Common properties of all spacetime metrics I understand that spacetime (in GR) has many possible metrics, all of which are calculated via the Einstein Field Equations. The Minkowski metric (from SR) is one solution of the EFEs; it is a solution of the EFEs for a flat spacetime in a vacuum.
Assuming I got the above right, my question is: what unites all possible spacetime metrics and makes them different from other kinds of metrics (e.g., from the Euclidean metric)? Of course, here I have in mind non-trivial properties of the spacetime metrics.
Background:
I am self-learning both the special and the general theory of relativity. Apologies if this question has an obvious answer.
 A: From a mathematical point of view, all you need to describe a 3+1D spacetime solution to the Einstein equations is a pseudo-Riemannian metric with signature $(3, 1)$ or $(1, 3)$ (depending on your convention). This is because any Lorentzian manifold defines a spacetime by simply declaring the associated energy-momentum tensor to be the Einstein tensor divided by $\kappa$:
$$
T_{\mu\nu}\equiv\underbrace{\frac{1}{\kappa}G_{\mu\nu}}_\text{geometric}\Rightarrow G_{\mu\nu}=\kappa T_{\mu\nu}
$$
For a vacuum solution, the Einstein equations simplify greatly:
$$
R_{\mu\nu}=\frac{\Lambda}{n-2}g_{\mu\nu}=kg_{\mu\nu}
$$
i.e. the Ricci tensor must be proportional to the metric, defining the so-called "Einstein manifold". With no cosmological constant, this reduces to the well-studied Ricci-flat manifold.
For a given energy-momentum tensor $T_{\mu\nu}$, the uniting property of all spacetime metrics is, somewhat tautologically, that they must satisfy the Einstein field equations.
$$
G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu}
$$
where the left-hand side of the equation can be computed directly from the metric alone. For completely general $T_{\mu\nu}$, this places no constraints on the metric that are simpler than the Einstein field equations themselves - defining the space of metrics as the intersection of the solution space to 10 nonlinear, second-order partial differential equations. However, we can often use energy conditions or exploit the symmetry of the energy-momentum distribution to, for example, generate ansätze.
For physically realisable (and useful) scenarios however, you may want additional "nice" global spacetime features of various strengths, e.g.

*

*no closed chronal/causal curves, various other notions of causality

*global hyperbolicity to make the Cauchy problem well-posed

*no isometric embeddings

*asymptotic flatness

*geodesic incompleteness to allow singularities - and by extension, the Penrose-Hawking singularity theorems

*a plethora of other conditions that you will likely find in The Large Scale Structure of Space-Time by Hawking and Ellis.

These constraints amount to identifying a subset of the space of metrics satisfying the EFE. Unfortunately the imposition of most of these conditions is notoriously difficult to analyse analytically in terms of the components of the metric without heavily simplifying assumptions.
