  We calculate light curves produced by a hot spot of a rapidly rotating
neutron star, assuming that the spot is perturbed by a core $r$-mode, which is
destabilized by emitting gravitational waves. To calculate light curves, we
take account of relativistic effects such as the Doppler boost due to the rapid
rotation and light bending assuming the Schwarzschild metric around the neutron
star. We assume that the core $r$-modes penetrate to the surface fluid ocean to
have sufficiently large amplitudes to disturb the spot. For a $l'=m$ core
$r$-mode, the oscillation frequency $\omega\approx2m\Omega/[l'(l'+1)]$ defined
in the co-rotating frame of the star will be detected by a distant observer,
where $l'$ and $m$ are respectively the spherical harmonic degree and the
azimuthal wave number of the mode, and $\Omega$ is the spin frequency of the
star. In a linear theory of oscillation, using a parameter $A$ we parametrize
the mode amplitudes such that ${\rm
max}\left(|\xi_\theta|,|\xi_\phi|\right)/R=A$ at the surface, where
$\xi_\theta$ and $\xi_\phi$ are the $\theta$ and $\phi$ components of the
displacement vector of the mode and $R$ is the radius of the star. For the
$l'=m=2$ $r$-mode with $\omega=2\Omega/3$, we find that the fractional Fourier
amplitudes at $\omega=2\Omega/3$ in light curves depend on the angular distance
$\theta_s$ of the spot centre measured from the rotation axis and become
comparable to or even larger than $A\sim0.001$ for small values of $\theta_s$.
