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 the electrical contributions:
\[ ds^{2} = c^{2} d\tau^{2} = \left( 1 - \frac{2GM}{c^{2} r} + \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} \right) c^{2} dt^{2} - \left( 1 - \frac{2GM}{c^{2} r} + \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} \right)^{-1} dr^{2} \] \[ - r^{2} d\theta^{2} - r^{2} \sin^{2}\theta\, d\phi^{2} \]or
\[ ds^{2} = c^{2} d\tau^{2} = \left( 1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) c^{2} dt^{2} - \left( 1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right)^{-1} dr^{2} \] \[ - r^{2} d\theta^{2} - r^{2} \sin^{2}\theta\, d\phi^{2} \]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 distortion of the vacuum as in the Schwarzschild solution.
- The additional term \( \frac{r_{Q}^{2}}{r^{2}} \) describes the repulsive (for like charges) electromagnetic effect within general relativity.
- This metric reduces to the standard Schwarzschild solution when \( Q = 0 \) (no charge).
- For rotating or additionally charged objects (such as electrons), even more general solutions exist, such as the Kerr–Newman metric.
In classical Newtonian physics there exists a gravitational field in the vacuum. According to Einstein, however, there is no gravitational field as such; instead spacetime itself is curved due to gravitation. 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: there is no matter or field present there, so Einstein’s field equations reduce to the vacuum equations.
- In the Reissner–Nordström case, however, there is still an electromagnetic field in the vacuum surrounding the central charge; this electromagnetic field carries energy and momentum, and therefore produces a non-zero stress–energy tensor.
- Specifically, \( T_{\mu\nu} \) in this case describes the energy–momentum of the radial electric field. This means that Einstein’s equation has an electromagnetic field as a source, even if no additional matter (such as mass or dust) is present outside the central object.
In summary: in the Reissner–Nordström metric, \( T_{\mu\nu} \neq 0 \) in the “vacuum” because the electromagnetic field of the charge is physically real and contains energy.
Derivation of the Reissner–Nordström Metric
Below follows a step-by-step derivation of the Reissner–Nordström metric starting from the Einstein–Maxwell equations. This is the standard procedure in general relativity for determining the spacetime of a spherically symmetric, charged mass.
Step 1: Assumptions — metric and source
We seek a static, spherically symmetric solution of the form (spherical coordinates):
\[ ds^{2} = c^{2} d\tau^{2} = A(r)\, c^{2} dt^{2} - B(r)\, dr^{2} - r^{2} d\theta^{2} - r^{2} \sin^{2}\theta\, d\phi^{2} \]where \( A(r) \) and \( B(r) \) are unknown functions of the radius \( r \).
- The source is an electromagnetic field of a point charge \( Q \).
- The stress–energy tensor of the electromagnetic field is (in natural units): \[ T_{\mu\nu} = \frac{1}{\mu_{0}} \left( F_{\mu\alpha} F_{\nu}{}^{\alpha} - \frac{1}{4} g_{\mu\nu} F_{\alpha\beta} F^{\alpha\beta} \right) \]
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:
\[ A_{\mu} = \left( \frac{\Phi}{c},\, -\mathbf{A} \right) \quad \text{or} \quad A^{\mu} = \left( \frac{\Phi}{c},\, \mathbf{A} \right) \]- \( \Phi \) : 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 \( F_{\mu\nu} \) contains all information about the electric field \( \mathbf{E} \) and the magnetic field \( \mathbf{B} \).
In matrix form:
\[ F_{\mu\nu} = \begin{pmatrix} 0 & -E_{x}/c & -E_{y}/c & -E_{z}/c \\ E_{x}/c & 0 & -B_{z} & B_{y} \\ E_{y}/c & B_{z} & 0 & -B_{x} \\ E_{z}/c & -B_{y} & B_{x} & 0 \end{pmatrix} \]For a purely radial electric field of a point charge \( Q \), this becomes:
\[ F_{\mu\nu} = \begin{pmatrix} 0 & \frac{Q}{r^{2}} & 0 & 0 \\ -\frac{Q}{r^{2}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \]The only non-zero component of \( F_{\mu\nu} \) in this radial field is:
\[ F_{tr} = -F_{rt} = \frac{Q}{r^{2}} \]Step 2: Einstein–Maxwell Equations
The Einstein equation with an electromagnetic source is:
\[ G_{\mu\nu} = 8\pi\, T_{\mu\nu} \]where \( G_{\mu\nu} \) is the Einstein tensor of the metric.
