Two-body problem in general relativity

General relativity
Introduction Mathematical formulation Resources
Fundamental concepts Special relativity Equivalence principle World line · Riemannian geometry
Phenomena Kepler problem · Lenses · Waves Frame-dragging · Geodetic effect Event horizon · Singularity Black hole
Equations Linearized Gravity Post-Newtonian formalism Einstein field equations Geodesic equation Friedmann equations ADM formalism BSSN formalism
Advanced theories Kaluza–Klein Quantum gravity
Solutions Schwarzschild Reissner–Nordström · Gödel Kerr · Kerr–Newman Kasner · Taub-NUT · Milne · Robertson–Walker pp-wave · van Stockum dust
Scientists Einstein · Minkowski · Eddington Lemaître · Schwarzschild Robertson · Kerr · Friedman Chandrasekhar · Hawking others

The two-body problem in general relativity is to determine the motion and gravitational field of two bodies interacting with one another by gravitation, as described by the field equations of general relativity. Solving the Kepler problem is essential to calculate the bending of light by gravity and the motion of a planet orbiting its sun. Solutions are also used to describe the motion of binary stars around each other, and estimate their gradual loss of energy through gravitational radiation. It is customary to assume that both bodies are point-like, so that tidal forces and the specifics of their material composition can be neglected.

General relativity describes the gravitational field by curved space-time; the field equations governing this curvature are nonlinear and therefore difficult to solve in a closed form. Only one exact solution, the Schwarzschild solution, has been found for the Kepler problem; this solution pertains when the mass M of one body is overwhelmingly greater than the mass m of the other. If so, the larger mass may be taken as stationary and the sole contributor to the gravitational field. This is a good approximation for a photon passing a star and for a planet orbiting its sun. The motion of the lighter body (called the "particle" below) can then be determined from the Schwarzschild solution; the motion is a geodesic ("shortest path between two points") in the curved space-time. Such geodesic solutions account for the 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, 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; this causes them to lose energy and angular momentum gradually, as illustrated by the binary pulsar PSR B1913+16.

Historical context

Classical Kepler problem

The Kepler problem derives its name from Johannes Kepler, who worked as an assistant to the Danish astronomer Tycho Brahe. Brahe took extraordinarily accurate measurements of the motion of the planets of the Solar System. From these measurements, Kepler was able to formulate Kepler's laws, the first modern description of planetary motion:

The orbit of every planet is an ellipse with the Sun at one of the two foci.
A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis 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 and Copernicus. Kepler's laws apply only in the limited case of the two-body problem. Voltaire and Émilie du Châtelet were the first to call them "Kepler's laws".

Nearly a century later, Isaac Newton 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, which states that the force between a mass M and another mass m is given by the formula

$F = G \frac{M m}{r^2}$ ,

where r is the distance between the masses and G is the gravitational constant. Given this force law and his equations of motion, Newton was able to show that two point masses attracting each other would each follow perfectly elliptical orbits. The ratio of sizes of these ellipses is m/M, with the larger mass moving on a smaller ellipse. If M is much larger than m, then the larger mass will appear to be stationary at the focus of the elliptical orbit of the lighter mass m. This model can be applied approximately to the Solar System. Since the mass of the Sun is much larger than those of the planets, the force acting on each planet is principally due to the Sun; the gravity of the planets for each other can be neglected to first approximation.

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 for one another. The apsides are the two points of closest and furthest distance of the orbit (the periapsis and apoapsis, respectively); apsidal precession corresponds to the rotation of the line joining the apsides. It also corresponds to the rotation of the Laplace–Runge–Lenz vector, which points along the line of apsides.

Newton's law of gravitation soon became accepted because it gave very accurate predictions of the apsidal precessions of all the planets. These calculations were carried out initially by Pierre-Simon Laplace in the late 18th century, and refined by Félix Tisserand in the later 19th century. Conversely, if Newton's law of gravitation had not predicted the apsidal precessions of the planets accurately, it would have to be discarded as a theory of gravitation. Such an anomalous precession was observed in the second half of the 19th century, and it led to the overthrow of Newtonian model of gravity and the development of general relativity.

