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Action in Minkowski Spacetime

Lagrangian mechanics is very powerful in that it can be used to describe the dynamics of a system in a very general way. We will show here that it can be used to describe relativistic systems as well. In the future, we will extend this to general relativity, as well as string theory, and then quantum field theory.

Table of Contents​

Minkowski Geometry Review​

Spacetime is a four-dimensional pseudo-Riemannian manifold. The metric tensor is used to define the geometry of spacetime. By the equivalence principle, locally, spacetime is flat, and general relativity does not come into play.

When spacetime is flat, the metric tensor is given by the Minkowski metric:

(where we have used the mostly-minus convention).

For a straight line in Minkowski spacetime, the metric tells us how to measure distances. A straight line can be modeled as a spacetime separation vector . Then, its magnitude is given by:

As one can see, the metric tensor is defined in terms of the inner product of the basis products:

If the line is curved, then we can imagine that it is made up of many infinitesimal straight lines. Each of these infinitesimal straight lines has a length of:

Then, the total length of the curve is given by the sum (integral) of all these infinitesimal lengths:

By the chain rule, we can write this as:

When we have an observer moving along the curve, one question is to measure how much time the observer measures. This is known as the proper time . In the observer's frame, the observer is at rest, meaning that . Then, the infinitesimal length is given by:

where we have replaced with to denote the proper time. Thus, the length and the proper time are related by:

Since we have also seen that , we can write:

Relativistic Action​

Notice in the expression for that we have the term . In the observer's frame, the observer is at rest, meaning that . However, if we have an observer that is moving differently from the curve, then . This causes the term to be less than , and so the proper time is less than the time measured by the observer (time dilation). In other words:

The proper time is maximized when the observer is at rest.

When we have situations like these, we can apply action principles to predict the motion of the system. The action can be then defined as the integral of the length of the curve:

where the negative sign is a convention and helps to ensure that the action has dimensions of energy times time.