Generalized Spacetime Geometries and Causal Structures

While it is often argued that 70% of the universe’ energy content is of yet unknown nature, an alternative interpretation of the astrophysical observations is provided by generalized spacetime theories. In addition, these theories are highly valuable for gaining deeper understanding of General Relativity. However, the field of generalized spacetime theories has grown and branched out extensively in the last decades, with a myriad of different theories to choose from [1]. 

Within my PhD supervisor’s group, we developed a novel framework to describe, categorize and restrict generalized spacetime theories based on the dispersion relations of matter systems [2]. The advantage of our approach is that dispersion relations are operationally and experimentally directly accessible/testable, in contrast to spacetime geometries. There is also a direct relation to quantum theory as the generalized dispersion relations must have a representation in the quantum dynamics of matter. Furthermore, generalized dispersion relations are often motivated by quantum gravity phenomenology and can be derived from Quantum Electrodynamics in curved spacetimes. In particular, we have shown that

1.   a dispersion relation must satisfy three simple algebraic properties to serve as a physical dispersion relation, based on inescapable physical requirements (local matter field dynamics must be predictive and allow for an observer-independent notion of positive energy),

2.   modifications of the standard relativistic dispersion relation are severely restricted, and

3.   dispersion relations passing the simple algebraic checks that we have found correspond to physically admissible Finslerian refinements of Lorentzian geometry.

Our initial results were the basis for the new research field of Constructive Gravity [3].

[1] Saridakis, Emmanuel N., et al. Modified Gravity and Cosmology. Springer International Publishing, 2021. Preprint: arxiv.org/abs/2105.12582 

[2] Rätzel D., Rivera S., Schuller F. P., "Geometry of physical dispersion relations" Phys. Rev. D 83, 044047 (2011), doi.org/10.1103/PhysRevD.83.044047, Preprint: arXiv.org/abs/1010.1369v2

[3] Schuller, Frederic P. "Constructive gravity: Foundations and applications." The Fifteenth Marcel Grossmann Meeting: On Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories (In 3 Volumes). 2022. Preprint: arxiv.org/abs/2003.09726