  We present the calculation of the Lyman-alpha (Lyman-$\alpha$) transmitted
flux fluctuations with full relativistic corrections to the first order. Even
though several studies exist on relativistic effects in galaxy clustering, this
is the first study to extend the formalism to a different tracer of underlying
matter at unique redshift range ($z=2-5$). Furthermore, we show a comprehensive
application of our calculations of the Quasar-Lyman-$\alpha$ cross-correlation
function. Our results indicate that the signal of relativistic effects is
sizeable at Baryonic Acoustic Oscillation (BAO) scale mainly due to the large
difference in density bias factors of our tracers. We construct an observable,
the anti-symmetric part of the cross-correlation function, that is dominated by
the relativistic signal and offers a new way to measure the relativistic terms
at relatively small scales. The analysis shows that relativistic effects are
important when considering cross-correlations between tracers with very
different biases, and should be included in the data analysis of the current
and future surveys. Moreover, the idea presented in this paper is highly
complementary to other techniques and observable trying to isolate the effect
of the relativistic corrections and thus test the validity of the theory of
gravity beyond the Newtonian regime.
