot be less than "e"_{1}and hence increases inexorably to infinity.**Quasi-elliptical orbits**Solutions $zeta\; =\; wp(phi\; -\; phi\_\{0\})$ in which φ

_{0}equals ω_{2}give a real value of ζ provided that the energy "E" satisfies the inequality "E"^{2}< "m"^{2}"c"^{4}. For such solutions, the variable ζ is confined between "e"_{3}≤ "ζ" ≤ "e"_{2}. If both roots are greater than -1/12, then ζ never becomes -1/12, at which point the radius "r" goes to infinity. Hence, such solutions correspond to a gradually precessing elliptical orbit. As the particle (or planet) revolves around the origin, its radius oscillates between a minimum radius:$r\_\{min\}\; =\; frac\{3r\_\{s\{1\; +\; 12e\_\{2$

and a maximum radius

:$r\_\{max\}\; =\; frac\{3r\_\{s\{1\; +\; 12e\_\{1$

corresponding to the value of "ζ" at the two extrema of radius. The real periodicity of Weierstrass' elliptic function is 2ω

_{1}; thus the particle returns to the same radius after revolving an angle 2ω_{1}, which may not equal 2π. Hence, the orbit is in general precessing. In general, the amount of precession per orbit (2ω_{1}- 2π) is quite small.**table circular orbits**A special case occurs when 2"e"

_{2}= 2"e"_{3}= −"e"_{3}, i.e., two of the roots of "G"("ζ") are equal and negative, while the third is positive. In this case there is a solution with "ζ" constant, equal to the repeated root, "e" = "e"_{2}= "e"_{3}. This corresponds to circular orbits, specifically, the classical "r"_{outer}solution derived above; as shown there, the radii of these orbits must be greater than 3"r"_{s}. Such circular orbits are stable, because a small perturbation of the parameters will separate the repeated roots, resulting in a quasi-elliptical orbit. For example, giving a small radial "kick" to a particle in a classical circular orbit pushes it into an elliptical orbit that gradually precesses, as derived above.**Unbounded orbits**An unbounded orbit occurs when "r" goes to infinity, corresponding to "ζ" = −

^{1}⁄_{12}. Unbounded orbits correspond to an oscillatory orbit in which -1/12 falls between the two limiting roots, i.e., when "e"_{3}≤ −^{1}⁄_{12}≤ "ζ" ≤ "e"_{2}.**Asymptotically circular orbits**Another special case occurs when −"e"

_{3}= 2"e"_{2}= 2"e"_{1}, i.e., two of the roots of "G"("ζ") are equal and positive, whereas the third root "e"_{3}is negative. Denoting the repeated root by "e" = "n"^{2}/3, the orbits are asymptotically circular at positive and negative infinite "φ"::$zeta\; =\; frac\{r\_\{s\{4r\}\; -\; frac\{1\}\{12\}\; =\; e\; -\; frac\{n^\{2\{cosh^\{2\}\; nvarphi\}.$

as may be verified by substitution. As "φ" goes to positive or negative infinity, the orbit approaches asymptotically to the circle

:$frac\{r\_\{s\{4r\}\; -\; frac\{1\}\{12\}\; =\; e.$

In such cases, the radius of the orbit must remain between 2"r"

_{s}and 3"r"_{s}.The asymptotic formula may also be derived from the expression for

Weierstrass' elliptic function in terms ofJacobi's elliptic functions :$zeta\; =\; wp(phi\; -\; phi\_\{0\})\; =\; e\_\{1\}\; +\; left(e\_\{1\}\; -\; e\_\{3\}\; ight)\; frac\{mathrm\{cn\}^\{2\}\; w\}\{mathrm\{sn\}^\{2\}\; w\}$

where $w\; =\; (phi\; -\; phi\_\{0\})sqrt\{e\_\{1\}\; -\; e\_\{3$ and the modulus equals

:$k\; =\; sqrt\{frac\{e\_\{2\}-e\_\{3\{e\_\{1\}-e\_\{3\}$

In the limit as "e"

