# Construction of some Electrostatic Fields in General Relativity

01 Mar 1962-Vol. 79, Iss: 3, pp 657-658

About: The article was published on 1962-03-01. It has received 3 citation(s) till now. The article focuses on the topic(s): Four-force & Theory of relativity.

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Abstract: A number of theorems concerning non-null electrovac spacetimes, that is space-times whose metric satisfies the source-free Einstein-Maxwell equations for some non-null bivector Fij, are presented. Firstly, we suppose that the metric is invariant under a one-parameter group of isornetries with Killing vector field ξ. It is proved that the electromagnetic field tensor Fij is invariant under the group, in the sense that its Lie derivative with respect to ξ vanishes, if and only if the gradient αij of the complexion scalar is orthogonal to ξ. It is is also proved that if in addition ξ is hypersurface orthogonal, it is necessarily parallel to α,i. These results are used to generalize theorems of Perjes and Majumdar concerning static electrovac space-times. Secondly, we suppose that the metric is invariant under a two-parameter othogonally transitive Abelian group of isometries. It is proved that in this case Fij is necessarily invariant under the group. The above results can be used to simplify many derivations of exact solutions of the Einstein-Maxwell equations.

52 citations

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TL;DR: In this article, a fresh look on static gravitational fields in general relativity, eight new theorems have been derived, including invariant necessary integral conditions for the existence of a solution, which corresponds to the equilibrium of matter.

Abstract: In a fresh look on static gravitational fields in general relativity, eight new theorems have resulted. Also in the process of deriving theorems a new solution has emerged. In the first of these theorems an invariant necessary integral condition for the existence of a solution has been derived. Physically this condition corresponds to the equilibrium of matter. In the second theorem a scalar condition has been found which implies the flatness of the static gravitational universe. In the third theorem, it has been proved that there cannot occur any group of motion along ``the lines of forces.'' In Theorems 5 and 6, the questions of whether the spatial part of a static gravitational universe can be Einstein, projectively flat, or Stackel are investigated. In the seventh theorem, the static gravitational field equations have been reduced to the geometrized equations in a spatial universe. In the last theorem, all conformastat gravitational universes have been found. One of these is the universe due to ``an i...

31 citations

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TL;DR: In this article, it was shown from the field equations that a body admitting an arbitrary symmetry must satisfy an integral condition analogous to the equilibrium criterion, and it was proved that the vanishing of the scalar curvature of the associated space implies the flatness of the space-time metric.

Abstract: The stationary gravitational equations in vacuum are expressed in five different forms. A necessary integral condition on the twist potential φ is derived. The Papapetrou‐Ehlers class of stationary solutions is rederived in a different way. In the study of the complex potential theory it is proved from the field equations that a body admitting an arbitrary symmetry must satisfy an integral condition analogous to the equilibrium criterion. It is proved that the vanishing of the scalar curvature of the associated space implies the flatness of the space‐time metric. A proof is given for the fact that the only analytic functions of the complex potential F which preserve the field equations form a four‐parameter Mobius group. It is also shown that any differentiable function of F and F which preserves the field equations must either be an analytic function of F or the conjugate of such a function. Next the conformastationary vacuum metrics are classified. In the study of the axially symmetric stationary field...

28 citations

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TL;DR: In this paper, it was shown that the most general electrostatic field in which the two fields are functionally related can be represented by a line element of the form {(d{x}^{1})}^{2} = \ensuremath{-} 2log(1+v)$.

Abstract: The work of Weyl on the gravitational field occasioned by an axially symmetric distribution of matter and charge is generalized to the case in which ${g}_{44}$ and $\ensuremath{\varphi}$ for an electrostatic field are functionally related, with or without spatial symmetry. It is shown that the most general electrostatic field in which ${g}_{44}$ and $\ensuremath{\varphi}$ are related by an equation of the form ${g}_{44}=\frac{1}{2}{(\ensuremath{\varphi}+c)}^{2}$ can be represented by a line element of the form ${(\mathrm{ds})}^{2}=\ensuremath{-}{e}^{\ensuremath{-}w}[{(d{x}^{1})}^{2}+{(d{x}^{2})}^{2}+{(d{x}^{3})}^{2}]+{e}^{w}{(\mathrm{dt})}^{2}$. Certain of the field equations are then identically satisfied while the remaining ones reduce to a single equation for $w$. The substitution $w=\ensuremath{-}2log(1+v)$ transforms this into Laplace's equation for $v$, so that the solution can be expressed in terms of harmonic function.

524 citations

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TL;DR: In this article, the nature of a static line element in the relativity theory is investigated and it is shown that both the force and the torque on an isolated body must vanish if the field equations for empty space are to have a static solution in its neighbourhood.

Abstract: The nature of a static line element in the relativity theory is investigated and it is shown that both the force and the torque on an isolated body must vanish if the field equations for empty space are to have a static solution in its neighbourhood. It is also shown how a general solution for a static electro-gravitational field can be constructed by a method of successive approximation.

1 citations