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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.

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    $\begingroup$ Maybe I'm missing something, but isn't every metric locally flat? At least as long as it doesn't have a singularity. $\endgroup$ – Javier May 13 '16 at 22:54
  • $\begingroup$ At the origin, the coordinates are not continuous, so there's a question of whether or not it is flat right at the origin. $\endgroup$ – juacala May 13 '16 at 23:10
  • $\begingroup$ 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. $\endgroup$ – CuriousOne May 13 '16 at 23:59
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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 $R=R^{\rho\alpha}_{\,\,\,\,\rho\alpha}$.

An alternative perspective on local flatness involves understanding the behavior of geodesics near the point of interest. In highly symmetric cases, null geodesics can provide way to investigate points where metric components appear to be singular. If spacetime is actually regular near the point, affine parameters of these geodesics give local coordinates, with respect to which metric components are non-singular. Hence, a curvature singularity can also be thought of as a place where null geodesics behave erratically, no matter how closely you zoom in on the region of erratic behavior: the region near the singularity looks curved at any scale.

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.

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