Chapter 18

Problem $18.12$ (a) Show that in a pscudo-Riemannian space the action principle $$\delta {\int }_{t_1}^{b_2}L,\mathrm{d}t=0$$ where $L=g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }$ gives rise to geodesic equations with affine parameter $t$. (b) For the sphere of radius $a$ in polar coordinates, $${\mathrm{ds}}^2=a^2(\mathrm{d}{\theta }^2+{\sin }^2\theta d{\varphi }^2)$$ use this variation principle to write out the equations of geodesics, and read off from them the Christoffel symbols ${\mathrm{\Gamma }}_{v{\rho }^{\ast }}^{\mu }$ where $L=g_{kv}{\dot{x}}^{\mu }{\dot{x}}^v$ gives rise to geodesic equations with affine parameter $t$. (b) For the sphere of radius $a$ in polar coordinates, $${\mathrm{ds}}^2=a^2(\mathrm{d}{\theta }^2+{\sin }^2\theta \mathrm{d}{\varphi }^2)$$ use this variation principle to write out the equations of geodesics, and read off from them the Christoffel symbols ${\mathrm{\Gamma }}_{i{\rho }^{\ast }}^{\mu }$ (c) Verify by direct substitution in the geodcsic equations, that $L={\theta }^2+{\sin }^2\theta {\dot{\varphi }}^2$ is a constant along the geodesics and use this to show that the general solution of the geodesic equations is grven by $$b\cot \theta =-\cos (\varphi -{\varphi }_0)\text{ where }b,{\varphi }_0=\text{ const. }$$ (d) Show that these curves are great circles on the sphere

Show that a space is locally flat if and only if there exists a local basis of vector ficlds \(\left\{e_{t} \mid\right.\) that are absolutely parallel, $D{e}_1=0$.

Let $(M,\phi )$ be a surface of revolution defined as a submantfold of ${\mathbb{E}}^3$ of the form $$r=g(u)\cos \theta ,y=g(u)\sin \theta ,z=h(u)$$ Show that the induced metric (see Example 18.1) is $$\mathrm{d}{s}^2=(g^{\prime }(u{)}^2+h^{\prime }(u{)}^2)\mathrm{d}{u}^2+g^2(u)\mathrm{d}{\theta }^2$$ Picking the parameter $u$ such that $g^{\prime }(u{)}^2+h^{\prime }(u{)}^2=1$ (interpret this choice!), and setting the basis 1-forms to be ${\epsilon }^{\prime }=\mathrm{d}u,{\epsilon }^2=gd\theta$, calculate the connection I-forms ${\omega }^{\prime }$, the curvature 1 -forms ${\rho }_{\prime }^{\prime }$, and the curvature tensor component $R_{1212}$.

Show that every two-dimensional space-time metric (signature 0 ) can be expressed locally in confor mal coontinates $${\mathrm{ds}}^2={\mathrm{e}}^{2\phi }(\mathrm{d}{x}^2-\mathrm{d}{t}^2)\text{ where }\varphi =\varphi (x,t)$$ Calculate the Rucmann curvature tensor component $R_{1212}$, and writc out the two-dimensional Enstein vacuum equations $R_{uj}=0$. What is their general solunon?

(a) For a perfect flud in general relat?uty, $$T_{\mu v}=(\rho c^2+P)U_{\mu }{U}_v+P{g}_{\beta v}(U^{\mu }{U}_y=-1)$$ show that the conservation identities $T^{\mu v},v=0$ imply ${\rho }_v{U}^v+(\rho c^2+P)U_{v^2}^{\ast }$ $(\rho c^2+P)U_{,}^{\mu }{U}^v+P_v(g^{\mu \prime }+U^{\mu }{U}^{\nu })$ (c) In the Newtonian approximatuon where $$U_{\mu }=(\frac{1}{c},-1)+O\left({\beta }^2\right),P=O\left({\beta }^2\right)\rho c^2,(\beta =\frac{v}{c})$$ where $|\beta |\ll 1$ and $g_{\mu \nu }={\eta }_{h^x}+ϵh_{\mu \nu }$ with $ϵ\ll 1$, show that $$h_{+\mu }\approx -\frac{2\varphi }{c^2},h_{ij}\approx -\frac{2\varphi }{c^2}{\delta }_j\text{ where }{\mathrm{\nabla }}^2\varphi =4\pi G\rho$$ and $h_{t4}=O(\beta )h_{44}$ Show in this approxumation that the equations $T^{\mu t},=0$ approvimate to $$\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho v)=0,\rho \frac{dv}{dt}=-\mathrm{\nabla }P-\rho \mathrm{\nabla }\varphi$$

