# Is there any relativistic constraint on the rate of change of a scalar field?

Consider a scalar field $$\phi(t, x, y, z)$$ obeying the waves equation with an Higgs-like potential (the "mexican hat"): $$\tag{1} \mathcal{V}(\phi) = \frac{\lambda}{4} (\phi^2 - \phi_0^2)^2,$$ where $$\lambda > 0$$ and $$\phi_0$$ are constants. $$\phi_0$$ is the value of the "true vacuum" field, which can be positive or negative. The waves equation of motion of the self-interacting field is this (I'm using units such that $$c \equiv 1$$): $$\tag{2} \frac{\partial^2 \phi}{\partial t^2} - \frac{\partial^2 \phi}{\partial x^2} - \frac{\partial^2 \phi}{\partial y^2} - \frac{\partial^2 \phi}{\partial z^2} + \lambda (\phi^2 - \phi_0^2) \, \phi = 0.$$ To solve the waves equation (2), we need a consistent set of initial conditions and boundary conditions. For the initial conditions, we usually give the field values at time $$t = 0$$ and its rate of change: \begin{align} \phi(0, x, y, z) &= \mathcal{F}(x, y, z), \tag{3} \\[2ex] \frac{\partial \phi}{\partial t} \, \bigg|_{t = 0} &= \mathcal{G}(x, y, z). \tag{4} \end{align} As far as I know, there is no constraint on the values of $$\phi(0, x, y, z)$$, in the classical theory of relativistic fields. The function $$\mathcal{F}$$ can be anything.

But is there any constraint on the values of the field derivative? Can the function $$\mathcal{G}$$ be completely arbitrary too in relativity? (this time derivative is not the same as a real velocity, which of course is constrained by causality: $$v < 1$$).