# How can we prove that a non-linear equation of motion for a classical scalar field satisfies causality?

Let $$\phi$$ be a classical scalar field in $$1+D$$-dimensional spacetime with coordinates $$(t,\vec x)$$, and consder the equation of motion $$\newcommand{\pl}{\partial} (\pl_t^2-\nabla^2)\phi+m^2\phi+ g\phi^3=0 \tag{1}$$ where $$\nabla$$ is the gradient with respect to the spatial coordinates $$\vec x$$. If two solutions $$\phi_1$$ and $$\phi_2$$ and their first time-derivatives are equal to each other at $$t=0$$ for all $$|\vec x|>R$$, then presumably they must also be equal to each other for all $$|\vec x|>R+|t|$$ for every $$t$$. In other words, an "initial" difference between two solutions presumably cannot propagate faster than the speed of light. (I'm using units where the speed of light is $$1$$.) This is what I mean by causality in the title of the question.

How can we prove that equation (1) has this property?

Cross-posted

I originally posted this question on Math Stack Exchange, where a comment gave me a hint that helped me prove a weaker result, which I also posted there. The weaker result considers a single solution that is zero for all $$|\vec x|>R$$ at $$t=0$$ and proves that it remains zero for $$|\vec x|>R+|t|$$ for all $$t$$. In other words, the (smallest) region where the solution is non-zero cannot spread faster than the speed of light. This weaker result can be used to answer my question in the case $$g=0$$, because in that case, the difference between two solutions is another solution. However, my original question remains unanswered: when $$g\neq 0$$, I still don't know how to prove that the (smallest) region where two generic solutions differ cannot spread faster than the speed of light.