This work proposes an explanation of the Pioneer anomaly, the unmodelled and as yet unexplained blueshift detected in the microwave signal of the Pioneer 10 and other spaceships by Anderson { \it et al} in 1998. What they observed is similar to the effect that would have either (i) an anomalous acceleration $a_{ \rm P}$ of the ship towards the Sun or (ii) an acceleration of the clocks $a_{ \rm t}=a_{ \rm P}/c$. The second alternative is investigated here, with a phenomenological model in which the anomaly is an effect of the background gravitational potential $\Psi (t)$ that pervades all the universe and is increasing because of the expansion. It is shown that $2a_{ \rm t}={ \rm d}\Psi /{ \rm d}t ={ \rm d}^2\tau_{ \rm clocks} /{ \rm d}t^2$, evaluated at present time $t_0$, where $t$ and $\tau_{ \rm clocks}$ are the coordinate time and the time measured by the clocks, respectively. The result of a simple estimate gives the value $a_{ \rm t}\simeq 1.8\times 10^{ -18}{ s}^{ -1}$, while Anderson { \it et al} suggested $a_{ \rm t}= (2.9\pm 0.4)\times 10^{ -18}{ s}^{ -1}$ on the basis of their observations. The calculation are performed near the Newtonian limit but in the frame of general relativity.
Categories: gr-qc