(Solved): P23.5 (part e, continued) solution will take over and matching the two metrics at the surface requir ...
P23.5 (part e, continued) solution will take over and matching the two metrics at the surface requires that
m(surface)
=M, the star's gravitational mass. So we can think of
m(r)as being the gravitational mass inside
r, which will be the total mass minus some gravitational binding energy. f. The fastest way to find the condition on
Aturns out to be to use the energy conservation equation (which is implicit in the Einstein equation, as we have seen in previous chapters). Show that the
\mu =rcomponent of this equation implies that
(1)/(A)(dA)/(dr)=-(2)/(\rho +p)(dp)/(dr)(Hint: Remember that the absolute gradient of the metric is zero. You will have to calculate one Christoffel symbol and argue that others are zero.) g. Use equations
23.6c,23.44c,23.47a, and 23.48 to show that
(dp)/(dr)=-(\rho +p)/(r^(2))[(4\pi Gpr^(3)+Gm(r))/(1-2Gm(r)/(r))]This is the Oppenheimer-Volkoff equation for stellar structure. This equation, the equation
d(m)/(d)r=4\pi \rho r^(2), and an "equation of state" that specifies
p(\rho )provide three equations in the three unknowns
m(r),p(r), and
\rho (r). Except for the simplest cases, these equations are pretty hard to solve analytically, but they are pretty easy to solve numerically (as I personally did as part of my doctoral thesis) for a specified equation of state and central density
\rho (0). If you like, you can also integrate equations 23.48 along the way to find
Aand use
23.47ato find
Bat any
r. For more information about stellar interiors, see Hobson, Efstathiou, and Lasenby, General Relativity, Cambridge, 2006, Pp. 288-296, from which I have adapted this problem.
