General relativity, the cosmological constant and modular forms

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 and thus to the theory of Taniyama-Shimura.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 Λ 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, Λ = 0 and ii) a case with all three cosmological parameters non-zero, are associated with elliptic curves with the property of complex multiplication and absolute modular invariant j = 0, 1728 respectively. The possibilty that all non-linear solutions of General Relativity are expressed in terms of theta functions associated with Riemann-surfaces is discussed.

Precise relativistic orbits in Kerr and Kerr–(anti) de Sitter spacetimes

“…See also [33] for a different approach. 10 We note the small eccentricity (0.014) of the GP-B satellite. …”

“…Moreover, the above solution is also important for probing the strong field regime of general relativity [9]. This is significant, since general relativity has triumphed in large scale cosmology [5,4,51,10], and in predicting solar system effects like the perihelion precession of Mercury with a very high precision [1,3]. In a recent paper [3] Kraniotis and Whitehouse, provided the exact solution in closed analytic form, of the time-like geodesics that describe the motion of a test particle in the Schwarzschild gravitational field including the contribution from the cosmological constant.…”

Parametric Solution of Certain Nonlinear Differential Equations in Cosmology

“…(3.1) is mathematically the same as Eq. (8.9) of [18], and since its solution is presented and discussed in [15] we greatly limit our remarks here.…”

“…of [15], which is associated with the metric (1.4) in the introduction. Since the functions K(z), M(z) are independent of t, Eq.…”

“…Also see [17], where one can consider D = −Q 2 = 0 (with yet E = 0), Q being a constant electric charge. The Friedmann-Lemaître-Robertson-Walker (FLRW) metric can be obtained from ds 2 in (1.4) (as a "limit"), and a known formula (see formula (42) of [15], for example) for the scale factor (the "radius" of the FLRW universe) also follows from the general formulas presented here. Further remarks on this, as well as the computation of scale factors in anisotropic models (in the Bianchi V and IX models, for example) are taken up in Sec.…”