Anomalous precession of Mercury

Einstein's theory of general relativity

Around 1904–1905, the works of Hendrik Lorentz, Henri Poincaré and finally Albert Einstein's special theory of relativity, exclude the possibility of propagation of any effects faster than the speed of light. It followed that Newton's law of gravitation would have to be replaced with another law, compatible with the principle of relativity, while still obtaining the newtonian limit for circumstances where relativistic effects are negligible. Such attempts were made by Henri Poincaré (1905), Hermann Minkowski (1907) and Arnold Sommerfeld (1910).^[4] In 1907 Einstein came to the conclusion that to achieve this a successor to special relativity was needed. From 1907 to 1915, Einstein worked towards a new theory, using his Equivalence Principle as a key concept to guide his way. According to this principle, a uniform gravitational field acts equally on everything within it and, therefore, cannot be detected by a free-falling observer. Conversely, all local gravitational effects should be reproducible in a linearly accelerating reference frame, and vice versa. Thus, gravity acts like a fictitious force such as the centrifugal force or the Coriolis force, which result from being in an accelerated reference frame; all fictitious forces are proportional to the inertial mass, just as gravity is. To effect the reconciliation of gravity and special relativity and to incorporate the equivalence principle, something had to be sacrificed; that something was the long-held classical assumption that our space obeys the laws of Euclidean geometry, e.g., that the Pythagorean theorem is true experimentally. Einstein used a more general geometry, pseudo-Riemannian geometry, to allow for the curvature of space and time that was necessary for the reconciliation; after eight years of work (1907–1915), he succeeded in discovering the precise way in which space-time should be curved in order to reproduce the physical laws observed in Nature, particularly gravitation. Gravity is distinct from the fictitious forces centrifugal force and coriolis force in the sense that the curvature of spacetime is regarded as physically real, whereas the fictitious forces are not regarded as forces. The very first solutions of 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, triangles obey the Pythagorean theorem, which states that the square distance ds² between two points in space is the sum of the squares of its perpendicular components

$ds^{2} = dx^{2} %2B dy^{2} %2B dz^{2} \,\!$

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

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

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

$ds^{2} = dr^{2} %2B r^{2} d\theta^{2} %2B r^{2} \sin^{2} \theta d\varphi^{2} \,\!$

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

$ds^{2} = g_{xx} dx^{2} %2B g_{xy} dx dy %2B g_{xz} dx dz %2B \cdots %2B g_{zy} dz dy %2B g_{zz} dz^{2} \,\!$

where the nine functions g_xx, g_xy constitute the metric tensor, which defines the geometry of the space in Riemannian geometry. In the spherical-coordinates example above, there are no cross-terms; the only nonzero metric tensor components are g_rr = 1, g_θθ = r² and g_φφ = r² sin² 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 in space-time — called "proper time" and denoted with the symbol dτ — that is invariant; in other words, it doesn't depend on the motion of the observer.

$c^{2} d\tau^{2} = c^{2} dt^{2} - dx^{2} - dy^{2} - dz^{2} \,\!$

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 and similarly holds only when there is no curvature in space-time. In general relativity, however, space and time may have curvature, so this distance formula must be modified to a more general form

$c^{2} d\tau^{2} = g_{\mu\nu} dx^{\mu} dx^{\nu} \,\!$

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. 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

See also: Geodesic and Christoffel symbols

According to Einstein's theory of general relativity, particles of negligible mass travel along geodesics 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^[5]

