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
A
turns 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 =r
component 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
A
and use
23.47a
to find
B
at 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.