Two-body problem in general relativity
General relativity |
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The two-body problem in general relativity (or relativistic two-body problem) is the determination of the motion and
General relativity describes the gravitational field by curved space-time; the
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
For binary black holes, the numerical solution of the two-body problem was achieved after four decades of research in 2005 when three groups devised breakthrough techniques.[1][2][3]
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 semi-major axisof 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
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
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 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.[dubious ] These calculations were carried out initially by Pierre-Simon Laplace in the late 18th century, and refined by Félix Tisserand in the later 19th century. Conversely, if Newton's law of gravitation did not predict the apsidal precessions of the planets accurately, it would have to be discarded as a theory of gravitation. Such an anomalous precession was observed in the second half of the 19th century.
Anomalous precession of Mercury
In 1859,
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'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.[note 1] To avoid those problems, between 1870 and 1900 many scientists used the electrodynamic laws of Wilhelm Eduard Weber, Carl Friedrich Gauss, Bernhard Riemann to produce stable orbits and to explain the perihelion shift of Mercury's orbit. In 1890, Maurice Lévy succeeded in doing so by combining the laws of Weber and Riemann, whereby the speed of gravity is equal to the speed of light in his theory. And in another attempt Paul Gerber (1898) even succeeded in deriving the correct formula for the perihelion shift (which was identical to that formula later used by Einstein). However, because the basic laws of Weber and others were wrong (for example, Weber's law was superseded by Maxwell's theory), those hypotheses were rejected.[8] Another attempt by Hendrik Lorentz (1900), who already used Maxwell's theory, produced a perihelion shift which was too low.[5]
Einstein's theory of general relativity
Around 1904–1905, the works of
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
where dx, dy and dz represent the infinitesimal differences between the x, y and z coordinates of two points in a Cartesian coordinate system. Now imagine a world in which this is not quite true; a world where the distance is instead given by
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
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
where the nine functions gxx, gxy, ..., gzz 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 grr = 1, gθθ = r2 and gφφ = r2 sin2 θ.
In his
which may be written in spherical coordinates as
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
just as we generalized the formula to measure distance on the surface of the Earth. The exact form of the metric gμν 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
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[10]
where Γ represents the
Schwarzschild solution
An exact solution to the Einstein field equations is the Schwarzschild metric, which corresponds to the external gravitational field of a stationary, uncharged, non-rotating, spherically symmetric body of mass M. It is characterized by a length scale rs, known as the Schwarzschild radius, which is defined by the formula
In practice, this ratio is almost always extremely small. For example, the Schwarzschild radius rs of the Earth is roughly 9
Orbits about the central mass
The orbits of a test particle of infinitesimal mass about the central mass is given by the equation of motion
where is the
where, for brevity, two length-scales, and , 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
Effective radial potential energy
The equation of motion for the particle derived above
can be rewritten using the definition of the Schwarzschild radius rs as
which is equivalent to a particle moving in a one-dimensional effective potential
The first two terms are well-known classical energies, the first being the attractive Newtonian
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 of the classical closed orbit.
The third term is attractive and dominates at small r values, giving a critical inner radius rinner 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:
Circular orbits are possible when the effective force is zero:
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 rinner and router:
which are obtained using the quadratic formula. The inner radius rinner 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 rinner (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 rs (the classical case), these formulae become approximately
Substituting the definitions of a and rs into router yields the classical formula for a particle of mass m orbiting a body of mass M.
The following equation
where ωφ is the orbital angular speed of the particle, is obtained in non-relativistic mechanics by setting the centrifugal force equal to the Newtonian gravitational force:
Where is the reduced mass.
In our notation, the classical orbital angular speed equals
At the other extreme, when a2 approaches 3rs2 from above, the two radii converge to a single value
The quadratic solutions above ensure that router is always greater than 3rs, whereas rinner lies between 3⁄2 rs and 3rs. Circular orbits smaller than 3⁄2 rs are not possible. For massless particles, a goes to infinity, implying that there is a circular orbit for photons at rinner = 3⁄2 rs. 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 router will oscillate in a stable manner with an angular frequency
which equals
Taking the square root of both sides and expanding using the binomial theorem yields the formula
Multiplying by the period T of one revolution gives the precession of the orbit per revolution
where we have used ωφT = 2π and the definition of the length-scale a. Substituting the definition of the Schwarzschild radius rs gives
This may be simplified using the elliptical orbit's semi-major axis A and eccentricity e related by the formula
to give the precession angle
Since the closed classical orbit is an ellipse in general, the quantity A(1 − e2) is the semi-latus rectum l of the ellipse.
Hence, the final formula of angular apsidal precession for a unit complete revolution is
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 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
The post-Newtonian expansion is a calculational method that provides a series of ever more accurate solutions to a given problem.[12] 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.
Modern computational approaches
Einstein's equations can also be solved on a computer using sophisticated numerical methods.[1][2][3] 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
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.[13] The rate of losing energy (averaged over a complete orbit) is given by[14]
where e is the
The rate of period decrease is given by[13][15]
where Pb 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.
-
Experimentally observed changes in the time of thePSR B1913+16 (red dots) matches the change due to the reduction in orbital period predicted by general relativity(blue curve) almost exactly.
-
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
- Center of mass (relativistic)
- Gravitational two-body problem
- Kepler problem
- Newton's theorem of revolving orbits
- Schwarzschild geodesics
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: e.g., see Gravitoelectromagnetism.
References
- ^ S2CID 24225193.
- ^ S2CID 5954627.
- ^ S2CID 23409406.
- Comptes Rendus. 49: 379–383.
- ^ a b c Pais 1982, pp. 253–256.
- ^ Pais 1982, p. 254.
- ^ Kopeikin, Efroimsky & Kaplan 2011, p. 177.
- ^ Roseveare 1982, p. [page needed].
- ^ Walter 2007, p. [page needed].
- ^ Weinberg 1972, p. [page needed].
- S2CID 119083668.
- ^ Kopeikin, Efroimsky & Kaplan 2011, p. [page needed].
- ^ .
- ^ Landau & Lifshitz 1975, pp. 356–357.
- Bibcode:2005ASPC..328...25W.
Bibliography
- Adler, R; Bazin M; Schiffer M (1965). Introduction to General Relativity. New York: McGraw-Hill Book Company. pp. 177–193. ISBN 978-0-07-000420-7.
- ISBN 978-0-691-02352-6.
- ISSN 0368-346X.
- Kopeikin, Sergei; Efroimsky, Michael; Kaplan, George (25 October 2011). Relativistic Celestial Mechanics of the Solar System. John Wiley & Sons. ISBN 978-3-527-63457-6.
- ISBN 978-0-486-65067-8.
- ISBN 978-0-08-018176-9.
- ISBN 978-0-7167-0344-0. (See Gravitation (book).)
- ISBN 0-19-520438-7.
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- ISBN 978-0-387-10090-6.
- Roseveare, N. T. (1982). Mercury's perihelion, from Leverrier to Einstein. Oxford: University Press. ISBN 0-19-858174-2.
- ISBN 978-0-7204-0066-3.
- ISBN 978-0-226-87032-8.
- Walter, S. (2007). "Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910". In Renn, J. (ed.). The Genesis of General Relativity. Vol. 3. Berlin: Springer. pp. 193–252. Archived from the original on 2009-01-30. Retrieved 2008-07-26.
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