Two-body problem in general relativity

The two-body problem in general relativity (or relativistic two-body problem) is the study of how two objects move under the influence of gravity, as described by the theory of general relativity. It helps us understand important phenomena like the bending of light by gravity and the orbits of planets around stars. This problem is also crucial for describing the behavior of binary stars as they orbit each other and lose energy over time through gravitational radiation.

General relativity explains gravity by describing space-time as a flexible fabric that curves under the influence of mass. Solving the equations for this curvature is very challenging, especially for two moving objects. While there is no exact solution for the general case, an important approximation called the Schwarzschild solution works well when one object, like the Sun, is much more massive than the other, like a planet. This solution helps explain the unusual orbit of the planet Mercury and the bending of light near massive objects, both of which support Einstein’s theory.

When both objects have significant mass, such as in binary star systems, solutions are more complex and usually approximate. Recent advances allow scientists to use computers to simulate these interactions. An exciting example is the study of binary black holes, where scientists finally solved the problem numerically in 2005 after many years of research. These studies are important for understanding events like black hole mergers detected by gravitational wave observatories.

Historical context

Classical Kepler problem

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.

The Kepler problem is named after Johannes Kepler, who worked with the astronomer Tycho Brahe. Brahe made very careful measurements of how planets move in our Solar System. From these measurements, Kepler discovered three important laws that describe how planets orbit the Sun:

Each planet’s path is an ellipse with the Sun at one end.
A line connecting a planet to the Sun covers the same area in equal times.
The time it takes for a planet to go around the Sun is related to its distance from the Sun.

Kepler shared these ideas in the early 1600s. Later, Isaac Newton used his ideas about motion and gravity to explain why planets follow these paths. Newton showed that if two objects pull each other with a force that gets weaker as they move farther apart, they will travel in paths just like the ones Kepler described.

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.

Apsidal precession

Sometimes the shape of a planet’s path isn’t a perfect ellipse. If the pull between two objects isn’t exactly what Newton’s law predicts, the orbit can slowly turn over time. This turning is called apsidal precession. It happens because factors like the Sun’s shape or the pull from other planets can change the orbit slightly.

The perihelion precession of Mercury, contributed by in the course of Mercury's orbit at times moving deeper into the Sun's gravity field, which is significantly stronger at Mercury than at other planets.

Newton’s ideas worked very well for most planets. But one planet, Mercury, didn’t quite fit. Its path turned a little more than Newton’s laws predicted.

Anomalous precession of Mercury

Einstein's theory of general relativity

Around 1905, new ideas about physics, called relativity, showed that nothing can go faster than light. This meant Newton’s rules for gravity needed updating. From 1907 to 1915, Albert Einstein worked on a new theory. He used a clever idea: gravity feels the same as being pushed in an accelerating vehicle. To make his theory work, he had to change some basic ideas about space and time.

Einstein’s new theory explained why Mercury’s orbit acted differently and even predicted that light would bend when it passed near a heavy object like the Sun. These ideas were checked later and found to be true.

General relativity, special relativity and geometry

In normal Euclidean geometry, triangles follow the Pythagorean theorem, which helps us calculate distances in flat space. However, the world isn’t always flat — think of the curved surface of the Earth, where flat maps can’t show everything perfectly. This idea extends to space and time, where distances and shapes can curve due to gravity.

Albert Einstein’s special theory of relativity shows that distance isn’t always the same for every observer — it depends on how they’re moving. In his general theory of relativity, Einstein explained that space and time can curve around massive objects. This curvature changes how we calculate distances and how objects move, affecting everything from the path of light to the orbits of planets.

Schwarzschild solution

Main article: Schwarzschild geodesics

Comparison between the orbit of a test particle in Newtonian (left) and Schwarzschild (right) spacetime. Please click for high resolution animated graphics.

The Schwarzschild solution describes the space around a simple, non-spinning object. It helps us understand how gravity affects light and the paths of objects orbiting a star or planet. This solution is based on Einstein's theory of general relativity, which explains how mass and energy shape the space around them.

The solution includes a special distance called the Schwarzschild radius, which is very small for most objects but becomes important near very dense objects like neutron stars or black holes. It shows how orbits and the bending of light differ slightly from what we expect in simpler theories of gravity. These differences are tiny for planets orbiting stars but become significant near very strong gravitational fields.

Beyond the Schwarzschild solution

The Schwarzschild solution describes how a large, still object influences smaller objects moving around it, which works well for things like light bending near the Sun or planets orbiting. But when two objects of similar size, like binary stars, orbit each other, this solution isn’t enough.

Diagram of the parameter space of compact binaries with the various approximation schemes and their regions of validity.

Scientists use special math methods, called post-Newtonian approximations, to estimate how these objects move. They also use powerful computers to solve Einstein’s equations directly, which can give even more precise answers. One famous example is understanding how two black holes merge.