Riemann surface
matlab
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Problem 1
Hydrostatic Equilibrium in GR. Model a neutron star atmosphere with a simple equation of state: $P=K \rho^\gamma$, where $P$ is pressure, $\rho$ is mass density, $\gamma$ is the adiabatic index and $K$ is a constant. Assume that $g_{00}=-\left(1-2 G M / r c^2\right)$, where $M$ is the mass of the star and $r$ is radius. If $\rho=\rho_0$ at the surface $r=R_0$, solve the equation of hydrostatic equilibrium to show that
$$
\frac{1+K \rho^{\gamma-1} / c^2}{1+K \rho_0^{\gamma-1} / c^2}=\left(\frac{1-R_S / r_0}{1-R_S / r}\right)^\alpha
$$
where $R_S=2 G M / c^2$ is the so-called Schwarzschild radius, and $2 \alpha \gamma=\gamma-1$. (Hint: See $\S 4.6$ of the notes.) What is the Newtonian limit of the above equation? Express your answer in terms of the speed of sound $a, a^2=\gamma P / \rho$ and the potential $\Phi(r)=-G M / r$. (OPTIONAL: For those who have studied fluids, what quantity is being conserved in the Newtonian limit?)
Solution: Here we encounter fluid dynamics in curved space-time. Recall that fluid dynamics is the dynamics of densities of conserved charges. In the relativistic case, these are $T^{00}$ and $T^{0 i}$ components of the conserved energy-momentum tensor. In flat space, the conservation law is simply
$$
\partial_\mu T^{\mu \nu}=0
$$
Other components of $T^{\mu \nu}$ are related to $T^{00}$ and $T^{0 i}$ via the constitutve relations
$$
T^{\mu \nu}=P \eta^{\mu \nu}+\left(\rho+P / c^2\right) u^\mu u^\nu+\cdots,
$$
where the infinite tail involving derivatives of $u^\mu(x)$ is omitted (we assume here that the gradients of $u^\mu(x)$ are small which of course may not always be the case). This (covariant) form of $T^{\mu \nu}$ follows either by applying Lorentz transformation (with the velocity encoded in $u^\mu=(\gamma c, \gamma \mathbf{v})$ ) to $T_0^{\mu \nu}$ in fluid’s rest frame,
$$
T_0^{\mu \nu}=\left(\begin{array}{rrrr}
\rho c^2 & 0 & 0 & 0 \
0 & P & 0 & 0 \
0 & 0 & P & 0 \
0 & 0 & 0 & P
\end{array}\right),
$$
or by writing the most general covariant expression involving all the relevant ingredients ( $\eta_{\mu \nu}$ and $u^\mu$ – but not $\partial_\nu u^\mu$ if we assume small gradients),
$$
T^{\mu \nu}=A \eta^{\mu \nu}+B u^\mu u^\nu,
$$
and then comparing this expression in fluid’s rest frame (where $u^\mu=(c, 0)$ ) to eq. (3) to read off the coefficients $A$ and $B$.
In curved space-time, eqs. (1) and (2) are replaced by
$$
\begin{aligned}
& T^{\mu \nu}=P g^{\mu \nu}+\left(\rho+P / c^2\right) u^\mu u^\nu+\cdots, \
& \nabla_\mu T^{\mu \nu}=0,
\end{aligned}
$$
where $g_{\mu \nu} u^\mu u^\nu=-c^2$. Explicitly, the equation (5) with $\nu=0$ is
$$
\frac{\partial P}{\partial x^\mu}+\left(\rho c^2+P(\rho)\right) \frac{\partial \ln \left|g_{00}\right|^{1 / 2}}{\partial x^\mu}=0
$$
which is supposed to be supplemented by the equation of state $P=P(\rho)$.
Problem 2
Bondi Accretion: go with the flow. To get some practise working with the equations of GR as well as some insight into relativistic dynamics in a practical problem in astrophysics, consider what is known as (relativistic) Bondi Accretion, the spherical flow of gas into a black hole. (The original Bondi accretion problem was Newtonian accretion onto an ordinary star.) We assume a Schwarzschild metric in the usual spherical coordinates:
$$
g_{00}=-\left(1-2 G M / r c^2\right), g_{r r}=\left(1-2 G M / r c^2\right)^{-1}, g_{\theta \theta}=r^2, g_{\phi \phi}=r^2 \sin ^2 \theta .
$$
2a) First, let us assume that particles are neither created or destroyed. So particle number is conserved. If $n$ is the particle number density in the local rest frame of the flow, then the particle flux is $J^\mu=n U^\mu$, where $U^\mu$ is the flow 4-velocity. Justify this statement, and using $\S 4.5$ in the notes, show that particle number conservation implies:
$$
J_{; \mu}^\mu=0
$$
If nothing depends upon time, show that this integrates to
$$
n u^r\left|g^{\prime}\right|^{1 / 2}=\text { constant }
$$
where $g^{\prime}$ is the determinant of $g_{\mu \nu}$ divided by $\sin ^2 \theta$ and $U^r$ is… well, you tell me what $U^r$ is.
Solution:
2b) We move on to energy conservation, $T_{; \nu}^{t \nu}=0$. (Refer to $\S 4.6$ in the notes.) Show that the only nonvanishing affine connection that we need to use is
$$
\Gamma_{t r}^t=\Gamma_{r t}^t=\frac{1}{2} \frac{\partial \ln \left|g_{t t}\right|}{\partial r}
$$
Derive and solve the energy equation. Show that its solution may be written
$$
\left(P+\rho c^2\right) U^r U_t\left|g^{\prime}\right|^{1 / 2}=\text { constant }
$$
where $U_t=g_{t \mu} U^\mu$, and $\rho$ is the total energy density of the fluid in the rest frame, including any thermal energy.
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