Question about Lorentz scalar

Question

I have a simple question about Lorentz Scalars.

In my course they are introduced like that.

$\phi$ is a scalar of Lorentz if it follows the following property :

A function $\phi$ is a scalar of Lorentz if it follows the following rules :

$\phi(x)=\phi'(x')$ and $\phi'=\phi$

But what would mean $\phi'$ ? For me, for a scalar quantity $\phi'$ doesn't mean anything.

Indeed, as I have a scalar the only thing I can change of coordinates is the variable : $x=f(x')$.

And we have $\phi(x)=\phi(f(x'))$.

So, maybe I misunderstood something but what would $\phi'$ mean in a general case ?

Furthermore, do you agree with me if I say that in fact all scalar quantities in physics must be Lorentz scalar (because as I just wrote, the only thing we do is a change of variable so we don't need any "property" on the quantity described by the scalar).

Answer 1

Lorentz scalars are a subset of Lorentz invariant quantities. A Lorentz scalar is a scalar that is invariant under Lorentz transformation. For example, the dot product of a four-vector with itself is a Lorentz scalar. The 4-velocity is defined as: $$\textbf{U}=\gamma(c,v)$$ So the dot product of 4-velocity with itself is: $$\textbf{||U||}^2=U^{\mu}U_{\mu}=\gamma^2\left(c^2-v^2\right)=c^2$$ The 4-momentum is defined as: $$\textbf{P}=m\textbf{U}=\gamma\left(mc,mv\right)=\gamma\left(\frac{E}{c},\textbf{p}\right)$$ The dot product of 4-momentum with itself is: $$\textbf{||P||}^2=\frac{E^2}{c^2}-\textbf{p}^2=m^2c^2$$ A Lorentz scalar is a scalar that remains the same in every inertial frame of reference. Energy is a scalar quantity, but it can have different values in different frames of reference, so it is not a Lorentz scalar.