Ernst equation

In an oriented space-time with a time-like Killing vector field $X$, the twist $1$-form $\tau$ is defined by

\begin{equation*} * \tau = \xi \bigwedge d \xi \end{equation*}

where $\xi = X _ { a } d x ^ { a }$. It is always closed in vacuum solutions to the Einstein gravitational equations; that is, when the Ricci tensor $R _ { ab }$ vanishes.

In such space-times, one can write (locally) $\tau = d \psi$, where $\psi$ is constant along $X$. The Ernst potential is the complex quantity $\mathcal{E} = f + i \psi$, where $f = X _ { a } X ^ { a }$ and $i = \sqrt { - 1 }$. It is used in a number of different ways in finding explicit solutions to Einstein's equations (cf. also Einstein equations; [a8] provides a wide-ranging introduction to the most of the original work on this subject).

One use is in the generation of new solutions with one Killing vector from a known one. The idea here is to use $\cal E$ and the metric $h$ on the quotient space by the Killing vector action as dependent variables (both are functions of three variables). The vacuum equations for the space-time metric can then be derived from the action

\begin{equation*} \int \left( R _ { h} + \frac { 1 } { 2 } f ^ { - 2 } h ^ { \alpha \beta } \partial _ { \alpha } \mathcal{E}\partial _ { \beta } \overline { \mathcal{E} } \right) d \mu _ { h}, \end{equation*}

where $R_{h}$ and $d \mu _ { h }$ are the scalar curvature and the volume element of the $3$-metric $h$. There is a straightforward extension to the Einstein–Maxwell equations.

The symmetries of the action and its electro-magnetic generalization allow transformations of the solution that preserve $h$, but change the potential. They include solution-generation transformations discussed in [a6], [a7].

A second use is in finding stationary axi-symmetric gravitational fields (or by a straightforward modification to the formalism, solutions with other symmetries representing, for example, cylindrically symmetric gravitational waves and the interaction of colliding plane waves). Here one assumes the existence a second Killing vector $Y$ such that $X$ and $Y$ together generate a $2$-dimensional Lie algebra of infinitesimal isometries. In this case, there are two twist $1$-forms. Their inner products with the Killing vectors are constant when $R _ { ab } = 0$, and vanish if some combination of the Killing vectors has a fixed point.

When they do vanish, the space-time metric can be written in the Weyl canonical form

\begin{equation*} f ( d t ^ { 2 } - \omega d \theta ^ { 2 } ) - r ^ { 2 } f ^ { - 1 } d \theta ^ { 2 } - \Omega ^ { 2 } ( d r ^ { 2 } + d z ^ { 2 } ), \end{equation*}

where $X = \partial / \partial_{ t }$ and $Y = \partial / \partial \theta$. In this case, the Ernst potential associated with $X$ is a function of $r$ and $z$ alone, and the vacuum equations reduce to the Ernst equation

\begin{equation*} \operatorname { Re } ( \mathcal{E} ) \nabla ^ { 2 } \mathcal{E} = \nabla \mathcal{E} \cdot \nabla \mathcal{E}, \end{equation*}

where $\nabla$ is the gradient in the three-dimensional Euclidean space on which $r$, $\theta$, $z$ are cylindrical polar coordinates [a4]. Once $\cal E$ is known, $\Omega$ is found by quadrature. Again, there is a straightforward extension to the Einstein–Maxwell case [a5].

Although still non-linear, this reduction to a single scalar equation in Euclidean space for the complex potential $\cal E$ is a great simplification of the original vacuum equations $R _ { ab } = 0$. It has been widely exploited in the search for exact solutions. In particular, the solution-generation techniques provide a rich source of new solutions since one can combine the transformations of a metric with one Killing vector with linear transformations in the Lie algebra spanned by $X$ and $Y$.

Although it is non-linear, the Ernst equation is integrable, and its transformation properties can be seen as part of the wider theory of integrable systems (cf. also Integrable system); some of the connections are explained in [a3]. One can understand them from another point of view through the observation [a2] that the Ernst equation is identical to a form of the self-dual Yang–Mills equation (cf. also Yang–Mills field) for static axi-symmetric gauge fields. If one writes

\begin{equation*} J = \frac { 1 } { f } \left( \begin{array} { c c } { 1 } & { - \psi } \\ { - \psi } & { \psi ^ { 2 } + r ^ { 2 } f ^ { 2 } } \end{array} \right), \end{equation*}

then the Ernst equation is equivalent to

\begin{equation*} \partial _ { r } ( r J ^ { - 1 } \partial _ { r } J ) + \partial _ { z } ( r J ^ { - 1 } \partial _ { z } J ) = 0, \end{equation*}

which is a symmetry reduction of the Yang equation. Solutions can therefore be found by solving a Riemann–Hilbert problem [a10], and, more generally, by the twistor methods reviewed in [a9].

The space-time metric gives rise to a solution of this same equation in another way by writing

\begin{equation*} J ^ { \prime } = \left( \begin{array} { c c } { f \omega ^ { 2 } - f ^ { - 1 } r ^ { 2 } } & { - f \omega } \\ { - f \omega } & { f } \end{array} \right). \end{equation*}

The mapping $J \mapsto J ^ { \prime }$ is a discrete symmetry of the reduction of Yang's equation, and many of the solution transformations can be obtained by combining it with $J \mapsto M ^ { t } J M$, $J ^ { \prime } \mapsto M ^ { \prime t } J ^ { \prime } M ^ { \prime }$ for constant matrices $M$ and $M^{\prime}$. In [a1], these are seen to generate the action of a loop group (in fact a central extension when the action on the conformal factor $\Omega$ is included).

References

[a1]	P. Breitenlohner, D. Maison, "On the Geroch group" Ann. Inst. H. Poincaré Phys. Th. , 46 (1987) pp. 215–46
[a2]	L. Witten, "Static axially symmetric solutions of self-dual $\operatorname{SU} ( 2 )$ gauge fields in Euclidean four-dimensional space" Phys. Rev. , D19 (1979) pp. 718–20
[a3]	C. Cosgrove, "Relationships between group-theoretic and soliton-theoretic techniques for generating stationary axisymmetric gravitational solutions" J. Math. Phys. , 21 (1980) pp. 2417–47
[a4]	F. Ernst, "New formulation of the axially symmetric gravitational field problem" Phys. Rev. , 167 (1968) pp. 1175–8
[a5]	F. Ernst, "New formulation of the axially symmetric gravitational field problem. II" Phys. Rev. , 168 (1968) pp. 1415–17
[a6]	R. Geroch, "A method for generating solutions of Einstein's equations" J. Math. Phys. , 12 (1971) pp. 918–24
[a7]	W. Kinnersley, "Recent progress in exact solutions" G. Shaviv (ed.) and J. Rosen (ed.) , General Relativity and Gravitation , Wiley (1975)
[a8]	D. Kramer, H. Stephani, M. MacCallum, E. Herlt, "Exact solutions of Einstein's field equations" , Cambridge Univ. Press (1980)
[a9]	L. Mason, N. Woodhouse, "Integrability, self-duality, and twistor theory" , Oxford Univ. Press (1996)
[a10]	R. Ward, "Stationary axisymmetric space-times: a new approach" Gen. Rel. Grav. , 15 (1983) pp. 105–9