Geometric Algebra as a Unifying Language for Physics and Engineering and Its Use in the Study of Gravity

Abstract

Geometric Algebra (GA) is a mathematical language that aids a unified approach and understanding in topics across mathematics, physics and engineering. In this contribution, we introduce the Spacetime Algebra (STA), and discuss some of its applications in electromagnetism, quantum mechanics and acoustic physics. Then we examine a gauge theory approach to gravity that employs GA to provide a coordinate free formulation of General Relativity, and discuss what a suitable Lagrangian for gravity might look like in two dimensions. Finally the extension of the gauge theory approach to include scale invariance is briefly introduced, and attention drawn to the interesting properties with respect to the cosmological constant of the type of Lagrangians which are favoured in this approach. The intention throughout is to provide a survey accessible to anyone, equipped only with an introductory knowledge of GA, whether in maths, physics or engineering.Mathematics Subject Classification. 00A79, 83C02 (Physics and General Relativity).

Geometric Algebra, Gravity and Gravitational Waves

“…Note that geometrically, R(B) is a mapping of bivectors to bivectors. (Also note that in [3], the expression for the Riemann given there (equation (5.5)) unfortunately contains two typographic errors -the ∂ a and ∂ b given there should have been a·∇ and b·∇, as here.) The Ricci scalar is…”

“…Obviously in a contribution of this length it is not possible to give full details of either Geometric Algebra or Gauge Theory Gravity, so for those readers wanting a fuller account we refer to the book 'Geometric Algebra for Physicists' [1] by Doran & Lasenby, and the paper 'Gravity, Gauge Theories and Geometric Algebra' by Lasenby, Doran & Gull [2]. The recent review [3] could also be useful, since it emphasises some different aspects of GA in gravity, and also contains a description of some applications of GA to electromagnetism, which is only treated very briefly here (in the context of joint EM and gravitational waves). Finally, we should note for those readers interested primarily in the particular 'memory effect' for gravitational waves discussed here, that this has been independently discovered, at about the same time as the work reported here, and also related to the Brinkmann metric, by Gary Gibbons, Peter Horvathy and co-workers, and that the paper [4] would be good to consult on this, being the first in a series of papers by them on this topic.…”

Spin-1/2 one- and two- particle systems in physical space without eigen-algebra or tensor product

“…However, as pointed out in relation to expression (10) there is full symmetry under the exchange S (1) ⇆ S (2) .…”

Spin-1/2 one- and two- particle systems in physical space without eigen-algebra or tensor product

“…Key developments in Quantum Mechanics (QM), such as the first phenomenological description of spin 1/2 by Pauli [1] and the first quantum relativistic description of the electron by Dirac [2], 're-invented' Clifford algebras (in the matrix representation), seemingly unaware of Grassmann's [3] and Clifford's [4] works more than half a century earlier. The promotion of vector-based Clifford algebras in Physics and in particular the development of the spacetime algebra (STA) was undertaken by Hesteness [5,6], with more scientists joining in during the last 2 -3 decades [7][8][9][10]. Spin formalism is arguably the main application of STA in QM [7].…”

“…The same reason as for the one-particle case is valid here, the measurement projecting out two of each spin components. Applying the 'on lab' rule of transformation to the cross-product terms in (10), which is the part of the squared total spin affected by a measurement, one obtains the correlation expectation values:…”