Geometrical Interpretation of Multipoles and Moments on Differential Manifolds

Pinto, Paolo and Gratus, Jonathan (2023) Geometrical Interpretation of Multipoles and Moments on Differential Manifolds. PhD thesis, Lancaster University.

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Abstract

In this thesis, the construction of a specific family of linear functionals with support on a closed embedding c : R ,→ M upon a manifold is discussed. The construction is performed in a purely coordinate free fashion, based on the De Rham push-forward approach and generalised to define "tensorial currents" called "multipoles". Several geometrical and algebraic properties are investigated and two main useful classes of non-trivial coordinate representations are compared and related to the choices of some extra structures on the manifold (i.e. affine connection, foliation, adapted atlas, adapted frames). It is shown that in general, the transformation rules are not given by the action of the linear group, unless some information upon the "transverse" directions with respect to the closed embedding is provided. It is shown how the multipoles are the geometrical objects naturally arising when some specific one parameter families of compact support tensor fields are expanded asymptotically around the closed embedding. In case a one parameter family satisfies also an extra condition (i.e. self similarity) it is shown how to recover the well known standard definition of "moments", opening the door to a new completely covariant and coordinate free meaning of the concept of "multipole expansion" of functions and tensor field upon the differential manifolds. It is shown how these linear functionals admit a coordinates representation coinciding with the moments commonly defined to perform the Pole-Dipole approximation of an Energy-Momentum Tensor field in General Relativity, and when a Levi Civita connection is assumed on a pseudo-Riemmanian manifold, the first two multipoles related to an Energy Momentum tensor field expansion can easily satisfy the well known Mathisson-Papapetrou-Dixon equation. Since the proposed method of construction of the multipoles does not rely on a specific metric or a specific affine connection, a generalisation of the Pole-Dipole approximation for a non metric connection is easily achieved, casting the Mathisson-Papapetrou-Dixon equation in presence of a non null torsion. Because of this, there is hence the possibility to interpret the test particles and test charges within the Relativistic Theories (possibly beyond General Relativity) just as the multipole approximation of the regular sources of the interaction fields, with a new clear geometrical background.