Is there a limitation on the values ​that Einstein tensor $G_{\mu\nu}$ can take? Is there a limitation on the values ​​that Einstein tensor $G_{\mu\nu}$ can take? 
For example: 


*

*Is it always bigger than zero?

*What is the highest amount that can be taken by it?

*What is the smallest amount it can get? 

*Is it always positive and non-zero?
 A: In general, the answer is that there is no generic constraint in the sense mentioned above. For example, in the case of de Sitter metric, 
\begin{align}
G_{\mu\nu} = -\Lambda g_{\mu\nu}\,,
\end{align}
where $g_{\mu\nu}$ for static patch of de Sitter spacetime can be given by line element
\begin{equation}
ds^2 = -(1-\Lambda r^2/3)dt^2+\frac{dr^2}{1-\Lambda r^2/3} +r^2d\Omega^2\,.
\end{equation}
For this particular chart, where $r>0$, it is clear that $g_{tt}\in (-1,\infty)$ while $g_{rr} \in (-\infty,1)$. The Poincare chart of AdS spacetime has diagonal elements $g_{\mu\mu}\in (0,\infty)$ for all $\mu$. Also, since $G_{\mu\nu}$ is a tensor, its elements change value even by simple coordinate transformation, so the constraints (1)-(4) do not really make sense. On the other hand, there are constraints one can ask about e.g. $G_{\mu\nu}u^\mu u^\nu$, which is closely related to energy condition. This makes sense because the contraction with four-velocity is a scalar so its magnitude is coordinate-independent.
A: @Everiana Four-velocity of what? In general I don't understand the question. Is the OP assuming that Einstein's equations are at work? Then it's tantamount to put the same question on $T_{\mu\nu}$. And then I do understand what $u_\mu$ means. Otherwise it could be a generic question about a (1,3) semi-riemannian manifold. In that case
only invariant expressions like 
$g^{\mu \nu} G_{\mu\nu}$ or 
$G^{\mu\nu} G_{\mu\nu}$ could be constrained but I'm unaware of such constraints.
