Algebra of Physical Space vs. Spacetime Algebra

[closet mathemetician]

What is the difference between the Algebra of Physical Space (APS) and the Spacetime Algebra (STA), and why do we need them both?

 

[HallsofIvy]

This question does not appear to me to have anything to do with "Linear & Abstract Algebra". The terms "Physical Space" and "Spacetime" make me think it is about general relativity. Any objection to my moving it?

 

[LukeD]

This is a question about Clifford Algebra, so it might fit here in Linear & Abstract Algebra. However, APS and STA are specifically used for special/general relativity.

APS is the algebra of 3 euclidean vectors + 1 scalar (Also called Cl_3). As far as I can tell from looking at the Wikipedia article, they seem to arbitrarily decide that the scalar entry is time. This seems very similar to the Quaternions and probably suffers from similar problems.

STA is the algebra of 4 minkowski type vectors (+++- signature), also called Cl_{3,1}. There are 4 components for each vector. In addition to vectors, this algebra contains scalars, bi-vectors, tri-vectors, and quad-vectors. These higher vector types correspond to different types of geometric objects that can naturally appear in a theory.

Either APS or STA (or even standard vector calculus) can be used to describe special/general relativity. However, APS is an artificial construction of spacetime from the algebra Cl_3, while STA is a very natural way of talking about spacetime using Cl_{3,1}.
STA puts space and time on equal footing, but APS makes space into vector components but time into a scalar. Because of this, the equations of STA are generally easier to interpret geometrically and work with algebraically than those of APS.

 

[arkajad]

Adding to the above: Lorentz transformations come from the http://en.wikipedia.org/wiki/Spin_group" in APS.

 

[arkajad]

This question does not appear to me to have anything to do with "Linear & Abstract Algebra". The terms "Physical Space" and "Spacetime" make me think it is about general relativity. Any objection to my moving it?

Well, Clifford algebras do not belong to linear algebra and not exactly to abstract algebra. Yet they belong to algebra and even, perhaps, to multilinear algebra. Moving it to general relativity, however, may be not a bad idea.

 

[closet mathemetician]

LukeD - Thank you. That is exactly the sort of answer I was looking for. I have no objections to moving the post to SR/GR.