# Transforming a Cauchy foliation of $(1+1)$ Minkowski plane and building out the product manifold: Is the result a valid spacetime or component of one?

Consider a spacetime $$(\zeta^{3,1},g)$$

where $$g=\frac{dudv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2} \quad \forall u,v,w,r \in (0,1)$$ Now this is just Minkowski space in different coordinates (related to Dirac/Light cone coordinates/null coordinates). I'm looking to take a Cauchy foliation of $$\zeta^{3,1},$$ restrict the metric back to $$g'=dudv-dw^2-dr^2,$$ and use the induced measure from $$g'$$ by means of the volume form to transform the foliated past light cone region onto a new manifold $$(\Psi^{~3,1},d).$$ This seems messier because I don't know if it will be very simple to transform the past light cone. The Cauchy foliation might not be able to be expressed explicitly here. On the other hand it is straightforward to construct a $$(1+1)$$ dimensional example, and transform using an integral transform, and then define a product manifold to (potentially) achieve a $$(3+1)$$ dimensional spacetime. Instead of starting with a bonafide spacetime, restricting the metric, and then transforming the past light cone, I will start with a lower dimensional spacetime, restrict the metric, transform the past light cone, and finally scale up to a spacetime with the product manifold equipped with the product metric.

Let's look at the steps involved for the $$(1+1)$$ dimensional case $$(\zeta^{1,1},u)$$ where $$u=\frac{dxdq}{xq}.$$ The Cauchy foliation is simply $$\ln(b)\ln(y)=t$$ (solving for $$y$$ gives an explicit representation) which can then be transformed via Mellin-like transform:

$$\Phi_s(t)= \int_{S=(0,1)} \exp {\frac{t s}{\ln b}}~db = \int_0^1 tb^{t-1} \exp \frac{s}{\ln b}~db = 2\sqrt{ts}K_1(2\sqrt{ts})$$

Observe that this is an unnormalized K-distribution and it's inverse transform yields an unnormalized distribution call it the "Zeta-distribution" (our Cauchy foliation). Observe that the Fisher information metric of the Zeta-distribution is $$\Phi_s(t)$$ up to a factor. Observe that the Zeta-distribution yields solutions to the Killing field $$X=\langle x \ln x, -y\ln y \rangle.$$ It also provides a distributional solution to the backwards/forwards heat equation with diffusivity depending on both space and time:

$$\frac{\partial^2}{\partial t^2}\Phi(t,b)=\pm \frac{b}{t}\frac{\partial}{\partial b}\Phi(t,b)$$

The reason for the $$\pm$$ in the equation is because the Cauchy foliation of the light cone solves the equation for $$-$$ and the Cauchy foliation of the non-light cone solve the equation for $$+.$$

In short, $$\Phi_1(t)=v$$ is a Lorentzian metric on the smooth $$(1+0)$$ manifold $$(\Psi^{~1,0}, v).$$ We have that $$\Psi^{~3,1}=\Psi^{~1,1}\times \Psi^{~1,0} \times \Psi^{~1,0}$$ so we can define the product metric on $$\Psi^{~3,1}.$$

In the $$(1+1)$$ dimensional case, I transformed the Cauchy foliation using an integral transform and obtained a metric using the Fisher metric formulation, both of which happen to have the same form (a Bessel function).

Is anyone familiar with the product manifold $$(\Psi ^{3,1},\mathrm{prod})$$ where $$\mathrm{prod}$$ is the product metric? Is it a spacetime or a past light cone component of one?

For some more intuition about this, the Cauchy foliation (Zeta-distribution) of $$(\zeta^{1,1},u)$$ is extracted from the manifold and pasted or superposed onto the more recognizable form of the Minkowski plane (with light cone coordinates) in such a way that it now makes sense to "integrate under" these leaves of the foliation and achieve something that is not "infinity." Because if I were to try to integrate under the leaves of the Cauchy foliation of $$(\zeta^{1,1},u)$$ with the induced measure from $$u$$ by means of the volume form, clearly the result would be meaningless.