Null geodesic structure for the Barriola-Vilenkin spacetime via $k$-essence

Based on the work of Chandrasekhar [{\it The Mathematical Theory of Black Holes, Oxford Univ. Press (1992)}], we investigate the null geodesic structure of the emergent Barriola-Vilenkin spacetime in the context of {\bf k-}essence theory. For {\bf k-}essence, the emergent gravity metric is a one-to-one correspondence with the Barriola-Vilenkin (BV) metric connected to the Schwarzschild background, where the global monopole charge is replaced by the dark energy density. This equivalence holds specifically for a certain class of {\bf k-}essence scalar fields that have been constructed by Gangopadhyay and Manna [Euro. Phys. Lett., 100, 49001, (2012)]. We have traced out different trajectories for null geodesic in the presence of dark energy for the {\bf k-}essence emergent Barriola-Vilenkin spacetime. It is demonstrated that the outcomes deviate from the typical Schwarzchild spacetime owing to the fundamental configuration with a constant dark energy density.