Chapter 3
The Motion of Particles
Abstract This chapter deals with the motion of particles, their energy and
momentum and acceleration, and emphasizes the geometric view of motion in spacetime. In particular it demonstrates that special relativity is not limited to motion at
constant velocity.
3.1 Energy and Momentum
The previous chapter contained a lot of formalism and little discussion of the physical
world. Now it is time to see that the formalism we have developed can make physics
more clear and easier (Schwartz 1968; Taylor 1963). We will consider some examples
of 4-vectors in physics. As in classical mechanics we first consider the trajectory of a
particle. Its position can be described by giving the functions of time x(t), y(t), z(t) in
some inertial lab frame; we thereby have the position 4-vector (ct, x(t), y(t), z(t))
as a function of time in that frame. The trajectory is a curve in four-dimensional
spacetime and is also called the world-line of the particle. We illustrate it for two
space dimensions in Fig. 3.1. Since the particle moves at less than the velocity of
light the trajectory lies inside a light cone with vertex on any point of the trajectory,
called the local light cone.
First consider an inertial coordinate system centered on a uniformly moving
particle; recall that it is called the proper or rest frame of the particle. In this frame
the position 4-vector is x
μ
= (cτ, 0, 0, 0), where τ is the time that a clock attached
to the particle would measure, which we call the proper time. However, since x
μ x μ
is an invariant we may write a relation that gives the proper time in any frame
c
2
τ
2
= x
μ x μ = c
2 t
2
− −
x
2
.
(3.1)
We emphasize that the proper time is an invariant, as is obvious from this expression!
© 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_3
19
The Motion of Particles
Abstract This chapter deals with the motion of particles, their energy and
momentum and acceleration, and emphasizes the geometric view of motion in spacetime. In particular it demonstrates that special relativity is not limited to motion at
constant velocity.
3.1 Energy and Momentum
The previous chapter contained a lot of formalism and little discussion of the physical
world. Now it is time to see that the formalism we have developed can make physics
more clear and easier (Schwartz 1968; Taylor 1963). We will consider some examples
of 4-vectors in physics. As in classical mechanics we first consider the trajectory of a
particle. Its position can be described by giving the functions of time x(t), y(t), z(t) in
some inertial lab frame; we thereby have the position 4-vector (ct, x(t), y(t), z(t))
as a function of time in that frame. The trajectory is a curve in four-dimensional
spacetime and is also called the world-line of the particle. We illustrate it for two
space dimensions in Fig. 3.1. Since the particle moves at less than the velocity of
light the trajectory lies inside a light cone with vertex on any point of the trajectory,
called the local light cone.
First consider an inertial coordinate system centered on a uniformly moving
particle; recall that it is called the proper or rest frame of the particle. In this frame
the position 4-vector is x
μ
= (cτ, 0, 0, 0), where τ is the time that a clock attached
to the particle would measure, which we call the proper time. However, since x
μ x μ
is an invariant we may write a relation that gives the proper time in any frame
c
2
τ
2
= x
μ x μ = c
2 t
2
− −
x
2
.
(3.1)
We emphasize that the proper time is an invariant, as is obvious from this expression!
© 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_3
19