Two-body problem in 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.

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.

If both masses are considered to contribute to the gravitational field, as in binary stars, the Kepler problem can be solved only approximately.

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.

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.

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.

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.

[5] After these explanations were discounted, some physicists were driven to the more radical hypothesis that Newton's inverse-square law of gravitation was incorrect.

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.

[5] 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.

Conversely, all local gravitational effects should be reproducible in a linearly accelerating reference frame, and vice versa.

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.

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.

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 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 does not depend on the motion of the observer.

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.

According to Einstein's theory of general relativity, particles of negligible mass travel along geodesics in the space-time.

where Γ represents the Christoffel symbol and the variable q parametrizes the particle's path through space-time, its so-called world line.

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.

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

As shown below and elsewhere, this inverse-cubic energy causes elliptical orbits to precess gradually by an angle δφ per revolution

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.

This is a reasonable approximation for photons and the orbit of Mercury, which is roughly 6 million times lighter than the Sun.

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.

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.

Figure 1. Typical elliptical path of a smaller mass m orbiting a much larger mass M . The larger mass is also moving on an elliptical orbit, but it is too small to be seen because M is much greater than m . The ends of the diameter indicate the apsides , the points of closest and farthest distance.
In the absence of any other forces, a particle orbiting another under the influence of Newtonian gravity follows the same perfect ellipse eternally. The presence of other forces (such as the gravitation of other planets), causes this ellipse to rotate gradually. The rate of this rotation (called orbital precession) can be measured very accurately. The rate can also be predicted knowing the magnitudes and directions of the other forces. However, the predictions of Newtonian gravity do not match the observations, as discovered in 1859 from observations of Mercury.
Eddington 's 1919 measurements of the bending of star -light by the Sun 's gravity led to the acceptance of general relativity worldwide.
Comparison between the orbit of a test particle in Newtonian (left) and Schwarzschild (right) spacetime. Please click for high resolution animated graphics.
Effective radial potential for various angular momenta. At small radii, the energy drops precipitously, causing the particle to be pulled inexorably inwards to r = 0. However, when the normalized angular momentum a / r s = L / mcr s equals the square root of three, a metastable circular orbit is possible at the radius highlighted with a green circle. At higher angular momenta, there is a significant centrifugal barrier (orange curve) and an unstable inner radius, highlighted in red.
The stable and unstable radii are plotted versus the normalized angular momentum a / r s = L / mcr s in blue and red, respectively. These curves meet at a unique circular orbit (green circle) when the normalized angular momentum equals the square root of three. For comparison, the classical radius predicted from the centripetal acceleration and Newton's law of gravity is plotted in black.
In the non-relativistic Kepler problem , a particle follows the same perfect ellipse (red orbit) eternally. General relativity introduces a third force that attracts the particle slightly more strongly than Newtonian gravity, especially at small radii. This third force causes the particle's elliptical orbit to precess (cyan orbit) in the direction of its rotation; this effect has been measured in Mercury , Venus and Earth. The yellow dot within the orbits represents the center of attraction, such as the Sun .
Diagram of the parameter space of compact binaries with the various approximation schemes and their regions of validity.