Chapter 2
Lorentz Transformations
Abstract This chapter uses the mathematics of matrices to discuss the Lorentz
transformation and vectors and tensors in spacetime. One important goal is to prepare
the reader for the more general vector and tensor algebra and analysis to be used in
Part II.
2.1 The Lorentz Group
We have obtained the Lorentz transformation for motion in the x direction and
discussed some elementary applications. Now we are going to look at such transformations from a more sophisticated mathematical viewpoint, and with a more elegant
notation (Schutz 2009). This chapter is intended to orient you towards the geometric
viewpoint of general relativity, and to show that the notation can do much of the
algebraic work for you. Only cartesian coordinates will be used in this chapter.
We will first derive a more general definition of a Lorentz transformation. Recall
that special relativity is based on the following principles (Schwartz 1968).
I. The analytical form of physical laws is the same in all inertial reference frames
as described by systems of Cartesian coordinates.
II. The speed of light in vacuum is a universal constant.
A more sophisticated way to state principle II is that we wish to make the equation of an expanding spherical wave front of light invariant under the relevant
transformation of the space and time coordinates. We write the wave front as
c
2 t
2
− −
x
2
= 0,
(2.1)
and show a picture in Fig. 2.1 with the z coordinate suppressed. Because of the shape
of the surface in this picture it is called a light cone. Events in an inertial system
are points in four-dimensional spacetime or Minkowski space. They are labeled by
x
μ
= (ct, x, y, z), with ct taken as the zeroth coordinate. The set of coordinates
is also called the position 4-vector. We wish to find a transformation between such
coordinates in two systems, with the linear form
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_2
13
Lorentz Transformations
Abstract This chapter uses the mathematics of matrices to discuss the Lorentz
transformation and vectors and tensors in spacetime. One important goal is to prepare
the reader for the more general vector and tensor algebra and analysis to be used in
Part II.
2.1 The Lorentz Group
We have obtained the Lorentz transformation for motion in the x direction and
discussed some elementary applications. Now we are going to look at such transformations from a more sophisticated mathematical viewpoint, and with a more elegant
notation (Schutz 2009). This chapter is intended to orient you towards the geometric
viewpoint of general relativity, and to show that the notation can do much of the
algebraic work for you. Only cartesian coordinates will be used in this chapter.
We will first derive a more general definition of a Lorentz transformation. Recall
that special relativity is based on the following principles (Schwartz 1968).
I. The analytical form of physical laws is the same in all inertial reference frames
as described by systems of Cartesian coordinates.
II. The speed of light in vacuum is a universal constant.
A more sophisticated way to state principle II is that we wish to make the equation of an expanding spherical wave front of light invariant under the relevant
transformation of the space and time coordinates. We write the wave front as
c
2 t
2
− −
x
2
= 0,
(2.1)
and show a picture in Fig. 2.1 with the z coordinate suppressed. Because of the shape
of the surface in this picture it is called a light cone. Events in an inertial system
are points in four-dimensional spacetime or Minkowski space. They are labeled by
x
μ
= (ct, x, y, z), with ct taken as the zeroth coordinate. The set of coordinates
is also called the position 4-vector. We wish to find a transformation between such
coordinates in two systems, with the linear form
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_2
13