_{2}approaches "e"_{1}, the modulus goes to one and "w" goes to "n"(φ-φ_{0}). Finally choosing φ_{0}to be the imaginary number $iK^\{prime\}$ (a quarter-period) gives the asymptotic formula above.**Decaying orbits**Real solutions for $zeta\; =\; wp(phi\; -\; phi\_\{0\})$ in which φ

_{0}equals ω_{1}(or some otherreal number ) have the property that "ζ" is never less than "e"_{1}. Since the equation of motion:$left(\; frac\{dzeta\}\{dvarphi\}\; ight)^\{2\}\; =\; 4\; zeta^\{3\}\; -\; g\_\{2\}\; zeta\; -\; g\_\{3\}\; =\; 4\; left(\; zeta\; -\; e\_\{1\}\; ight)\; left(\; zeta\; -\; e\_\{2\}\; ight)\; left(\; zeta\; -\; e\_\{3\}\; ight)$

is positive for all values of "ζ">"e"

_{1}, "ζ" increases without bound, corresponding to the particle falling inexorably to the origin "r" = 0.**Corrections to the geodesic solutions**According to

general relativity , two bodies rotating about one another will emitgravitational radiation , causing the orbits to differ slightly from the geodesics calculated above. This has been observed indirectly in abinary star system known asPSR B1913+16 , for whichRussell Alan Hulse andJoseph Hooton Taylor, Jr. were awarded the 1993Nobel Prize in Physics . The twoneutron 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 eccentricity of 0.62 (62%). According to general relativity, the shortorbital 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.The formulae describing the loss of

energy andangular momentum due to gravitational radiation from the two bodies of the Kepler problem have been calculated. [*cite journal | author = Peters PC, Mathews J | date = 1963 | title = Unknown title | journal = Physical Review | volume = 131 | pages = 435–?*] The rate of losing energy (averaged over a complete orbit) is given by [*Landau and Lifshitz, p. 356–357.*]:$-langle\; frac\{dE\}\{dt\}\; angle\; =\; frac\{32G^\{4\}m\_\{1\}^\{2\}m\_\{2\}^\{2\}left(m\_\{1\}\; +\; m\_\{2\}\; ight)\}\{5c^\{5\}\; a^\{5\}\; left(\; 1\; -\; e^\{2\}\; ight)^\{7/2\; left(\; 1\; +\; frac\{73\}\{24\}\; e^\{2\}\; +\; frac\{37\}\{96\}\; e^\{4\}\; ight)$

where "e" is the

orbital eccentricity and "a" is the semimajor axis of the 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:$-langle\; frac\{dL\_\{z\{dt\}\; angle\; =\; frac\{32G^\{7/2\}m\_\{1\}^\{2\}m\_\{2\}^\{2\}sqrt\{m\_\{1\}\; +\; m\_\{2\}\{5c^\{5\}\; a^\{7/2\}\; left(\; 1\; -\; e^\{2\}\; ight)^\{2\; left(\; 1\; +\; frac\{7\}\{8\}\; e^\{2\}\; ight)$

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.

**Mathematical derivations of the orbit equation****Hamilton–Jacobi approach**The orbital equation can be derived from the

Hamilton–Jacobi equation . The advantage of this approach is that it equates the motion of the particle with the propagation of a wave, and leads neatly into the derivation of the deflection of light by gravity ingeneral relativity , throughFermat's principle . The basic idea is that, due to gravitational slowing of time, parts of a wave-front closer to a gravitating mass move more slowly than those further away, thus bending the direction of the wave-front's propagation (add Figure).Using general covariance, the

Hamilton–Jacobi equation for a single particle in arbitrary coordinates can be expressed as:$g^\{mu\; u\}\; frac\{partial\; S\}\{partial\; x^\{mu\; frac\{partial\; S\}\{partial\; x^\{\; u\; =\; m^\{2\}\; c^\{2\}.$

