Klein-Gordon signal propagation

[alle.fabbri]

Given Klein-Gordon equation for a particle of mass m in covariant notation
$$\left[ \partial_{\mu} \partial^{\mu} + \frac{m_0^2 c^2}{\hbar^2} \right] \phi = 0$$
show that the solution preserves causality, i.e. signals have a velocity not higher than c.

HINT: You can build up a quantity $$j^{\mu} = j^{\mu} \left( \phi \right)$$ which satisfy a continuity equation $$\partial_{\mu} j^{\mu} = 0$$. Then apply the Gauss Theorem to the lower layer of a light cone $$x_{\mu} x^{\mu} = 0$$ of finite height $$\overline{x^0}$$ delimited by two 3-sphere of radium R, lower, and $$R - \overline{x^0}$$, upper, and show that the space integral of $$j^0$$ can only decrease during the evolution.

I have built up a 4-vector that behaves like that but I can't realize how this justify the fact that Klein-Gordon is a causal theory.

Any ideas?

 

[BigJDubs]

The key is to use the continuity equation \partial_{\mu} j^{\mu} = 0 to show that the Klein-Gordon equation preserves causality. Applying the Gauss Theorem to the lower layer of a light cone x_{\mu} x^{\mu} = 0, of finite height \overline{x^0} delimited by two 3-spheres of radii R, lower, and R - \overline{x^0}, upper, we can show that the space integral of j^0 can only decrease during the evolution. This implies that any signal must propagate with a velocity less than or equal to the speed of light, thus preserving causality.