The Reissner-Nordström Metric Revisited

In 1916 Hans Jacob Reissner (1874 Jan 18 – 1967 Oct 02), a German aeronautical engineer, who pursued mathematical physics as an avocation, devised the precursor of what we now call the Reissner-Nordström metric equation. He showed in "Über die Eigengravitation des elektrischen Feldes der Einsteinschen Theorie" (Annalen der Physik (Leipzig), Vol. 50, pp 106 - 120) that an electrically charged point mass (an electron) has a naked singularity, but not an event horizon. In 1918 the Finnish theoretical physicist, Gunnar Nordström (1881 Mar 12 – 1923 Dec 24), extended Reissner’s result for a point charge to a spherically symmetrical, electrically charged body and reported his own result in "On the Energy of the Gravitational Field in Einstein’s Theory" (Verhandl. Koninkl. Ned. Akad. Wetenschap. Afdel. Natuurk., Amsterdam, Vol 26, pp 1201 - 1208). The metric equation that Nordström produced looks like Schwarzschild’s. For a body of mass M and electric charge Q the equation takes the form

(Eq’n 1)

in which the coefficient

(Eq’n 2)

When we worked out the virtual-velocity form of the
Schwarzschild solution of Einstein’s equation, we obtained the 2MG/rc^{2}
term by equating the square of the velocity of a body falling freely in a
Newtonian gravitational field to twice the gravitational potential. Substituting
that result into the equation for the Lorentz factor gave us the coefficient
that we needed in our transformation equations. Now we have

(Eq’n 3)

The electric charge has the effect of slowing the fall of a free body and, thus, of the reference frame that the body marks. It achieves that effect by diminishing the gravitational potential.

As we can see from the form in which I put the second term on the right side of Equation 3, we have an electrostatic potential energy divided by the square of lightspeed, which calculation gives us the equivalent of a mass. Multiplying that result by G/r gives the term the same form as the first term, though the second term diminishes the first and we now want to understand how it does that.

Imagine that we have created a uniform, spherical body of mass M with no electric charge on it. Far enough from the body that its gravitational pull comes arbitrarily close to zero (that is, at a point that we call pseudo-infinity) we assemble a numerous collection of minuscule bodies whose masses will not add a noticeable amount to M. Each of the bodies carries an electric charge dq, the same amount and kind of charge on all of the bodies so that dq=Q. One by one, each of those bodies descends onto the main body and deposits its charge there. As electric charge builds up on the main body, an electrostatic field emanates from it and grows ever stronger.

Because the electric field repels the small bodies from the main body, something must do work upon them to bring them down on the main body against the repulsion. That something must be the gravitational field of the main body, which provides the only other force existing in this system. We can calculate easily the amount of work that gravity must do in getting all of the small bodies down onto the main body. Integrating the work done on each small body over the full set of small bodies gives us

(Eq’n 4)

But now the conservation of energy theorem tells us that the electric charge gaining that much energy in coming down on the main body must be matched by an equal loss of energy from some other part of the system. That loss must come necessarily out of the gravitational potential of the main body’s field. When we express the potential in algebraic form it will look as if the mass of the main body has decreased slightly; that is, we will divide Equation 4 by the square of lightspeed and subtract the quotient from the mass in the formula for the basic gravitational potential energy.

If we drop a small, electrically neutral body of mass m onto the main body from pseudo-infinity, we can calculate its radial velocity at any point by equating the kinetic energy that it gains to the potential energy that it loses; that is, we have the equation

(Eq’n 5)

Dividing that equation by m and multiplying it by two gives us the squared speed at which the warped reference frame that the dropped body occupies and marks moves. That calculation enables us to modify the Lorentz factor between the reference frames occupied by two observers, one at pseudo-infinity and the other at some distance r from the center of the main body, who have no actual motion between them. We get

(Eq’n 6)

As we did with the Schwarzschild solution, we can now devise transformation equations that the two observers would use, each to translate the other’s measurements of two given events into what they themselves would measure. Because the system has perfect spherical symmetry, we ignore the latitudinal and longitudinal coordinates and focus our attention on the radial and temporal coordinates. Again, upper-case variables represent measurements made by the upper observer and lower-case variables represent measurements made by the lower observer.

