How can you tell if spherical-like coordinates are locally flat across the origin?

In general relativity, with spherical-like coordinates in a radial gauge, I have a metric that looks like:

$$-g_{tt}\mathrm{d}t^2 + g_{rr}\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\ \mathrm{d}\phi^2)$$

I want to know what are the weakest requirements on the time and radial component of the metric to guarantee local flatness at the origin (r=0). To give an example of the type of function I'm curious about, consider:

$$-r\mathrm{d}t^2 + 3\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\ \mathrm{d}\phi^2)$$

which is a (non-asymptotically) flat solution to Einstein's equations for a radiation fluid. Is it locally flat at the origin? I'm looking for a methodology rather than an answer to that specific situation, since there are other solutions I'm interested in as well.

• Maybe I'm missing something, but isn't every metric locally flat? At least as long as it doesn't have a singularity. – Javier May 13 '16 at 22:54
• At the origin, the coordinates are not continuous, so there's a question of whether or not it is flat right at the origin. – juacala May 13 '16 at 23:10
• It doesn't make any sense to talk about physics at a singularity (at least not at a real one). Near one, maybe, directly at one the only thing you get is nonsense as the theory breaks down. – CuriousOne May 13 '16 at 23:59

To find points where local flatness breaks down, a general strategy is to calculate the curvature tensor $R_{\mu\nu\rho\sigma}$, and find the locus of various singularities (points where $R_{\mu\nu\rho\sigma}$ becomes unbounded). Except in rare cases (i.e. unusual cancellation), singular behavior of $R_{\mu\nu\rho\sigma}$ is apparent in the Ricci scalar $R=R^{\rho\alpha}_{\,\,\,\,\rho\alpha}$.
With this in mind: when you're interested mainly in the region near $r=0$, it's useful to consider how the metric changes as you apply the coordinate transformation $(t',r')=(\alpha t,\alpha r)$ with $\alpha \gg 1$. The metric with respect to the new coordinates is $\alpha^{-2}(-g_{tt}(\alpha^{-1}x)dt'^2+g_{rr}(\alpha^{-1}x)dr'^2+r'^2d^2\Omega)$, and a necessary condition for local flatness is that the rescaled metric approach something proportional to the ordinary Minkowski metric, $-dt^2+dr^2+r^2d\Omega^2$. Hence, the form of the angular metric component you assumed strongly constrains the form of $g_{tt}$ and $g_{rr}$: in particular, in order for the manifold to be locally flat near the origin, it's necessary for $g_{tt}(0)=-g_{rr}(0)=1$ (this condition rules out your example, which has a cone-like singularity). You can also notice singular behavior in your example from the form of null geodesics: any function $(t(\tau),0,\dots)$ is a null geodesic, but so are functions of the form $r(t)=\frac{1}{12}t^2+C$ that pass through the origin.