Using the

Schwarzschild metric "g"^{μν}, this equation becomes:$frac\{1\}\{c^\{2\}\; left(1\; -\; frac\{r\_\{s\{r\}\; ight)\}\; left(\; frac\{partial\; S\}\{partial\; t\}\; ight)^\{2\}\; -\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; left(\; frac\{partial\; S\}\{partial\; r\}\; ight)^\{2\}\; -frac\{1\}\{r^\{2\; left(\; frac\{partial\; S\}\{partial\; varphi\}\; ight)^\{2\}\; =\; m^\{2\}\; c^\{2\}$

where we again orient the spherical coordinate system with the plane of the orbit. The time "t" and longitude "φ" are cyclic coordinates, so that the solution for Hamilton's principal function "S" can be written

:$S\; =\; -Et\; +\; Lvarphi\; +\; S\_\{r\}(r)\; ,$ where "E" and "L" again represent the particle's

energy andangular momentum , respectively. TheHamilton–Jacobi equation gives an integral solution for the radial part "S_{r}(r)":$S\_\{r\}(r)\; =\; int\; frac\{L\; dr\}\{1\; -\; frac\{r\_\{s\{r\; sqrt\{frac\{1\}\{b^\{2\; -\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; left(\; frac\{1\}\{a^\{2\; +\; frac\{1\}\{r^\{2\; ight)\}.$

Taking the derivative of Hamilton's principal function "S" in the usual way

:$frac\{partial\; S\}\{partial\; L\}\; =\; varphi\; +\; frac\{partial\; S\_\{r\{partial\; L\}\; =\; mathrm\{constant\}$

yields the orbital equation derived earlier

:$left(\; frac\{dr\}\{dvarphi\}\; ight)^\{2\}\; =\; frac\{r^\{4\{b^\{2\; -\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; left(\; frac\{r^\{4\{a^\{2\; +\; r^\{2\}\; ight).$

This approach can also be used to derive the rate of orbital precession in an elegant way. [

*Landau and Lifshitz (1975), pp. 307–308.*]In the limit of zero mass "m" (or, equivalently, infinite "a"), the radial part of Hamilton's principal function "S" becomes

:$S\_\{r\}(r)\; =\; frac\{E\}\{c\}\; int\; dr\; sqrt\{frac\{r^\{2\{left(\; r\; -\; r\_\{s\}\; ight)^\{2\; -\; frac\{b^\{2\{r\; left(\; r\; -\; r\_\{s\}\; ight)$

from which the equation for the deflection of light can be derived.

**Lagrangian approach**In

general relativity , free particles of negligible mass "m" followgeodesic s inspace-time , owing to theequivalence principle . Geodesics inspace-time are defined as curves for which small local variations in their coordinates (while holding their endpoints events fixed) make no significant change in their overall length "s". This may be expressed mathematically using thecalculus of variations :$0\; =\; delta\; s\; =\; delta\; int\; ds\; =\; delta\; int\; sqrt\{g\_\{mu\; u\}\; frac\{dx^\{mu\{d\; au\}\; frac\{dx^\{\; u\{d\; au\}\; \}\; d\; au\; =\; delta\; int\; sqrt\{2T\}\; d\; au$

where "τ" is the

proper time , "s"="cτ" is the arc-length inspace-time and "T" is defined as:$2T\; =\; c^\{2\}\; =\; left(\; frac\{ds\}\{d\; au\}\; ight)^\{2\}\; =\; g\_\{mu\; u\}\; frac\{dx^\{mu\{d\; au\}\; frac\{dx^\{\; u\{d\; au\}\; =\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; c^\{2\}\; left(\; frac\{dt\}\{d\; au\}\; ight)^\{2\}\; -\; frac\{1\}\{1\; -\; frac\{r\_\{s\{r\; left(\; frac\{dr\}\{d\; au\}\; ight)^\{2\}\; -\; r^\{2\}\; left(\; frac\{dvarphi\}\{d\; au\}\; ight)^\{2\}$

in analogy with

kinetic energy . If the derivative with respect to proper time is represented by a dot for brevity:$dot\{x\}^\{mu\}\; =\; frac\{dx^\{mu\{d\; au\}$

"T" may be written as

:$2T\; =\; c^\{2\}\; =\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; c^\{2\}\; left(\; dot\{t\}\; ight)^\{2\}\; -\; frac\{1\}\{1\; -\; frac\{r\_\{s\{r\; left(\; dot\{r\}\; ight)^\{2\}\; -\; r^\{2\}\; left(\; dot\{varphi\}\; ight)^\{2\}$

