As early as the 1930s, Einstein and Nathan Rosen discovered that some solutions to gravitational equations represented shortcuts through space. This spacetime structure is known as 'Einstein-Rosen Bridge'.
In 1985, well-known astronomer Carl Sagan completed writing a science fiction 'Contact' and asked his old friend Kip Thorne, an established expert in gravitational theory, to check whether the description on instant cosmic voyage was scientifically feasible. Who would have thought that such a casual incident would kick-start a new wave of researches in spacetime travel!
Thorne and his study group found that spacetime travel did not go against any laws of physics known today. They further proposed to use a wormhole as a vehicle for spacetime travels.
A wormhole is a hypothetical tunnel through the fabric of spacetime, allowing an object that passes through it to appear instantaneously in some other part of the Universe - not just in a different place, but also in a different time.
Unlike black holes which can be formed after the death of massive stars, wormholes do not occur naturally. Theoretically speaking, there are two 'feasible' strategies to construct wormholes.
If we had a super microscope (hypothetical) which enabled us to zoom into a scale of
10-35m (1010 times smaller than a nucleus atom), we would find that an apparently smooth spacetime were actually filled up with tiny, irregular and probabilistic spacetime distortions. Scientists call these distortions "quantum foam". Occasionally, extremely tiny wormholes will be formed. Provided that we had infinitely advanced civilization, we might able to locate, capture and finally enlarge these tiny wormholes to a usable size by adding exotic
materials* to them.
*Exotic material is a hypothetical material that has negative gravitational force as seen by a light beam travelling through it.
If we had infinitely advanced civilization, we might try to warp and twist space on macroscopic scales so as to make a wormhole where previously none existed. We could first create a "sock" in the curvature of space, fold gently the space outside the sock in hyperspace, tear holes in the toe of the sock and in the space just below it, and finally "sew" the holes together to form a wormhole.
As the discussion of wormhole and exotic materials requires the most advanced physical theories, it is very difficult to explain them in details here. For further reading, one excellent book on this subject is by Prof. Kip S. Thorne himself. The name of the book is "Black Holes & Time Warps".