Solving the Radius of a Collapsing Star in Schwarzschild Geometry

[latentcorpse]

I have that $\left( \frac{dR}{d \tau} \right)^2 = ( 1 - \epsilon)^2 ( \frac{R_{\text{max}}}{R}-1)$ describes the radius of the surface of a collapsing star in Schwarzschild geometry. I need to show it falls to R=0 in time $\tau = \frac{\pi M}{(1-\epsilon)^{3/2}}$

So far I have rearranged to get
$\int_{R_{\text{max}}}^0 \sqrt{\frac{R}{R_{\text{max}}-R}} dR = \int_0^\tau (1-\epsilon^2)^{1/2} = (1-\epsilon^2)^{1/2} \tau$

How do I do that R integral though?

Thanks.

 

[tiny-tim]

hi latentcorpse!

that π in the answer suggests you should go for a trig substitution

 

[latentcorpse]

hi latentcorpse!

that π in the answer suggests you should go for a trig substitution

I forgot to mention that $R_{\text{max}}=\frac{2M}{1-\epsilon^2}$

Making the substitution $R=R_{max} \sin^2{\theta}$

I get $\int_{\pi/2}^0 2 R_{max} \sin^2{\theta} d \theta = - R_{max} \frac{\pi}{2} = - \frac{M \pi}{(1-\epsilon^2)}$

equating to $(1-\epsilon^2)^{1/2}\tau$

we get $\tau=-\frac{\pi M}{(1-\epsilon)^{3/2}}$

i.e. an extra minus sign. and it can't take a negative amount of time for a star surface to collapse to r=0, i have messed up a sign. i think the limits on my integrals are sound but could you check please?

thanks.

 

[fzero]

You have to choose the sign when taking the square root to agree with $$dR/d\tau<0$$.

 

[paranormal]

Based on the information provided, it seems like you are on the right track in solving for the time at which the collapsing star reaches a radius of 0. To solve the integral for R, you can use the substitution u = R/Rmax - 1, which will then turn the integral into:

\int_{-1}^1 \sqrt{\frac{u+1}{u}} du

This integral can then be solved using a trigonometric substitution or by using a table of integrals. Once you have solved for the integral, you can plug in the limits of integration and solve for the time \tau.

However, it is important to note that this calculation assumes a simplified model of a collapsing star in Schwarzschild geometry and may not accurately represent the dynamics of a real collapsing star. It is always important to consider the assumptions and limitations of any mathematical model in scientific research.