In section 3.9 of A Relativist's Toolkit by E. Poisson, the author discusses the gravitational collapse of a thin uniform spherical shell.

The spacetime inside is assumed1 to be Minkowski and the spacetime outside is necessarily Schwartzschild by spherical symmetry. The shell is made of pressureless matter and its surface stress-energy tensor is constrained to have the following form.

$$S^{ab}=\sigma u^a u^b\tag{3.68}$$

The coordinates on the hypersurface are $\{\tau,\theta,\phi\}$. He continues to write the following components of the extrinsic curvature.

$$K^{\tau}_{\pm \tau}=\frac{\dot{\beta}_{\pm}}{\dot{R}} \text{ & }K^{\theta}_{\pm \theta}=K^{\phi}_{\pm \phi}=\frac{\beta_{\pm}}{R} \tag{1}$$ where $\beta_+=\sqrt{\dot{R}^2+1-2M/R}$ and $\beta_-=\sqrt{\dot{R}^2+1}$.

  1. What exactly are the equations of the hypersurface/shell as seen from both the spacetimes?
    Is it $r=R(\tau)$ and $t=T(\tau)$ as seen from both Minkowski and Schwartzschild? If this were true, then the first junction condition would be eq. $(2)$, which is not mentioned anywhere in the section. $$h_{\tau \tau}=F\dot{T}^2 - F^{-1}\dot{R}^2=\dot{T}^2-\dot{R}^2 \tag{2}$$ where, $F=1-\frac{2M}{R}$.
  2. The author mentions that he has borrowed the extrinsic curvature results from section 3.8, in which he discusses the Oppenheimer-Snyder collapse, to write eqs. $(1)$. But the first junction condition of FRW-Schwartzschild junction (eq. $(3)$) was assumed to derive the results given in section 3.8. $$F\dot{T}^2 - F^{-1}\dot{R}^2=1 \tag{3}$$ Eq. $(3)$ doesn't follow from eq. $(2)$. So, how is the author allowed to borrow the results?

I'd greatly appreciate any assistance that helps me better understand this section of the book.

1 Although, according to this answer, it must necessarily be Minkowski inside as well.

Edit 1

$$n_{+\mu}=\frac{1}{\sqrt{F\dot{R}^2-F^{-1}\dot{T_+}^2}}(-\dot{R},\dot{T_+},0,0)$$ $$K_{+\theta \theta}=\underbrace{-{\Gamma^t}_{\theta \theta}n_{+t}}_{=0}-{\Gamma^{r}}_{\theta \theta}n_{+r}=\frac{1}{2}g_{rr}\partial_rg_{\theta \theta} \frac{\dot{T_+}}{\sqrt{F\dot{R}^2-F^{-1}\dot{T_+}^2}}$$ $$\Rightarrow K_{+\theta \theta}=FR\frac{\dot{T_+}}{\sqrt{F\dot{R}^2-F^{-1}\dot{T_+}^2}}$$ $$K^{\theta}_{+\theta}=K_{+\theta \theta}h^{+\theta \theta}=\frac{F}{R}\frac{\dot{T_+}}{\sqrt{F\dot{R}^2-F^{-1}\dot{T_+}^2}}\neq \frac{\beta_+}{R}$$

  • 1
    $\begingroup$ It looks like you write equation (2) assuming that time coordinate with respect to which metric inside is flat is the same as Schwartzschild time outside. This might be the problem with this equation, as these two time coordinates are not the same. Instead, you should write metric on shell using proper time of comoving observer, and then I think you will obtain equation 3. $\endgroup$ – Aleksandr Artemev Feb 27 at 17:13
  • $\begingroup$ @AleksandrArtemev Assuming a different time coordinate as you've suggested ($t=T_{\pm}(\tau)$), gives the following first Israel junction condition: $h_{\tau \tau}=F_{+}\dot{T}^2_{+}-F^{-1}_{+}\dot{R}^2=F_{-}\dot{T}^2_{-}-F^{-1}_{-1}\dot{R}^2$. The notations are as defined in Qmechanic's answer. But, in the thin shell collapse section of the book, we can observe that $S^{\tau \tau}=\sigma$ and ${S^{\tau}}_{\tau}=-\sigma$: Together they imply that the author has used $h_{\tau \tau}=-1$. I'm unable to see how it follows from the first junction condition. $\endgroup$ – Ajay Mohan Feb 28 at 4:36
  • $\begingroup$ When you put $h_{\tau \tau} = - 1$, you use the fact that tau is proper time for comoving observer. It is the definition of what proper time is - geodesic distance measured on the hypersurface that shell covers during its movement in spacetime (or, rather, a section of this hypersurface with $(r, t) $ plane) . If tau is just a generalized parameter, of course this relation is no longer true. $\endgroup$ – Aleksandr Artemev Feb 28 at 6:29

The first Israel junction condition is essentially mentioned in the equation

$$ \mathrm{d}s_{\Sigma}^2~=~\underbrace{-(F_{\pm}\dot{T}^2_{\pm}-F^{-1}_{\pm}\dot{R}^2)}_{=h_{\tau\tau}}\mathrm{d}\tau^2+R^2(\tau)\mathrm{d}\Omega^2 $$

before eq. (3.62) in A Relativist's Toolkit by E. Poisson.

Here $F_-=1$ and $F_+=1-\frac{2M}{R}$ and are Minkowski/Schwarzschild structure functions from the inside/outside bulk metrics, respectively. Note that the inside/outside time coordinates $T_{\pm}(\tau)$ are different, cf. above comment by Aleksandr Artemev. Similarly let us define $$\beta_{\pm}~:=~\sqrt{F_{\pm}+\dot{R}^2}.\tag{3.63'}$$ The formulas (1) for extrinsic curvature follow from their definitions after relatively long calculations that we will not try to reproduce here.

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  • $\begingroup$ If we notice eq. (3.68) in the book and the eq. above (3.69), we observe that $S^{\tau \tau}=\sigma$ and ${S^{\tau}}_{\tau}=S^{\tau \tau}h_{\tau \tau}=-\sigma$ and they imply that $h_{\tau \tau}=-1$. This doesn't follow from the first junction condition that you have proposed: $-h_{\tau \tau}={F_+}^{-1}\dot{T_+}^2-F_+\dot{R}^2={F_-}^{-1}\dot{T_-}^2-F_-\dot{R}^2$. $\endgroup$ – Ajay Mohan Feb 27 at 19:04
  • $\begingroup$ Also, in "Edit 1" in my question, I've calculated the $K^{\theta}_{+\theta}$ component. But, it doesn't match with the result in the book. $\endgroup$ – Ajay Mohan Feb 27 at 19:10

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