Solving for metrics that are Einstein, i.e that satisfy $R_{\mu \nu} = \Lambda g_{\mu \nu}$ is highly non-trivial as soon as $g_{\alpha \beta}$ is allowed to have off-diagonal components. However, there have been some results relating to the existence of coordinate systems that diagonalise the metric (K. P. Tod, Class. Quantum Grav. 9 1992). In physics we expect certain components of the metric to obey certain kinds of symmetry relations (axisymmetry, existence of certain Killing vectors...) depending on the phenomenon that we are interested in. If one can show that these properties can be encoded in a diagonal metric under certain coordinate conditions then solving for the components in that particular basis becomes much simpler. The famous example here is the Newman-Janis application to the Schwarzschild solution to generate the Kerr metric via a complex coordinate transformation.

In classical matrix group theory one has the Jordan-Chevalley decomposition theorem which states (loosely) that any matrix can be written as the sum of a diagonalizable and nilpontent component: for any $M \in GL(n,\mathbb{C})$ we can write $M = D + N$. This result has extensions to semi-simple Lie algebras $\mathfrak{g}$.

I know that, at least for Banach algebras with unit, one can define a Riesz functional calculus analogous to the Cauchy integral formula to define functions of bounded, linear operators; let $a \in \mathfrak{A}$ with unit $I$ and $f$ holomorphic then:

$f(a) = \frac {1} {2 i \pi} \oint_{\Gamma} f(z) (zI - a)^{-1} dz$

With $\Gamma$ a rectifiable Jordan curve. One can think of the Einsteinian condition being a kind of differential equation with some differential operator $\mathfrak{D}$ lying in a Banach algebra $\mathfrak{A}$ since the Ricci tensor $R^{\mu}_{\nu}$ consists of derivatives of $g^{\mu}_{\nu}$.

Now for the question, can one decompose the metric tensor into a sum of diagonalizable (in the existence of coordinate sense) and another part? What results, if any, exist regarding such a decomposition? If the space-time does not permit a coordinate system $\{X^{\mu}\}$ under which the metric tensor is diagonal, can one trace this back to a physical property?

For simplicity we can take the dimension of the space-time to be 4.

One can formulate the theory of general relativity (and metric-affine gravity theories) in terms of gauge theory language, i.e, general relativity (in 4-dimensions) can be formulated as a gauge theory over $\mathfrak{su}(2,\mathbb{C})$. This is why I suspect these kinds of results to exist, but am unaware of any.

orthogonal systems. Darboux wrote a book about triply orthogonal systems, i.e. such coordinates, just for Euclidean space of dimension 3. But he wasn't looking at general metrics or decompositions, so I guess that isn't really relevant. – Ben McKay May 2 at 8:51