An astrophysical peek into Einstein's static universe

TL;DR

This paper derives Einstein's static universe metric from Einstein's equations, showing that a zero effective density implies the need for a dynamic dark energy component to explain the universe's observed properties.

Contribution

It provides a direct derivation of the ESU metric considering both Einstein tensor and energy-momentum tensor, highlighting the role of dark energy and time dependence.

Findings

Effective density in ESU must be zero for uniform pressure and acceleration.

A static universe with dark energy requires zero effective density, implying zero matter and cosmological constant under weak energy condition.

Time-dependent dark energy explains the small observed value of the cosmological constant.

Abstract

We derive here the metric for Einstein's static universe (ESU) directly from Einstein equation, i.e., by considering both $G_{ik}$ and $T_{ik}$. We find that in order that the fluid pressure and acceleration are {\em uniform} and finite despite the presence of a coordinate singularity, the effective density $\rho_e = \rho + \Lambda/8 \pi =0$, where $\Lambda$ is the cosmological constant. Under weak energy condition, this would imply $\rho = \Lambda =0$ for ESU. This means that if one would need to invoke a source of ``repulsive gravity'' in some model, (i) the model must be non-static, (ii) the repulsive gravity must be due to a ``quintessence'' or a ``dark energy'' fluid with negative pressure and appear on the right hand side (RHS) of the Einstein equation through $T_{ij}$ rather than through a fundamental constant residing on the LHS of the same equation, and (iii) energy density of both normal matter and the ``dark energy fluid'' should be time dependent. In fact, the repulsive gravity would be due to a time independent $\Lambda$, it would be extremely difficult to understand why the associated energy density should be approximately $10^{120}$ times lower than the value predicted by quantum gravity. On the other hand, for a dark energy fluid whose energy density is time dependent, it would be much easier to understand such an extremely low present energy density: the original initial value of the energy density of the fluid could be equal to the quantum gravity value while the present low value is due to decay with time.