What do worm holes have to do with black holes?

All worm holes are black holes, but not all black holes are worm holes. A black hole is a region of space in which space-time curvature has become so extreme that an event horizon is present. This horizon is a 3-D spherical surface separating the outside world from what is inside the black hole. Light signals from within this horizon cannot escape to infinity so that communication with the outside universe is impossible. Black holes are solutions to Einstein's general relativistic equation for gravity, and there are many individual black hole 'solutions' depending on the properties of the matter that went into forming the black hole.

Although all of these solutions contain an internal singularity, the mathematics seems to suggest that some of these singularities are 'avoidable' and others are not. If the black hole is not rotating ( i.e. has zero angular momentum) the solution is the famous, static 'Schwarschild Black Hole' whose geometric center contains the unavoidable singularity. If it is rotating, you get some version of the 'Kerr Black Hole' and the singularity is deformed into a ring, and for some paths entering the black hole, this kind of singularity is avoidable. The full mathematical description of this space-time geometry has the entry and exit through the ring plane sending a traveler from one space-time into another space-time. These may be parts of our own universe but shifted in time, or the joining point between two different universes entirely. This is often called the Kerr worm hole solution. The problem is that this solution is only a special case of a more general solution to the interior Kerr geometry which includes the effects of gravitational radiation. After all, the black hole has to form from a physical process, and this process will emit enormous amounts of gravitational radiation is it is not perfectly spherical. So, Kerr worm holes are not expected to exist in the physical world as usable tunnels to elsewhere.

There are 'vacuum solutions' to general relativity that allow worm holes to exist without, apparently, any central source or mass to generate them. It is hard to imagine how to build such curvature into space-time without some source for it, but these vacuum solutions are theoretically possible. From the outside they would, presumably, still have a Schwarschild-like event horizon and would be non-rotating, but inside there would be no singularity or gravitational radiation to contend with. In principle, they would be traversable if they were big enough so that their tidal gravitational field would not shred you into spaghetti.

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