Appendix 4 The Schwarzschild Formula Extended for Electric Charges | The General Theory of Relativity of Einstein

Appendix 4 – The Schwarzschild Formula Extended for Electric Charges

Reissner–Nordström metric

The correct solution within general relativity for a charged, non-rotating, spherically symmetric mass is the Reissner–Nordström metric (1918). This metric describes the spacetime interval around a charged mass and incorporates both the gravitational and electrical contributions:

\begin{equation} \begin{aligned} ds^{2} = c^{2} d\tau^{2} &= \left( 1 - \frac{2GM}{c^{2} r} + \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} \right) c^{2} dt^{2} - \left( 1 - \frac{2GM}{c^{2} r} + \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} \right)^{-1} dr^{2} - r^{2} d\theta^{2} - r^{2} \sin^{2}\theta\, d\phi^{2} \end{aligned} \end{equation}

\begin{equation} \begin{aligned} ds^{2} = c^{2} d\tau^{2} &= \left( 1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) c^{2} dt^{2} - \left( 1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right)^{-1} dr^{2} - r^{2} d\theta^{2} - r^{2} \sin^{2}\theta\, d\phi^{2} \end{aligned} \end{equation}

where:

\( r_{s} = \frac{2GM}{c^{2}} \) : Schwarzschild radius (mass effect),
\( r_{Q}^{2} = \frac{GQ^{2}}{4\pi \varepsilon_{0} c^{4}} \) : charge term,
\( Q \) : electric charge of the central object,
\( M \) : mass of the object,
\( G \) : gravitational constant,
\( c \) : speed of light.

Interpretation

The first term \( \frac{r_{s}}{r} \) represents the gravitational curvature of spacetime as in the Schwarzschild solution.
The additional term \( \frac{r_{Q}^{2}}{r^{2}} \) describes the repulsive electromagnetic effect (for like charges) within general relativity.
This metric reduces to the standard Schwarzschild solution when \( Q = 0 \).
For rotating or more general charged objects, more general solutions exist, such as the Kerr–Newman metric.

In classical Newtonian physics, there exists a gravitational field in vacuum. However, according to Einstein, there is no gravitational field, but rather spacetime is curved due to gravity. In that case, \( T_{\mu\nu} = 0 \).

In the case of the Reissner–Nordström solution, the stress-energy tensor \( T_{\mu\nu} \) is not zero everywhere, even though one speaks of a “vacuum.”

Explanation

In the classical Schwarzschild case (without charge), \( T_{\mu\nu} = 0 \) in the vacuum outside the mass.
In the Reissner–Nordström case, however, there remains an electromagnetic field in the vacuum, which carries energy and momentum.
Specifically, \( T_{\mu\nu} \) describes the energy-momentum of the radial electric field.

In summary: in the Reissner–Nordström metric, \( T_{\mu\nu} \neq 0 \) in the “vacuum” because the electromagnetic field is physically real and contains energy.

Derivation of the Reissner–Nordström metric

Below follows a step-by-step derivation of the Reissner–Nordström metric from the Einstein–Maxwell equations.

Step 1: Assumptions — metric and source

We seek a static, spherically symmetric solution:

\begin{equation} \begin{aligned} ds^{2} = c^{2} d\tau^{2} = A(r)\, c^{2} dt^{2} - B(r)\, dr^{2} - r^{2} d\theta^{2} - r^{2} \sin^{2}\theta\, d\phi^{2} \end{aligned} \end{equation}

where \( A(r) \) and \( B(r) \) are unknown functions.

The source is an electromagnetic field of a point charge \( Q \).
The stress-energy tensor of the electromagnetic field is (in natural units):
\begin{equation} \begin{aligned} T_{\mu\nu} = \frac{1}{\mu_{0}} \left( F_{\mu\alpha} F_{\nu}{}^{\alpha} - \frac{1}{4} g_{\mu\nu} F_{\alpha\beta} F^{\alpha\beta} \right) \end{aligned} \end{equation}

Here:

\( F_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} \) is the electromagnetic field tensor,
\( A_{\mu} \) is the four-dimensional electromagnetic potential,
\( g_{\mu\nu} \) is the metric,
\( \mu_{0} = 4\pi \times 10^{-7}\,\mathrm{H/m} \) is the magnetic permeability of the vacuum.

For the four-potential we take:

\begin{equation} \begin{aligned} A_{\mu} = \left( \frac{\Phi}{c},\, -\mathbf{A} \right) \quad \text{or} \quad A^{\mu} = \left( \frac{\Phi}{c},\, \mathbf{A} \right) \end{aligned} \end{equation}

\( \Phi \) : electric potential (in volts),
\( \mathbf{A} = (A_{x}, A_{y}, A_{z}) \) : magnetic vector potential (in weber per meter),
\( c \) : speed of light.

