to another part of that same universe very quickly or would allow travel from one universe to another. The possibility of traversable wormholes in general relativity was first demonstrated in a 1973 paper by Homer Ellis and independently in a 1973 paper by K. A. Bronnikov. Ellis analyzed the topology and the geodesics of the Ellis drainhole, showing it to be geodesically complete, horizonless, singularity-free, and fully traversable in both directions. The drainhole is a solution manifold of Einstein's field equations for a vacuum spacetime, modified by inclusion of a scalar field minimally coupled to the Ricci tensor with antiorthodox polarity (negative instead of positive). (Ellis specifically rejected referring to the scalar field as 'exotic' because of the antiorthodox coupling, finding arguments for doing so unpersuasive.) The solution depends on two parameters: , which fixes the strength of its gravitational field, and , which determines the curvature of its spatial cross sections. When is set equal to 0, the drainhole's gravitational field vanishes. What is left is the Ellis wormhole, a nongravitating, purely geometric, traversable wormhole. Kip Thorne and his graduate student Mike Morris independently discovered in 1988 the Ellis wormhole and argued for its use as a tool for teaching general relativity. For this reason, the type of traversable wormhole they proposed, held open by a spherical shell of exotic matter, is also known as a Morris–Thorne wormhole. Later, other types of traversable wormholes were discovered as allowable solutions to the equations of general relativity, including a variety analyzed in a 1989 paper by Matt Visser, in which a path through the wormhole can be made where the traversing path does not pass through a region of exotic matter. However, in the pure Gauss–Bonnet gravity (a modification to general relativity involving extra spatial dimensions which is sometimes studied in the context of brane cosmology) exotic matter is not needed in order for wormholes to exist—they can exist even with no matter. A type held open by negative mass cosmic strings was put forth by Visser in collaboration with Cramer et al., in which it was proposed that such wormholes could have been naturally created in the early universe. Wormholes connect two points in spacetime, which means that they would in principle allow travel in time, as well as in space. In 1988, Morris, Thorne and Yurtsever worked out how to convert a wormhole traversing space into one traversing time