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The two-body problem in general relativity is the determination of the motion and
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational pheno ...
of two bodies as described by the
field equation In theoretical physics and applied mathematics, a field equation is a partial differential equation which determines the dynamics of a physical field, specifically the time evolution and spatial distribution of the field. The solutions to the equ ...
s of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. Solving the
Kepler problem In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ei ...
is essential to calculate the bending of light by gravity and the motion of a
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
orbiting its sun. Solutions are also used to describe the motion of
binary star A binary star is a system of two stars that are gravitationally bound to and in orbit around each other. Binary stars in the night sky that are seen as a single object to the naked eye are often resolved using a telescope as separate stars, in ...
s around each other, and estimate their gradual loss of energy through
gravitational radiation Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
. General relativity describes the gravitational field by curved space-time; the field equations governing this curvature are
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
and therefore difficult to solve in a closed form. No exact solutions of the Kepler problem have been found, but an approximate solution has: the
Schwarzschild solution In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assu ...
. 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 In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...
("shortest path between two points") in the curved space-time. Such 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 bending of light in a gravitational field, another prediction famously used as evidence for general relativity. 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 In general relativity, the post-Newtonian expansions (PN expansions) are used for finding an approximate solution of the Einstein field equations for the metric tensor. The approximations are expanded in small parameters which express orders of ...
, 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 radiation Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
; this causes them to lose energy and angular momentum gradually, as illustrated by the binary pulsar
PSR B1913+16 PSR may refer to: Organizations * Pacific School of Religion, Berkeley, California, US * Palestinian Center for Policy and Survey Research * Physicians for Social Responsibility, US ;Political parties: * Revolutionary Socialist Party (Portugal) ( ...
. For
binary black hole A binary black hole (BBH) is a system consisting of two black holes in close orbit around each other. Like black holes themselves, binary black holes are often divided into stellar binary black holes, formed either as remnants of high-mass binar ...
s numerical solution of the two body problem was achieved after four decades of research, in 2005, when three groups devised the breakthrough techniques.


Historical context


Classical Kepler problem

The Kepler problem derives its name from
Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
, who worked as an assistant to the Danish astronomer
Tycho Brahe Tycho Brahe ( ; born Tyge Ottesen Brahe; generally called Tycho (14 December 154624 October 1601) was a Danish astronomer, known for his comprehensive astronomical observations, generally considered to be the most accurate of his time. He was ...
. 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 In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbi ...
, the first modern description of planetary motion: # The
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
of every
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
is an
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
with the Sun at one of the two foci. # A
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
joining a planet and the Sun sweeps out equal
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
s during equal intervals of time. # The
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of the
orbital period The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting pla ...
of a planet is directly proportional to the
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...
of the
semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lon ...
of its orbit. 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 Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
and
Copernicus Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, Niklas Koppernigk, german: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic canon, who formulat ...
. Kepler's laws apply only in the limited case of the two-body problem.
Voltaire François-Marie Arouet (; 21 November 169430 May 1778) was a French Enlightenment writer, historian, and philosopher. Known by his '' nom de plume'' M. de Voltaire (; also ; ), he was famous for his wit, and his criticism of Christianity—e ...
and
Émilie du Châtelet Gabrielle Émilie Le Tonnelier de Breteuil, Marquise du Châtelet (; 17 December 1706 – 10 September 1749) was a French natural philosopher and mathematician from the early 1730s until her death due to complications during childbirth in 1749. ...
were the first to call them "Kepler's laws". Nearly a century later,
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
had formulated his 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
law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
, which states that the force between a mass ''M'' and another mass ''m'' is given by the formula : F = G \frac, where ''r'' is the distance between the masses and ''G'' is the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
. 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.


