3D Geometric Algebra and Special Relativity

Without the belief that it is possible to grasp reality with our theoretical constructions, without the belief in the inner harmony of the world, there can be no science. This belief has and always will be the fundamental motive for all scientific creation.[1]

Project SummaryEdit

Currently this page is just notes that i'll clean up eventually.

Project GoalsEdit

This learning project aims to:

Assumed background knowledge/skillsEdit

For this learning project, if you are familiar with:

IntroductionEdit

Most applications of Geometric Algebra to Special Relativity (from Hestenes' work) start by defining the space as 4D Minkowskian space. Eg.   (re-phrase using more familiar x,y,z and w. But instead of starting with this assumption, start by investigating some of the properties of rotations in normal 3D geometric algebra.

Example RotationsEdit

Setting up a rotation by  

 

 

Therefore, rotor   (need diagrams to visualise... and lots of them!)

Imagine a unit cube (again, need to draw). If we apply normal 3D rotation as follows, for each point   of the cube:

  (What's latex for tilda?? can't display the inverse of r)

and the cube is rotated by  . Note that only the   and   components are affected by rotation.   component of all vectors remains unchanged.

Now taking the same unit cube, apply the following rotation for each point:

  (without the inverse of r this time)

we find that for all points of the cube the   and   components remain unchanged, but the   and, a new component of our multivector,   are modified. It's still a rotation, as  , but not one that we're used to.

Seems as those cube is compressed along z-axis (only 0.5 in length along z-axis), not dissimilar to Lorentz contraction, but the   component of the points p4 to p7 has been changed from zero to  . If this component is a time-like component, could provide interesting interpretation of such rotations.

For eg, rotating one unit of   results in:

 

That is, for each unit of time in the unrotated frame, only half a unit passes in the rotated frame. And, furthermore, for each unit of time in the unrotated frame, the rotated frame shifts a distance along the z-axis. Again, not dissimilar to time-dilation and velocity.

Finally, to calculate the shape of the actual cube in the rotated frame, want to get a snapshot for all the points when they have the same temporal value of  . If we translate the original p4-p7 back in time by   (again, diagram very necessary here) then it turns out that the cube is elongated along the z-axis (in fact it's doubled in length) as it moves along.

Next, try rotation of   as this corresponds to point of singularity in SR (c=1), but behaves nicely here...

ReferencesEdit

  1. Albert Einstein, The Evolution of Physics, (New York: Simon & Shuster, 1938), p. 313.