$\frac{d^2x^{\mu}}{d q^2} %2B \Gamma^{\mu}_{\nu\lambda} \frac{dx^{\nu}}{d q} \frac{dx^{\lambda}}{dq} = 0$

where Γ represents the Christoffel symbol and the variable q parametrizes the particle's path through space-time, its so-called world line. The Christoffel symbol depends only on the metric tensor g_μν, or rather on how it changes with position. The variable q is a constant multiple of the proper time τ for timelike orbits (which are traveled by massive particles), and is usually taken to be equal to it. For lightlike (or null) orbits (which are traveled by massless particles such as the photon), the proper time is zero and, strictly speaking, cannot be used as the variable q. Nevertheless, lightlike orbits can be derived as the ultrarelativistic limit of timelike orbits, that is, the limit as the particle mass m goes to zero while holding its total energy fixed.

Schwarzschild solution

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

$\left( \frac{dr}{d\tau} \right)^{2} = \frac{E^{2}}{m^{2}c^{2}} - \left( 1 - \frac{r_{s}}{r} \right) \left( c^{2} %2B \frac{h^{2}}{r^{2}} \right).$

which can be converted into an equation for the orbit

$\left( \frac{dr}{d\varphi} \right)^{2} = \frac{r^{4}}{b^{2}} - \left( 1 - \frac{r_{s}}{r} \right) \left( \frac{r^{4}}{a^{2}} %2B r^{2} \right)$

where, for brevity, two length-scales, a and b, have been introduced. They are constants of the motion and depend on the initial conditions (position and velocity) of the test particle. Hence, the solution of the orbit equation is

$\varphi = \int \frac{dr}{r^{2} \sqrt{\frac{1}{b^{2}} - \left( 1 - \frac{r_{s}}{r} \right) \left( \frac{1}{a^{2}} %2B \frac{1}{r^{2}} \right)}}.$

Bending of light by gravity

See also: Gravitational lens and Shapiro delay

The orbit of photons and particles moving close to the speed of light (ultrarelativistic particles) is obtained by taking the limit as the length-scale a goes to infinity. In this limit, the equation for the orbit becomes

$\varphi = \int \frac{dr}{r^{2} \sqrt{\frac{1}{b^{2}} - \left( 1 - \frac{r_{s}}{r} \right) \frac{1}{r^{2}}}}$

Expanding in powers of r_s/r, the leading order term in this formula gives the approximate angular deflection δφ 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, the length-scale b can be interpreted as the distance of closest approach. Although this formula is approximate, it is accurate for most measurements of gravitational lensing, due to the smallness of the ratio r_s/r. For light grazing the surface of the sun, the approximate angular deflection is roughly 1.75 arcseconds, roughly one millionth part of a circle.^[6]

Effective radial potential energy

The equation of motion for the particle derived above

$\left( \frac{dr}{d\tau} \right)^{2} = \frac{E^2}{m^2 c^2} - c^{2} %2B \frac{ r_{s} c^2}{r} - \frac{h^2}{ r^2 } %2B \frac{ r_{s} h^2 }{ r^3 }$

can be rewritten using the definition of the Schwarzschild radius r_s as

$\frac{1}{2} m \left( \frac{dr}{d\tau} \right)^{2} = \left[ \frac{E^2}{2 m c^2} - \frac{1}{2} m c^2 \right] %2B \frac{GMm}{r} - \frac{ L^2 }{ 2 \mu r^2 } %2B \frac{ G(M%2Bm) L^2 }{c^2 \mu r^3}$

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

$V(r) = -\frac{GMm}{r} %2B \frac{ L^2 }{ 2 \mu r^2 } - \frac{ G(M%2Bm) L^2 }{ c^2 \mu r^3 }$

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

$\delta \varphi \approx \frac{ 6\pi G(M%2Bm) }{ c^2 A \left( 1- e^{2} \right)}$

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

$V(r) = \frac{mc^{2}}{2} \left[ - \frac{r_{s}}{r} %2B \frac{a^{2}}{r^{2}} - \frac{r_{s} a^{2}}{r^{3}} \right]$

