Metric for accreting spherically symmetric spacetime? I know this may be an over-simplification of the system but can we treat the metric below as something that represents accretion?
$$\mathrm ds^2 = - \left(1-\frac{2m(t)}{r}\right) \mathrm dt^2 + \left(1-\frac{2m(t)}{r}\right)^{-1}\mathrm dr^2 +r^2 \mathrm d\Omega^2$$
where $m(t)$ is taken to increase with $t$. Correct me please if I'm wrong, but I interpret this as a Schwarzschild spacetime where the mass of the central object is increasing (for example through accretion).
 A: No, the OP's metric could not be described as representing accretion.
To see this, take the function $m(t)$ to have the form $m(t)=M+\delta M \, H (t)$, where $H(t)$ is a step-like function with the properties 
$$H(t)=\begin{cases} 0,& t < -\epsilon \\ 1, & t>  + \epsilon \end{cases},$$
for some small $\epsilon$. The metric then is  Schwarzschild metric with the mass $M$ for $t<-\epsilon$ and  Schwarzschild metric with the mass $M+\delta M$ for $t>\epsilon$, and so it  satisfies vacuum Einstein field equations for $|t|>\epsilon$ while around $t=0$ Einstein tensor is non-zero. But we could not interpret this region of spacetime as having any sort of realistic matter, since the slice $t=0$ is spacelike (for $ r > 2 M $). In other words, the mass in the OP's metric propagates with superluminal velocities.
A more realistic yet quite simple model of an accreting spacetime is the (ingoing) Vaidya metric that corresponds to a null dust matter:
$$
ds^{2}=-{\Big (}1-{\frac {2M(v)}{r}}{\Big )}dv^{2}+2dvdr+r^{2}(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}),
$$
which is obtained from the Schwarzshild solution written in Eddington–Finkelstein coordinates, by replacing constant $M$ with a function $M(v)$. The difference from OP's metric is that here the matter propagates at precisely the speed of light along null ingoing hypersurface $v=\mathrm{const}$.
A few other simple models of accretion and gravitational collapse  could be found here:


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*Adler, R. J., Bjorken, J. D., Chen, P., & Liu, J. S. (2005). Simple analytical models of gravitational collapse. American journal of physics, 73(12), 1148-1159, doi:10.1119/1.2117187, arXiv:gr-qc/0502040.