Constant factors (such as "c" or the square root of two) don't affect the answer to the variational problem; therefore, taking the variation inside the integral yields

Hamilton's principle :$0\; =\; delta\; int\; sqrt\{2T\}\; d\; au\; =\; int\; frac\{delta\; T\}\{sqrt\{2T\; d\; au\; =\; frac\{1\}\{c\}\; delta\; int\; T\; d\; au.$

The solution of the variational problem is given by Lagrange's equations

:$frac\{d\}\{d\; au\}\; left(frac\{partial\; T\}\{partial\; dot\{x\}^\{sigma\; ight)\; =\; frac\{partial\; T\}\{partial\; x^\{sigma.$

When applied to "t" and "φ", these equations reveal two constants of motion

:$frac\{d\}\{d\; au\}\; left\; [\; r^\{2\}\; frac\{dvarphi\}\{d\; au\}\; ight]\; =\; 0,$

:$frac\{d\}\{d\; au\}\; left\; [\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; frac\{dt\}\{d\; au\}\; ight]\; =\; 0,$

which may be written as the equations for "L" and "E"

:$r^\{2\}\; frac\{dvarphi\}\{d\; au\}\; =\; frac\{L\}\{m\},$

:$left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; frac\{dt\}\{d\; au\}\; =\; frac\{E\}\{mc^\{2.$

As shown above, substitution of these equations into the definition of the

Schwarzschild metric yields the equation for the orbit.**Hamilton's principle**The action integral for a particle affected only by gravity is

:$S\; =\; int\{\; -\; m\; c^2\; d\; au\}\; =\; -\; m\; c\; int\{\; c\; frac\{d\; au\}\{dq\}\; dq\}\; =\; -\; m\; c\; int\{\; sqrt\{g\_\{mu\; u\}\; frac\{dx^\{mu\{dq\}\; frac\{dx^\{\; u\{dq\}\; \}\; dq\}$

where "τ" is the

proper time and "q" is any smooth parameterization of the particle's world line. If one applies thecalculus of variations to this, one again gets the equations for a geodesic. The calculations are simplified, if we first take the variation of the square of the integrand. For the metric and coordinates of this case, that square is:$left(c\; frac\{d\; au\}\{dq\}\; ight)^2\; =\; g\_\{mu\; u\}\; frac\{dx^\{mu\{dq\}\; frac\{dx^\{\; u\{dq\}\; =\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; c^\{2\}\; left(\; frac\{dt\}\{dq\}\; ight)^\{2\}\; -\; frac\{1\}\{1\; -\; frac\{r\_\{s\{r\; left(\; frac\{dr\}\{dq\}\; ight)^\{2\}\; -\; r^\{2\}\; left(\; frac\{dvarphi\}\{dq\}\; ight)^\{2\}$

Taking variation of this, we get

:$delta\; left(c\; frac\{d\; au\}\{dq\}\; ight)^2\; =\; 2\; c^\{2\}\; frac\{d\; au\}\{dq\}\; delta\; frac\{d\; au\}\{dq\}\; =\; delta\; left\; [\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; c^\{2\}\; left(\; frac\{dt\}\{dq\}\; ight)^\{2\}\; -\; frac\{1\}\{1\; -\; frac\{r\_\{s\{r\; left(\; frac\{dr\}\{dq\}\; ight)^\{2\}\; -\; r^\{2\}\; left(\; frac\{dvarphi\}\{dq\}\; ight)^\{2\}\; ight]$

If we vary with respect to longitude "φ" only, we get

:$2\; c^\{2\}\; frac\{d\; au\}\{dq\}\; delta\; frac\{d\; au\}\{dq\}\; =\; -\; 2\; r^\{2\}\; frac\{dvarphi\}\{dq\}\; delta\; frac\{dvarphi\}\{dq\}$

we divide by $2\; c\; frac\{d\; au\}\{dq\}$ to get the variation of the integrand itself