The Maxwell equations in vacuum are:
\[ \nabla_{\mu} F^{\mu\nu} = 0, \qquad \nabla_{[\alpha} F_{\beta\gamma]} = 0 \]For the static, spherically symmetric situation this yields:
\[ F^{tr} = \frac{Q}{r^{2}} \sqrt{\frac{1}{A(r) B(r)}} \]Step 3: Computing the Einstein Tensor
The Einstein tensor components for the general metric \[ ds^{2} = A(r)c^{2}dt^{2} - B(r)dr^{2} - r^{2}d\theta^{2} - r^{2}\sin^{2}\theta\, d\phi^{2} \] are:
\[ G_t^t = \frac{B'}{r B^{2}} + \frac{1}{r^{2}}\left(1 - \frac{1}{B}\right) \] \[ G_r^r = \frac{A'}{r A B} - \frac{1}{r^{2}}\left(1 - \frac{1}{B}\right) \] \[ G_\theta^\theta = G_\phi^\phi = \frac{1}{4AB} \left( 2A'' - A' \frac{B'}{B} + \frac{A'^2}{A} \right) - \frac{1}{2rB} \left( \frac{A'}{A} - \frac{B'}{B} \right) \]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:
\[ T_t^t = T_r^r = -\frac{Q^{2}}{8\pi r^{4}}, \qquad T_\theta^\theta = T_\phi^\phi = \frac{Q^{2}}{8\pi r^{4}} \]Step 5: Coupling and Solving the Equations
The Einstein equations become explicitly:
\[ \frac{B'}{r B^{2}} + \frac{1}{r^{2}}\left(1 - \frac{1}{B}\right) = -\frac{Q^{2}}{r^{4}} \] \[ \frac{A'}{r A B} - \frac{1}{r^{2}}\left(1 - \frac{1}{B}\right) = -\frac{Q^{2}}{r^{4}} \]Solving this system yields:
\[ A(r) = \frac{1}{B(r)} = 1 - \frac{2M}{r} + \frac{Q^{2}}{r^{2}} \]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:
\[ ds^{2} = \left( 1 - \frac{2GM}{c^{2} r} + \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} \right)c^{2} dt^{2} - \left( 1 - \frac{2GM}{c^{2} r} + \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} \right)^{-1} dr^{2} - r^{2} d\Omega^{2} \]where \[ d\Omega^{2} = d\theta^{2} + \sin^{2}\theta\, d\phi^{2}. \]
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 interactions are incorporated into 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 omitted, 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:
or …
Schwarzschild metric
\[ ds^{2} = c^{2} d\tau^{2} = \left(1 - \frac{r_{s}}{r}\right)c^{2} dt^{2} - \left(1 - \frac{r_{s}}{r}\right)^{-1} dr^{2} - r^{2} d\Omega^{2} \]where \( r_{s} = \frac{2GM}{c^{2}} \) and \[ d\Omega^{2} = d\theta^{2} + \sin^{2}\theta\, d\phi^{2}. \]
Full Einstein equation
\[ R_{\mu\nu} - \frac{1}{2} R\, g_{\mu\nu} + \lambda g_{\mu\nu} = 8\pi T_{\mu\nu} \]In the Schwarzschild derivation, \( \lambda = 0 \) is assumed (i.e. no cosmological constant).
If \( \lambda \neq 0 \): the Schwarzschild–de Sitter metric
When the \( \lambda \)-term is included, the Schwarzschild–de Sitter (or Kottler) metric arises:
\[ ds^{2} = \left(1 - \frac{r_{s}}{r} - \frac{\lambda r^{2}}{3}\right)c^{2} dt^{2} - \left(1 - \frac{r_{s}}{r} - \frac{\lambda r^{2}}{3}\right)^{-1} dr^{2} - r^{2} d\Omega^{2} \]This is also an exact solution, but now explicitly includes the cosmological constant and describes, for example, a black hole in an expanding universe.
Conclusion
- The Schwarzschild metric is exact, but only for the case \( \lambda = 0 \).
- For \( \lambda \neq 0 \), the effect is fully incorporated in the Schwarzschild–de Sitter solution.
- Neglecting \( \lambda \) is usually justified for stars or planets, since \( \lambda \) 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 the Schwarzschild–de Sitter metric is combined with the Reissner–Nordström metric, the Reissner–Nordström–de Sitter solution emerges:
\[ ds^{2} = \left( 1 - \frac{2GM}{c^{2} r} + \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} - \frac{\lambda r^{2}}{3} \right)c^{2} dt^{2} \] \[ - \left( 1 - \frac{2GM}{c^{2} r} + \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} - \frac{\lambda r^{2}}{3} \right)^{-1} dr^{2} - r^{2} d\Omega^{2} \]Explanation of the terms
- \( \frac{2GM}{c^{2} r} \): gravitational (mass) term — attractive due to mass.
- \( \frac{GQ^{2}}{4\pi\varepsilon_{0} c^{4} r^{2}} \): 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
- \( \lambda = 0 \) → ordinary Reissner–Nordström metric
- \( Q = 0 \) → Schwarzschild–de Sitter (or Kottler) metric
- \( Q = 0,\ \lambda = 0 \) → classical Schwarzschild metric