# Relativistic Action: Lorentz Transform to Get Action A

• actionintegral
In summary: No, I am not confusing them. It is just that the proper time is a more fundamental quantity than the coordinate time.Is it possible, that you are confusing the coordinate time t and the proper time \tau, which is by definition Lorentz invariant (as coalquay404 explained) as it is the square root of the spacetime interval?It is possible. I will try to convince myself that this is the case.It is possible. I will try to convince myself that this is the case.
actionintegral
I am at rest. I see an object with action A. Someone moves by with
velocity v. How should I use the Lorentz transform to get the action A' in his frame?

What do you mean by "I see an object with action A"? This doesn't make much sense to me.

Do you mean an "I see an object which follows a path that minimises the action integral A"?

Jheriko said:
Do you mean an "I see an object which follows a path that minimises the action integral A"?

Not really - I am trying to treat action on it's own merit. Let's keep the path (dt) or (dx) infinitesimally small.

I'm still confused. The action is just a number formed from an integral of some quantity:

$$S = \int L(q,\dot{q}).$$

If you're interested in an action which is appropriate for, say, a relativistic point particle of mass $m$, then the obvious candidate is

$$S = -m\int d\tau$$

where $d\tau$ is the infinitesimal proper time along the world-line. The point here is that Lorentz invariance is built in to the action from the start since $d\tau$ is a Lorentz invariant. Because of this you're guaranteed that the equation of motion that you get from varying this action will automatically be Lorentz invariant also.

Apologies if I'm missing something here, but the question seems trivial.

coalquay404 said:
the obvious candidate is
...
the question seems trivial.

What I am missing is the following: I am at rest. I see an object with energy E. I watch it for time dt. I calculate E*T. A moving person will see the same object with energy E' and the time interval in question is dt'. It is not obvious to me that Edt = E'dt' but I will try to convince myself of that.

actionintegral said:
What I am missing is the following: I am at rest. I see an object with energy E. I watch it for time dt. I calculate E*T. A moving person will see the same object with energy E' and the time interval in question is dt'. It is not obvious to me that Edt = E'dt' but I will try to convince myself of that.

Is it possible, that you are confusing the coordinate time t and the proper time $$\tau$$, which is by definition Lorentz invariant (as coalquay404 explained) as it is the square root of the spacetime interval?

## 1. What is relativistic action?

Relativistic action is a concept in physics that describes the motion of a particle or system in terms of its energy and momentum. It takes into account the effects of special relativity, such as time dilation and length contraction, which occur at high velocities.

## 2. What is the Lorentz transform?

The Lorentz transform is a mathematical equation that describes how the measurements of time and space change between two inertial frames of reference that are moving relative to each other at a constant velocity. It is a key component of special relativity theory.

## 3. How is the Lorentz transform used to get action A?

In order to obtain the relativistic action for a particle or system, the Lorentz transform is applied to the classical action, which is a measure of the energy and motion of a system in Newtonian mechanics. This results in a modified action that takes into account the effects of special relativity.

## 4. Why is relativistic action important?

Relativistic action is important because it provides a way to describe the motion of particles and systems at high velocities, where the effects of special relativity become significant. It allows for more accurate predictions and calculations in situations where classical mechanics would not be applicable.

## 5. How is relativistic action related to quantum mechanics?

Relativistic action is a crucial component in the development of quantum field theory, which combines the principles of special relativity and quantum mechanics. The relativistic action for a particle or system is used to derive the corresponding quantum field equations, which describe the behavior of particles at the quantum level.

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