The Reissner-Nordström Metric

In 1916 and 1918 respectively Hans Jacob Reissner (1874 Jan 18 – 1967 Oct 02), a German aeronautical engineer (who pursued mathematical physics as an avocation), and Gunnar Nordström (1881 Mar 12 – 1923 Dec 24), a Finnish theoretical physicist, devised a modified version of Schwarzschild’s solution of Einstein’s Equation of space-time warping. They produced the metric equation

(Eq’n 1)

in which

(Eq’n 2)

for a radially symmetric body pondering a mass M and bearing an electric charge Q. If the body were to shed its electric charge, Equation 1 would revert to the metric equation describing the Schwarzschild solution.

Now we want to deduce the Reissner-Nordström solution as we did with the Schwarzschild solution. To that end we recall to mind the equivalence principle: if we have two points stationary relative to each other in a gravitational field, they occupy two different warped inertial frames. Suppose the existence of two points, point B lying deep inside a gravitational field and point A lying far enough above it to lie effectively "at infinity". In that case, point A lies arbitrarily close to a set of inertial frames that, occupied and marked by freely falling bodies, pass point B at speeds arbitrarily close to the speed determined by the gravitational potential between the two points, the speed derived from

(Eq’n 3)

We then substitute that velocity into a restricted version of the Lorentz Transformation, one that consists simply of a time dilation and a Lorentz-Fitzgerald contraction, thereby obtaining transformation equations equivalent to the Schwarzschild metric equation. So how does the electric force come into play?

Consider how the warpage of space and time gets
established. Imagine that a mass M exists extremely far from the origin of our
coordinate grid, where our gravitating body will eventually exist. It exists in
the form of a great spherical shell that surrounds the origin and, one by one,
component shells, each pondering a differential mass of dM, peel off of the main
shell and collapse onto the origin. As each shell collapses it warps the space
and time through which it passes into a gravitational field. The next shell to
come down thus occupies and marks an inertial frame deformed by that
gravitational field, while warping the space and time even further. We know that
the work done on each shell by the gravitational field increases the mass-energy
of the shell, but that work gets converted into heat when the shell comes down
on the growing body at the origin and that heat radiates back into space. Thus
the body ends up pondering a total mass of M= dM, which then surrounds itself
with the Schwarzschild metric of Equation 1 with f(r)=1-2MG/rc^{2}.

Now imagine adding an electric charge Q to the above imaginary experiment. The electric force, because it does not conform to the equivalence principle, does not contribute directly to the warping of the inertial frame. But the shell-by-shell assembly of the charged body necessitates that the gravitational force do work to overcome the electric repulsion between the growing body and the next component shell coming down onto it. That work must come out of the final mass of the fully assembled body. If we add an electric charge Q to our gravitating body, then each shell carries dM and dQ onto the growing body, so we calculate the work necessary to overcome the electric repulsion as

(Eq’n 4)

Thus when we come to the distance R from the center of mass of the body that much work has been taken out of the gravitational field. The effective mass of the body generating the gravitational field thus becomes

(Eq’n 5)

We simply substitute that into the Schwarzschild solution to obtain the Reissner-Nordström solution. We get both the metric equation and the equations of what we call the Reissner-Nordström Transformation:

(Eq’ns 6)

Note that for convenience I have left the latitudinal and longitudinal parts of the transformation out of the metric equation.

But this only gives us the semi-classical solution, which only remains valid in those regions of space from which escape velocity stands substantially less than the speed of light. However, since we have already derived the equations pertaining to the fully-relativistic Schwarzschild solution, we need only substitute the reduced mass from Equation 5 into the equations for that solution to get:

(Eq’ns 7)

Thus we have the fully relativistic Reissner-Nordström solution.

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