There is presented a particularly simple transformation of the Schwarzschild metric into new coordinates, whereby the "spherical singularity" is removed and the maximal singularity-free extension is clearly exhibited.

Maximal Extension of Schwarzschild Metric

Exact solutions are obtained for the anisotropic Bianchi type‐I model in normal gauge for Lyra’s geometry. The physical behavior of the models is examined in vacuum and in the presence of perfect fluids.

Bianchi type‐I cosmological models in Lyra’s geometry

A generalization of Einstein's theory is discussed in which the gravitation is described by a tensor and a scalar field. The theory is more consistent with Mach's principle and less reliant on absolute properties of space. The modification involves a violation of the “strong principle of equivalence” on which Einstein's theory is based. In the original version of this new theory, the “constant” of gravitationG is varying and particle masses are fixed. Later on another version of the theory was given in whichG is truly a constant and the particle masses vary. The two versions are related by a conformal transformation. The physical and mathematical foundations of this theory have been discussed and the field equations have been derived. The astrophysical and cosmological consequences of the theory have been elaborately reviewed.

Scalar-tensor theories of gravitation: Foundations and prospects

We present a class of nonsingular cosmological models with a variable cosmological term described by the second-rank symmetric tensor Λνμ, evolving from Λδνμ to λδνμ with λ < Λ. All Λνμ-dominated cosmologies belong to Lemaitre type models for an anisotropic perfect fluid. The expansion starts from a nonsingular non-simultaneous de Sitter bang, with Λ on the scale responsible for the earliest accelerated expansion, which is followed by an anisotropic Kasner type stage. For a certain class of observers these models can also be identified as Kantowski–Sachs models with regular R regions. For Kantowski–Sachs observers the cosmological evolution starts from horizons with a highly anisotropic 'null bang' where the volume of the spatial section vanishes. We study in detail the spherically symmetric case and consider the general features of cosmologies with planar and pseudospherical symmetries. Nonsingular Λνμ-dominated cosmologies are Bianchi type I in the planar case and hyperbolic analogues of the Kantowski–Sachs models in the pseudospherical case. At late times all models approach a de Sitter asymptotic with small λ.

https://iopscience.iop.org/article/10.1088/0264-9381/20/16/317/pdf

Nonsingular vacuum cosmologies with a variable cosmological term

The spherically symmetric dust model of Lema\^{\i}tre and Tolman can describe wormholes, but the causal communication between the two asymptotic regions through the neck is even less than in the vacuum (Schwarzschild-Kruskal-Szekeres) case. We investigate the anisotropic generalization of the wormhole topology in the Szekeres model. The function $E(r,p,q)$ describes the deviation from spherical symmetry if ${\ensuremath{\partial}}_{r}E\ensuremath{\ne}0,$ but this requires the mass to be increasing with radius, ${\ensuremath{\partial}}_{r}Mg0,$ i.e. nonzero density. We investigate the geometrical relations between the mass dipole and the loci of the apparent horizon and shell crossings. We present the various conditions that ensure physically reasonable quasispherical models, including a regular origin, regular maxima and minima in the spatial sections, and the absence of shell crossings. We show that physically reasonable values of ${\ensuremath{\partial}}_{r}E\ensuremath{\ne}0$ cannot compensate for the effects of ${\ensuremath{\partial}}_{r}Mg0$ in any direction, so that communication through the neck is still worse than in the vacuum. We also show that a handle topology cannot be created by identifying hypersufaces in the two asymptotic regions on either side of a wormhole, unless a surface layer is allowed at the junction. This impossibility includes the Schwarzschild-Kruskal-Szekeres case.

You cannot get through Szekeres wormholes: Regularity, topology, and causality in quasispherical Szekeres models

In connection with the problem of the initial singularity in the scalar-tensor anisotropic cosmology of Jordan-Brans-Dicke, the dynamics of homogeneous models of Bianchi type I is examined on the basis of the general analytic solutions in vacuo and in the presence of gravitating matter with state equations P=nɛ (0 ⩽ n ⩽ 1). It is shown that the scalar homogeneous ϑ-field, as an effective source of the V4geometry, has to influence essentially the dynamics of the early anisotropic stage of the Universe's expansion, and significantly modifies the character of the initial singularity. At negative ω (ω < −6), the sourceless scalar ϑ-field may remove the singularity and provide regular ‘bouncing’ in the models with matter (P ⩽ ɛ/3) if it prevails over the tensor anisotropic mode of the vacuum gravitational field.

V. A. Ruban

Papers

On the initial singularity in the scalar-tensor anisotropic cosmology

Spherically Symmetric T-Models in the General Theory of Relativity

T-models of ''sphere'' in general relativity theory.

Spherically Symmetric T-models in the General Theory of Relativity