A signature invariant geometric algebra framework for spacetime physics and its applications in relativistic dynamics of a massive particle and gyroscopic precession

Bofeng Wu

doi:10.1038/s41598-022-06895-0

Abstract

A signature invariant geometric algebra framework for spacetime physics is formulated. By following the original idea of David Hestenes in the spacetime algebra of signature (+,-,-,-), the techniques related to relative vector and spacetime split are built up in the spacetime algebra of signature (-,+,+,+). The even subalgebras of the spacetime algebras of signatures (pm ,mp ,mp ,mp ) share the same operation rules, so that they could be treated as one algebraic formalism, in which spacetime physics is described in a signature invariant form. Based on the two spacetime algebras and their “common” even subalgebra, rotor techniques on Lorentz transformation and relativistic dynamics of a massive particle in curved spacetime are constructed. A signature invariant treatment of the general Lorentz boost with velocity in an arbitrary direction and the general spatial rotation in an arbitrary plane is presented. For a massive particle, the spacetime splits of the velocity, acceleration, momentum, and force four-vectors with the normalized four-velocity of the fiducial observer, at rest in the coordinate system of the spacetime metric, are given, where the proper time of the fiducial observer is identified, and the contribution of the bivector connection is considered, and with these results, a three-dimensional analogue of Newton’s second law for this particle in curved spacetime is achieved. Finally, as a comprehensive application of the techniques constructed in this paper, a geometric algebra approach to gyroscopic precession is provided, where for a gyroscope moving in the Lense-Thirring spacetime, the precessional angular velocity of its spin is derived in a signature invariant manner.

Highlights

By following the original idea of David Hestenes in the spacetime algebra of signature (+, −, −, −), the techniques related to relative vector and spacetime split are built up in the spacetime algebra of signature (−, +, +, +)
The even subalgebras of the spacetime algebras of signatures (±, ∓, ∓, ∓) share the same operation rules, so that they could be treated as one algebraic formalism, in which spacetime physics is described in a signature invariant form
By following the original idea of David Hestenes, we will build up these techniques in the spacetime algebra (STA) of signature (−, +, +, +) so that a more convenient approach to relativistic physics could be given in the language of geometric algebra (GA)

Summary

Introduction

The key point is to write down signature invariant expression of the bivector field Ω(τ ) and the spacetime split of the gyroscope’s four-acceleration a with the normalized four-velocity γ0 of the fiducial observer based on the “common” even subalgebra of the two STAs. According to Refs.[33,35], the bivector connection ω(u) associated with {γα} can be directly derived, and by recasting it in terms of the relative vectors {σ k} , its signature invariant expression and those of its electric part ω(E)(u) and magnetic part ω(B)(u) are obtained.

Results

Conclusion