A Domain Wall Geometries: Continuous Coordinates, Discontinuous Metric

[Roy_1981]

Hi all, I need some help regarding domain wall geometries, essentially a bubble of some spacetime (say deSiiter or flat) inside another kind, say anti-deSitter. For simplicity it is spherical symmetric situation and we are intent on using Schwarzschild like static coordinates. So now I am used to the set up where the metric is continuous across the (infinitely thin) domain wall/bubble wall, while the static time coordinate is discontinuous across while the radial coordinate is a global (continuous) coordinate.

But I have heard that one can also set up coordinates so that the metric is discontinuous while BOTH the static coordinates "r" and "t" are continuous. My problem is I can't seem to locate a reference where this is done or illustrated. All the references I can locate use continuous metric and discontinuous coordinates.

I would tremendously appreciate if any of you forum members/user can suggest a reference on "continuous coordinates/ discontinuous metric" choice.

 

[PeterDonis]

the static time coordinate is discontinuous

I don't understand what this means. If you are using a single coordinate chart, the coordinate values must change continuously from event to event everywhere in the chart. So a discontinuous coordinate appears to violate a basic requirement of a coordinate chart.

I have heard that one can also set up coordinates so that the metric is discontinuous

This would also violate a basic requirement of a coordinate chart.

 

[George Jones]

Just as idealized situations with surface charge layers and distributional (delta function) volume charge densities, and with electric field discontinuities are useful in undergrad electromagnetism, mass hypersurface layers with metric component discontinuities and distributional stress-energy tensors are useful in general relativity, e.g., for domain walls. This is called the the thin shell/junction condition formalism.

 

[Roy_1981]

Hi George Jones,

Indeed you are in the right direction. In case of gravity you can integrate Einstein equation across the shell to obtain junction conditions, relating metric derivatives to matter on thin shell. But what I asked is a bit different - its about a choice of gauge (coordinates) and as I mentioned there are two choices (perhaps more) to parametrize the geometry. atm I just can't seem to locate any reference which uses the second gauge choice.