Apsidal precession

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 to 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. Newton's law of gravitation soon became accepted because it gave very accurate predictions of the motion of all the planets. These calculations were carried out initially by
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarize ...
in the late 18th century, and refined by
Félix Tisserand François Félix Tisserand (13 January 1845 – 20 October 1896) was a French astronomer. Life Tisserand was born at Nuits-Saint-Georges, Côte-d'Or. In 1863 he entered the École Normale Supérieure, and on leaving he went for a month as profes ...
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.


Anomalous precession of Mercury

In 1859,
Urbain Le Verrier Urbain Jean Joseph Le Verrier FRS (FOR) HFRSE (; 11 March 1811 – 23 September 1877) was a French astronomer and mathematician who specialized in celestial mechanics and is best known for predicting the existence and position of Neptune using ...
discovered that the orbital
precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In oth ...
of the 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. The effect is small (roughly 43
arcsecond A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The n ...
s of rotation per century), but well above the measurement error (roughly 0.1
arcsecond A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The n ...
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. After these explanations were discounted, some physicists were driven to the more radical hypothesis that Newton's
inverse-square law In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be unders ...
of gravitation was incorrect. For example, some physicists proposed a
power law In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one q ...
with an
exponent Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to r ...
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 Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
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 enough allowed him to put a lower limit on the
speed of gravity In classical theories of gravitation, the changes in a gravitational field propagate. A change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational field which it produces. In ...
that turned out to be many orders of magnitude faster than the speed of light. Laplace's estimate for the speed of gravity is not correct in a field theory which respects the principle of relativity. Since electric and magnetic fields combine, the attraction of a point charge which is moving at a constant velocity is towards the extrapolated instantaneous position, not to the 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 potential The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these desc ...
s which are used in the
Wheeler–Feynman absorber theory The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is an interpretation of electrodynamics derived from the assu ...
. Presumably the same holds in linearized gravity: e.g., see Gravitoelectromagnetism.
To avoid those problems, between 1870 and 1900 many scientists used the electrodynamic laws of
Wilhelm Eduard Weber Wilhelm Eduard Weber (; ; 24 October 1804 – 23 June 1891) was a German physicist and, together with Carl Friedrich Gauss, inventor of the first electromagnetic telegraph. Biography of Wilhelm Early years Weber was born in Schlossstrasse i ...
,
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
,
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
to produce stable orbits and to explain the perihelion shift of Mercury's orbit. In 1890,
Maurice Lévy Maurice Lévy (February 28, 1838, Ribeauvillé – September 30, 1910, Paris) was a French engineer and member of the Institut de France. Lévy was born in Ribeauvillé in Alsace. Educated at the École Polytechnique, where he was a student ...
succeeded in doing so by combining the laws of Weber and Riemann, whereby the
speed of gravity In classical theories of gravitation, the changes in a gravitational field propagate. A change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational field which it produces. In ...
is equal to the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
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. Another attempt by
Hendrik Lorentz Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorent ...
(1900), who already used Maxwell's theory, produced a perihelion shift which was too low.


Einstein's theory of general relativity

Around 1904–1905, the works of
Hendrik Lorentz Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorent ...
,
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
and finally
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
's
special theory of relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
, exclude the possibility of propagation of any effects faster than the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
. 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 Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
(1905),
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
(1907) and
Arnold Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretic ...
(1910). In 1907 Einstein came to the conclusion that to achieve this a successor to special relativity was needed. From 1907 to 1915, Einstein worked towards a new theory, using his
equivalence principle In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (su ...
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 A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame. It is related to Newton's second law of motion, which trea ...
such as the
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
or the
Coriolis force In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
, which result from being in an accelerated reference frame; all fictitious forces are proportional to the
inertial mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementa ...
, just as gravity is. To effect the reconciliation of gravity and
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
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 Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, e.g., that the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
is true experimentally. Einstein used a more general geometry,
pseudo-Riemannian geometry In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which th ...
, 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 In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
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.