The electromagnetic field tensor

The electromagnetic field tensor (electromagnetic field strength tensor) \( F_{\mu\nu} \) contains all information about the electric field \( \mathbf{E} \) and the magnetic field \( \mathbf{B} \).

In matrix form:

\begin{equation} \begin{aligned} F_{\mu\nu} = \begin{pmatrix} 0 & -E_{x}/c & -E_{y}/c & -E_{z}/c \\ E_{x}/c & 0 & -B_{z} & B_{y} \\ E_{y}/c & B_{z} & 0 & -B_{x} \\ E_{z}/c & -B_{y} & B_{x} & 0 \end{pmatrix} \end{aligned} \end{equation}

For a purely radial electric field of a point charge \( Q \), this becomes:

\begin{equation} \begin{aligned} F_{\mu\nu} = \begin{pmatrix} 0 & \frac{Q}{r^{2}} & 0 & 0 \\ -\frac{Q}{r^{2}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \end{aligned} \end{equation}

The only non-zero component of \( F_{\mu\nu} \) in this radial field is:

\begin{equation} \begin{aligned} F_{tr} = -F_{rt} = \frac{Q}{r^{2}} \end{aligned} \end{equation}

Step 2: Einstein–Maxwell equations

The Einstein equation with an electromagnetic source is:

\begin{equation} \begin{aligned} G_{\mu\nu} = 8\pi\, T_{\mu\nu} \end{aligned} \end{equation}

where \( G_{\mu\nu} \) is the Einstein tensor of the metric.

The Maxwell equations in vacuum are:

\begin{equation} \begin{aligned} \nabla_{\mu} F^{\mu\nu} = 0, \qquad \nabla_{[\alpha} F_{\beta\gamma]} = 0 \end{aligned} \end{equation}

For the static, spherically symmetric case, it follows that:

\begin{equation} \begin{aligned} F^{tr} = \frac{Q}{r^{2}} \sqrt{\frac{1}{A(r) B(r)}} \end{aligned} \end{equation}

Step 3: Computing the Einstein tensor

The Einstein tensor components for the general metric

\begin{equation} \begin{aligned} ds^{2} = A(r)c^{2}dt^{2} - B(r)dr^{2} - r^{2}d\theta^{2} - r^{2}\sin^{2}\theta\, d\phi^{2} \end{aligned} \end{equation}

are:

\begin{equation} \begin{aligned} G_t^t = \frac{B'}{r B^{2}} + \frac{1}{r^{2}}\left(1 - \frac{1}{B}\right) \end{aligned} \end{equation}

\begin{equation} \begin{aligned} G_r^r = \frac{A'}{r A B} - \frac{1}{r^{2}}\left(1 - \frac{1}{B}\right) \end{aligned} \end{equation}

\begin{equation} \begin{aligned} G_\theta^\theta = G_\phi^\phi = \frac{1}{4AB} \left( 2A'' - A' \frac{B'}{B} + \frac{A'^2}{A} \right) - \frac{1}{2rB} \left( \frac{A'}{A} - \frac{B'}{B} \right) \end{aligned} \end{equation}

where a prime (′) denotes differentiation with respect to \( r \).

Step 4: Stress-energy tensor components

The stress-energy tensor of the electric field is diagonal with:

\begin{equation} \begin{aligned} T_t^t = T_r^r = -\frac{Q^{2}}{8\pi r^{4}}, \qquad T_\theta^\theta = T_\phi^\phi = \frac{Q^{2}}{8\pi r^{4}} \end{aligned} \end{equation}

Step 5: Coupling and solving the equations

The Einstein equations become explicitly:

\begin{equation} \begin{aligned} \frac{B'}{r B^{2}} + \frac{1}{r^{2}}\left(1 - \frac{1}{B}\right) = -\frac{Q^{2}}{r^{4}} \end{aligned} \end{equation}

\begin{equation} \begin{aligned} \frac{A'}{r A B} - \frac{1}{r^{2}}\left(1 - \frac{1}{B}\right) = -\frac{Q^{2}}{r^{4}} \end{aligned} \end{equation}

Solving this system yields:

\begin{equation} \begin{aligned} A(r) = \frac{1}{B(r)} = 1 - \frac{2M}{r} + \frac{Q^{2}}{r^{2}} \end{aligned} \end{equation}

where \( M \) is an integration constant representing the mass (in geometrized units).

