Relativity: An Introduction to Special and General Relativity

Hans Stephani

Chapter 18 The covariant derivative and parallel transport - all with Video Answers

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Chapter Questions

Problem 1

Use equations $(18.8),(16.21)$ and (16.31) to show that $\left(T^a{ }_{; n}\right)^{\prime}$ $=\partial T^{a^{\prime}} / \partial x^{n^{\prime}}+\Gamma_{n^{\prime} m^{\prime}}^{a^{\prime}} T^{m^{\prime}}$ transforms like a tensor, i.e. that $\left(T^a{ }_{; n}\right)^{\prime}=A_a^{a^{\prime}} A_{n^{\prime}}^n T^a{ }_{; n}$ holds.

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Problem 2

Apply $h_{(a) i}=g_{i \bar{n}} h_{(a)}^{\bar{n}}$ to a tetrad system (17.40) with (17.32) to show that $g_{i \bar{n}}=g_{\bar{n} i}$ holds.

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Problem 3

Show that the scalar product of any two vectors does not change under Fermi-Walker transport (18.19).

Dominador Tan

Numerade Educator

Problem 4

Show that the Lie derivative really has properties (a), (b), and (d).

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Problem 5

In a space with a given metric $g_{a b}$, a covariant derivative is defined by $T^a{ }_{\| n}=T^a{ }_{, n}+D_{n m}^a T^m$. Calculate $f_{, n m}-f_{, m \| n}$ and show that $S_{n m}^a=D_{n m}^a-D_{m n}^a$ is a tensor! Can $S_{n m}^a$ be determined by demanding $g_{a b \| n}=0$ ?

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Problem 6

To any vector $\mathbf{a}$, with components $a^n$, an operator $\mathbf{a}=a^n \partial / \partial x^n$ can be assigned. Use this notation to give the Lie derivative of the vector $T^n$ a simple form.

Victor Salazar

Numerade Educator