  In this paper we investigate the possibility of measuring the general
relativistic gravitoelectric contribution P^(GE) to the orbital period P of the
transiting exoplanet HD 209458b 'Osiris'. It turns out that the predicted
magnitude of such an effect is \sim 0.1 s, while the most recent determinations
of the orbital period of HD 209458b with the photometric transit method are
accurate to \sim 0.01 s. The present analysis shows that the major limiting
factor is the \sim 1 m s^-1 sensitivity in the measurement of the projected
semiamplitude of the star's radial velocity K. Indeed, it affects the
determination of the mass m of the planet which, in turn, induces a systematic
error in the Keplerian period P^(0) of \sim 8 s. It is of crucial importance
because P^(0) should be subtracted from the measured period in order to extract
the relativistic correction. The present-day uncertainty in $m$ does not yet
make necessary the inclusion of relativistic corrections in the data-reduction
process of the determination of the system's parameters. The present situation
could change only if improvements of one-two orders of magnitude in the
ground-based Doppler spectroscopy technique occurred.

General Relativity and Quantum Cosmology;Astrophysics