Reissner–Nordström black hole

In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M.

The metric was discovered by Hans Reissner and Gunnar Nordström.

In spherical coordinates (t, r, θ, φ), the line element for the Reissner–Nordström metric is

where c is the speed of light, t is the time coordinate (measured by a stationary clock at infinity), r is the radial coordinate, ${\displaystyle \textstyle d\Omega _{(2)}^{2}}$ is a 2-sphere defined by

r_S is the Schwarzschild radius of the body given by

and r_Q is a characteristic length scale given by

Here 1/4πε₀ is Coulomb force constant.

In the limit that the charge Q (or equivalently, the length-scale r_Q) goes to zero, one recovers the Schwarzschild metric. The classical Newtonian theory of gravity may then be recovered in the limit as the ratio r_S/r goes to zero. In that limit that both r_Q/r and r_S/r go to zero, the metric becomes the Minkowski metric for special relativity.

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