General Relativity, the Cosmological Constant and Modular Forms

TL;DR

This paper derives exact inhomogeneous solutions to Einstein's equations with a Cosmological Constant, linking general relativity to modular forms and elliptic curves, and explores their cosmological implications.

Contribution

It provides the first full exact solutions within the Szekeres-Szafron family connecting general relativity with modular forms and elliptic curves, including solutions with complex multiplication.

Findings

Exact solutions connect GR with modular forms and elliptic curves.

Special cases correspond to elliptic curves with complex multiplication.

Cosmological implications of non-linear solutions are systematically analyzed.

Abstract

Strong field (exact) solutions of the gravitational field equations of General Relativity in the presence of a Cosmological Constant are investigated. In particular, a full exact solution is derived within the inhomogeneous Szekeres-Szafron family of space-time line element with a nonzero Cosmological Constant. The resulting solution connects, in an intrinsic way, General Relativity with the theory of modular forms and elliptic curves. The homogeneous FLRW limit of the above space-time elements is recovered and we solve exactly the resulting Friedmann Robertson field equation with the appropriate matter density for generic values of the Cosmological Constant %Lambda and curvature constant K. A formal expression for the Hubble constant is derived. The cosmological implications of the resulting non-linear solutions are systematically investigated. Two particularly interesting solutions i) the case of a flat universe K=0, Lambda not= 0 and ii) a case with all three cosmological parameters non-zero, are described by elliptic curves with the property of complex multiplication and absolute modular invariant j=0 and 1728, respectively. The possibility that all non-linear solutions of General Relativity are expressed in terms of theta functions associated with Riemann-surfaces is discussed.