Introduction to General Relativity

Solutions to Problems

John Dirk Walecka

• Publisher: WSPC
• Year: 2017
• ISBN:
  9789813227712
• Topic: Sciences physiques
• Subtopic: Physique mathématique et informatique

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It is important for every physicist today to have a working knowledge of Einstein's theory of general relativity. Introduction to General Relativity published in 2007 was aimed at first-year graduate students, or advanced undergraduates, in physics. Only a basic understanding of classical lagrangian mechanics is assumed; beyond that, the reader should find the material to be self-contained.

The mechanics problem of a point mass constrained to move without friction on a two-dimensional surface of arbitrary shape serves as a paradigm for the development of the mathematics and physics of general relativity. Special relativity is reviewed. The basic principles of general relativity are then presented, and the most important applications are discussed. The final special topics section takes the reader up to a few areas of current research. An extensive set of accessible problems enhances and extends the coverage.

As a learning and teaching tool, this current book provides solutions to those problems. This text and solutions manual are meant to provide an introduction to the subject. It is hoped that these books will allow the reader to approach the more advanced texts and monographs, as well as the continual influx of fascinating new experimental results, with a deeper understanding and sense of appreciation.

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--> Contents:

• Preface
• Introduction
• Particle on a Two-Dimensional Surface
• Curvilinear Coordinate Systems
• Particle on a Two-Dimensional Surface–Revisited
• Some Tensor Analysis
• Special Relativity
• General Relativity
• Precession of Perihelion
• Gravitational Redshift
• Neutron Stars
• Cosmology
• Gravitational Radiation
• Special Topics
• Appendices:
  • Reduction of g μν δR μν to Covariant Divergences
  • Robertson-Walker Metric with k ≠ 0
• Bibliography
• Index

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--> Readership: Advanced undergraduate or graduate students in mathematical and theoretical physics. -->
General Relativity;Special Relativity;Riemannian Geometry;Precession of Perihelion;Gravitational Redshift;Neutron Stars;Black Holes;Cosmology;Gravitational Waves;Cosmological Constant;Inflation0

Chapter 1

Introduction

There are no problems associated with the Introduction.

Chapter 2

Particle on a Two-Dimensional Surface

Problem 2.1 Consider two parallel flat surfaces interconnected through a smooth circular cylindrical tube to make a single unified surface. Give a qualitative discussion of the particle orbits and geodesics on the unified surface.

Solution to Problem 2.1

The easiest way to track the geodesics is to imagine a string connecting two points on the surface (which we assume is smooth), and then just pull the string tight. We sketch three geodesics in Fig. 2.1: one is unperturbed on the lower surface, one starts on the upper surface, dips into the cylinder, and then returns on the upper surface, and one starts on the upper surface, dives down the cylinder, and returns on the lower surface. There are also, of course, orbits with different winding numbers around the cylinder.¹

Fig. 2.1 Sketch of three geodesics on the given two-sided surface (see text).

¹ Readers are urged to envision a few of these.

Chapter 3

Curvilinear Coordinate Systems

Problem 3.1 Given the definition of δⁱ_j in Eq. (2.13), the fact that the metric raises and lowers indices, and Eq. (2.20), show δ_i^j = δⁱ_j.

Solution to Problem 3.1

The basis vectors in the reciprocal basis satisfy [see Eq. (2.13)]

The metric in the reciprocal basis is defined in Eqs. (2.17)

It is also symmetric, and satisfies Eq. (2.20)

Since the metric raises and lowers indices, one has

where the last equality follows from the definition of δⁱ_j.

Problem 3.2 Suppose a contraction v_iwⁱ is invariant under coordinate transformations v_iwⁱ =

_i

ⁱ for an arbitrary vector v_i, which transforms as

. Show that wⁱ must then also be a vector transforming as

.

Solution to Problem 3.2

We are given that

for an arbitrary vector v_i, which transforms as

_i =

_i^jv_j. Substitute this relation in the above, and re-label indices on the r.h.s.

Since v_i is arbitrary, we can equate coefficients

where the second equality follows from Eqs. (3.36). Now invert this relation using Eqs. (3.32)

Hence, with a re-labeling of indices

This is the desired result.

Problem 3.3 Choose cartesian coordinates (q¹, q², q³) = (x, y, z) in three-dimensional euclidian space and write the line element as

Here (ê_x, ê_y, ê_z) are the set of global, orthonormal, cartesian unit vectors.

(a) Show the metric is

(b) Show the reciprocal basis is identical to the original basis in this case, and hence there is no need t...