  In this paper the application of the $M$-projective curvature tensor in the
general theory of relativity has been studied. Firstly, we have proved that an
$M$-projectively flat quasi-Einstein spacetime is of a special class with
respect to an associated symmetric tensor field, followed by the theorem that a
spacetime with vanishing $M$-projective curvature tensor is a spacetime of
quasi-constant curvature. Then we have proved that an $M$-projectively flat
quasi-Einstein spacetime is infinitesimally spatially isotropic relative to the
unit timelike vector field $\xi$. In the next section we have proved that an
$M$-projectively flat Ricci semi-symmetric quasi-Einstein spacetime satisfying
a definite condition is an $N(\frac{2l-m}{6})$-quasi Einstein spacetime. In the
last section, we have firstly proved that an $M$-projectively flat perfect
fluid spacetime with torse-forming vector field $\xi$ satisfying Einstein field
equation with cosmological constant represents an inflation, then we have found
out the curvature of such spacetime, followed by proving the theorem that the
spacetime also becomes semi-symmetric under these conditions. Lastly, we have
found out the square of the length of the Ricci tensor in this type of
spacetime and also proved that if an $M$-projectively flat perfect fluid
spacetime satisfying Einstein field equation with cosmological constant, with
torse-forming vector field $\xi$ admits a symmetric $(0,2)$ tensor $\alpha$
parallel to $\nabla$ then either $\lambda = \frac{k}{2}(p-\sigma)$ or $\alpha$
is a constant multiple of $g$.
