Fundamentals of Time and Relativity

Covariant Action

  • According to Hamilton’s action principle, the motion of a mechanical system can be described by an action which is a functional of the path of the system.
  • The action functional becomes extremal for the actual motion when its variation vanishes.

As the simplest example, we first consider the motion of a free relativistic particle. Because of the principle of relativity, the action should not depend on the choice of inertial system; it must be invariant with respect to Lorentz transformations. This requirement guarantees that different inertial observers will agree about the stationarity of the action. Thus, the same set of equations of motion will be valid in all the inertial frames.

The proper time, or equivalently the timelike distance, between two world points in Minkowski space, is a good action candidate, because:

  1. it is a quantity that all observers will agree on;
  2. it has a maximal value for a straight worldline between two events; due to time dilatation and Lorentz contraction, the lapse of time is always less along any other, curved, time-like worldline.

These considerations lead to the following 'Ansatz' for the covariant action integral of a free particle moving between two world points x 1 = x 1 ν e ν and x 2 = x 2 ν e ν :

10.1

S free [x]:=−m ∫ 1 2 ds =−m ∫ 1 2 g μν d x μ d x ν =−m ∫ 1 2 dx⋅dx

The negative sign in (10.1) has the effect that the maximum along a straight worldline becomes a minimum, and the mass factor ensures the correct non-relativistic limit; it also gives the action its proper dimension [S]= energy×time .

To make the connection with Hamilton’s principle of classical mechanics, we note that in any given inertial frame we can use (4.10) to write the action (10.1) as the time integral

10.2

S free [x(t)]=−m ∫ 1 2 dt 1− v 2 = ∫ 1 2 dt L(v(t))

The integrand at the right-hand side is the (non-covariant) relativistic Lagrangian which, in the case of a free particle, is a function of velocity only. In the limit of small velocities the Lagrangian becomes L=m v 2 /2−m ; the appearance of the constant rest energy is of no consequence in non-relativistic mechanics and can be ignored.

  • The action (10.2) gives the relativistic equation of motion in a particular reference frame with a particular choice of time coordinate t .
  • The action principle is also referred to as the principle of least action, although the extremum can be either a minimum or a maximum of the action.