# Complex four vector algebra in relativity

Complex four vector spacetime algebras in relativity

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

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George Jones
Staff Emeritus
Gold Member
Does anyone have any experience or references as to the usefulness and applicability of this in relativity?
Null tetrads use complexified Minkowski spaces. See d'Inverno or de Felice and Clarke.

Jonathan Scott
Gold Member
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.

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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.
Very interesting paper Jonathan.
Thanks!

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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.

robphy
Homework Helper
Gold Member
(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(?).
Are you referring to the gradient-type operator?

pervect
Staff Emeritus
Null tetrads use complexified Minkowski spaces. See d'Inverno or de Felice and Clarke.
Should anything in this null tetrad formulation encourage MeJennnifer to believe that there exists vectors which are neither timelike, spacelike, or null?

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

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 :-(

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