As in the Schwarzschild case, we can devise our transformation equations by little more than considering a pulse of light passing from one observer to the other. Because light and matter are, in concept, interchangeable, we know that light descending in a gravitational field gains energy and light rising through a gravitational field loses energy; thus, gravity alters the frequency and wavelength of light. Physicists know this phenomenon as the gravitational redshift.

Both observers use pulses of light as both clocks and rulers in making measurements of the two given events (whatever they may be) and they use lasers to communicate with each other. The upper observer knows that when the lower observer’s laser beam reaches the upper observation post it has a longer wavelength and a lower frequency than it had when it left the lower observation post. That knowledge enables the upper observer to infer that the lower observer’s rulers are putting too few centimeters between two events separated from each other in the radial direction and that the lower observer’s clocks are putting too few seconds between the events. Conversely, the lower observer infers that the upper observer puts too many centimeters and too many seconds between the two events. But those facts, by themselves, don’t give us the transformation that we want. We need to get more mathematical.

The two events occur separated by (dr, dt) as measured by the lower observer and separated by (dR, dT) as measured by the upper observer. We need to consider how the observers measure those quantities. Let’s assume that both events occur close enough to the body M that the lower observer can measure the distance and duration between them directly.

Using the flashing of colored lights to mark the events, the lower observer calculates the interval that elapses between the flashes and then adds or subtracts a correction due to the time it takes the light from one of the flashes to cross the distance to the location of the other flash. The upper observer also detects the flashes and measures the interval that elapses between them. That observer also knows that the light has been subjected to something like a Doppler shift due to the virtual motion that gravity puts between them and the lower observer, so, using a form of the relativistic Doppler shift, they write

(Eq’n 7)

In that equation the primes indicate the actual measurement of the time interval uncorrected for the distance between the events. Also the equation reflects the fact that the actual velocity (V) between the observers equals zero. If we include the correction for distance, assuming the distance is only radial and has no longitudinal or latitudinal components, we have

(Eq’n 8)

Clearly events separated by a short time in the lower observer’s experience are separated by a longer elapse of time in the upper observer’s experience.

Our previous statements about the gravitational redshift of light imply that

(Eq’n 9)

which is also consistent with Equation 8. But that is not what the observers would actually measure. In order to justify that redshift, the upper observer must treat the points at which the events occur as if they were moving away from the upper observation post. If we take a positive temporal difference as indicating that the uppermore event occurred before the lowermore event, then we must have

(Eq’n 10)

in which v represents the virtual velocity between the observers. By the same reasoning that we use in Special Relativity, we must infer the occurrence of a temporal offset between the clocks recording the events, one such that

(Eq’n 11)

The observers won’t actully measure the offsets depicted
in Equations 10 and 11; instead, they observe an effect that appears as a result
of setting dT=0 (that is, the upper observer has contrived to measure the
distance between the events as if they had occurred simultaneously in their
frame). Substituting dt=-vdr/c^{2} into Equation 10 gives us

(Eq’n 12)

which leads us to

(Eq’n 13)

That dilation of space, as measured by the upper observer, makes the transformation equations asymmetrical. In the Schwarzschild case it also explains several observed phenomena, such as the Shapiro delay of and the deflection of light passing a gravitating body.

So now we can combine Equations 8 and 13 with Equation 6 to obtain the equations of the Reissner-Nordström transformation:

(Eqns 14)

We can also write those equations in terms of a transformation matrix and get

(Eq’n 15)

In that equation I multiplied the temporal components by the square root of minus one so that the following calculations will put a minus sign in the correct place in the metric tensor. If we multiply that equation, in the manner of a dot product, by itself, we will obtain the corresponding metric equation;

(Eq’n 16)

In that calculation, in going to the third line, I replaced the square root
of minus one on the temporal terms with a minus one on the 4,4 term in the
matrix in order to get the right coefficient on the temporal term in the metric
equation. In the second step I transposed the first two factors so that I could
multiply the transformation matrix and its transpose together to obtain the
metric tensor. But the metric tensor on the third line doesn’t look right. In
polar coordinates the g_{22} and g_{33} elements should not be
equal to one. Thus, we pull the coefficients off the angular terms in the
four-vectors and transfer them into the appropriate places in the metric tensor.
We have then

(Eq’n 17)

Finally we substitute from Equation 6 into the expression on the last line of Equation 16 to get the full Reissner-Nordström metric equation in all of its coruscating glory:

(Eq’n 18)

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