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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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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface of Revolution
A surface of revolution is created by rotating a curve around an axis in space. This surface has a symmetrical shape, reminiscent of objects like spheres or cones. In the exercise, the surface of revolution is defined through equations specifying the coordinates:
  • r, which represents the x-coordinate as a function of u and θ: \(r = g(u) \cos \theta\)
  • y, corresponding to the y-coordinate: \(y = g(u) \sin \theta\)
  • z, the height, given by: \(z = h(u)\)
This means that by changing θ, we revolve the curve formed by g(u) and h(u) around the z-axis to generate the surface.

Importantly, the functions g(u) and h(u) define the shape of the revolved curve, impacting the form and character of the resulting surface. A real-world example might be a vase or a bell shape, where the profile of the vase forms g(u) and h(u), while the circular symmetry is represented by rotating it around an axis.
Induced Metric
The induced metric tells us about distances on the surface of revolution.
  • It allows us to measure how far apart two points are on the surface, with emphasis on the real, curved nature of the shape, instead of just linear distances.
  • The metric is derived from the surface itself; in other words, when we place the surface in a space like \(\mathbb{E}^{3}\), it "inherits" the metric from this ambient space.
The exercise provides the induced metric as:\[\mathrm{d} s^{2}=\left(g^{\prime}(u)^{2}+h^{\prime}(u)^{2}\right) \mathrm{d} u^{2}+g^{2}(u) \mathrm{d} \theta^{2}\]

This formula accounts for both changes along the curve (with respect to u) and around the curve (involving θ).

Choosing u such that \(g^{\prime}(u)^{2}+h^{\prime}(u)^{2}=1\) simplifies calculations by making u a natural parameter: the arc length. This means u directly measures distance along the curve, akin to measuring with a ruler laid along the shape.
Connection Forms
Connection forms help in understanding how vectors move on a surface, which is key in the analysis of curvature and direction. With connection 1-forms, represented in this context as ω, we can compute how vectors evolve when they shift across the surface.
  • The basis 1-forms \(\varepsilon^{1} = \mathrm{d}u\) and \(\varepsilon^{2} = g \mathrm{d}\theta\) are used to express changes on the surface in terms of a coordinate system.
  • The connection forms relate to the 'rules' or mathematical structures that describe these changes.
These components enable the calculation of how vectors transported along the surface undergo rotation or distortion, a crucial requirement for formulating the concept of covariant differentiation. This ensures consistency, allowing vectors to be compared meaningfully as they move from one point to another on the surface.

Overall, connection forms reveal the geometric structures underlying the surface's shape. They help in determining how the surface is embedded in the larger space without tearing or stretching excessively.
Curvature Tensor
The curvature tensor is like the DNA of a surface's shape, providing vital information about how the surface bends and twists.
  • The power of the curvature tensor lies in its ability to reveal intrinsic properties of the surface, independent of the surrounding space.
  • The component often discussed is \(R_{1212}\), a particular part of the curvature tensor where its indices describe the relationship between the relevant directions and deformations within the surface.
The process of calculating this component involves:
  • Using connection forms to determine how the surface 'curves' internally.
  • Developing curvature 1-forms as intermediaries in the structural calculus equations.
Why is this important? By understanding the curvature tensor, one can predict how the surface behaves under various physical forces, like how a sheet of metal might bend when heated or cooled.

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Most popular questions from this chapter

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

Compute the Euler-Lagrange equations and energy-stress tensor for a scalar ficld Lagrangian in general relatwity given by $$ L_{5}=-\psi_{-\mu} \psi_{1} g^{n}-m^{2} \psi^{2} $$ Verify \(T^{n v},=0\)

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.

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.

(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 $$ \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho v)=0, \quad \rho \frac{d v}{d t}=-\nabla P-\rho \nabla \phi $$
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