**(Anti) de SITTER RELATIVITY, MODIFIED NEWTONIAN DYNAMICS, NONCOMMUTATIVE PHASE SPACES AND THE PIONEER ANOMALY**

by Carlos Castro

Center for Theoretical Studies of Physical Systems

Clark Atlanta University, Atlanta GA. 3031, castro@ctsps.cau.edu

September, 2007

**Abstract **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^2/*R_H* ~8.74×10^−10 m/s^2, where *c* is the speed of light and *R_H* is the present Hubble scale. We display the explicit isomorphism among Yang’s Noncommutative space-time algebra, the 4*D* 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 lambda (ultraviolet cutoff) and a minimum momentum *p = ¯h*/*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 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.

*Received:* 9 October 2007. *Submitted to:* JMP.