Step 6: Result — Reissner–Nordström metric

The solution is now the metric line element:

\begin{equation} \begin{aligned} ds^{2} = \left( 1 - \frac{2GM}{c^{2} r} + \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} \right)c^{2} dt^{2} - \left( 1 - \frac{2GM}{c^{2} r} + \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} \right)^{-1} dr^{2} - r^{2} d\Omega^{2} \end{aligned} \end{equation}

where

\begin{equation} \begin{aligned} d\Omega^{2} = d\theta^{2} + \sin^{2}\theta\, d\phi^{2}. \end{aligned} \end{equation}

Conclusion

The Reissner–Nordström metric is the unique static, spherically symmetric solution of the Einstein–Maxwell equations with a point mass and electric charge. This means that both gravitational and electromagnetic effects are incorporated in the spacetime description.

Remark on the cosmological constant

The classical Schwarzschild solution is an exact solution of Einstein’s field equations, but under the explicit assumption that the cosmological constant \( \lambda = 0 \). In the original Schwarzschild derivation, this \( \lambda \)-term is therefore neglected or absent, meaning that the metric does not account for cosmological expansion or repulsion that would be caused by a non-zero \( \lambda \).

To what extent is \( \lambda \) included?

The “standard” Schwarzschild metric describes a static vacuum solution without a cosmological constant:

\begin{equation} \begin{aligned} ds^{2} = c^{2} d\tau^{2} = \left(1 - \frac{2GM}{c^{2} r}\right)c^{2} dt^{2} - \left(1 - \frac{2GM}{c^{2} r}\right)^{-1} dr^{2} - r^{2} d\theta^{2} - r^{2} \sin^{2}\theta\, d\phi^{2} \end{aligned} \end{equation}

or …

Schwarzschild metric

\begin{equation} \begin{aligned} ds^{2} = c^{2} d\tau^{2} = \left(1 - \frac{r_{s}}{r}\right)c^{2} dt^{2} - \left(1 - \frac{r_{s}}{r}\right)^{-1} dr^{2} - r^{2} d\Omega^{2} \end{aligned} \end{equation}

where \( r_{s} = \frac{2GM}{c^{2}} \) and

\begin{equation} \begin{aligned} d\Omega^{2} = d\theta^{2} + \sin^{2}\theta\, d\phi^{2}. \end{aligned} \end{equation}

Full Einstein equation

\begin{equation} \begin{aligned} R_{\mu\nu} - \frac{1}{2} R\, g_{\mu\nu} + \lambda g_{\mu\nu} = 8\pi T_{\mu\nu} \end{aligned} \end{equation}

In the Schwarzschild derivation, \( \lambda = 0 \) is assumed.

If \( \lambda \neq 0 \): the Schwarzschild–de Sitter metric

When the \( \lambda \)-term is included, one obtains the Schwarzschild–de Sitter (or Kottler) metric:

\begin{equation} \begin{aligned} ds^{2} = \left(1 - \frac{r_{s}}{r} - \frac{\lambda r^{2}}{3}\right)c^{2} dt^{2} - \left(1 - \frac{r_{s}}{r} - \frac{\lambda r^{2}}{3}\right)^{-1} dr^{2} - r^{2} d\Omega^{2} \end{aligned} \end{equation}

This is also an exact solution, now explicitly including the cosmological constant, and describing, for example, a black hole in an expanding universe.

Conclusion

The Schwarzschild metric is exact, but only for \( \lambda = 0 \).
For \( \lambda \neq 0 \), the effect is fully included in the Schwarzschild–de Sitter solution.
Neglecting \( \lambda \) is usually justified for stars or planets, because \( \lambda \) is extremely small compared to local gravitational fields.

Thus, the classical Schwarzschild metric is exact, but only under the assumption that the cosmological constant plays no role.

Reissner–Nordström–de Sitter metric

When combining the Schwarzschild–de Sitter metric with the Reissner–Nordström metric, one obtains the Reissner–Nordström–de Sitter solution:

\begin{equation} \begin{aligned} ds^{2} = \left( 1 - \frac{2GM}{c^{2} r} + \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} - \frac{\lambda r^{2}}{3} \right)c^{2} dt^{2} \\ \notag - \left( 1 - \frac{2GM}{c^{2} r} + \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} - \frac{\lambda r^{2}}{3} \right)^{-1} dr^{2} - r^{2} d\Omega^{2} \end{aligned} \end{equation}

Explanation of the terms

\( \frac{2GM}{c^{2} r} \): gravitational (mass) term — attractive.
\( \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} \): electromagnetic (charge) term — repulsive for like charges.
\( \frac{\lambda r^{2}}{3} \): cosmological term — repulsive for \( \lambda > 0 \) (de Sitter), attractive for \( \lambda < 0 \) (anti-de Sitter).

Special cases

\( \lambda = 0 \) → standard Reissner–Nordström metric
\( Q = 0 \) → Schwarzschild–de Sitter (Kottler) metric
\( Q = 0,\ \lambda = 0 \) → classical Schwarzschild metric