# Star collapse in general relativity — pressure as a function of star radius

• Lilian Sa

#### Lilian Sa

Homework Statement
How can I find the pressure as a function of the radius of a star that have a constant energy density, spherical symmetric, its initial radius is R and the total mass is M?
Relevant Equations
TOV equations.
##ds^2=-e^{\nu(r)}dt^2+r^{\lambda(r)}dr^2+r^2d\Omega^2##
What I've done is using the TOV equations and I what I found at the end is:
##e^{[\frac{-8}{3}\pi G\rho]r^2+[\frac{16}{9}(G\pi\rho)^{2}]r^4}-\rho=P(r)##
so I am sure that this is not right, if someone can help me knowing it I really apricate it :)

How can I find the pressure as a function of the radius of a star that have a constant energy density, spherical symmetric, its initial radius is R and the total mass is M?
I haven't worked this through. The TOV equations require gravitational equilibrium, so I presume that's what you're interested and not the actual collapse (which I have no idea how to deal with)? You'd need a barotropic equation of state ##p = p(\rho)##; does the problem statement supply one? Then you solve the coupled system\begin{align*}

\frac{dm}{dr} &= 4\pi r^2 \rho \\

\frac{dp}{dr} &= \frac{-(p+\rho)(m + 4\pi r^3 p)}{r(r-2m)}

\end{align*}in ##m(r)## and ##p(r)##. You can set ##m(0) = 0##