NationMaster - Encyclopedia: Kepler problem in general relativity

In general relativity, the Kepler problem involves solving for the motion of a particle of negligible mass in the external gravitational field of another body of mass M. This gravitational field is described by the Schwarzschild solution to the vacuum Einstein equations of general relativity, and particle motion is described by the space-time geodesics of this solution. General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... The gravitational field is a field (physics), generated by massive objects, that determines the magnitude and direction of gravitation experienced by other massive objects. ... Introduction In Einsteins theory of general relativity, the Schwarzschild metric is the most general static, spherically symmetric solution of the vacuum field equations. ... Look up Vacuum in Wiktionary, the free dictionary. ... This article or section is in need of attention from an expert on the subject. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ...

The solution accounts for the anomalous precession of the planet Mercury, which is a key piece of evidence supporting the theory of general relativity. It also describes the deflection of light in a gravitational field, which was another prediction famously used to provide evidence for the theory. Tests of Einsteins general theory of relativity did not provide an experimental foundation for the theory until well after it was introduced in 1915. ... This article is about the planet. ... General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

1 Derivation from the Lagrangian
2 Orbit types
3 Derivation from the Hamilton–Jacobi equation
4 Bending of light
5 Rate of precession
6 See also
7 Notes
8 References

Derivation from the Lagrangian

The main ingredient needed to solve the Kepler problem in general relativity is an explicit formula for the Schwarzschild metric. In order to analyse the solution, it suffices to work in a spherical coordinate system aligned with the plane of the orbit of the particle, so that the zenith angle θ (also known as the colatitude or the altitude angle) may be eliminated from consideration. The Schwarzschild metric in this plane is then given by It has been suggested that Deriving the Schwarzschild solution be merged into this article or section. ... A point plotted using the spherical coordinate system In mathematics, the spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the zenith angle from the positive z-axis, and the azimuth angle... In broad terms, the zenith is the direction pointing directly above a particular location (perpendicular, orthogonal). ... In spherical coordinates, colatitude is the complementary angle of the latitude. ...

$ds^{2} = left( 1 - frac{r_{s}}{r} right) c^{2} dt^{2} - frac{dr^{2}}{1 - frac{r_{s}}{r}} - r^{2} dphi^{2}$

where φ is the azimuthal angle, r is the radial coordinate, and r_s is the Schwarzschild radius of the massive body, which is related to its mass M by Azimuth is the horizontal component of a direction (compass direction), measured around the horizon, from the north toward the east (i. ...

$r_{s} = frac{2GM}{c^{2}}$

where G is the gravitational constant.^[1] and c is the speed of light. According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation...

The orbit of a particle about a mass M can be determined directly using the geodesic equation^[2]

$frac{dx^{mu}}{dq^{2}} + Gamma^{mu}_{nulambda} frac{dx^{nu}}{dq} frac{dx^{lambda}}{dq} = 0$

where the variable q parametrizes the orbit and is usually taken as the proper time. However, it is swifter and more elegant to derive the orbit from a Lagrangian approach.^[3] (A third approach using the Hamilton-Jacobi equation is described below.) Using the Schwarzschild metric, the Lagrangian can be formed from the kinetic energy T, given by A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... The Hamilton-Jacobi-Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... The kinetic energy of an object is the extra energy which it possesses due to its motion. ...

$2 T = left( 1 - frac{r_{s}}{r} right) c^{2} left( frac{dt}{ds} right)^{2} - frac{1}{1 - frac{r_{s}}{r}} left( frac{dr}{ds} right)^{2} - r^{2} left( frac{dphi}{ds} right)^{2},$

since there is no gravitational potential energy in general relativity. The first two Euler–Lagrange equations are Potential energy is the energy that is by virtue of the relative positions (configurations) of the objects within a physical system. ... General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

$r^{2} frac{dphi}{ds} = frac{-L}{mc},$

$left( 1 - frac{r_{s}}{r} right) frac{dt}{ds} = frac{-E}{mc^{3}}.$

where E and L are the particle's energy and angular momentum, respectively. Substituting these equations into the definition of ds² yields the initial equation for the orbit This gyroscope remains upright while spinning due to its angular momentum. ...

