Approximate solutions representing the gravitational-electrostatic balance of two arbitrary point sources in general relativity have led to contradictory arguments in the literature with respect to the condition of balance. Up to the present time, the only known exact solutions which can be interpreted as the non-linear superposition of two spherically symmetric (Reissner-Nordström) bodies without an intervening strut has been for critically charged masses,In the present paper, an exact electrostatic solution of the EinsteinMaxwell equations representing the exterior field of two arbitrary charged Reissner-Nordström bodies in equilibrium is studied. The invariant physical charge for each source is found by direct integration of Maxwell's equations. The physical mass for each source is invariantly defined in a manner similar to which the charge was found. It is shown through numerical methods that balance without tension or strut can occur for non-critically charged bodies. It is demonstrated that other authors have not identified the correct physical parameters for the mass and charge of the sources. Further properties of the solution, including the multipole structure and comparison with other parameterizations, are examined.

Recent work on gravitational geons is extended to examine the stability properties of gravitational and electromagnetic geon constructs. All types of geons must possess the property of regularity, self-consistency and quasistability on a time-scale much longer than the period of the comprising waves. Standard perturbation theory, modified to accommodate timeaveraged fields, is used to test the requirement of quasi-stability. It is found that the modified perturbation theory results in an internal inconsistency.The time-scale of evolution is found to be of the same order in magnitude as the period of the comprising waves. This contradicts the requirement of slow evolution. Thus not all of the requirements for the existence of electromagnetic or gravitational geons are met though perturbation theory.From this result it cannot be concluded that an electromagnetic or a gravitational geon is a viable entity. The broader implications of the result are discussed with particular reference to the problem of gravitational energy.

The Brill-Hartle gravitational geon construct as a spherical shell of small amplitude, high frequency gravitational waves is reviewed and critically analyzed. The Regge-Wheeler formalism is used to represent gravitational wave perturbations of the spherical background as a superposition of tensor spherical harmonics and an attempt is made to build a non-singular solution to meet the requirements of a gravitational geon. High-frequency waves are seen to be a necessary condition for the geon and the field equations are decomposed accordingly. It is shown that this leads to the impossibility of forming a spherical gravitational geon. The attempted constructs of gravitational and electromagnetic geons are contrasted. The spherical shell in the proposed Brill-Hartle geon does not meet the regularity conditions required for a non-singular source and hence cannot be regarded as an adequate geon construct. Since it is the high frequency attribute which is the essential cause of the geon non-viability, it is argued that a geon with less symmetry is an unlikely prospect. The broader implications of the result are discussed with particular reference to the problem of gravitational energy.

It is shown that no spherically symmetric gravitational geon composed of small amplitude gravitational waves can be in consistent with general relativity. The necessity for high frequency gravitational waves in turn prevents their confinement. This suggests that no gravitational geon can exist.

We illustrate how Ehlers' formal mathematical definition of the Newtonian limit requires additional information to successfully determine the Newtonian limit. This information can be obtained through the physical arguments of Cooperstock's definition. We show that Ehlers' formalism is equivalent to Cooperstock's derivation of the Newtonian limit when the physical arguments are included in the former definition. PACS number: 04.20.C~. 04.20. Jb Many papers have been written on how t o calculate relativistic covariant multipole moments for axially symmetric space-times [I-71. In a recent paper [8], Quevedo uses Ehlers' [9] definition to calculate the Newtonian multipole moments in the Weyl class of axially symmetric space-times. There is a set of parameters {q,) that are important in determining the Newtonian limit. The formal mathematical definition presented by Ehlers does not explicitly take into account the inherent structure of the {q,), which is revealed only through physical arguments. The physical motivation has been discussed by Cooperstock [lo] in his method for determining the Newtonian limit of an axially symmetric space-time. Quevedo discusses the fact that {q,) must be bounded above in order for the Newtonian multipole expansion to converge. An explicit form for these bounds is found in Cooperstock's method. Finally, we show that the methods of Cooperstock and Ehlers are equivalent when proper consideration is given t o (9,).Ehlers' definition of the Newtonian limit is where X = C -' (C = speed of light) and $(p, z, A) is the metric function $ in Weyl's canonical coordinates containing the parameter A. Specifically, Weyl's metric is [Ill where $J,Y are functions of p and z . The asymptotically flat solution $ of the Einstein field equations (EFE's) in Weyl canonical coordinates is where and r* = dp2 + ( z + GMX)2. Expressing Q, (x) as a descending power series in x [12], with ( n + s)! (n + 2s)! b(n, s ) = s! (2n + 2s + I)! ' one can rewrite Eq. (3) as It can be shown that [8] Hence, from Eq. ( I ) , m lirn s x b ( n , s ) A-0 s=o This equation is equivalent t o Eq. (52) of Ref. [8] except that in Ref.[8], the coefficient q, is placed outside of the limit. This implies that q, is independent of A. The X dependence is subsequently introduced into q, through the new parameter Gn = qn(GX)n after the limit has presumably been taken. It would have been appropriate to make the substitution where it is explicitly stated that 4, is X independent.