(a) Compute the components of the Ricer tensor } R_{\mu v} \text { for a space-tume that has a }\end{array}$metricoftheform\$\$\mathrm{d}{s}^2={\mathrm{dx}}^2+\mathrm{d}{v}^2-2\mathrm{d}u\mathrm{d}v+2H\mathrm{d}{v}^2(H=H(\mathrm{x},y,u,v))\$\$(b)Showthatthespace-timeisavacuumifandonlyif$H=\alpha(x, y, v)+f(v) u$where$f(v)$isanarbitraryfunctionand$\alpha$sat?sfiesthetwo-dimensionalLaplaceequation\$\$\frac{{\partial }^2\alpha }{\partial x^2}+\frac{{\partial }^2\alpha }{\partial y^2}=0\$\$andshowthatitispossibletoset$f(v)=0$byacoordunatetransformation$u^{\prime}=u g(v), v^{\prime}=h(v)$.(c)Showthat$R_{\text {taj } 4}=-H_{v}$for$i, j=1,2$.

Consider an oscillator at $r=r_0$ emitting a pulse of light (null geodesic) at $t=t_0$. If this is received by an observer at $r=r_1$ at $t=t_1$, show that $$t_1=t_0+{\int }_{r_0}^{r_1}\frac{dr}{c(I-2m/r)}$$ By considering a signal emitted at $t_0+\mathrm{\Delta }{t}_0$, received at $t_1+\mathrm{\Delta }{t}_1$ (assuming the radial positions $r_0$ and $r_1$ to be constant), shou that $t_0=t_1$ and the gravitational redsbift found by comparing proper times at cmission and reception is given by $$1+z=\frac{\mathrm{\Delta }{t}_1}{\mathrm{\Delta }{\tau }_0}=\sqrt{\frac{1-2m/r_1}{1-2m/r_0}}$$ Show that for two clocks at different heights $h$ on the Earth's surface, this reduces to $$z\approx \frac{2GM}{c^2}\frac{h}{R}$$ where $M$ and $R$ are the mass and radius of the Earth.

In the Schwarzschild solution show the only possible closed photon path is a circular orbit at $r=3m$, and show that it is unstable.

(a) A particle falls radially inwards from rest at in finity in a Schwarzschild solution. Show that it will arrive at $r=2m$ m a finite proper time after crossing some fixed reference position $r_0$, but that coordinate time $t\to \mathrm{\infty }$ as $r\to 2m$. (b) On an infalling extended body compute the tidal force in a radual direction, by parallel propagating a tetrad (only the radial spacelike unit vector need he considered) and calculating $R_{1414}$. (c) Estimate the total tidal force on a person of height $1.8\mathrm{m}$, weighing $70\mathrm{kg}$, fallang head-first into a solar mass black hole $(M_3=2\times {10}^{10}\mathrm{kg})$, as he crosses $r=2\mathrm{m}$.

Show that the rodiation filled universe, $P=\frac{1}{3}\rho$ has $\rho \propto a^{-4}$ and the time evolution for $k=0$ is given by $a\propto t^{1/2}$. Assuming the radration is black body, $\rho =a_3{T}^4$, where $a_5=7.55\times$ ${10}^{-15}\mathrm{erg}{\mathrm{cm}}^{-3}{\mathrm{K}}^{-4}$, show that the temperature of the unverse evolves with time as $$T={\left(\frac{3c^2}{32\pi G{a}_{\mathrm{s}}}\right)}^{1+}{t}^{-1/2}=\frac{1.52}{\sqrt{t}}\mathrm{K}(t\text{ in seconds })$$