Necessity for action to be a Lorentz scalar

In summary, the conversation discusses the relationship between the Lorentz scalar of an action and the invariance of equations of motion in different frames. While it is agreed that a Lorentz scalar action ensures the same equations of motion in all frames, it is questioned whether it is necessary for the action to be invariant in order for the equations of motion to be invariant. An example is given to demonstrate that the action itself does not need to be invariant for the equations of motion to be invariant.
  • #1
HJ Farnsworth
128
1
Hello,

On p.573 of Jackson 2nd Ed. (section 12.1), he says, "From the first postulate of special relativity the action integral must be a Lorentz scalar because the equations of motion are determined by the extremum condition, [itex]\delta A=0[/itex]."

I agree that if the action is a Lorentz scalar, then that is sufficient to assure that the equations of motion are the same in all frames: Lorentz scalar, so Lorentz invariant, so action is minimized in all frames when it is minimized in one frame, since it is the same in all frames as it is in that one frame.

However, Jackson seems to imply not only that it is sufficient, but that it is necessary as well. I do not see why this is the case - it seems that the action could vary from frame to frame, but still be minimized in all frames, resulting in the same equation of motion in all frames.

Is inter-frame invariance of the action necessary, and if so why?

Thanks very much for any help that you can give.

-HJ Farnsworth
 
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  • #2
You are completely right! In order for the equations of motion to be invariant around the stationary point, not the action itself must be invariant but only its variation.

Example: Take the electromagnetic potentials in Coulomb gauge and write down the corresponding action by working in the constraint with a Lagrange multiplier. This action is not Lorentz invariant, but the equations of motion, Maxwell's equations, are!
 
  • #3
Excellent, thank you very much!
 

1. What is a Lorentz scalar?

A Lorentz scalar is a physical quantity that remains the same under a Lorentz transformation, which is a mathematical transformation used to describe the relationship between space and time in special relativity.

2. Why is it necessary for action to be a Lorentz scalar?

In physics, action is a fundamental concept that describes the dynamics of a system. In order for this concept to be consistent with the principles of special relativity, it must be invariant under a Lorentz transformation. This means that the value of action should not change when viewed from different reference frames, which is only possible if it is a Lorentz scalar.

3. How does the concept of Lorentz scalar relate to special relativity?

The concept of Lorentz scalar is closely related to special relativity, as it is one of the fundamental principles of this theory. In special relativity, the laws of physics should remain the same in all inertial reference frames, and Lorentz scalar quantities allow for this consistency.

4. What are some examples of Lorentz scalar quantities?

Some examples of Lorentz scalar quantities include mass, electric charge, and scalar potential. These quantities do not change when viewed from different reference frames and are therefore consistent with the principles of special relativity.

5. How does the concept of Lorentz scalar impact our understanding of the universe?

The concept of Lorentz scalar is an essential part of our understanding of the universe, as it allows for the consistency of physical laws in different reference frames. Without this concept, our understanding of special relativity and the fundamental principles of physics would be incomplete.

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