1
$\begingroup$

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).

$\endgroup$
  • $\begingroup$ That's the Vaidya metric. $\endgroup$ – Slereah Apr 9 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. $\endgroup$ – Ben Crowell Apr 9 at 23:04
0
$\begingroup$

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.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.