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Noteworthy examples of vacuum solutions, electrovacuum solutions, and so forth, are listed in specialized articles (see below). These solutions contain at most one contribution to the energy–momentum tensor, due to a specific kind of matter or field. However, there are some notable exact solutions which contain two or three contributions ...
The no-hair theorem (which is a hypothesis) states that all stationary black hole solutions of the Einstein–Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three independent externally observable classical parameters: mass, angular momentum, and electric charge.
The sources of any gravitational field (matter and energy) is represented in relativity by a type (0, 2) symmetric tensor called the energy–momentum tensor. It is closely related to the Ricci tensor. Being a second rank tensor in four dimensions, the energy–momentum tensor may be viewed as a 4 by 4 matrix.
But if the exact solution is required or a solution describing strong fields, the evolution of the metric and the stress–energy tensor must be solved for together. To obtain solutions, the relevant equations are the above quoted EFE (in either form) plus the continuity equation (to determine evolution of the stress–energy tensor):
For example, Christoffel symbols cannot be tensors themselves if the coordinates do not change in a linear way. In general relativity, one cannot describe the energy and momentum of the gravitational field by an energy–momentum tensor. Instead, one introduces objects that behave as tensors only with respect to restricted coordinate ...
No exact solutions of the Kepler problem have been found, but an approximate solution has: the Schwarzschild solution. This solution pertains when the mass M of one body is overwhelmingly greater than the mass m of the other. If so, the larger mass may be taken as stationary and the sole contributor to the gravitational field.
is the gravitational constant and the big O notation is used to indicate that terms of order c −4 or beyond have been omitted. The coordinates used here are harmonic. The first term on the right hand side is the Newtonian gravitational acceleration at A; in the limit as c → ∞, one recovers Newton's law of motion.
A common misconception occurs between centre of mass and centre of gravity.They are defined in similar ways but are not exactly the same quantity. Centre of mass is the mathematical description of placing all the mass in the region considered to one position, centre of gravity is a real physical quantity, the point of a body where the gravitational force acts.