Ehlers and Harrison Transformations

Ehlers and Harrison transformations act on reduced gravitational potentials rather than directly on arbitrary metric components. An Ehlers map mixes a Killing norm with its twist and can add dual mass, or NUT charge, to a static vacuum seed. A Harrison map in stationary Einstein–Maxwell theory can add electric or magnetic charge; under an axial reduction a related group element can instead immerse a seed in an external electromagnetic field. Which charge is generated depends on the chosen Killing field, potential normalization, and asymptotic compensation.

Required background. Killing Fields and Isometry Groups supplies the non-null symmetry used in the reduction.

Helpful background. Einstein’s Equation supplies vacuum and electrovac equations; The Riemann Curvature Tensor supplies invariant checks; Exact Solutions of Einstein’s Equations supplies Schwarzschild, Reissner–Nordström, and Taub–NUT seeds or targets; Formulations of General Relativity supplies Killing reduction; PDEs and Green Functions supplies boundary conditions for the reconstruction equations.

Scope. The Ehlers map below acts on four-dimensional Ricci-flat vacuum solutions, and the Harrison map acts on source-free Einstein–Maxwell solutions; both require $\Lambda=0$. Twist and Ernst Potentials fixes the earlier Ernst normalization used at first reduction.

Ehlers action on norm and twist

Use a timelike stationary reduction with $f=-\xi^2>0$, twist convention $d\chi=f^2*_h dA$, and vacuum Ernst potential

$$\mathcal E=f+i\chi.$$

One finite Ehlers representative is

$$\mathcal E'= \frac{\mathcal E}{1+ic\mathcal E}, \qquad c\in\mathbb R.$$

For a twist-free seed, $\mathcal E=f$, this gives

$$f'=\frac{f}{1+c^2f^2}, \qquad \chi'=-\frac{cf^2}{1+c^2f^2}.$$

The three-base $h_{ij}$ is unchanged in this convention, but the four-metric is not known until $A'$ has been reconstructed from $d\chi'=f'^2*_h dA'$. The integrability of that first-order equation follows from the transformed Ernst equation; its integration constants can affect global identifications.

The raw map does not preserve $\mathcal E\to1$ at spatial infinity. A twist shift, constant rescaling, and corresponding normalization of $t$ must be composed with it before asymptotic coefficients are interpreted as charges. After that compensation, the complex monopole $M+iN$ undergoes an $SL(2,\mathbb R)$ rotation, with the sign of $N$ fixed by the declared twist orientation. Schwarzschild is thereby mapped locally to Taub–NUT. The result has NUT charge and a stringlike axis obstruction or periodic-time alternative; it is not an ordinary asymptotically flat rotating black hole. The explicit Schwarzschild construction is worked through in Contopoulos et al. 2016, §§3–3.1, arXiv v3 pp. 7–11, Eqs. (44)–(75), PDF.

Harrison charging map

For stationary Einstein–Maxwell fields define

$$\Phi=v+iu, \qquad \mathcal E=f-|\Phi|^2+i\chi.$$

In this normalization, a Harrison transformation with complex parameter $b$ can be written

$$\mathcal E'=\frac{\mathcal E}{D}, \qquad \Phi'=\frac{\Phi+b\mathcal E}{D},$$ $$D=1-2\bar b\Phi-|b|^2\mathcal E.$$

Changing the normalization of $\Phi$ moves the factors $2$ and $|b|^2$; formulas from different conventions cannot be mixed. Reality of the reconstructed metric follows from $f'=\operatorname{Re}\mathcal E'+|\Phi'|^2$, but a zero of $D$ can create a singular surface or leave the chosen reduction patch.

For this map, $m$ and $M$ denote geometrized mass lengths and $Q$ denotes the charge length: $m=G M_{\mathrm{seed}}$, $M=G M_{\mathrm{final}}$, and $Q^2=G Q_{\mathrm{phys}}^2$ in the displayed Einstein–Maxwell normalization. Starting from Schwarzschild with seed mass length $m$ and $\Phi=0$, then restoring zero electromagnetic potential and unit lapse at infinity, a real charging parameter $q$ may be chosen so that the resulting Reissner–Nordström parameters are

$$M=m\frac{1+q^2}{1-q^2}, \qquad Q=\frac{2mq}{1-q^2}, \qquad |q|<1.$$

They obey

$$M^2-Q^2=m^2.$$

A phase of the complex parameter rotates electric into magnetic charge. The transformation reaches nonextremal solutions from a positive-mass Schwarzschild seed; the extremal limit $|Q|=M$ lies at $|q|\to1$ together with a singular parameter rescaling and must be taken as a controlled limit. Harrison’s original construction and examples are in Harrison 1968, pp. 1744–1752, while the $SU(2,1)$ potential symmetry is stated in Kinnersley 1973, pp. 651–653.

Charge generation is not physical selection

Before and after either map, record the seed’s field equations, Killing vector, nonzero-norm region, gauge at infinity, and parameter domain. Then verify the lifted Einstein or Einstein–Maxwell equations independently. Finally test signature, zeros of every transformation denominator, axis regularity, asymptotic frame, flux-normalized charges, horizons, and singularities. A map can preserve the local reduced equations while spoiling asymptotic flatness, introducing a NUT identification, magnetizing infinity, or producing a naked singularity.

Exercise

Verify the invariant relation for the charged Schwarzschild map and identify its extremal limit.

Solution

Direct substitution gives

$$M^2-Q^2 =m^2\frac{(1+q^2)^2-4q^2}{(1-q^2)^2} =m^2.$$

For fixed final $M$, $|Q|/M=2|q|/(1+q^2)$ approaches one as $|q|\to1$. Keeping $M$ finite then requires the seed parameter $m=M(1-q^2)/(1+q^2)$ to tend to zero. Extremal Reissner–Nordström is therefore a joint seed-and-transformation limit, not the image of a fixed positive $m$ under a finite regular $q$.

References

• Contopoulos, I. G., F. P. Esposito, K. Kleidis, D. B. Papadopoulos, and L. Witten. “Generating Solutions to the Einstein Field Equations.” International Journal of Modern Physics D 25, no. 2 (2016): 1650022. DOI; Open PDF.
• Harrison, B. Kent. “New Solutions of the Einstein–Maxwell Equations from Old.” Journal of Mathematical Physics 9 (1968): 1744–1752. DOI.
• Kinnersley, William. “Generation of Stationary Einstein–Maxwell Fields.” Journal of Mathematical Physics 14 (1973): 651–653. DOI.