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).

  • $\begingroup$ That's the Vaidya metric. $\endgroup$
    – Slereah
    Apr 9, 2019 at 18:29
  • $\begingroup$ For collapse of a cloud of material particles, see the Oppenheimer-Snyder model. However, the interpretation of this is complicated and controversial. $\endgroup$
    – user4552
    Apr 9, 2019 at 23:04

1 Answer 1


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:

  • 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.

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