# Prove Lorentz invariance for momentum 4-vector

#### flintbox

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

2. Homework Equations
I am given that $F^{\mu\nu}$ is a tensor of rank two.

3. 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.

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#### Fightfish

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

#### flintbox

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.