Chapter 18: Problem 12

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, $$ \mathrm{ds}^{2}=a^{2}\left(\mathrm{~d} \theta^{2}+\sin ^{2} \theta d \phi^{2}\right) $$ use this variation principle to write out the equations of geodesics, and read off from them the Christoffel symbols $\Gamma_{v \rho^{}}^{\mu}$ where $L=g_{k v} \dot{x}^{\mu} \dot{x}^{v}$ gives rise to geodesic equations with affine parameter $t$. (b) For the sphere of radius $a$ in polar coordinates, $$ \mathrm{ds}^{2}=a^{2}\left(\mathrm{~d} \theta^{2}+\sin ^{2} \theta \mathrm{d} \phi^{2}\right) $$ use this variation principle to write out the equations of geodesics, and read off from them the Christoffel symbols $\Gamma_{i \rho^{}}^{\mu}$ (c) Verify by direct substitution in the geodcsic equations, that $L=\theta^{2}+\sin ^{2} \theta \dot{\phi}^{2}$ is a constant along the geodesics and use this to show that the general solution of the geodesic equations is grven by $$ b \cot \theta=-\cos \left(\phi-\phi_{0}\right) \text { where } b, \phi_{0}=\text { const. } $$ (d) Show that these curves are great circles on the sphere

Short Answer

Expert verified

The action principle gives rise to geodesic equations in a pseudo-Riemannian space. These equations, when applied to the sphere of radius 'a' in polar coordinates, yield specific Christoffel symbols. Additionally, it can be shown that $L=\theta^2+\sin^2\theta \d\phi^2$ remains constant along geodesics and the general solution is $b \cot \theta=-\cos (\phi-\phi_0)$, where $b$ and $\phi_0$ are constants. These geodesics correspond to great circles on a sphere.

Step by step solution

Apply the action principle to derive geodesic equations

Using the variational principle, one can obtain the geodesic equations from the action principle. Start with the initial action integral $\delta \int_{t_{1}}^{b_{2}} L \, dt=0$, vary the action with respect to $x^\mu(t)$ and set that equal to 0 to get the geodesics.

Use variation principle for the spherical coordinates

Now plug the metric for the sphere of radius $a$ in polar coordinates into the Lagrangian $\ds^2=a^2\left(\d\theta^2+\sin^2\theta \,\d\phi^2\right)$. Then apply the variation principle again to obtain the equations of the geodesic and consequently the Christoffel symbols.

Verification and finding general solution

Next, substitute $L=\theta^2+\sin^2\theta \dot{\phi}^2$ directly into the geodesic equations to confirm that it is a constant along the geodesics. Then solve the geodesic equations to find the general solution, resulting in $b \cot \theta=-\cos (\phi-\phi_0)$, where $b$ and $\phi_0$ are constants. This can be done by separating variables and integrating using standard techniques.

Demonstrating great circles

Finally, prove that the geodesics you derived correspond to the great circles on a sphere. This can be done by showing that they represent paths of shortest distance, as any geodesic in geometry is defined to be the shortest path between two points.