General relativity, special relativity and geometry

In the normal
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, triangles obey the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
, 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 ''x'', ''y'' and ''z'' coordinates of two points in a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
(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 earth is curved, which is why it is impossible to make a perfectly accurate flat map of the earth. 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\theta^2 + r^2 \sin^2 \theta \, d\varphi^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_ \,dx^2 + g_ \, dx \, dy + g_ \, dx \, dz + \cdots + g_ \, dz \, dy + g_ \, dz^2 \,\! where the nine functions ''g''xx, ''g''xy, …, ''g''zz constitute the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
, which defines the geometry of the space in
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
. 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 sin2 θ. In his
special theory of relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
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 In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
— called "proper time" and denoted with the symbol dτ — that ''is'' invariant; in other words, it does not depend on the motion of the observer. : c^2 \, d\tau^2 = c^2 \, dt^2 - dx^2 - dy^2 - dz^2 \,\! which may be written in spherical coordinates as : c^2 \, d\tau^2 = c^2 \, dt^2 - dr^2 - r^2 \, d\theta^2 - r^2 \sin^2 \theta \, d\varphi^2 \,\! This formula is the natural extension of the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
and similarly holds only when there is no curvature in space-time. In
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, however, space and time may have curvature, so this distance formula must be modified to a more general form : c^2 \, d\tau^2 = g_ dx^\mu \, dx^\nu \,\! 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 the
Einstein field equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...
. 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.


Geodesic equation

According to Einstein's theory of general relativity, particles of negligible mass travel along
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...
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 : \frac + \Gamma^\mu_ \frac \frac = 0 where Γ represents the
Christoffel symbol In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing dist ...
and the variable ''q'' parametrizes the particle's path through
space-time In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
, its so-called
world line The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from c ...
. The Christoffel symbol depends only on the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
''g''μν, or rather on how it changes with position. The variable ''q'' is a constant multiple of the
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
''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 A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
), 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 In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light . The expression for the relativistic energy of a particle with rest mass and momentum is given by :E^2 = m^2 c^4 + p^2 c^2. The energy of ...
of timelike orbits, that is, the limit as the particle mass ''m'' goes to zero while holding its total
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
fixed.


Schwarzschild solution

An exact solution to the
Einstein field equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...
is the
Schwarzschild metric In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assump ...
, 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''s, known as the
Schwarzschild radius The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteri ...
, which is defined by the formula :: r_s = \frac where ''G'' is the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
. 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 that defined by
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
. In practice, this ratio is almost always extremely small. For example, the Schwarzschild radius ''r''s of the Earth is roughly 9  mm
inch Measuring tape with inches The inch (symbol: in or ″) is a unit of length in the British imperial and the United States customary systems of measurement. It is equal to yard or of a foot. Derived from the Roman uncia ("twelfth ...
); 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''s/''r'' is roughly 4 parts in a million. A
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes ...
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 A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
s (where the ratio is roughly 50%) and
black hole A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...
s.


Orbits about the central mass

The orbits of a test particle of infinitesimal mass m about the central mass M is given by the equation of motion : \left( \frac \right)^2 = \frac - \left( 1 - \frac \right) \left( c^2 + \frac \right). where h is the
specific relative angular momentum In celestial mechanics, the specific relative angular momentum (often denoted \vec or \mathbf) of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative positi ...
, h = r \times v = and \mu is the
reduced mass In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
. This can be converted into an equation for the orbit : \left( \frac \right)^2 = \frac - \left( 1 - \frac \right) \left( \frac + r^2 \right), where, for brevity, two length-scales, a = \frac and b = \frac, 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 : \varphi = \int \frac \left frac - \left(1 - \frac\right) \left(\frac + \frac \right)\right \, dr.


