Joining Spacetimes on Fractal Hypersurfaces

TL;DR

This paper extends the Israel formalism for joining spacetimes by applying fractional calculus, enabling smoother connections without matter stress conditions, which could impact gravitational physics and spacetime structure understanding.

Contribution

The paper introduces a fractional calculus-based generalization of the Israel formalism, allowing seamless spacetime joining without traditional matter energy condition constraints.

Findings

Spacetimes can be joined smoothly without matter stress conditions.

Fractional calculus leads to new spacetime matching results.

Implications for gravitational physics and spacetime structure.

Abstract

The theory of fractional calculus is attracting a lot of attention from mathematicians as well as physicists. The fractional generalisation of the well-known ordinary calculus is being used extensively in many fields, particularly in understanding stochastic process and fractal dynamics. In this paper, we apply the techniques of fractional calculus to study some specific modifications of the geometry of submanifolds. Our generalisation is applied to extend the Israel formalism which is used to glue together two spacetimes across a timelike, spacelike or a null hypersurface. In this context, we show that the fractional extrapolation leads to some striking new results. More precisely we demonstrate that, in contrast to the original Israel formalism, where many spacetimes can only be joined together through an intermediate thin hypersurface of matter satisfying some non- standard energy conditions, the fractional generalisation allows these spacetimes to be smoothly sewed together without any such requirements on the stress tensor of the matter fields. We discuss the ramifications of these results for spacetime structure and the possible implications for gravitational physics.