Questions on Static Cosmology

Ongoing research in cosmology has produced static solutions to Einstein’s field equations which are consistent with cosmic redshift and do not require an expanding universe. The various metrics associated with these solutions differ from the ΛCDM Friedman-Lemaitre-Robertson-Walker (FLRW) metric in that the time component g⁰⁰ of the metric is a function of r, while the radial component g¹¹ is also a function of r and does not include the dynamical scale factor a(t).

It is possible for a static metric to obey a redshift-distance relation by allowing the time component g⁰⁰ to be a function of r, as was proved in the paper entitled A Static Cosmological Model, MOND, and the Galactic Rotation Curve found on the home page of this website. Einstein, in his original static cosmology, set the time component equal to unity. His theory was thus unable accomodate the Hubble redshift. In response to this, he adopted the expanding universe or big bang model, based on the FLRW metric first introduced by Friedman in 1922.

Einstein’s rejection of static models has had a major influence on the development of cosmological theory. As a result of his work, few physicists have investigated static models. Notable exceptions are Fritz Zwicky, who suggested that cosmic redshift might be explained in a static universe by a phenomenon called tired light. A suitable mechanism for tired light, however, was never found, and Zwicky’s theory was subsequently abandoned.

A second exception is Robert H. Dicke, who in the 1950’s proposed a static model in which cosmic redshift was explained by a variable speed of light (VSL). One complication related to metric VSL theories, however, is that variations in the speed of light affect the g⁰⁰ component of the metric, and thus the rate of time flow. Light velocity can therefore be made constant through a transformation of the time coordinate. To accommodate a varying speed of light in a nontrivial way, an extra dimension or additional free variable, such as a scalar potential, must be introduced. Perhaps this explains why Dicke abandoned his VSL model in favor of the Brans-Dicke theory, a scalar-tensor theory in which includes a scalar potential.

That static theories have been neglected seems a serious oversight, especially at a time when hundreds of theories of gravity are being investigated. The majority of current models, even while modifying or rejecting Einstein’s field equations, retain the FLRW metric as a dynamic representation of an expanding universe. Yet the FLRW metric raises question from a mathematical standpoint. One feature of the FLRW metric that may raise doubts is the fact that the co-moving coordinate r, in conjuction with an orthogonal conformal time coordinate t, requires the time axes of receeding galaxies to diverge. Indeed, beyond a certain distance, the conformal time axes of remote galaxies become superluminal or spacelike, even though the sign of g⁰⁰ in the line element does not change. This is in contrast to the static Shwarzschild line element, in which g⁰⁰ becomes positive and g¹¹ becomes negative in the region r<2m, where timelike trajectories become spacelike.

In contrast, on the surface of the earth, the latitudinal axes, or geodesics of constant longitude, diverge at various angles from the poles. Yet here there is no contradiction. Divergence is possible because the surface of the earth is described by Riemannian geometry. General relativistic metrics, in contrast, are pseudo-Riemannian, and thus only one dimension in the line element ds²  can have a signature of opposite sign. If the region of opposite sign is one-dimensional, can the axes in that dimension be non-parallel? Would two dimensions not be necessary to accomodate this divergence? Interestingly, in the context of 5-dimensional variable mass theories, there are arguably two dimensions of time, t and s, where s is proper time τ.