$dr^{2} = left[ frac{E^{2}r^{4}}{L^{2} c^{2}} - left( 1 - frac{r_{s}}{r} right) left( frac{m^{2} c^{2}}{L^{2}} r^{4} + r^{2} right) right] dphi^{2}.$

Changing variables to u = 1/r gives an intermediate equation for the orbit:

$left( frac{du}{dphi} right)^{2} = frac{E^{2}}{L^{2} c^{2}} - left( 1 - r_{s} u right) left( frac{m^{2} c^{2}}{L^{2}} + u^{2} right).$

Changing then to a dimensionless variable ζ

$zeta = frac{r_{s} u}{4} - frac{1}{12}$

yields the final equation for the orbit:

$left( frac{dzeta}{dphi} right)^{2} = 4 zeta^{3} - g_{2} zeta - g_{3},$

where the constant, dimensionless coefficients g₂ and g₃ are defined by

$g_{2} = frac{1}{12} - frac{r_{s}^{2} m^{2} c^{2}}{4 L^{2}}$

and

$g_{3} = frac{1}{216} + frac{r_{s}^{2} m^{2} c^{2}}{24 L^{2}} - frac{r_{s}^{2} E^{2}}{16 L^{2} c^{2}}.$

The solution of the final orbital equation for ζ is an elliptic function

$frac{r_{s}}{4r} = frac{1}{12} + J(phi - phi_{0}).$

Orbit types

Quasi-elliptical orbits

There are three roots e₁, e₂, and e₃ at which the derivative of the inverse radius u with respect to φ is zero

$frac{dzeta}{dphi} = sqrt{4zeta^{3} - g_{2} zeta - g_{3}} = 0.$

We can define the change in angle between two such nodes (which is real)

$Delta phi = int_{e_{1}}^{infty} frac{dzeta}{sqrt{4zeta^{3} - g_{2} zeta - g_{3}}}.$

(For the planets in the solar system apart from Mercury, this will be undetectably close to π radians.) The particle (or planet) moves between a minimum radius The eight planets and three dwarf planets of the Solar System. ... Major features of the Solar System (not to scale; from left to right): Pluto, Neptune, Uranus, Saturn, Jupiter, the asteroid belt, the Sun, Mercury, Venus, Earth and its Moon, and Mars. ... Look up Mercury in Wiktionary, the free dictionary. ...

$r_{min} = frac{3r_{s}}{1 + 12e_{3}}$

and a maximum radius

$r_{max} = frac{3r_{s}}{1 + 12e_{2}}$

corresponding to the value of ζ at the two extrema of radius.

Transiently unstable circular orbits

If two of the three roots are equal and positive, the orbits are asymptotically circular at positive and negative infinite φ. Let the two repeated roots be called e which we can also call n²/3; the third, unrepeated root is –2e. The solution is then

$zeta = frac{r_{s}}{4r} - frac{1}{12} = e - frac{n^{2}}{cosh^{2} nphi}.$

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 2r_s and 3r_s.

Stable circular orbits

If two roots are equal and negative, the orbits are circular, periodic orbits.

The radii of these orbits must be greater than 3r_s.

Derivation from the Hamilton–Jacobi equation

The same 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 in general relativity. The basic idea is that, due to gravitational slowing of time, parts of a wave-front closer to the gravitating mass move more slowly than those further away, thus bending the direction of the front's propagation (add Figure). In physics and mathematics, the Hamiltonâ€“Jacobi equation (HJE) is a reformulation of classical mechanics and, thus, equivalent to other formulations such as Newtons laws of motion, Lagrangian mechanics and Hamiltonian mechanics. ... General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

Using general covariance, the Hamilton–Jacobi equation for a single particle in arbitrary coordinates can be expressed as In physics and mathematics, the Hamiltonâ€“Jacobi equation (HJE) is a reformulation of classical mechanics and, thus, equivalent to other formulations such as Newtons laws of motion, Lagrangian mechanics and Hamiltonian mechanics. ...

$g^{munu} frac{partial S}{partial x^{mu}} frac{partial S}{partial x^{nu}} = m^{2} c^{2}$

Using Schwarzschild's solution for the metric g^μν, this equation becomes

$frac{1}{c^{2} left(1 - frac{r_{s}}{r} right)} left( frac{partial S}{partial t} right)^{2} - left( 1 - frac{r_{s}}{r} right) left( frac{partial S}{partial r} right)^{2} - frac{1}{r^{2}} left( frac{partial S}{partial phi} right)^{2} = m^{2} c^{2}$

where we again orient the spherical coordinate system into the plane of the orbit. The time and the azimuthal angle are cyclic coordinates, so that the solution for S can be written

S = - E t + L φ + S r (r)

where E and L again represent the particle's energy and angular momentum, respectively. The Hamilton–Jacobi equation therefore gives the integral solution This gyroscope remains upright while spinning due to its angular momentum. ... In physics and mathematics, the Hamiltonâ€“Jacobi equation (HJE) is a reformulation of classical mechanics and, thus, equivalent to other formulations such as Newtons laws of motion, Lagrangian mechanics and Hamiltonian mechanics. ...

