We study relativistic effects, arising from the light propagation in an
inhomogeneous universe. We particularly investigate the effects imprinted in a
cross-correlation function between galaxy positions and intrinsic galaxy shapes
(GI correlation). Considering the Doppler and gravitational redshift effects as
major relativistic effects, we present an analytical model of the GI
correlation function, from which we find that the relativistic effects induce
non-vanishing odd multipole anisotropies. Focusing particularly on the dipole
anisotropy, we show that the Doppler effect dominates at large scales, while
the gravitational redshift effect originated from the halo potential dominates
at the scales below $10$ -$30{\textstyle \phantom{\rule{0.167em}{0ex}}}\mathrm{M}\mathrm{p}\mathrm{c}/h$ , with the amplitude of the dipole
GI correlation being positive over all the scales. Also, we newly derive the
covariance matrix for the modelled GI dipole. Taking into account the full
covariance, we estimate the signal-to-noise ratio and show that the GI dipole
induced by the relativistic effects is detectable in future large-volume galaxy
surveys. We discuss how the measurement of dipole GI correlation could be
helpful to detect relativistic effects in combination with the conventional
galaxy-galaxy cross correlation.