Schwarzschild geodesics

In general relativity, Schwarzschild geodesics describe the motion of particles of infinitesimal mass in the gravitational field of a central fixed mass M {displaystyle M} . Schwarzschild geodesics have been pivotal in the validation of Einstein's theory of general relativity. For example, they provide accurate predictions of the anomalous precession of the planets in the Solar System, and of the deflection of light by gravity. In general relativity, Schwarzschild geodesics describe the motion of particles of infinitesimal mass in the gravitational field of a central fixed mass M {displaystyle M} . Schwarzschild geodesics have been pivotal in the validation of Einstein's theory of general relativity. For example, they provide accurate predictions of the anomalous precession of the planets in the Solar System, and of the deflection of light by gravity. Schwarzschild geodesics pertain only to the motion of particles of infinitesimal mass m {displaystyle m} , i.e., particles that do not themselves contribute to the gravitational field. However, they are highly accurate provided that m {displaystyle m} is many-fold smaller than the central mass M {displaystyle M} , e.g., for planets orbiting their sun. Schwarzschild geodesics are also a good approximation to the relative motion of two bodies of arbitrary mass, provided that the Schwarzschild mass M {displaystyle M} is set equal to the sum of the two individual masses m 1 {displaystyle m_{1}} and m 2 {displaystyle m_{2}} . This is important in predicting the motion of binary stars in general relativity. The Schwarzschild solution was found by Karl Schwarzschild shortly after Einstein published his field equations. The Schwarzschild metric is named in honour of its discoverer Karl Schwarzschild, who found the solution in 1915, only about a month after the publication of Einstein's theory of general relativity. It was the first exact solution of the Einstein field equations other than the trivial flat space solution. An exact solution to the Einstein field equations is the Schwarzschild metric, which corresponds to the external gravitational field of an uncharged, non-rotating, spherically symmetric body of mass M {displaystyle M} . The Schwarzschild solution can be written as