View source for Ernst equation

In an oriented [[Space-time|space-time]] with a time-like [[Killing vector|Killing vector]] field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e1201601.png" />, the twist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e1201603.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e1201604.png" /> is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e1201605.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e1201606.png" />. It is always closed in vacuum solutions to the Einstein gravitational equations; that is, when the [[Ricci tensor|Ricci tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e1201607.png" /> vanishes.

In such space-times, one can write (locally) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e1201608.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e1201609.png" /> is constant along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016010.png" />. The Ernst potential is the complex quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016013.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016014.png" /> and the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016015.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016016.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016018.png" /> are the scalar curvature and the volume element of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016019.png" />-metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016020.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016021.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016024.png" /> together generate a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016025.png" />-dimensional [[Lie algebra|Lie algebra]] of infinitesimal isometries. In this case, there are two twist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016026.png" />-forms. Their inner products with the Killing vectors are constant when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016027.png" />, 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016028.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016030.png" />. In this case, the Ernst potential associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016031.png" /> is a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016033.png" /> alone, and the vacuum equations reduce to the Ernst equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016034.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016035.png" /> is the [[Gradient|gradient]] in the three-dimensional Euclidean space on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016038.png" /> are cylindrical polar coordinates [[#References|[a4]]]. Once <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016039.png" /> is known, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016040.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016041.png" /> is a great simplification of the original vacuum equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016042.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016044.png" />.

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016045.png" /></td> </tr></table>

then the Ernst equation is equivalent to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016046.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016047.png" /></td> </tr></table>

The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016048.png" /> is a discrete symmetry of the reduction of Yang's equation, and many of the solution transformations can be obtained by combining it with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016050.png" /> for constant matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016052.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016053.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120160/e12016054.png" /> 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>