Inverted time translations

by Willie Wong

Plot of the vector field K_0 and its stream function

In the study of the global properties of wave-type equations, a well-developed method is the vector field method due to Sergiu Klainerman and Demetrios Christodoulou. Maybe in another day I will write a more detailed treatise on what the vector field method is and how to apply it; I won’t do it now. The method is crucial in many proofs of nonlinear stability for wave-type problems, and with perhaps the most striking application the global nonlinear stability of Minkowski space. The main idea behind the vector field method is to construct a tensor that measures the local energy content of the solution to our equations, and exploit the properties of this tensor via vector fields. Examples of this tensor includes the Einstein-Hilbert stress for electromagnetism, as well as the Bel-Robinson tensor for spin-2 (graviton) fields. To exploit the fine properties of this tensor field, one applies the divergence theorem to the tensor field contracted against suitable vector fields. For vector fields associated to the symmetries of the problem, this procedure will produce conservation laws, which will give control of the physical solution at a later time based on control at the present.

As it turns out, the useful symmetries of the equation, in the geometrical case, are closedly related to the conformal symmetries of Minkowski space. These include the true symmetries (translations, rotations, and Lorentzian boosts), as well as the conformal scaling and, what we will discuss here, the inverted time translation, which lies at the heart of decay estimates for spin-1 and spin-2 fields on Minkowski space.

The inverted time translation, often denoted K_0, is the vector field given in radial coordinates K_0 = (t^2 + r^2)\partial_t + 2tr \partial_r. In the picture to the upper left, the vector field is plotted along with its stream function. This vector field is a conformal symmetry of Minkowski space. The name of the vector field indicates the fact that it is associated to a conformal inversion (which is also used in the conformal compactification of Minkowski space). On Minkowski space, the inversion map x^\mu \mapsto \frac{x^\mu}{\langle x,x\rangle} is a conformal isometry. The vector field K_0 can be checked to be the vector field \partial_t conjugated by the inversion map. As such, it has a very nice property compared with the other symmetry vector fields. The time translation \partial_t and the inverted time translation K_0 are essentially (up to Lorentz boosts) the only globally causal conformal vector fields of Minkowski space. As such, with a dominant-energy type condition, they are the ones associated to which we have nonnegative energy controls.