Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

The Einstein-Weyl equations in 2+1 dimensions contain, as a special class, the Toda field equations for the group SU( infinity ). A family of solutions belonging to this class, and depending on an arbitrary function of one variable, is exhibited.

https://iopscience.iop.org/article/10.1088/0264-9381/7/4/003/pdf

Einstein-Weyl spaces and SU(∞) Toda fields

Riemannian space-times with self-dual curvature and which admit at least one Killing vector field (stationary) are examined. Such space-times can be classified according to whether a certain scalar fieldψ (which is the difference between the Newtonian and NUT potentials) reduces to a constant or not. In the former category (called here KSD) are the multi-TaubNUT and multi-instanton space-times. Nontrivial examples of the latter category have yet to be discovered. It is proved here that the static self-dual metrics are flat. It is also proved that each stationary metric for which the Newtonian and nut potentials are functionally related admits a Killing vector field relative to which the metric is KSD. It has also been proved that the regularity of theψ field everywhere implies that the metric is KSD. Finally it is proved that for non-KSD space-times every regular compact level surface of theψ field encloses the total NUT charge, which must be proportional to the Euler number of the surface.

Stationary Riemannian space-times with self-dual curvature

U(1)×U(1) quaternionic metrics from harmonic superspace

The classical gravitational equations of Einstein are investigated for an anisotropic fluid body in the case of spherical symmetry. An equation of state T (4)(4)+k2T (1)(1)=0 is imposed. The junction conditions [T ba]nb=0 of Synge are required to be satisfied on the boundary of the fluid body. A class of exact, analytical solutions depending on four parameters is obtained. The solutions satisfy the equation of state and the weak energy conditions prior to the collapse of the boundary inside the event horizon. However, in the interior of the event horizon, the matter undergoes a transition into an exotic state. A portion of the fluid turns tachyonic with T44<0, a second portion has complex eigenvalues for [T ba], and a third part has signature +4.

Spherically symmetric collapse of an anisotropic fluid body into an exotic black hole

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...

M. M. Som

Papers

On the stationary gravitational fields