# foliation of a globally hyperbolic spacetime by Cauchy hypersurfaces

While studying semi-Riemannian geometry I thought:

There is a globally hyperbolic spacetime $$\zeta^{3,1}:=\zeta^{1,0}\times \zeta^{1,0} \times \zeta^{1,1}$$, where $$\zeta^{1,0}\simeq \Bbb R^{1,0},$$ and $$f:\Bbb R^{1,0}\to \zeta^{1,0}$$ via $$f(x)=e^x.$$ I'm interested in defining a foliation of Cauchy surfaces of $$\big(\zeta^{3,1},g\big).$$ Needed is a smooth Cauchy temporal function (the gradient is everywhere timelike, not just causal, and each level set is a Cauchy surface that is necessarily spacelike).

What is a foliation of $$\big(\zeta^{3,1},g\big)$$ by Cauchy surfaces?

The metric of $$\zeta^{1,1}$$ is $$g=\frac{dxdy}{xy}.$$

For $$\zeta^{1,1}$$ I have $$\textbf{Grad}\big(f(x,y)\big)=\textbf{Grad}\big(\frac{x^2}{2}\ln x - \frac{x^2}{4} - \frac{y^2}{2}\ln y + \frac{y^2}{4} + C\big)=\big\langle x\ln x,-y \ln y \big\rangle$$ for some constant $$C.$$

This I think gives the projection of the vector field from the gradient function onto the x-y plane in $$(0,1)^2.$$ Then the integral curves of this vector field foliate $$\zeta^{1,1}.$$

I'm stuck on how to generalize this to higher dimensions.