On Stationary Distributions of Charged Dust

Abstract

The correct value of the ratio of charge density to mass density is obtained for a stationary distribution of charged dust with vanishing Lorentz force. It is found that the ratio is truly an arbitrary constant, in disagreement with the apparently. incorrect result obtained by Misra et al. (1972).In recent communications Misra et al. (1972, 1973) have investigated the Einstein-Maxwell equations for a stationary distribution of charged dust in order to find the relation between mass and charge density. Unfortunately their calculations appear to be in error owing to an incorrect choice for the sign of 8 in their use of the field equations given by Harrison (1968). We show in this note that with a correct formulation of the field equations the value of I (J 11m, the ratio of the charge density to the mass density, turns out to be truly an arbitrary constant, allowing both extremes of a massless charge distribution and an uncharged dust distribution. This is in contrast to the ratio obtained by Misra et al. which had an upper limit that was less than unity.For stationary charged incoherent matter without any spatial symmetry, we use the line element (1) where Latin indices assume values from 1 to 3 and the metric is independent of time t. In what follows Greek indices may have values from 1 to 4.The field equations are now (2) where (3) which is a combination of the energy tensors due to the matter and the electromagnetic field. Assuming that the Lorentz force F/Lv VV vanishes everywhere and using comoving coordinates, one gets from the Bianchi identityThis relation yields goo = const. and without loss of generality we may assume