Gravitating Tubes Beyond World Line Paradigm In General Relativity

Abstract

The simplest point-particle description of classical matter is incompatible with Einstein's General Relativity because the stress-energy tensor of a point particle is distributional and concentrated on a one-dimensional worldline. For such higher-codimension sources, smooth spacetime solutions generally do not exist. This obstruction was established by Geroch and Traschen for sources of codimension $\geq2$. Motivated by this result, this thesis proposes codimension-zero tubes as a fundamental description of gravitating matter. Timelike tubes are constructed within the tubular neighbourhood of an auxiliary timelike curve. The tube interior is foliated by timelike codimension-one hypersurfaces whose dynamics are governed by a brane-like action. The resulting collective stress-energy tensor is smooth, unlike that of a point particle. For a broad class of tension and potential profiles, the strong energy condition is violated inside the tube, while the null and weak energy conditions remain satisfied. In the ultraviolet limit, where the tube radius vanishes, an appropriate rescaling of the Lagrangian density reduces the tube action to the point-particle action together with a canonical self-force-like term. The particle's rest mass then emerges as an effective quantity rather than a fundamental localized parameter. Perturbative stability is analysed at two levels. Field perturbations yield an infinite squared sound speed, showing that the foliation-generating scalar is non-dynamical and cuscuton-like. Small deformations of the leaves lead to the Jacobi equation for timelike hypersurface congruences, further constraining admissible tension and potential profiles. These results establish gravitating tubes as a geometrically and dynamically consistent description of matter that respects the Geroch--Traschen obstruction.