What is a Cauchy surface and its significance in spacetime?

[Ratzinger]

Would someone care explaining to me what a Cauchy surface is?

thanks

 

[Chris Hillman]

Your holiness:

The idea is that in the theory of wave equations (in which disturbances in some field propagate at a certain speed--- in our case, the speed of light), a small change in initial conditions confined some a small location can only affect a certain region of spacetime, an inverted triangle like this:

    *****
     ***
------*-------

Dually, the shaded area completely determined by initial conditions on a piece of initial time slice is a "triangle" like this:

     *
    ***
---*****----

(Edit: unfortunately, my attempt to obtain fixed pitch font ASCII diagrams failed, but perhaps you can see what I have in mind anyway--- this is meant to look like a triangle whose left, right sides have slope 1, -1 respectively.)

In gtr, this suggests trying to reformulate the EFE (in particular, the vacuum EFE) in terms of initial data on a spatial hyperslice (satisfying certain constraint equations) coupled with evolution equations which enables us to extend the original hyperslice to an entire family of spatial hyperslices, and thus to describe the spacetime in terms of Riemannian three-manifolds which evolve "over time". Naturally, there is tremendous freedom in choice of such hyperslicings, so to obtain a unique evolution (for purposes of numerical simulation, say), one must impose additional conditions, often = called "gauge fixing" conditions. If this program is successful, the initial hyperslice is called a Cauchy hyperslice. This terminology honors the fact that Cauchy provided a similar initial value formulation of the standard wave equation in euclidean three space.

In curved Lorentzian manifolds, it is possible that some regions cannot causally affect others (in particular, what happens inside an event horizon cannot, classically speaking, affect what happens outside). Such spacetimes do not have Cauchy surfaces; there is no hyperslice such that giving initial conditions all along that slice will determine the entire spacetime by evolving according to the ADM prescription.

 

[smallphi]

My intuitive understanding of it is that you consider a region of the given spacetime and all timelike or null curves (not necessarlity geodesics) from which that region is built. Choosing one point on each curve in a 'smooth way' you get a spacelike hypersurface intersecting all the curves, the Cauchy surface of those curves.

If you specify conditions on a Cauchy surface, you can calculate what happens in regions that are causally connected with it through the timelike/null curves. It's something like initial condition surface or surface of initial time.

If the spacetime admits a global Cauchy surface, it is called globally hyperbolic. The most basic example of such spacetime is Minkowski. You can take the hypersurface t=const as Cauchy surface. Of course the choise of such surface is not unique.