Algebra of Physical Space vs. Spacetime Algebra

In summary, the conversation discusses the difference between the Algebra of Physical Space (APS) and the Spacetime Algebra (STA) and why they are both needed. APS is an artificial construction of spacetime using Cl_3, while STA is a natural way of describing spacetime using Cl_{3,1}. STA puts space and time on equal footing and is generally easier to interpret and work with than APS. Both APS and STA can be used to describe special/general relativity, but STA is more commonly used due to its ease of use.
  • #1
closet mathemetician
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What is the difference between the Algebra of Physical Space (APS) and the Spacetime Algebra (STA), and why do we need them both?
 
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  • #2
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?
 
  • #3
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 [tex]Cl_3[/tex]). 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 [tex]Cl_{3,1}[/tex]. 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 [tex]Cl_3[/tex], while STA is a very natural way of talking about spacetime using [tex]Cl_{3,1}[/tex].
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.
 
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  • #4
Adding to the above: Lorentz transformations come from the http://en.wikipedia.org/wiki/Spin_group" [Broken] in APS.
 
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  • #5
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.
 
  • #6
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.
 

What is the difference between Algebra of Physical Space and Spacetime Algebra?

The Algebra of Physical Space deals with the mathematical representation of physical objects and their movements in three-dimensional space. It uses vectors and matrices to describe the position, velocity, and acceleration of objects. On the other hand, Spacetime Algebra is a mathematical framework that combines space and time into a single entity, known as spacetime. It is used in the field of relativity to describe the relationships between space and time.

How is Algebra of Physical Space used in physics?

Algebra of Physical Space is used to describe the motion of physical objects in three-dimensional space. It is an essential tool in classical mechanics, which deals with the motion of macroscopic objects. It is also used in other branches of physics, such as electromagnetism, thermodynamics, and fluid mechanics.

What are the advantages of using Spacetime Algebra?

Spacetime Algebra allows for a more elegant and efficient way of describing the relationships between space and time. It also simplifies the equations used in relativity, making it easier to understand and calculate. Additionally, it allows for the incorporation of spinors, which are essential in describing the behavior of particles with spin.

Why is Spacetime Algebra important in the theory of relativity?

In the theory of relativity, space and time are not separate entities, but rather intertwined in a four-dimensional spacetime. Spacetime Algebra provides a mathematical framework that allows for the description of this spacetime and its properties, such as curvature and motion. It is crucial in understanding the principles of relativity and making accurate predictions in this field.

Can Spacetime Algebra be applied to other branches of physics?

Yes, Spacetime Algebra can be applied to other branches of physics, such as quantum mechanics and particle physics. In these fields, it is used to describe the behavior of particles and their interactions in spacetime. It is also used in cosmology to study the evolution of the universe and its large-scale structure.

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