In general relativity, Schwarzschild geodesics describe the motion of test particles in the gravitational field of a central fixed mass that is, motion in the Schwarzschild metric. 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 masses so small they contribute little to the gravitational field. However, they are highly accurate in many astrophysical scenarios provided that is many-fold smaller than the central mass , 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 is set equal to the sum of the two individual masses and . This is important in predicting the motion of binary stars in general relativity. Historical context 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. In 1931, Yusuke Hagihara published a paper showing that the trajectory of a test particle in the Schwarzschild metric can be expressed in terms of elliptic functions. Schwarzschild metric 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 . The Schwarzschild solution can be written as where , in the case of a test particle of small positive mass, is the proper time (time measured by a clock moving with the particle) in seconds, is the speed of light in meters per second, is, for , the time coordinate (time measured by a stationary clock at infinity) in seconds, is, for , the radial coordinate (circumference of a circle centered at the star divided by ) in meters, is the colatitude (angle from North) in radians, is the longitude in radians, and is the Schwarzschild radius of the massive body (in meters), which is related to its mass by where is the gravitational constant. The classical Newtonian theory of gravity is recovered in the limit as the ratio goes to zero.