I should preface my comments by saying that I tried to get through a graduate level course in general relativity but "crashed and burned". However I was a bit stupid by taking the course without first taking the prerequisite in electrodynamics (essential!).

Anyway, disclaimer aside, I'm very sceptical of any physical solution based upon negative mass or negative energy. When solving a differential equation based upon a physical model, it is not uncommon to come up with two solutions, viz. a physical along with a nonphysical solution. It's normally trivial to determine and discard the nonphysical solution because it won't match boundary conditions or it violates a more basic physical law, i.e. the second law of thermodynamics. One can take the equations of motion derived from the special theory of relativity and come up with all sorts of solutions for vehicles travelling faster than the speed of light. Of course these solutions are nonsense since to reach those speeds one would have to violate at least one of Einstein's Postulates used in deriving the original equations of motion.

One should always keep in mind that the mathematical equations used to describe physics are models of the physical laws (they are NOT in themselves the actual physical laws). Those t-shirts that undergraduates in physics wear showing the Maxwell Equations with the text "Let there be light" actually show a basic misunderstanding of physics. There maybe a way to model the Laws of Physics that has a one-to-one correspondance with their full truth. I have no clue how one would do that. This gets us into the rarified world of meta-mathematics. The first step in that journey is to ask which is more fundamental: "Physics or mathematics". One might promptly answer that mathematics is more fundamental. However remember that mathematics is ultimately based upon counting. There is an awful lot of deep physics involved in the process of counting.

Gary