The evolution of the Weyl and Maxwell fields in curved space-times

Andreas de Vries, published in: Mathem. Nachr. 179 (1996), 27-45 (Preprint)

This article studies a new approach to apply linear functional analytic methods in general relativity theory. It offers a way to consider field equations as evolution equation. One of the basic problems to introduce evolution into general relativity is the the narrow tissue formed by space and time: From the relativistic viewpoint, they are a single continuum and cannot be considered independent from each other and curved in a unified way by gravitation - evolution equations in functional analysis, however, are based necessarily on the strict separation of time and space dimensions!

The solution provided in the article is as simple as effective: Restrict the region of a general space-time so that you can apply a single "coordinate chart" (where time and space are different kinds of coordinate!). Then every field equation, restricted to that region, is a true evolution equation. A central result is that the criterion for an arbitrary coordinate chart is very simple: The timelike component g⁰⁰ of the "metric tensor" g only has to be positive,

g⁰⁰ > 0.

(For the experts: g is a Lorentz metric with signature -2). That's all! By the way, the apparently equivalent condition g₀₀ > 0 fails, as the counterexample of the ergosphere of a rotating black hole shows...

The benefits are astonishing: In this chart the notion energy is well-defined, and so standard energy considerations are applicable. In particular, new estimates for the energy change rates for general charged rotating black holes are achieved, forming limits on the energy production of rotation (known as superradiance). The results can be generalized to Dirac fields (massive spinors). Also there can be deduced some fundamental limits for the mass of the universe.