Consider the Heaviside function $\Theta(k^{0})$.
This function is Lorentz invariant if $\text{sign}\ (k^{0})$ is invariant under a Lorentz transformation.
I have been told that only orthochronous Lorentz transformations preserve $\text{sign}\ (k^{0})$ under the condition that $k$ is a time-like vector.
I would like to prove this explicitly, to convince myself of the fact. How do I start?
N.B.: Lorentz-invariance of step function contains a sketch of the proof but it does not contain the complete proof in explicit steps, though.
