• PhD Dissertation

Official version: Finsler Geometry, Spacetime & Gravity - From Metrizability of Berwald Spaces to Exact Vacuum Solutions in Finsler Gravity, 2024 (Supervisors: Andrea Fuster, Luc Florack)
arXiv version: arXiv:2404.09858

• Popular summary

A little over a hundred years ago, Albert Einstein revolutionized our understanding of gravity with his general theory of relativity (GR). In this theory, gravity is not a force but a manifestation of the geometry of the four-dimensional spacetime continuum—spacetime, for short. Massive objects like the sun warp the spacetime around them, and other objects then move along the straightest possible paths—'geodesics'—through this curved spacetime. Due to the curvature, however, these geodesics are not straight in the usual sense at all. In the case of the planets in the solar system, for instance, they describe elliptical orbits around the sun. In this way, GR elegantly explains the movement of the planets as well as numerous other gravitational phenomena, ranging from everyday situations to black holes and the big bang. As such it remains a cornerstone of modern physics.

However, GR clashes with quantum mechanics (QM), another fundamental theory of physics. QM suggests that particles typically exist in multiple states at once until they are measured, at which point they randomly select a state due to a mysterious event often dubbed the 'collapse of the wave function'. All matter in our universe seems to be governed by these strange probabilistic laws of QM. And the same is true for all forces of nature—except for gravity. This discrepancy leads to deep philosophical and mathematical paradoxes, and resolving these is one of the primary challenges in fundamental physics today.

One approach to the problem is to modify GR by expanding its underlying mathematical framework. Traditionally, GR is formulated in terms of pseudo-Riemannian geometry, a mathematical language capable of describing most types of spacetime geometries. Recent findings suggest, however, that our universe's spacetime might be of a type that is outside the scope of pseudo-Riemannian geometry and can only be described by Finsler geometry, a more advanced mathematical language that extends beyond the traditional framework. Reformulating GR in the language of Finsler geometry leads to a much richer theory known as Finsler gravity, which captures all phenomena explained by GR and potentially many more.

In this dissertation, we have investigated the mathematical structure and the physical implications of Pfeifer and Wohlfarth's vacuum field equation of Finsler gravity. This equation governs the potential shapes that the geometry of spacetime may take in the absence of matter. To a good approximation, this includes all interstellar space between stars and galaxies, as well as the empty space surrounding stellar objects such as the Sun and the Earth. By carefully analyzing the field equation, we have identified and mathematically classified several new types of spacetime geometries and provided physical interpretations for them.

One particularly exciting discovery involves a class of spacetime geometries that represent gravitational waves—ripples in the fabric of spacetime that propagate at the speed of light. The first direct detection of gravitational waves on September 14, 2015, marked the dawn of a new era in astronomy, allowing scientists to explore the universe in an entirely new way. Since then, many observations of gravitational waves have been made and our research indicates that these are all consistent with the hypothesis that our spacetime exhibits a Finslerian nature.

While our results are promising, they merely scratch the surface of the implications of the field equation of Finsler gravity. The field is still young and further research in this direction is actively ongoing. We are optimistic, though, that our results will prove instrumental in deepening our understanding of gravity and we hope that, eventually, they may even shine light on the reconciliation of gravity with quantum mechanics.