Wormholes in Euclidean Space?

JPBenowitz

As it would appear the universe is spatially flat, a Euclidean Plane. If this is true then how could black holes exist? Doesn't this necessitate that if black holes are embedded in flat space that the mean curvature must be zero and thus all black holes are minimal surfaces? So, if black holes are catenoids on the surface of a euclidean plane then where the heck would all of the matter go?? Puzzling indeed. Then again the torus has zero Gaussian Curvature and classifies as a flat surface... What is the possibility that the universe is a torus with catenoids scattered all over it?

PeterDonis

Only when averaged on a very large scale (hundreds of millions to billions of light years and larger).

First of all, because on smaller scales the universe is not spatially flat. Second, because a black hole is a feature of *spacetime*, not just space. Spacetime can be curved even if spatial slices cut out of it are flat.

JPBenowitz

Can you link any papers dealing with curved spacetime with flat spatial features?

PeterDonis

Two quick examples:

In FRW spacetime when the density is equal to the critical density, the spatial slices of constant "comoving" time are flat.

In Schwarzschild spacetime, the spatial slices of constant Painleve time, which is the time experienced by observers falling into the black hole from rest "at infinity", are flat.

Both of the above statements should be obvious from looking at the line elements in the appropriate coordinate charts. These are given, for example, on the Wikipedia pages:

http://en.wikipedia.org/wiki/Friedma...3Walker_metric

http://en.wikipedia.org/wiki/Gullstr...A9_coordinates

Remember that whether a spacelike slice is flat or not depends on how you "cut" it out of the spacetime. I started a thread on PF some time ago about what conditions a general spacetime must satisfy in order for it to be possible to "cut" a set of flat spatial slices out of it:

http://www.physicsforums.com/showthread.php?t=446589

Unfortunately we didn't really come up with a conclusive answer.

JPBenowitz

Hmmm an area where the mean curvature vanishes everywhere satisfying some boundary conditions?

PeterDonis

Not sure what you're asking here. Can you please clarify?