Relativistic Action

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.

Table of Contents

• Table of Contents
• Proper Time for a Curved Worldline
• The Relativistic Action
  • Derivation of Geodesic Equation

Proper Time for a Curved Worldline

We shall first show how to define the proper time for a curved worldline. This will be important for the Lagrangian formalism, as we will need to define the action in terms of the proper time. Recall that the proper time is defined as the time measured by a clock that is moving along with the object.

I will use pink letters for the coordinates in a rest frame, and blue letters for the coordinates in the moving frame. The proper time is defined as the time measured by a clock that is moving along with the object, i.e. the time coordinate in the moving frame:

To begin with a simpler case, consider the object moving with a constant velocity in the direction. The beta coefficient is then . The Lorentz boost from the moving frame to the rest frame is given by:

However, in the moving frame, the worldline has because the object is at rest in its own frame. Applying this to the first equation gives:

Hence:

For a curved worldline, we can use identical reasoning. First, split the worldline into infinitesimal segments, each of them being straight lines. Then, for each segment, the proper time is simply the differential of Equation :

We can now integrate this equation to obtain the proper time for the entire worldline:

The Relativistic Action

The action is defined as the integral of the Lagrangian over time. However, we do not currently have an expression for the Lagrangian, nor do we want to use "time" as the parameter (because it is not invariant). We will instead use the proper time as the parameter.

An object travels along a path that maximizes the proper time. Given that we have a problem involving extreminizing a functional, we can apply action principles. In this case, we can define the action as the proper time, with some prefactor to give it the same dimensions as the nonrelativistic action:

We also attach a minus sign to the action, so that the action is minimized for a particle moving in a straight line (instead of maximized). There are two things to note here:

1. Because we did not assert a specific form of , this is regardless of the metric, and hence applies to any spacetime.
2. The quantity is invariant, so the entire action is invariant.

Because we define as the integral of the Lagrangian, it must be the case that . This means that the Lagrangian is given by:

It is a good exercise to show that applying the Euler-Lagrange equations to this Lagrangian gives the relativistic equation of motion, given that we use the Minkowski metric.

Derivation of Geodesic Equation

Our next task is to show that this action principle leads to the geodesic equation. A geodesic is a curve that locally minimizes the distance between two points. Geodesics are the "straightest" possible curves in a given space. The key is that the principle regarding maximizing proper time still holds. Thus, we can use the same action principle as before. Expanding the action, we have:

Because extremizing the action is independent of multiplying by a constant, we can drop the prefactor . We can also drop the square root; extremizing is equivalent to extremizing if is constant along . Choosing gives the quantity in the square root as , which is just because . Thus, the quantity in the square root is constant and we can drop it. The Lagrangian is then given by:

where we have added a factor of for convenience. Our action is:

We can now apply the Euler-Lagrange equations to this Lagrangian. Recall that the Euler-Lagrange equations are given by:

where the dot signifies a derivative with respect to . We can now compute the partial derivatives:

1. . Since only the metric explicitly depends on , we have:

2. The inner derivative :

   We can rename the dummy index on the very right from to , giving:

3. : we can use the value from the previous step. Both and are functions of , so we can use the product rule. But first, we need to compute the derivative of with respect to using the chain rule:

   We can now compute the derivative:

Plugging all of this into the Euler-Lagrange equations gives:

To make the equation more suggestive, we can first isolate the second derivative term:

Renaming the dummy index to in the rightmost term, then swapping and gives:

Next, because of the symmetry of the metric tensor, , then and naturally . Then, making the swap on the left, and and in the rightmost term gives .

This means that we can write:

The quantity inside the parentheses should look familiar—they are the connection coefficients for the Levi-Civita connection;

Rearranging gives:

Finally, we can raise the index using the inverse metric :

Using , we have:

This is the geodesic equation, which describes the motion of a free particle in a curved spacetime.