Worked Solutions to Hobson, Efstathiou, and Lasenby’s General
Relativity: An Introduction for Physicists
M. Haddad
May 25, 2016
2
Contents
1 The
... [Show More] Spacetime of Special Relativity 5
1.1 Deriving the Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Invariance of Intervals under Galilean and Lorentz transformations . . . . . . . . . . . . . . . . . . . . 7
1.3 Relative Velocities of Particles Moving Orthogonally . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Addition of Rapidity for Boosts in the Same Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 The General Lorentz Boost Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Composition of Two Non-Collinear Boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 The Relativistic Rotating Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.8 Boosting to Achieve Simultaneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.9 Derivation of the Relativistic (Longitudinal) Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . 11
1.10 The Transverse Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.11 Radio 4 Flyby . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.12 Spacetime Diagram for a Lorentz Boost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.13 Geometric Derivation for Length Contraction and Time Dilation . . . . . . . . . . . . . . . . . . . . . 13
1.14 The Twin Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.15 The Limit on Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.16 The Infinite Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Manifolds and Coordinates 15
2.1 Three-Dimensional Spherical-Cartesian Coordinate Transformation . . . . . . . . . . . . . . . . . . . . 15
2.2 The Three-Dimensional Line Element in Cylindrical and Spherical Coordinates . . . . . . . . . . . . . 16
2.3 A Non-Orthogonal Coordinate System in 3D Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Stereographic Projection Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Earth Coordinates and The Mercator Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Angle- and Area-Preserving Map Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Null Curves and Conformal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.8 Curve on a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.9 Line Element of a 3-Sphere in Four Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.10 Some 3D Space Calculations in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.11 Numerology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.12 Embedding a 2D Geometry into 3D Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.13 Non-Orthogonal Coordinates in Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3
4 CONTENTS
Chapter 1
The Spacetime of Special Relativity
1.1 Deriving the Lorentz Transformation
Starting with two frames S and S
0
that are in standard configuration (coinciding origins at t
0 = t = 0 and S
0 moving
in the x direction at speed v relative to S), we can write the general linear transformation from the unprimed frame
to the primed frame as: [Show Less]