:$c\; ,\; delta\; frac\{d\; au\}\{dq\}\; =\; -\; frac\{r^\{2\{c\}\; frac\{dvarphi\}\{d\; au\}\; delta\; frac\{dvarphi\}\{dq\}\; =\; -\; frac\{r^\{2\{c\}\; frac\{dvarphi\}\{d\; au\}\; frac\{d\; delta\; varphi\}\{dq\}$

Thus we have

:$0\; =\; delta\; int\; \{\; c\; frac\{d\; au\}\{dq\}\; dq\; \}\; =\; int\; \{\; c\; delta\; frac\{d\; au\}\{dq\}\; dq\; \}\; =\; int\; \{\; -\; frac\{r^\{2\{c\}\; frac\{dvarphi\}\{d\; au\}\; frac\{d\; delta\; varphi\}\{dq\}\; dq\; \}$

integrating by parts gives

:$0\; =\; -\; frac\{r^\{2\{c\}\; frac\{dvarphi\}\{d\; au\}\; delta\; varphi\; -\; int\; \{\; frac\{d\}\{dq\}\; left\; [\; -\; frac\{r^\{2\{c\}\; frac\{dvarphi\}\{d\; au\}\; ight]\; delta\; varphi\; dq\; \}$

The variation of the longitude is assumed to be zero at the end points, so the first term disappears. The integral can be made nonzero by a perverse choice of δφ unless the other factor inside is zero everywhere. So we get the equation of motion

:$frac\{d\}\{dq\}\; left\; [\; -\; frac\{r^\{2\{c\}\; frac\{dvarphi\}\{d\; au\}\; ight]\; =\; 0$

If we vary with respect to time "t" only, we get

:$2\; c^\{2\}\; frac\{d\; au\}\{dq\}\; delta\; frac\{d\; au\}\{dq\}\; =\; 2\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; c^\{2\}\; frac\{dt\}\{dq\}\; delta\; frac\{dt\}\{dq\}$

we divide by $2\; c\; frac\{d\; au\}\{dq\}$ to get the variation of the integrand itself

:$c\; delta\; frac\{d\; au\}\{dq\}\; =\; c\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; frac\{dt\}\{d\; au\}\; delta\; frac\{dt\}\{dq\}=\; c\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; frac\{dt\}\{d\; au\}\; frac\{d\; delta\; t\}\{dq\}$

Thus we have

:$0\; =\; delta\; int\; \{\; c\; frac\{d\; au\}\{dq\}\; dq\; \}=\; int\; \{\; c\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; frac\{dt\}\{d\; au\}\; frac\{d\; delta\; t\}\{dq\}\; dq\; \}$

integrating by parts gives

:$0\; =\; c\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; frac\{dt\}\{d\; au\}\; delta\; t\; -\; int\; \{\; frac\{d\}\{dq\}\; left\; [\; c\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; frac\{dt\}\{d\; au\}\; ight]\; delta\; t\; dq\; \}$

giving the equation of motion

:$frac\{d\}\{dq\}\; left\; [\; c\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; frac\{dt\}\{d\; au\}\; ight]\; =\; 0$

If we integrate these equations of motion and determine the constants of integration, we again get the equations

:$r^\{2\}\; frac\{dvarphi\}\{d\; au\}\; =\; frac\{L\}\{m\},$

:$left(\; 1\; -\; frac\{r\_\{s\{r\}\; ight)\; frac\{dt\}\{d\; au\}\; =\; frac\{E\}\{mc^\{2.$

These two equations for the constants of motion "L" and "E" can be combined to form one equation that is true even for

photon s and other massless particles for which theproper time along a geodesic is zero.:$frac\{r^\{2\{bc\}\; frac\{dvarphi\}\{dt\}\; =\; 1\; -\; frac\{r\_\{s\{r\}$

**ee also***

Kepler problem

*Schwarzschild metric

*Tests of general relativity

*Laplace–Runge–Lenz vector

*Deriving the Schwarzschild solution **Notes****References***

*

*

*

*

* (See

Gravitation (book) .)*

*

*

*Cite book

author=Roseveare, N. T

year=1982

title=Mercury's perihelion, from Leverrier to Einstein

location=Oxford

publisher=University Press

isbn=0198581742*

*

*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/*

*

*Wikimedia Foundation.
2010.*