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General relativity is often used in context of Lorentzian manifolds. But the texts which describe Einstein Field Equations discuss them in context of general pseudo-Riemannian manifolds.

It seems natural that general relativity be restricted to Lorentzian manifolds because of time problems, but it is not clear whether there is and should be explicit restriction to Lorentzian manifolds.

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The spacetime in general relativity is defined as an $n$-dimensional Lorentzian manifold. Lorentzian manifolds are a type of pseudo-Riemannian manifold (Since they are manifolds of signature $(p,q)$, a Lorentzian manifold is just a pseudo-Riemannian manifold with $p = 1$), which is why you find the term occasionally in textbooks.

It is indeed easy to generalize general relativity to an arbitrary signature $p,q$, but this is not general relativity then. The case $(0,n)$ is referred to as Euclidian gravity, but there isn't really a term for other cases $(p,q)$, the ultrahyperbolic case. Due to various reasons (see the comments), it's not a terribly attractive theory.

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