Action for a relativistic free particle

[Higgsono]

The action for a relativistic point particle is baffling simple, yet I don't really understand why it is written as,

$$S = -m\int ds $$

I know it's right because we get the right equations of motion from it, but can one understand it in a more intuitive way?

 

[Orodruin]

I know it's right because we get the right equations of motion from it, but can one understand it in a more intuitive way?

What would you consider intuitive? Whether it gives you empirically sound equations of motion is the only relevant issue when it comes down to judging the choice.

 

[Higgsono]

What would you consider intuitive? Whether it gives you empirically sound equations of motion is the only relevant issue when it comes down to judging the choice.

Maybe intuitive is not the right word, but what I meant is some deeper explanation.

 

[Orodruin]

You could ask the same question about any Lagrangian.

 

[Dale]

can one understand it in a more intuitive way?

This Lagrangian minimizes the path length, the shortest distance between two points is a straight line, a free particle moves in a straight line.

You can say it the other way too. A free particle travels in a straight line in spacetime, a straight line is the shortest distance, so we will minimize (extremize) the interval.

 

[Ibix]

Isn't it basically saying "we will extremise the path length" (where length should be read as the interval)? So "stuff will follow the longest or shortest path available to it"?

 

[haushofer]

You could similarly ask: why does Newton tell us that a free particle follows a straight line in space(time)? Why don't we need a net force to keep a particle going? We don't know. Science doesn't dictate why nature is as it is; it only gives a method to understand how you reliably can obtain knowledge about it (maybe that's the lesson we should learn from all failed attempts to understand the "why" question from mathematical consistency, unification, string theory etc.)

The intuitive way of understanding that action is as other people above me said: it's proportional to the length of the path in spacetime, and apparently nature extremizes this path for free particles. This is just Newton's laws 2.0 and Einstein's geodesic postulate 0.5.

 

[vanhees71]

Maybe intuitive is not the right word, but what I meant is some deeper explanation.

The "deeper explanation" is the answer to the question "why does the Lagrangian look as it looks?". The somewhat "stupid" answer of course is, "because it works", which in physics means that it describes an aspect of nature in accordance with the observations associated with it.

A somewhat more satisfactory answer is "because of symmetry principles". I think it's Einstein's most important general merit to modern physics to have introduced symmetry principles into physics model building. The mathematical formulation in a pretty comprehensive way is due to Emmy Noether, formulating among other important things (related to local gauge symmetries) theorems about the relation between Lie symmetries and conservation laws. One way to derive the Lagrangian for a free relativistic particle is to build an action that is invariant under the proper orthochronous Lorentz group.

Another argument is given in

https://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[sweet springs]

I know it's right because we get the right equations of motion from it, but can one understand it in a more intuitive way?

The length of world line is the only scalar for all the coordinates. It should be used to describe the motion. Coefficient mc assures that dimension of S is action. Intuitively the equation means the law of inertial motion.

 

[pervect]

Maybe intuitive is not the right word, but what I meant is some deeper explanation.

This idea has been called the principle of extremal aging, or sometimes (less accurately) the principle of maximal aging. But that's really just a name for the same explanation - maybe you'll like the new name batter, though.