$S_{r}(R) = int dr sqrt{frac{E^{2}}{J^{2} c^{2}} - frac{1}{J} left( m^{2} c^{2} + frac{L^{2}}{r^{2}} right)}$

where we introduce the function J(r) for brevity

$J(r) = 1 - frac{r_{s}}{r}$

The orbit is given by the normal technique

$frac{partial S}{partial L} = phi + frac{partial S_{r}}{partial L} = mathrm{constant}$

giving the same equation as derived earlier

$dr^{2} = left[ frac{E^{2}r^{4}}{L^{2} c^{2}} - J left( frac{m^{2} c^{2}}{L^{2}} r^{4} + r^{2} right) right] dphi^{2}.$

which has the solution

$phi = int frac{L dr}{r^{2} sqrt{frac{E^{2}}{c^{2}} - J left( m^{2} c^{2} + frac{L^{2}}{r^{2}} right)}}$

Bending of light

In the limit as the particle mass m goes to zero (appropriate for the photon), this equation becomes

$phi = int frac{dr}{r^{2} sqrt{frac{1}{rho^{2}} - frac{J}{r^{2}}}}$

where the length-scale ρ is introduced for brevity

$rho = frac{cL}{E}$

This distance is the distance of closest approach. Expanding in powers of r_s/r, the angular deflection for a massless particle coming from infinity and going to infinity equals

$delta phi = frac{2r_{s}}{rho} = frac{4GM}{c^{2}rho}$

For completeness, the radial part of Hamilton's principal function becomes in the limit of zero mass m

$S_{r}(R) = frac{E}{c} int dr sqrt{frac{r^{2}}{left( r - r_{s} right)^{2}} - frac{rho^{2}}{r left( r - r_{s} right)}}$

from which the orbital equation can be derived as above.

Rate of precession

To calculate the rate of precession for a planet such as Mercury, we use the equation This article is about the planet. ...

$Deltaphi = - frac{partial Delta S_{r}}{partial L}$

where Δφ and ΔS_r represent the changes over one orbit. The function S_r may be again expanded in powers of r_s/r, to obtain

$Delta S_{r} = Delta S_{r}^{0} - frac{3m^{2} c^{2} r_{s}^{2}}{4L} frac{partial Delta S_{r}^{0}}{partial L}$

In the Newtonian limit, the rotation angle is simply one revolution

$Deltaphi^{0} = - frac{partial Delta S_{r}^{0}}{partial L} = 2pi$

Therefore, the precession angle equals

$delta phi = Deltaphi - 2pi = frac{3pi m^{2} c^{2} r_{s}^{2}}{2L^{2}} = frac{6pi G^{2} m^{2} M^{2}}{c^{2} L^{2}}$

after substituting for the Schwarzschild radius r_s. The formula for the elliptical orbit states

$frac{L^{2}}{G M m^{2}} = a left( 1- e^{2} right)$

where e is the eccentricity and a the major semiaxis. This gives the final formula

$delta phi = frac{6pi G M}{c^{2} a left( 1- e^{2} right)}$

Notes

^ Landau 1975
^ Weinberg 1972
^ Whittaker 1937

References

Hagihara, Y (1931). "Unknown title". Japanese Journal of Astrophysics and Geophysics 8: 67–176.

Landau, LD (1975). The Classical Theory of Fields. New York: Pergamon Press, 299–309. ISBN 0-08-025072-6.

Weinberg, S (1972). <Gravitation and Cosmology. New York: John Wiley and Sons. ISBN 0-471-92567-5.

Whittaker, ET (1937). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies, 4th ed., New York: Dover Publications, pp. 389–393.

Category: Exact solutions in general relativity Lev Davidovich Landau Lev Davidovich Landau (Russian language: Ð›ÐµÌÐ² Ð”Ð°Ð²Ð¸ÌÐ´Ð¾Ð²Ð¸Ñ‡ Ð›Ð°Ð½Ð´Ð°ÌÑƒ) (January 22, 1908 â€“ April 1, 1968) was a prominent Soviet physicist, who made fundamental contributions to many areas of theoretical physics. ... Steven Weinberg (born May 3, 1933) is an American physicist. ... Edmund Taylor Whittaker (24 October 1873 - 24 March 1956) was an English mathematician, who contributed widely to applied mathematics, mathematical physics and the theory of special functions. ...

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