General Relativity, the Cosmological Constant and Modular Forms

Details General Relativity, the Cosmological Constant and Modular Forms General Relativity, the Cosmological Constant and Modular Forms

2001-05-05

arxiv.org/abs/gr-qc/0105022v4

Class.Quant.Grav.19:5073-5100,2002

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

LaTeX file, 34 pages plus 9 EPS figures, Accepted for Publication in Classical and Quantum Gravity

Scientific paper

10.1088/0264-9381/19/20/304

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.

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