Master Presentation: Rosann van Wijk
Master Presentation: Rosann van Wijk
When:
8 July, 10:00 - 11:00
Where:
Feringa Building, 5612.0142
Title:
Local Symmetries of Partial Differential Equations
Abstract:
This thesis studies the symmetry structures of various partial differential equations, beginning with Lie point symmetries of classical equations on a fixed spacetime and ending with General Relativity, where the Einstein equations determine spacetime geometry itself. The focus lies on how the symmetry groups change across these settings.
First, classical equations such as the heat equation are considered. Their symmetry groups do not necessarily exponentiate to globally well-defined transformation groups. Rather, their symmetry actions may only be defined locally, which introduces the notion of a local group of transformations.
The thesis also considers relativistic equations in flat spacetime, such as the Klein-Gordon and Dirac equations, whose symmetry structures are governed by the Poincaré group. These examples show that Poincaré-invariant equations avoid the globalization issues encountered by the classical equations.
As the final part of the thesis considers General Relativity, the previously fixed background spacetime becomes dynamical. The Einstein equations now determine the geometry itself, which introduces a second interpretation for the notion of locality. Geometrically, locality refers to the fact that when one zooms in far enough, spacetime becomes Minkowskian. This results in the Poincaré group appearing as the associated symmetry group. Algebraically, locality refers to the infinitesimal symmetry generators themselves, which is the same notion of locality encountered for the local groups of transformations. For $\mathrm{Diff}(M)$, this forms an infinite-dimensional Lie algebra. This thesis clarifies how this structure relates to the finite-dimensional Poincaré algebra, and why the relation between $\mathrm{Diff}(M)$ and “local Poincaré” depends on the notion of locality involved. This distinction is essential for interpreting the symmetry structure of the Einstein equations.