Complex four vector algebra in relativity

1. May 25, 2007

MeJennifer

Complex four vector spacetime algebras in relativity

Does anyone have any experience or references as to the usefulness and applicability of this in relativity?

Last edited: May 25, 2007
2. May 25, 2007

George Jones

Staff Emeritus
Null tetrads use complexified Minkowski spaces. See d'Inverno or de Felice and Clarke.

3. May 25, 2007

Jonathan Scott

Your question doesn't make it clear whether you mean special or general relativity.

For special relativity, then "complex four-vector algebra" is a very natural way of looking at things; it is of course equivalent to the Pauli Algebra but with a slightly different approach. I wrote my own informal summary of the subject some time ago, which you can find at this URL, although Clifford Algebra enthusiasts (such as the late Pertti Lounesto) use slightly different notation from mine:

http://pws.prserv.net/jonathan_scott/physics/cfv.pdf

William Baylis likes to call this algebra the "Algebra of Physical Space" (APS), and you may find further information by searching on that subject. One thing I find most interesting about this notation is that it allows one to write the Dirac equation directly in this algebra (that is, the Pauli Algebra) rather than needing to use the Dirac Algebra.

Last edited by a moderator: Apr 22, 2017
4. May 25, 2007

MeJennifer

Very interesting paper Jonathan.
Thanks!

Last edited by a moderator: Apr 22, 2017
5. May 25, 2007

Bret Danfoss

Hi :-)

I've had a look and I think it's great! ..... From what I've read so far, you've written it very well (IMO).

Two things though;
(1) Many of the opertors within the document appear as "boxes" (e.g. pg 31). I think you need to do something about the font embedding(?).
(2) I (personally), "disagree" with the "the Neutrino Equation" section (as is) because as you say, recent experiments imply mass. You do state this, but to me it comes across as a bit of a contradiction. A paragraph stating why it's "OK" (as a techique) to take that approach should be included (IMO) and would "fix" the issue about it.

Well done overall though!

Thanks.

6. May 25, 2007

robphy

Are you referring to the gradient-type operator?

7. May 26, 2007

pervect

Staff Emeritus
Should anything in this null tetrad formulation encourage MeJennnifer to believe that there exists vectors which are neither timelike, spacelike, or null?

8. May 26, 2007

George Jones

Staff Emeritus
Neither the null tetrad formalism nor the formalism of Jonathan Scott's paper (which can be extended to GR by using tetrads) changes the fact that spacetime is modeled by a real 4-dimensional differentiable manifold that has a real 4-dimensional vector space as tangent space at each event. As you say, every tangent vector has a real "length" that is used to characterize it as timelike, lightlike, or spacelike.

9. May 27, 2007

Bret Danfoss

Yep, that's them :-) <--- disregard please.

Sorry, I don't know how it came to pass, but I posted a reply in the wrong place .... I was talking about something in a another forum.

OOPS :-(

Last edited: May 27, 2007
10. May 29, 2007

shoehorn

No you weren't. Don't try to compound your error by backtracking.