  The general relativistic precession rate of periastra in close-in exoplanets
can be orders of magnitude larger than the magnitude of the same effect for
Mercury. The realization that some of the close-in exoplanets have significant
eccentricities raises the possibility that this precession might be detectable.
We explore in this work the observability of the periastra precession using
radial velocity and transit light curve observations. Our analysis is
independent of the source of precession, which can also have significant
contributions due to additional planets and tidal deformations. We find that
precession of the periastra of the magnitude expected from general relativity
can be detectable in timescales of <~ 10 years with current observational
capabilities by measuring the change in the primary transit duration or in the
time difference between primary and secondary transits. Radial velocity curves
alone would be able to detect this precession for super-massive, close-in
exoplanets orbiting inactive stars if they have ~100 datapoints at each of two
epochs separated by ~20 years. We show that the contribution to the precession
by tidal deformations may dominate the total precession in cases where the
relativistic precession is detectable. Studies of transit durations with Kepler
might need to take into account effects arising from the general relativistic
and tidal induced precession of periastra for systems containing close-in,
eccentric exoplanets. Such studies may be able to detect additional planets
with masses comparable to that of Earth by detecting secular variations in the
transit duration induced by the changing longitude of periastron.
