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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

$$R_{\mu\nu}-\dfrac{1}{2}{g_{\mu\nu}}R=\dfrac{8\pi G}{c^4}T_{\mu\nu}$$

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?

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    $\begingroup$ General relativity does not require spacetime to be time orientable. Modern sensitivity to the issue did not arise until van Stockum in 1937 and especially Gödel in 1949 found solutions to the Einstein equations with closed timelike curves. But even now time-orientable spacetimes (i.e. with a global timelike vector field) are usually treated as a proper subclass of all spacetimes. $\endgroup$ – Conifold Mar 6 '17 at 19:08
  • $\begingroup$ The metric and the connection are not really two separate pieces of information. GR assumes no torsion, so the metric and the connection are completely redundant. $\endgroup$ – Ben Crowell Mar 6 '17 at 20:32
  • $\begingroup$ @BenCrowell Did you mean "specifying both the metric and the connection is completely redundant"? By the way, the Levi-Civita connection (which you are most likely referring to) is not uniquely specified by stipulating that there should be no torsion: You have to require metric compatibility too. $\endgroup$ – Danu Mar 7 '17 at 10:29
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I believe the question started being studied in earnest in the 1950s, with formal definition and first results usually attributed to Markus (1955, p. 412) — by, e.g., Hawking in his famous 1966 prize essay (emphasis added):

5.2 Time orientability

In our own neighbourhood, there is a clear division of non-spacelike vectors into future- and past-directed. However, it is not so obvious that we could extend this to give a continuous division at all points of $M$. This problem has been dealt with by Markus (1955): as there is a Lorentz metric on $M$, we can find a continuous line-element field $(X,−X)$, where $X$ is a timelike vector. The manifold $\smash{\tilde M}$ is defined as the set of all pairs $\smash{(p,X|_p)}$ and $\smash{(p,-X|_p)}$ with the natural structure. Clearly, $\smash{\tilde M}$ covers $M$ twice. There are two possibilities: either $\smash{\tilde M}$ is connected, in which case it is impossible to introduce a continuous division into future- and past-directed vectors in $M$, but one exists in $\smash{\tilde M}$, or $\smash{\tilde M}$ consists of two disconnected components, in which case one can find a continuous division in $M$. Physically it would seem very reasonable to expect $M$ to be time orientable, but even if it was not, we could apply all the following theorems to the time orientable covering manifold $\smash{\tilde M}$.

As for Einstein himself, Gödel’s model containing closed timelike curves (1949) prompted him to at least some memories in (1949, pp. 687-688) — where he is careful to distinguish the local and global question (emphasis added):

The problem here involved disturbed me already at the time of the building up of the general theory of relativity, without my having succeeded in clarifying it (...):

If $P$ is a world-point, a “light-cone” ($\smash{ds^2 = 0}$) belongs to it. We draw a “time-like” world-line through $P$ and on this line observe the close world-points $B$ and $A$, separated by $P$. Does it make any sense to provide the world-line with an arrow, and to assert that $B$ is before $P$, $A$ after $P$?

(...) If, therefore, $B$ and $A$ are two, sufficiently neighboring, world-points, which can be connected by a time-like line, then the assertion: “$B$ is before $A$,” makes physical sense. But does this assertion still make sense, if the points, which are connectable by the time-like line, are arbitrarily far separated from each other? Certainly not, if (...) those paradoxes, regarding the direction of the causal connection, arise, of which Mr. Gödel has spoken.

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