Algebra of Physical Space vs. Spacetime Algebra

Click For Summary

Discussion Overview

The discussion centers around the differences between the Algebra of Physical Space (APS) and Spacetime Algebra (STA), exploring their definitions, applications, and the necessity of both in the context of special and general relativity.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants propose that APS is the algebra of 3 Euclidean vectors plus 1 scalar, while STA is the algebra of 4 Minkowski-type vectors, highlighting their structural differences.
  • One participant notes that APS may be seen as an artificial construction of spacetime, whereas STA is viewed as a more natural representation.
  • It is suggested that STA allows for easier geometric interpretation and algebraic manipulation compared to APS.
  • There is mention of Lorentz transformations being derived from the spin group in APS.
  • Some participants express uncertainty about the categorization of these algebras within the broader fields of linear and abstract algebra, suggesting a possible connection to multilinear algebra.

Areas of Agreement / Disagreement

Participants express differing views on the classification of APS and STA, with some advocating for their relevance to general relativity while others see them fitting within linear and abstract algebra. The discussion remains unresolved regarding the best categorization and the implications of each algebra.

Contextual Notes

There are limitations in the definitions and interpretations of APS and STA, particularly regarding the assumptions made about their structures and applications in relativity. The discussion reflects a variety of perspectives without reaching a consensus.

closet mathemetician
Messages
44
Reaction score
0
What is the difference between the Algebra of Physical Space (APS) and the Spacetime Algebra (STA), and why do we need them both?
 
Physics news on Phys.org
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?
 
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.
 
Last edited:
Adding to the above: Lorentz transformations come from the http://en.wikipedia.org/wiki/Spin_group" in APS.
 
Last edited by a moderator:
HallsofIvy said:
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.
 
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.
 

Similar threads

Graduate Spectrum of an algebra (Alain Connes paper)
  • · Replies 19 ·
Replies
19
Views
4K
High School What unique contributions to math did linear algebra make?
  • · Replies 19 ·
Replies
19
Views
4K
Insights Lie Algebras: A Walkthrough - The Basics
  • · Replies 8 ·
Replies
8
Views
3K
Graduate About semidirect product of Lie algebra
  • · Replies 19 ·
Replies
19
Views
3K
Insights Lie Algebras: A Walkthrough - The Structures
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Solving algebraic equations that cannot be solved in radicals
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad ##\mathbb{C}\oplus\mathbb{C}\cong\mathbb{C}\otimes\mathbb{C}##
  • · Replies 14 ·
Replies
14
Views
6K
Undergrad Meaning of "passage to the quotient"?
  • · Replies 3 ·
Replies
3
Views
1K
Graduate Infinite-Dimensional Lie Algebra
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Proving inequality about dimension of quotient vector spaces
  • · Replies 25 ·
Replies
25
Views
4K