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A particle orbiting in the Schwarzschild metric can have a stable circular orbit with r > 3r s. Circular orbits with r between 1.5r s and 3r s are unstable, and no circular orbits exist for r < 1.5r s. The circular orbit of minimum radius 1.5r s corresponds to an orbital velocity approaching the speed of light.
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. [1] Samuil Kaplan in 1949 has shown that there is a minimum radius for the circular orbit to be stable in Schwarzschild metric. [2]
For example, the meaning of "r" is physical distance in that classical law, and merely a coordinate in General Relativity.] The Schwarzschild metric can also be derived using the known physics for a circular orbit and a temporarily stationary point mass. [1] Start with the metric with coefficients that are unknown coefficients of :
For example, the Schwarzschild radius r s of the Earth is roughly 9 mm (3 ⁄ 8 inch); at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The Schwarzschild radius of the Sun is much larger, roughly 2953 meters, but at its surface, the ratio r s /r is roughly 4 parts in a
In the Schwarzschild metric, free-falling objects can be in circular orbits if the orbital radius is larger than (the radius of the photon sphere). The formula for a clock at rest is given above; the formula below gives the general relativistic time dilation for a clock in a circular orbit: [11] [12]
Since a Schwarzschild black hole has spherical symmetry, all possible axes for a circular photon orbit are equivalent, and all circular orbits have the same radius. This derivation involves using the Schwarzschild metric , given by
The Kerr metric is commonly expressed in one of two forms, the Boyer–Lindquist form and the Kerr–Schild form. It can be readily derived from the Schwarzschild metric, using the Newman–Janis algorithm [6] by Newman–Penrose formalism (also known as the spin–coefficient formalism), [7] Ernst equation, [8] or Ellipsoid coordinate ...
A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle. In this case, ... In Schwarzschild metric, ...