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m (AUTOMATIC EDIT (latexlist): Replaced 53 formulas out of 53 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
 
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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
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If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
<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>
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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.
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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
  
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).
+
\begin{equation*} * \tau = \xi \bigwedge d \xi \end{equation*}
  
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
+
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.
  
<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>
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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).
  
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]].
+
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
  
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]]].
+
\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*}
  
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.
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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
 
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>
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\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 <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
+
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
  
<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>
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\begin{equation*} \operatorname { Re } ( \mathcal{E} ) \nabla ^ { 2 } \mathcal{E} = \nabla \mathcal{E} \cdot \nabla \mathcal{E}, \end{equation*}
  
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]]].
+
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 <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 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
 
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>
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\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
 
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>
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\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]]].
 
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]]].
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The space-time metric gives rise to a solution of this same equation in another way by writing
 
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>
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\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 <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).
+
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====
 
====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>
+
<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>

Latest revision as of 16:57, 1 July 2020

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
How to Cite This Entry:
Ernst equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ernst_equation&oldid=50182
This article was adapted from an original article by N.M.J. Woodhouse (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article