Construction of sources for Majumdar-Papapetrou spacetimes.

*(English)*Zbl 1033.83010Majumdar and Papapetrou (MP) showed that assuming a static spacetime with conformastatic metric \(ds^2 = -V^2\, dt^2 +\frac{1}{V^2}d \vec c\cdot d\vec x\), where \(V\!= V(x^1, x^2, x^3)\), the problem of finding solutions for the \(3+1\) Einstein-Maxwell equations can be greatly simplified. In fact, assuming a linear relationship between the time-like component of the electromagnetic potential and \(V\), they found that \(\frac{1}{V}\) is harmonic, i.e. it is a solution of the Laplace equation.

The author considers fundamental facts of the MP approach and reviews the proof that the relationship between the electrostatic potential and \(V\) is actually a consequence of the field equations. This is followed by a discussion of the electrovac MP solution in the spherically symmetric case. The spherically symmetric solution of Gürses is considered in detail. The author introduces new parameters that simplify the construction of class \(C^1\), singularity-free geometries. The arising sources are bounded or unbounded, and the redshift of light signals allows an observer at spatial infinity to distinguish these cases. Unbounded dust sources are examined, when solutions of the non-linear potential equation are matched to the Gürses internal geometry. An interesting affinity between the conformastatic metric and some homothetic and matter collineations are found out. MP sources with geometric symmetries are investigated. These symmetries are formulated in terms of Lie derivatives of the metric and energy-momentum tensors, and provide us with distinctive solutions that can be used to construct non-singular sources for MP spacetimes. This approach dispenses with the need for ad-hoc functionals \(\rho =\rho(V)\) and confronts us with distinctive classes of MP solutions.

The author considers fundamental facts of the MP approach and reviews the proof that the relationship between the electrostatic potential and \(V\) is actually a consequence of the field equations. This is followed by a discussion of the electrovac MP solution in the spherically symmetric case. The spherically symmetric solution of Gürses is considered in detail. The author introduces new parameters that simplify the construction of class \(C^1\), singularity-free geometries. The arising sources are bounded or unbounded, and the redshift of light signals allows an observer at spatial infinity to distinguish these cases. Unbounded dust sources are examined, when solutions of the non-linear potential equation are matched to the Gürses internal geometry. An interesting affinity between the conformastatic metric and some homothetic and matter collineations are found out. MP sources with geometric symmetries are investigated. These symmetries are formulated in terms of Lie derivatives of the metric and energy-momentum tensors, and provide us with distinctive solutions that can be used to construct non-singular sources for MP spacetimes. This approach dispenses with the need for ad-hoc functionals \(\rho =\rho(V)\) and confronts us with distinctive classes of MP solutions.

Reviewer: Serguey M. Pokas (Odessa)

##### MSC:

83C22 | Einstein-Maxwell equations |

##### Keywords:

\(3+1\) Einstein-Maxwell equations; static spacetime; Laplace equation; Majumdar-Papapetrou electrovac solutions; spherically symmetric solution of Gürses; charged dust; conformastatic metric; singularity-free geometries; linear model; sine-Gordon model; sources with geometric symmetries
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\textit{V. Varela}, Gen. Relativ. Gravitation 35, No. 10, 1815--1831 (2003; Zbl 1033.83010)

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