Why is it obvious that this Lagrangian is Lorentz invariant?

[Kara386]

We've just been introduced to Langrangians, and my lecturer has told us that the Lagrangian density $\mathcal{L} = \frac{1}{2} (\partial ^{\mu}) (\partial_{\mu}) -\frac{1}{2} m^2\phi^2$ is obviously Lorentz invariant. Why? Yes it's a scalar, but I can't see why it obviously has to be a Lorentz invariant one. Thanks for any help!

 

[Orodruin]

Did you try performing a Lorentz transformation to see how it transforms?

$\phi$ is a scalar field and its derivative $\partial_\mu\phi$ therefore transforms as a dual vector.