Gauss-Bonnet Theorem for Analysis of Warp Metric Topologies

The Gauss-Bonnet Theorem (GBT) relates the geometry of a manifold, such as a wormhole or Alcubierre warped spacetime, to the manifold’s Euler characteristic chi = 2 (1 – g), which is a topological invariant. (The genus g denotes the number of handles/throats of the manifold). GBT specifies the volume integral of the Gaussian curvature k (= 8 mu + ½ ||h||²) as the lower limit to 2 pi chi. Here, k is expressed in terms of the energy density u and the trace of the 2^nd fundamental form h [1]. Wormholes have an Euler characteristic of at least 1 and the specific Euler characteristics for many wormholes are well known. We apply the GBT to each of three representative warp drive metrics (Alcubierre, Van Den Broeck, and Natário) to determine (i) which, if any, of these warp metrics produce a local change in the topology of spacetime, and (ii) for those that do produce topological change, which wormholes possess matching topology.

[1] Ida, D., and Hayward, S. A., “How much negative energy does a wormhole need?,” Phys. Lett. A, Vol. 260 (1999) pp. 175-181.