On Tensor Multi-Scalar Theories in a Post-Newtonian Setting

Abstract

The first experimental detection of gravitational waves in 2015 established a fundamentally new way of testing General Relativity 100 years after its inception in 1915 by Albert Einstein. This novel approach allows us to test General Relativity against alternative theories in the strong field regime. During the past century, numerous tests of Relativity have been conducted in the gravity regime of our solar system. Einstein's theory amazingly passed these tests with very high accuracy. Hence, testing General Relativity in regimes much more bound to gravity than our weak field solar system is essential.

We establish the mathematical foundation of General Relativity and viable alternatives in the second chapter of this thesis after a brief introduction in Chapter 1. We explicitly detail various ways to append Relativity to more complex theories viably. The fact that the theory of Albert Einstein is already excellent in multiple tests, some of them explained in Chapter 2, of course, puts quite strong constraints on new theories, and most will have a specific limit towards General Relativity. The protagonist theory discussed in this dissertation, known as Tensor Multi-Scalar Theory of Gravity, is a viable alternative that allows for differences to General Relativity in the strong field regime. The theory adds multiple scalar fields to Relativity already on the level of the action in a covariant way. Hence, we keep several geometrical properties of Relativity that we can utilize to our advantage later on.

We detail the mathematical machinery of Direct Integration of the Relaxed Field Equations in Chapter 3. This toolkit, designed to ultimately end up with a family of gravitational wave templates via post-Newtonian analysis, is very well suited to our use case since it is theory agnostic in the sense that every theory motivated by an action can be analyzed using this specific framework. Approaches adding a single scalar field to the Einstein-Hilbert action of General Relativity have already utilized this mathematical setup to significant effect , and we aim to generalize this even further in the following chapters.

The bulk of this dissertation is the calculations in Chapter 4. We adapt the previously mentioned formalism to the generalized theory of Tensor Multi-Scalar Gravitation. Extra flat wave equations append the field equations for the multiple scalar fields, which need to be carefully evaluated at every step in the goal of calculating the post-Newtonian metric to some order accurate enough for the analysis in this dissertation. We find out that the geometry target space, a manifold equipped with a Riemannian metric, plays a crucial role in the near zone dynamics of compact objects as we calculate the equation of motion of a broad class Tensor Multi-Scalar Theories through 2.5 post-Newtonian order. This effect differs from pure General Relativity as no extra scalar fields are added, and hence no target space exists. As expected from the analysis of single Scalar Field Theories we also find a 1.5 post-Newtonian contribution to the motion absent in classical Relativity.

We end the dissertation with an Outlook to future work and explain how this work fits in with the brighter goal of having an extensive library of gravitational wave templates to compare to experimental data to test General Relativity right down to its core indeed.