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.

related: extrapolate vector 3-flow in $(0,1)^3$ from boundary vector flows


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