(Anti) de Sitter Relativity, Modified Newtonian Dynamics, Noncommutative Phase Spaces and the Pioneer Anomaly

It is shown how the de-Sitter Relativistic behaviour of the hyperbolic trajectory of Pioneer, due to the expansion of the Universe (non-vanishing cosmological constant), is the underlying physical reason for the observed anomalous acceleration of the Pioneer spacecraft of the order of c/RH ∼ 8.74 × 10−10 m/s2, where c is the speed of light and RH is the present Hubble scale. We display the explicit isomorphism among Yang’s Noncommutative space-time algebra, the 4D Conformal algebra SO(4, 2) and the area-bi-vector-coordinates algebra in Clifford spaces. The former Yang’s algebra involves noncommuting coordinates and momenta with a minimum Planck scale λ (ultraviolet cutoff) and a minimum momentum p = h̄/R (maximal length R, infrared cutoff ). It is shown how Modified Newtonian dynamics is also a consequence of Yang’s algebra resulting from the modified Poisson brackets. To finalize we study the deformed Kepler and free motion resulting from the modified Newtonian dynamics due to the Leznov-Khruschev noncommutative phase space algebra and which stems also from the Conformal algebra SO(4, 2) in four dimensions. Numerical examples are found which yield results close to the experimental observations, but only in very extreme special cases and which seem to be consistent with a Machian view of the Universe.