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

Andreas de Vries, published in:
*Mathem. Nachr.* **179** (1996), 27-45
[DOI: 10.1002/mana.19961790103]
(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
*g*^{00} of the "metric tensor" *g* only
has to be positive,

*g*^{00} > 0.

(For the experts: *g* is a Lorentz metric with signature -2).
That's all! By the way, the apparently equivalent condition
*g*_{00} > 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.