Circular orbits are possible when the effective force is zero

$F = -\frac{dV}{dr} = -\frac{mc^{2}}{2r^{4}} \left[ r_{s} r^{2} - 2a^{2} r %2B 3r_{s} a^{2} \right] = 0$

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

$r_{\mathrm{outer}} = \frac{a^{2}}{r_{s}} \left( 1 %2B \sqrt{1 - \frac{3r_{s}^{2}}{a^{2}}} \right)$

$r_{\mathrm{inner}} = \frac{a^{2}}{r_{s}} \left( 1 - \sqrt{1 - \frac{3r_{s}^{2}}{a^{2}}} \right) = \frac{3a^{2}}{r_{\mathrm{outer}}}$

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-relativistic Kepler 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 the centrifugal force equal to the Newtonian gravitational force:

$\frac{GMm}{r^{2}} = \mu \omega_{\varphi}^{2} r$

Where $\mu$ is the reduced mass.
In our notation, the classical orbital angular speed equals

$\omega_{\varphi}^{2} \approx \frac{GM}{r_{\mathrm{outer}}^{3}} = \left( \frac{r_{s} c^{2}}{2r_{\mathrm{outer}}^{3}} \right) = \left( \frac{r_{s} c^{2}}{2} \right) \left( \frac{r_{s}^{3}}{8a^{6}}\right) = \frac{c^{2} r_{s}^{4}}{16 a^{6}}$

At the other extreme, when a² approaches 3r_s² 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 3r_s, whereas r_inner lies between ³⁄₂ r_s and 3r_s. Circular orbits smaller than 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 = ³⁄₂ 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 stably with an angular frequency

$\omega_{r}^{2} = \frac{1}{m} \left[ \frac{d^{2}V}{dr^{2}} \right]_{r=r_{\mathrm{outer}}}$

which equals

$\omega_{r}^{2} = \left( \frac{c^{2} r_{s}}{2 r_{\mathrm{outer}}^{4}} \right) \left( r_{\mathrm{outer}} - r_{\mathrm{inner}} \right) = \omega_{\varphi}^{2} \sqrt{1 - \frac{3r_{s}^{2}}{a^{2}}}$

Taking the square root of both sides and expanding using the binomial theorem yields the formula

$\omega_{r} = \omega_{\varphi} \left( 1 - \frac{3r_{s}^{2}}{4a^{2}} %2B \cdots \right)$

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

$\delta \varphi = T \left( \omega_{\varphi} - \omega_{r} \right) \approx 2\pi \left( \frac{3r_{s}^{2}}{4a^{2}} \right) = \frac{3\pi m^{2} c^{2}}{2L^{2}} r_{s}^{2}$

where we have used ω_φT = 2п and the definition of the length-scale a. Substituting the definition of the Schwarzschild radius r_s gives

$\delta \varphi \approx \frac{3\pi m^{2} c^{2}}{2L^{2}} \left( \frac{4G^{2} M^{2}}{c^{4}} \right) = \frac{6\pi G^{2} M^{2} m^{2}}{c^{2} L^{2}}$

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

$\frac{ h^2 }{ G(M%2Bm) } = A \left( 1 - e^2 \right)$

to give the precession angle

$\delta \varphi \approx \frac{6\pi G(M%2Bm)}{c^2 A \left( 1 - e^{2} \right)}$

Corrections to 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 stars, in which the masses may be of similar magnitude.

The metric for the case of two comparable masses cannot be solved in closed form and therefore one has to resort to approximation techniques such as the post-Newtonian approximation or numerical approximations. In passing, we mention one particular exception in lower dimensions (see R=T model for details). In (1+1) dimensions, i.e. a space made of one spatial dimension and one time dimension, the metric for two bodies of equal masses can be solved analytically in terms of the Lambert W function^[7]. 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 is a calculational method that provides a series of ever more accurate solutions to a given problem. The method is iterative; an initial solution for particle motions is used to calculate the gravitational fields; from these derived fields, new particle motions can be calculated, from which even more accurate estimates of the fields can be computed, and so on. This approach is called "post-Newtonian" because the Newtonian solution for the particle orbits is often used as the initial solution.

