Research Note on Static Cosmology:
My ongoing research in mathematical 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 LambdaCDM Friedman-Robertson Walker (FRW) metric in that the tt component of g is not unity but rather a function of r, while the rr component is a function of r rather than the dynamical scale factor a(t).
It is possible to inject a redshift-distance relation into a static metric by allowing the tt component of g to be a function of r, as I proved in my paper on the home page of this website. Einstein, in his original static theory, set the tt component equal to unity. This was why his model could not accomodate the cosmic Hubble redshift, and why he subsequently adopted the expanding universe Big Bang model based on the FRW metric, first introduced by Friedman in 1922.
Einstein’s rejection of static models has had a major influence on the development of cosmology over the past century. As a result, few prominent physicists have investigated static models. Notable exceptions are Fritz Zwicky, who suggested that cosmic redshift might be explained in a static universe by postulating a phenomenon called “tired light”. A suitable mechanism for tired light, however, could not be found, and Zwicky’s theory was abandoned.
As an aside, there is after all a possible mechanism for tired light which I intend to investigate in the near future. The mechanism depends on the assumption that the bending of light around a star or galaxy should induce a transverse momentum, and hence a kinetic energy, in that star or galaxy, albeit very small. The induced kinetic energy would have to be balanced by a decrease in energy, and hence frequency, of the photon. The only question is, is this effect of the right order of magnitude? Of course, a photon-deflection tired light mechanism would produce a redshift that increases linearly with distance, rather than the observed standard t to the 2/3 power relation, not to mention more recent observations such as that of cosmic acceleration, discovered in the 1990’s. These discrepancies would have to be explained.
A second exception is Robert (?) Dicke, who in the 1950’s proposed a static model in which cosmic redshift was explained by a variable speed of light (VSL). The problem with metric VSL theories, however, is that any change in the speed of light also affects the rate of the flow of time, so that light velocity can be made constant through a trivial transformation of the time coordinate. The only way to accommodate a nontrivial varying speed of light is to introduce either a fifth dimension or another free variable, such as a scalar potential. Indeed, Dicke soon abandoned his VSL theory in favor of the Brans-Dicke theory, a scalar-tensor theory which introduces a scalar potential in addition to the metric.
That static theories have been so thoroughly rejected they are rarely if ever seen in the literature seems a profound oversight, especially in an era when literally hundreds of gravitational theories are being investigated. The vast majority of current theories, even while modifying Einstein’s Field Equations, retain the FRW metric, which is a dynamic representation of an expanding universe. Yet the FRW metric, from a mathematical standpoint, raises many questions. Is it a valid metric? Maybe not. I have an intuition that it is not valid, but despite several attempts, have not been able to prove this conjecture. I intend to devote more time to the question. The main feature of the FRW metric that raises my doubts is the fact that the co-moving coordinate r, in conjuction with an orthogonal time coordinate t, requires the time axes of receeding galaxies to diverge. By “time axis” here, I mean the geodesic r=constant of the galaxy’s trajectory through spacetime.
Now, on the surface of the earth, for example, the “latitudinal axes”, ie the geodesics of constant longitude, diverge from the south pole, and there is no contradiction in this case. This is because the surface of the earth is described by true Riemannian geometry. Relativistic metrics, in contrast, describe pseudo-Riemannian geometry, in which one—but only one—dimension, normally time, has a length of opposite sign. If only one dimension has a length of opposite sign, how can axes in that dimension diverge? Would there not need to be two dimensions of opposite sign to accomodate this divergence? My sense is yes. Of course, in a different context (the context of 5-dimensional variable mass theories), I suspect there may be two dimensions of time, namely t and s, where s is proper time and also the measure of relativistic length. But this is not pertinent to the validity of the FRW metric.
Yet, this is not the only reason to doubt the FRW metric. After all, FRW was touted as an exact solution to Einstein’s Field Equations (EFE). So how can modified gravities, such as f(R) gravities, retain FRW but throw out EFE? Are the two not intrinsically connected?
It is also the case that FRW metric only trivially solves EFE, in the sense that the equation of state carries far more information, involving many more assumptions, than the field equations themselves. But that is a topic for another discussion.