Effective radial potential energy

The equation of motion for the particle derived above : \left( \frac \right)^2 = \frac - c^2 + \frac - \frac + \frac can be rewritten using the definition of the
Schwarzschild radius The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteri ...
''r''s as : \frac m \left( \frac \right)^2 = \left \frac - \frac m c^2 \right+ \frac - \frac + \frac which is equivalent to a particle moving in a one-dimensional effective potential : V(r) = -\frac + \frac - \frac The first two terms are well-known classical energies, the first being the attractive Newtonian
gravitational potential energy Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conver ...
and the second corresponding to the repulsive "centrifugal" potential energy; however, the third term is an attractive energy unique to
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. 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 where ''A'' is the semi-major axis and ''e'' is the eccentricity. Here ''δφ'' is ''not'' the change in the ''φ''-coordinate in (''t'', ''r'', ''θ'', ''φ'') coordinates but the change in the
argument of periapsis The argument of periapsis (also called argument of perifocus or argument of pericenter), symbolized as ''ω'', is one of the orbital elements of an orbiting body. Parametrically, ''ω'' is the angle from the body's ascending node to its periap ...
of the classical closed orbit. 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 length ''a'' = ''h''/''c'': : V(r) = \frac \left - \frac + \frac - \frac \right Circular orbits are possible when the effective force is zero: : F = -\frac = -\frac \left r_ r^2 - 2a^2 r + 3r_s a^2 \right= 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: :\begin r_ &= \frac \left( 1 + \sqrt \right) \\ r_ &= \frac \left( 1 - \sqrt \right) = \frac, \end which are obtained using the
quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, ...
. 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-relativistic
Kepler problem In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ei ...
. When ''a'' is much greater than ''r''s (the classical case), these formulae become approximately :\begin r_ &\approx \frac \\ r_ &\approx \frac r_s \end Substituting the definitions of ''a'' and ''r''s into ''r''outer yields the classical formula for a particle of mass ''m'' orbiting a body of mass ''M''. The following equation : r_^3 = \frac where ''ω''''φ'' is the orbital angular speed of the particle, is obtained in non-relativistic mechanics by setting the
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
equal to the Newtonian gravitational force: : \frac = \mu \omega_\varphi^2 r Where \mu is the
reduced mass In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
. In our notation, the classical orbital angular speed equals : \omega_\varphi^2 \approx \frac = \left( \frac \right) = \left( \frac \right) \left( \frac\right) = \frac At the other extreme, when ''a''2 approaches 3''r''s2 from above, the two radii converge to a single value : r_ \approx r_ \approx 3 r_s The quadratic solutions above ensure that ''r''outer is always greater than 3''r''s, whereas ''r''inner lies between  ''r''s and 3''r''s. Circular orbits smaller than  ''r''s are not possible. For massless particles, ''a'' goes to infinity, implying that there is a circular orbit for photons at ''r''inner =  ''r''s. The sphere of this radius is sometimes known as the photon 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 in a stable manner with an angular frequency : \omega_r^2 = \frac \left \frac \right which equals : \omega_r^2 = \left( \frac \right) \left( r_ - r_ \right) = \omega_\varphi^2 \sqrt Taking the square root of both sides and expanding using the binomial theorem yields the formula : \omega_r = \omega_\varphi \left( 1 - \frac + \cdots \right) Multiplying by the period ''T'' of one revolution gives the precession of the orbit per revolution : \delta \varphi = T(\omega_\varphi - \omega_r) \approx 2\pi \left( \frac \right) = \frac r_s^2 where we have used ''ω''''φ''''T'' = 2 and the definition of the length-scale ''a''. Substituting the definition of the
Schwarzschild radius The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteri ...
''r''s gives : \delta \varphi \approx \frac \left( \frac \right) = \frac This may be simplified using the elliptical orbit's semi-major axis ''A'' and eccentricity ''e'' related by the
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
: \frac = A\left(1 - e^2\right) to give the precession angle : \delta \varphi \approx \frac Since the closed classical orbit is an ellipse in general, the quantity ''A''(1 − ''e''2) is the semi-
latus rectum In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a sp ...
''l'' of the ellipse. Hence, the final formula of angular apsidal precession for a unit complete revolution is : \delta \varphi \approx \frac


