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I have some confusion regarding solutions of Einstein's field equations. I have read in several places that an exact solution to the field equations is a Lorentzian manifold. Now given a stress-energy tensor [tex]T_{\mu\nu}[/tex] the equations [tex]G_{\mu\nu} = 8\pi T_{\mu\nu}[/tex] determine the metric tensor [tex]g_{\mu\nu}[/tex]. But a Lorentzian manifold [tex](M,g)[/tex] is a differential manifold [tex]M[/tex] together with a pseudo-Riemannian metric [tex]g[/tex], so it's not clear to me how that manifold is related to the metric tensor [tex]g_{\mu\nu}[/tex].

There might exist several differential manifolds which could be given the same metric, so in which sense do we say that a Lorentzian manifold is a solution to the field equations?