From: harke@acm.org Newsgroups: sci.math.research,sci.math,sci.physics.research Subject: Re: Geometric Algebra and/or Differential Topology Date: 9 Apr 1998 21:07:22 -0700 In article <6fseq1$6td$1@agate.berkeley.edu>, FILTER.biggus@colorado.edu (jeff) writes: >Anyone know the relationship between geometric algebra and differential >topology? I've been reading up on the former, and am learning the latter, >but am unclear about their ultimate connection. Does one subsume the other >in some way? > >Some have suggested just learning differential topology instead of >geometric algebra, because it can do all that the other can, but this came >from one not too familiar with geometric algebra. Any suggestions? I cannot not say much about differential topology but I have been studying geometric algebra and believe that is a very excellent system for many areas of physics including gravitation. You do not mention what works you are reading so I will go ahead and mention some refences. First and foremost: "Clifford Algebra to Geometric Calculus" by David Hestenes and Garret Sobczyk, D. Reidel Publishing -- The hardcover is pricey but there is a soft cover edition. I regard this as an essential book for the mathemtical foundations. Also there is: "New Foundations for Classical Mechanics" by David Hestenes, D. Reidel Publishing This book demonstrates the use of geometric algebra in formulating classical mechanics. >Also, does anyone know a good source for geometric algebra in gravitation >theory? I would be very interested in reformulations of general relativity >and post-newtonian mechanics. You will certainly want to visit HTTP:www.mrao.cam.ac.uk/~clifford They have quite a number of papers that can be downloaded as well as links to other related sites. For gravitation, be sure to see: "Gravity, Gauge Theories and Geometric Algebra" by Anthony Lasenby, Chris Doran and Stephen Gull. This paper is also available in the Philosophical Transactions of the Royal Society, Series A, Volume 356, Number 1737 pp 487-582 (1998 March 15) This is a very exciting theory of gravity in that, for those predictions of General Relativity THAT CAN BE TESTED CURRENTLY, it gives just the same answers as GR. But it does give different results when a horizon is present. It also provides some advantage in dealing with matter with spin. This may also lead to testable differences. It may also provide important insights in the direction of unifying gravitation with quantum mechanics. Richard Harke ============================================================================== From: Ed Gerck Newsgroups: sci.math.research,sci.math Subject: Re: Geometric Algebra and/or Differential Topology Date: Fri, 10 Apr 1998 13:58:25 -0600 In article <6gjpaq$rnu@hopper.ACM.ORG>, harke@acm.org wrote: > > In article <6fseq1$6td$1@agate.berkeley.edu>, FILTER.biggus@colorado.edu > (jeff) writes: > > >Anyone know the relationship between geometric algebra and differential > >topology? I've been reading up on the former, and am learning the latter, > >but am unclear about their ultimate connection. Does one subsume the other > >in some way? > > > >Some have suggested just learning differential topology instead of > >geometric algebra, because it can do all that the other can, but this came > >from one not too familiar with geometric algebra. Any suggestions? > What has been called Grassmann, quaternionic, Dirac, Pauli, vector, multivector, and geometric algebras are all usually considered particular cases of a basic structure known as Clifford algebra. Their applications in physics cover a wide range of topics from classical mechanics to general relativity, electromagnetism, elementary particle physics and many aspects of quantum mechanics. One important feature of Clifford (ie, geometric) algebra is the direct sum over different vector spaces, allowing multivectors to be defined. Another feature is the exterior product and the eye-opening so-called "Clifford's choices" for the elementary results. Both allow true vector division to be defined and eliminate the problems of pseudo-vectors and pseudo-scalars found in differential topology, while also making operations self-contained in the algebra... unlike the vector product of two vectors in DT. Besides the references given below, there are also some given under the recent thread on Grassmann geometry, at sci.math. Thanks to Clifford, Grassmann`s geometric algebra was not only better understood but also expanded. Unfortunately, the same did not happen until today to the more geometric aspects of Grassmann's Ausdehnungslehre ... but the algebraic aspects are finding more and more uses in the physical sciences. More enthusiats in math research are clearly needed. Some additional web pointers: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Clifford.html http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Grassmann.html http://www.ime.unicamp.br/~walrod/ http://dvworld.nwu.edu/kleber/Grassmann/node1.html (Portuguese) http://www.physiology.uwo.ca/TweedWeb/6MirrorAlgebra.htm#Introduction http://www.hit.fi/~lounesto/counterexamples.htm Cheers, Ed Gerck ______________________________________________________________________ Dr.rer.nat. E. Gerck egerck@novaware.cps.softex.br http://novaware.cps.softex.br --- Meta-Certificate Group member, http://www.mcg.org.br --- [quote of previous article deleted -- djr] -----== Posted via Deja News, The Leader in Internet Discussion ==----- http://www.dejanews.com/ Now offering spam-free web-based newsreading ============================================================================== From: saw24@hermes.cam.ac.uk (Stephen Wells) Newsgroups: sci.math.research Subject: Re: Geometric Algebra and/or Differential Topology Date: Sun, 19 Apr 1998 19:54:35 +0100 Hi, I'm a student at Cambridge university, and I'm currently doing a literature review on geometric algebra and relativity, supervised by the guys who wrote the paper you mention below. One of the things I've found while searching for references is that the impact of GA on mainstream research in GR seems to be rather small- which is a shame. Would you mind if I mention your post in my review, just as evidence that someone outside the Cavendish lab has heard of the subject? Yours, Stephen Wells. In article <6gjpaq$rnu@hopper.ACM.ORG>, harke@acm.org wrote: > > You will certainly want to visit HTTP:www.mrao.cam.ac.uk/~clifford > > They have quite a number of papers that can be downloaded as well as links > to other related sites. For gravitation, be sure to see: > "Gravity, Gauge Theories and Geometric Algebra" by Anthony Lasenby, > Chris Doran and Stephen Gull. This paper is also available in the > Philosophical Transactions of the Royal Society, Series A, Volume 356, > Number 1737 pp 487-582 (1998 March 15) > This is a very exciting theory of gravity in that, for those predictions > of General Relativity THAT CAN BE TESTED CURRENTLY, it gives just the > same answers as GR. But it does give different results when a horizon > is present. It also provides some advantage in dealing with matter > with spin. This may also lead to testable differences. It may also > provide important insights in the direction of unifying gravitation > with quantum mechanics. > > Richard Harke -- Mieulx est de ris que de larmes escrire Pour ce que rire est le propre de l'homme. - Alcofribas Nasier