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This is unfounded because that law has relativistic corrections. 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]
For example, the Schwarzschild radius () of the Earth is roughly 8.9 mm, while the Sun, which is 3.3 × 10 5 times as massive [6] has a Schwarzschild radius () of approximately 3.0 km. The ratio becomes large only in close proximity to black holes and other ultra-dense objects such as neutron stars .
The simplest example of a Lorentzian manifold is flat spacetime, which can be given as R 4 with coordinates (,,,) and the metric = + + + =. These coordinates actually cover all of R 4 . The flat space metric (or Minkowski metric ) is often denoted by the symbol η and is the metric used in special relativity .
The symbol was introduced originally in 1770 by Nicolas de Condorcet, who used it for a partial differential, and adopted for the partial derivative by Adrien-Marie Legendre in 1786. [3] It represents a specialized cursive type of the letter d , just as the integral sign originates as a specialized type of a long s (first used in print by ...
for the nth derivative. When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are: [6]
Solving the geodesic equations is a procedure used in mathematics, particularly Riemannian geometry, and in physics, particularly in general relativity, that results in obtaining geodesics. Physically, these represent the paths of (usually ideal) particles with no proper acceleration , their motion satisfying the geodesic equations.
Examples of important exact solutions include the Schwarzschild solution and the Friedman-Lemaître-Robertson–Walker solution. The EIH approximation plus other references (e.g. Geroch and Jang, 1975 - 'Motion of a body in general relativity', JMP, Vol. 16 Issue 1).
The gradient of a function is obtained by raising the index of the differential , whose components are given by: =; =; =, = = The divergence of a vector field with components is