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In an oriented [[Space-time|space-time]] with a time-like [[Killing vector|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|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|Einstein equations]]; [[#References|[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|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 [[#References|[a6]]], [[#References|[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|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|gradient]] in the three-dimensional Euclidean space on which $r$, $\theta$, $z$ are cylindrical polar coordinates [[#References|[a4]]]. Once $\cal E$ is known, $\Omega$ is found by quadrature. Again, there is a straightforward extension to the Einstein–Maxwell case [[#References|[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|Integrable system]]); some of the connections are explained in [[#References|[a3]]]. One can understand them from another point of view through the observation [[#References|[a2]]] that the Ernst equation is identical to a form of the self-dual Yang–Mills equation (cf. also [[Yang–Mills field|Yang–Mills field]]) for static axi-symmetric gauge fields. If one writes

\begin{equation*} J = \frac { 1 } { f } \left( \begin{array} { c c } { 1 } &amp; { - \psi } \\ { - \psi } &amp; { \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|Riemann–Hilbert problem]] [[#References|[a10]]], and, more generally, by the twistor methods reviewed in [[#References|[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 } } &amp; { - f \omega } \\ { - f \omega } &amp; { 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 [[#References|[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====
<table><tr><td valign="top">[a1]</td> <td valign="top">  P. Breitenlohner,   D. Maison,   "On the Geroch group"  ''Ann. Inst. H. Poincaré Phys. Th.'' , '''46'''  (1987)  pp. 215–46</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  C. Cosgrove,   "Relationships between group-theoretic and soliton-theoretic techniques for generating stationary axisymmetric gravitational solutions"  ''J. Math. Phys.'' , '''21'''  (1980)  pp. 2417–47</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  F. Ernst,   "New formulation of the axially symmetric gravitational field problem"  ''Phys. Rev.'' , '''167'''  (1968)  pp. 1175–8</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  F. Ernst,   "New formulation of the axially symmetric gravitational field problem. II"  ''Phys. Rev.'' , '''168'''  (1968)  pp. 1415–17</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  R. Geroch,   "A method for generating solutions of Einstein's equations"  ''J. Math. Phys.'' , '''12'''  (1971)  pp. 918–24</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  W. Kinnersley,   "Recent progress in exact solutions"  G. Shaviv (ed.)  and J. Rosen (ed.) , ''General Relativity and Gravitation'' , Wiley  (1975)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  D. Kramer,   H. Stephani,   M. MacCallum,   E. Herlt,   "Exact solutions of Einstein's field equations" , Cambridge Univ. Press  (1980)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  L. Mason,   N. Woodhouse,   "Integrability, self-duality, and twistor theory" , Oxford Univ. Press  (1996)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  R. Ward,   "Stationary axisymmetric space-times: a new approach"  ''Gen. Rel. Grav.'' , '''15'''  (1983)  pp. 105–9</td></tr></table>