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Chapter 18: Problem 18

Let \((M, \varphi)\) be a surface of revolution defined as a submantfold of \(\mathbb{E}^{3}\) of the form $$ r=g(u) \cos \theta, \quad y=g(u) \sin \theta, \quad z=h(u) $$ Show that the induced metric (see Example 18.1) is $$ \mathrm{d} s^{2}=\left(g^{\prime}(u)^{2}+h^{\prime}(u)^{2}\right) \mathrm{d} u^{2}+g^{2}(u) \mathrm{d} \theta^{2} $$ 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 \(\varepsilon^{\prime}=\mathrm{d} u, \varepsilon^{2}=g d \theta\), calculate the connection I-forms \(\omega^{\prime}\), the curvature 1 -forms \(\rho_{\prime}^{\prime}\), and the curvature tensor component \(R_{1212}\).

Short Answer

Expert verified
The induced metric is confirmed to be \(\mathrm{d}s^{2} = (g^{\prime}(u)^{2} + h^{\prime}(u)^{2})\mathrm{d}u^{2} + g^{2}(u)\mathrm{d}\theta^{2}\). The parameter \(u\) is interpreted as the arclength parameter due to the Pythagorean differential relation. And after picking the basis 1-forms as given, the connection 1-forms, curvature 1-forms, and the curvature tensor component \(R_{1212}\) are calculated using their standard definitions and equations from calculus.

Step by step solution

01

Calculate the differential

Given the form of the surface of revolution, we have to calculate the differential \(\mathrm{d}s\) to determine the metric. By substituting the given equations into the equation for \(\mathrm{d}s^2\), we get: \(\mathrm{d}s^{2} = \mathrm{d}x^{2} + \mathrm{d}y^{2} + \mathrm{d}z^{2}\). Using which, we can calculate the corresponding metric.
02

Verify the Induced Metric

The induced metric can be verified by plugging in the functions \(g(u)\) and \(h(u)\) into the metric formula calculated in Step 1. The expression should match with the given metric \(\mathrm{d}s^{2} = (g^{\prime}(u)^{2} + h^{\prime}(u)^{2})\mathrm{d}u^{2} + g^{2}(u)\mathrm{d}\theta^{2}\).
03

Interpret the Parameter Choice

The equation \(g^{\prime}(u)^{2} + h^{\prime}(u)^{2} = 1\) is an instance of the Pythagorean differential relation. This means the path of the function on the surface is a unit-speed curve, i.e., our parameter \(u\) can be interpreted as the arclength parameter.
04

Calculate the Connection 1-forms

Given the basis 1-forms \(\varepsilon^{1} = \mathrm{d}u, \varepsilon^{2} = g d\theta\), we can calculate the connection 1-forms by using the concept of covariant differentiation.
05

Calculate Curvature 1-forms and Curvature Tensor Component

Using the structure equations from calculus, we develop the expressions for curvature 1-forms and calculate them. Similarly, the curvature tensor component \(R_{1212}\) can be evaluated based on the connection forms and curvature forms since \(R_{abc}^d = d\omega_{abc}^d + \omega_{ae}^d \wedge \omega_{bc}^e\).

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Most popular questions from this chapter

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

(a) For a perfect flud in general relat?uty, $$ T_{\mu v}=\left(\rho c^{2}+P\right) U_{\mu} U_{v}+P g_{\beta v} \quad\left(U^{\mu} U_{y}=-1\right) $$ show that the conservation identities \(T^{\mu v}, v=0\) imply \(\rho_{v} U^{v}+\left(\rho c^{2}+P\right) U_{v^{2}}^{*}\) \(\left(\rho c^{2}+P\right) U_{,}^{\mu} U^{v}+P_{v}\left(g^{\mu \prime}+U^{\mu} U^{\nu}\right)\) (c) In the Newtonian approximatuon where $$ U_{\mu}=\left(\frac{1}{c},-1\right)+O\left(\beta^{2}\right), \quad P=O\left(\beta^{2}\right) \rho c^{2}, \quad\left(\beta=\frac{v}{c}\right) $$ where \(|\beta| \ll 1\) and \(g_{\mu \nu}=\eta_{h^{x}}+\epsilon h_{\mu \nu}\) with \(\epsilon \ll 1\), show that $$ h_{+\mu} \approx-\frac{2 \phi}{c^{2}}, \quad h_{i j} \approx-\frac{2 \phi}{c^{2}} \delta_{j} \quad \text { where } \quad \nabla^{2} \phi=4 \pi G \rho $$ and \(h_{t 4}=O(\beta) h_{44}\) Show in this approxumation that the equations \(T^{\mu t},=0\) approvimate to $$ \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho v)=0, \quad \rho \frac{d v}{d t}=-\nabla P-\rho \nabla \phi $$

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 $$ t_{1}=t_{0}+\int_{r_{0}}^{r_{1}} \frac{d r}{c(I-2 m / r)} $$ By considering a signal emitted at \(t_{0}+\Delta t_{0}\), received at \(t_{1}+\Delta t_{1}\) (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 $$ 1+z=\frac{\Delta t_{1}}{\Delta \tau_{0}}=\sqrt{\frac{1-2 m / r_{1}}{1-2 m / r_{0}}} $$ Show that for two clocks at different heights \(h\) on the Earth's surface, this reduces to $$ z \approx \frac{2 G M}{c^{2}} \frac{h}{R} $$ where \(M\) and \(R\) are the mass and radius of the Earth.

If a Lagrangian depends on second and higher order derivatives of the fields, \(L=\) \(L\left(\Phi_{\lambda}, \Phi_{A \mu}, \Phi_{A, \mu \nu}, \ldots\right)\) derive the generalized Euler-Lagrange equations $$ \frac{\delta L \sqrt{-g}}{\delta \Phi_{A}} \equiv \frac{\partial L \sqrt{-g}}{\partial \Phi_{A}}-\frac{\partial}{\partial x^{\mu}}\left(\frac{\partial L \sqrt{-g}}{\partial \Phi_{A, \mu}}\right)+\frac{\partial^{2}}{\partial x^{\mu} \partial x^{\prime}}\left(\frac{\partial L \sqrt{-g}}{\partial \Phi_{A_{L} w}}\right)-\cdots=0 $$

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
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