Finite Energy Geodesic Rays in Big Cohomology Classes

For a big class represented by $\theta$, we show that the metric space $({\mathcal{E}}^p(X,\theta ),d_p)$ for $p\ge 1$ is Buseman convex. This allows us to construct a chordal metric $d_p^c$ on the space of geodesic rays in ${\mathcal{E}}^p(X,\theta )$. We also prove that the space of finite $p$-energy geodesic rays with the chordal metric $d_p^c$ is a complete geodesic metric space. With the help of the metric $d_p$, we find a characterization of geodesic rays lying in ${\mathcal{E}}^p(X,\theta )$ in terms of the corresponding test curves via the Ross-Witt Nyström correspondence. This result is new even in the Kähler setting.