×

Geometric algebra as a unifying language for physics and engineering and its use in the study of gravity. (English) Zbl 1368.83015

Summary: 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.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
00A79 Physics
83E05 Geometrodynamics and the holographic principle
78A25 Electromagnetic theory (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Born, M., Wolf, E.: Principles of optics: electromagnetic theory of propagation, interference and diffraction of light. CUP Archive (2000) · Zbl 0086.41704
[2] Doran, C., Lasenby, A.N.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003) · Zbl 1078.53001
[3] Doran C., Lasenby A., Gull S., Somaroo S., Challinor A.: Spacetime algebra and electron physics. Adv. Imaging Electron Phys. 95, 271 (1996) · doi:10.1016/S1076-5670(08)70158-7
[4] Gull S., Lasenby A., Doran C.: Imaginary numbers are not real—the geometric algebra of spacetime. Found. Phys. 23, 1175 (1993) · doi:10.1007/BF01883676
[5] Gull S., Lasenby A., Doran C.: Electron paths, tunnelling, and diffraction in the spacetime algebra. Found. Phys. 23, 1329 (1993) · doi:10.1007/BF01883782
[6] Grumiller, D.: http://quark.itp.tuwien.ac.at/ grumil/pdf/belgium (2011)
[7] Hestenes, D.: Space-Time Algebra. Birkhauser/Springer, Basel/Berlin (1966/2015) · Zbl 0183.28901
[8] Hestenes D.: Local observables in the Dirac theory. J. Math. Phys. 14, 893 (1973) · doi:10.1063/1.1666413
[9] Hestenes, D.: New Foundations for Classical Mechanics. Kluwer, New York (2002) · Zbl 0932.70001
[10] Hestenes D.: Zitterbewegung in quantum mechanics. Found. Phys. 40, 1 (2010) · Zbl 1193.81020 · doi:10.1007/s10701-009-9360-3
[11] King B.T.: An Einstein addition law for nonparallel boosts using the geometric algebra of space-time. Found. Phys. 25, 1741 (1995) · doi:10.1007/BF02057886
[12] Lasenby, A., Doran, C., Gull, S.: Gravity, gauge theories and geometric algebra. Proc. R. Soc. Lond. Philos. Trans. Ser. A 356, 487 (1998). arXiv:gr-qc/0405033 · Zbl 0945.83023
[13] Lasenby, A., Hobson, M.: Scale-invariant gauge theories of gravity: theoretical foundations (2015, arXiv e-prints). arXiv:1510.06699 · Zbl 1350.83003
[14] Macdonald, A.: A Survey of Geometric Algebra and Geometric Calculus, This Proceedings · Zbl 1367.15038
[15] Rienstra, S., Hirschberg, A.: An Introduction to Acoustics. Eindhoven University of Technology (2013, online). http://www.win.tue.nl/ sjoerdr/papers/boek. Accessed 18 July 2016
[16] Vold T.: An introduction to geometric calculus and its application to electrodynamics. Am. J. Phys. 61, 505 (1993) · Zbl 1219.78033 · doi:10.1119/1.17202
[17] Weyl, H.: Gravitation und Elektrizität. Sitzungsber. Preuss. Akad. Wiss., Berlin, p. 465 (1918) · JFM 46.1300.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.