The theory of General Relativity as usually presented currently defines the relativistic spacetime as a tuple $(M,g,\nabla, T)$ where $(M,g)$ is a four dimensional smooth lorentzian manifold, $\nabla$ is the Levi-Civita connection of the metric $g$ and $T$ is a time orientation.
In other words, $T$ is a nowhere vanishing vector field on $M$ such that $g(T,T)>0$ always.
Well, by what I know about the history of General Relativity, since its begining it invoked the idea of curvature of spacetime. More than that, Einstein's equations
already requires the smooth manifold structure, the lorentzian metric and the Levi-Civita connection in order to have the curvature tensor.
So the structure $(M,g,\nabla)$ for spacetime was already included in Einstein's original formulation of the theory, even if in a different language than the one used today.
However the time orientation is something that I still don't know where it historically came from.
Did Einstein already needed and used somehow the idea of time orientation in his work on General Relativity? If not, who introduced time orientation as a part of the structure of spacetime and why he did so?