Bronnikov observed that the permanence of Lorentzian wormholes can be explored by analyzing the eigenmodes of the wave equation (Bronnikov, K.A., 1973) (Battye, string theory, 2018). The wave equation for a scalar field in Lorentzian wormhole spacetime may be separated and expanded into the product of a part describing propagation in the wormhole ("inside") and a part describing propagation in the surrounding spacetime ("outside"). Because the wormhole spacetime is made static by the presence of a classical matter source, one may express the wormhole part of the solution in terms of what would be found in Minkowski space. The separation of variables, though analogous to the way it is applied to bound (null) geodesic motion of test particles in curved spacetimes, has no real counterparts with the usual Penrose (1959) method in general relativity. For example, we could never separate variables and introspect the wormhole part by applying the Penrose method to wave equations in general relativity.
Bronnikov found that in one-wave mode (with no splitting of variables), absorption of the wave in the wormhole's throat is not a bad thing because the wormhole acts as a spherical mirror. In the particular case of a spherically symmetric wormhole, only odd transverse modes are normalizable, and therefore the wormhole can only absorb a particular form of the incoming wave. By comparing the behavior of odd and even modes, Bronnikov is able to use the energy conservation law to draw conclusions about traversability. However, he wrote that in more than one-mode situation the analytical results are complicated, and he refrained from investigating traversability. This was one of the few attempts to quantify the traversability of a wormhole.
To date, wave propagation in Lorentzian wormholes has been an open problem which has entwined both characterizations of wormholes and traversability. But even though there is no known complete classification, there is a definite trend in traversable wormhole characterizations. The nature of the wave description of spacetime in physics is of great interest in its own right. d2c66b5586