Beyond the Schwarzschild solution


Post-Newtonian expansion

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 A binary star is a system of two stars that are gravitationally bound to and in orbit around each other. Binary stars in the night sky that are seen as a single object to the naked eye are often resolved using a telescope as separate stars, in ...
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 In general relativity, the post-Newtonian expansions (PN expansions) are used for finding an approximate solution of the Einstein field equations for the metric tensor (general relativity), metric tensor. The approximations are expanded in small ...
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. However, the gravitational energy between the two bodies is exchanged via dilatons rather than gravitons which require three-space in which to propagate. The
post-Newtonian expansion In general relativity, the post-Newtonian expansions (PN expansions) are used for finding an approximate solution of the Einstein field equations for the metric tensor. The approximations are expanded in small parameters which express orders of ...
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. The theory can be divided into two parts: first one finds the two-body effective potential that captures the GR corrections to the Newtonian potential. Secondly, one should solve the resulting equations of motion. 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''''s'' and the precession angle per revolution δφ.


Modern computational approaches

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.


Gravitational radiation

If there is no incoming gravitational radiation, according to
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, two bodies orbiting one another will emit
gravitational radiation Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
, causing the orbits to gradually lose energy. The formulae describing the loss of
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
and
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
due to gravitational radiation from the two bodies of the Kepler problem have been calculated. The rate of losing energy (averaged over a complete orbit) is given by : -\left\langle \frac \right\rangle = \frac \left( 1 + \frac e^2 + \frac e^4 \right) where ''e'' is the
orbital eccentricity In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values bet ...
and ''a'' is the
semimajor axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lo ...
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 : -\left\langle \frac \right\rangle = \frac \left( 1 + \frac e^2 \right) The rate of period decrease is given by : -\left\langle \frac \right\rangle = \frac \left( 1 + \frac e^2 + \frac e^4 \right) \left(\frac\right)^ where ''P''''b'' 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. File:PSR B1913+16 period shift graph.svg, Experimentally observed decreases of the
orbital period The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting pla ...
of the
binary pulsar A binary pulsar is a pulsar with a binary companion, often a white dwarf or neutron star. (In at least one case, the double pulsar PSR J0737-3039, the companion neutron star is another pulsar as well.) Binary pulsars are one of the few objects ...
PSR B1913+16 PSR may refer to: Organizations * Pacific School of Religion, Berkeley, California, US * Palestinian Center for Policy and Survey Research * Physicians for Social Responsibility, US ;Political parties: * Revolutionary Socialist Party (Portugal) ( ...
(blue dots) match the predictions of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
(black curve) almost exactly. File:Wavy2.gif, Two neutron stars rotating rapidly around one another gradually lose energy by emitting gravitational radiation. As they lose energy, they orbit each other more quickly and more closely to one another.


See also

*
Binet equation The Binet equation, derived by Jacques Philippe Marie Binet, provides the form of a central force given the shape of the orbital motion in plane polar coordinates. The equation can also be used to derive the shape of the orbit for a given force ...
*
Center of mass (relativistic) In physics, relativistic center of mass refers to the mathematical and physical concepts that define the center of mass of a system of particles in relativistic mechanics and relativistic quantum mechanics. Introduction In non-relativistic phys ...
* Gravitational two-body problem *
Kepler problem In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ei ...
*
Newton's theorem of revolving orbits In classical mechanics, Newton's theorem of revolving orbits identifies the type of central force needed to multiply the angular speed of a particle by a factor ''k'' without affecting its radial motion (Figures 1 and 2). Newton applied his ...
* Schwarzschild geodesics


Notes


References


Bibliography

* * * * * * * (See
Gravitation (book) ''Gravitation'' is a widely adopted textbook on Albert Einstein's general theory of relativity, written by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. It was originally published by W. H. Freeman and Company in 1973 and repr ...
.) * * * * * * * * *


External links


Animation
showing relativistic precession of stars around the Milky Way supermassive black hole

from ''Reflections on Relativity'' by Kevin Brown. {{Relativity Exact solutions in general relativity