When this method is applied to the two-body problem without restriction on their masses, the result is remarkably simple. To the lowest order, the relative motion of the two particles is equivalent to the motion of an infinitesimal particle in the field of their combined masses. In other words, the Schwarzschild solution can be applied, provided that the M + m is used in place of M in the formulae for the Schwarzschild radius r_s and the precession angle per revolution δφ.

Modern computational approaches

Gravitational radiation

See also: Gravitational radiation and PSR B1913+16

If there is no incoming gravitational radiation, according to general relativity, two bodies revolving about one another will emit gravitational radiation, causing the orbits to gradually lose energy. This has been observed indirectly in a binary star system known as PSR B1913+16, for which Russell Alan Hulse and Joseph Hooton Taylor, Jr. were awarded the 1993 Nobel Prize in Physics. The two neutron stars 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 short orbital period and high eccentricity should make the system an excellent emitter of gravitational radiation, thereby losing energy and decreasing the orbital period still further. The observed decrease in the orbital period over thirty years matches the predictions of general relativity within even the most precise measurements. General relativity predicts that, in another 300 million years, these two stars will spiral into one another.

The formulae describing the loss of energy and angular momentum due to gravitational radiation from the two bodies of the Kepler problem have been calculated.^[8] The rate of losing energy (averaged over a complete orbit) is given by^[9]

$-\Bigl\langle \frac{dE}{dt} \Bigr\rangle = \frac{32G^{4}m_{1}^{2}m_{2}^{2}\left(m_{1} %2B m_{2}\right)}{5c^{5} a^{5} \left( 1 - e^{2} \right)^{7/2}} \left( 1 %2B \frac{73}{24} e^{2} %2B \frac{37}{96} e^{4} \right)$

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

$-\Bigl\langle \frac{dL_{z}}{dt} \Bigr\rangle = \frac{32G^{7/2}m_{1}^{2}m_{2}^{2}\sqrt{m_{1} %2B m_{2}}}{5c^{5} a^{7/2} \left( 1 - e^{2} \right)^{2}} \left( 1 %2B \frac{7}{8} e^{2} \right)$

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.

Notes

^ Feynman Lectures on Physics vol. II gives a thorough treatment of the analogous problem in electromagnetism. Feynman shows that for a moving charge, the non-radiative field is an attraction/repulsion not toward the apparent position of the particle, but toward the extrapolated position assuming that the particle continues in a straight line in a constant velocity. This is a notable property of the Liénard-Wiechert Potentials which are used in the Wheeler-Feynman absorber theory. Presumably the same holds in linearized gravity.

References

^ Le Verrier, UJJ (1859). "Unknown title". Comptes Revues d'Academie de la Science de Paris 49: 379–?.
^ ^a ^b ^c Pais 1982
^ Roseveare 1982
^ Walter 2007
^ Weinberg 1972.
^ http://www.mathpages.com/rr/s6-03/6-03.htm - 6.3 Bending Light
^ T. Ohta and R.B. Mann (1997). "Exact solution for the metric and the motion of two bodies in (1+1)-dimensional gravity", Phys. Rev. D. 55: 4723-4747. [1] Arxiv article [2].
^ Peters PC, Mathews J (1963). "Gravitational Radiation from Point Masses in a Keplerian Orbit". Physical Review 131: 435–440. Bibcode 1963PhRv..131..435P. doi:10.1103/PhysRev.131.435.
^ Landau and Lifshitz, p. 356–357.

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External links

Animation showing relativistic precession of stars around the Milky Way supermassive black hole
Excerpt from Reflections on Relativity by Kevin Brown.