Prove Lorentz invariance for momentum 4-vector

• flintbox
In summary, the conversation revolves around showing that the given equation is manifestly Lorentz invariant. The person asking the question is unsure of how to approach it and suggests trying a Lorentz transformation. The other person explains that to show Lorentz invariance, the equation must hold true in all inertial reference frames accessible via a Lorentz transformation. The person asking the question realizes this and thanks the other person for their help.
flintbox

Homework Statement

I am meant to show that the following equation is manifestly Lorentz invariant:
$$\frac{dp^{\mu}}{d\tau}=\frac{q}{mc}F^{\mu\nu}p_{\nu}$$

Homework Equations

I am given that ##F^{\mu\nu}## is a tensor of rank two.

The Attempt at a Solution

I was thinking about doing a Lorents transformation to this four vector, but I don't know what this would yield.

flintbox said:
I was thinking about doing a Lorentz transformation to this four vector, but I don't know what this would yield.
Well then why not try it? When we say that an equation is Lorentz invariant, what we mean is that it holds true in all (inertial) reference frames accessible via a Lorentz transformation. So, that means that you have to show that if
$$\frac{dp^{\mu}}{d\tau}=\frac{q}{mc}F^{\mu\nu}p_{\nu}$$
then
$$\frac{dp^{\mu'}}{d\tau}=\frac{q}{mc}F^{\mu'\nu'}p_{\nu'}$$
is also true.

flintbox
Fightfish said:
Well then why not try it? When we say that an equation is Lorentz invariant, what we mean is that it holds true in all (inertial) reference frames accessible via a Lorentz transformation. So, that means that you have to show that if
$$\frac{dp^{\mu}}{d\tau}=\frac{q}{mc}F^{\mu\nu}p_{\nu}$$
then
$$\frac{dp^{\mu'}}{d\tau}=\frac{q}{mc}F^{\mu'\nu'}p_{\nu'}$$
is also true.
You're right! Thank you so much.

1. What is Lorentz invariance?

Lorentz invariance is a fundamental principle in physics that states that the laws of physics must remain the same for all observers moving at a constant velocity. This means that the physical properties of objects, such as mass and energy, are the same regardless of the observer's frame of reference.

2. Why is it important to prove Lorentz invariance for the momentum 4-vector?

The momentum 4-vector is a mathematical representation of an object's momentum in special relativity. By proving Lorentz invariance for the momentum 4-vector, we are showing that the laws of physics related to momentum are consistent for all observers, which is essential for understanding the behavior of objects at high speeds.

3. How is Lorentz invariance for the momentum 4-vector proven?

Lorentz invariance for the momentum 4-vector can be proven using mathematical equations and transformations derived from special relativity. This involves showing that the momentum 4-vector remains unchanged under certain transformations, such as a change in velocity or direction of motion.

4. What evidence supports the idea of Lorentz invariance for the momentum 4-vector?

There is a vast amount of experimental evidence that supports Lorentz invariance for the momentum 4-vector. For example, the motion of particles in high-energy accelerators is consistent with the predictions of special relativity, and the behavior of particles at high speeds, such as in cosmic ray collisions, also aligns with the principles of Lorentz invariance.

5. Are there any exceptions to Lorentz invariance for the momentum 4-vector?

While Lorentz invariance is a fundamental principle in physics, there are certain circumstances where it may not hold true, such as in the presence of strong gravitational fields. However, these exceptions are typically explained by more complex theories, such as general relativity, which incorporate the principles of Lorentz invariance. In general, Lorentz invariance for the momentum 4-vector has been consistently supported by experimental evidence and remains a crucial concept in modern physics.

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