Proper time, velocity, momentum, and invariants in STA (Geometric algebra's Space Time Algebra)

In this video I introduce geometric algebra using the Dirac basis, referred to as the Space Time Algebra, or STA.

• We show how to represent a vector length in coordinates,
• How to transition between upper and lower indexes,
• Introduce the reciprocal frame for the standard basis,
• Introduce the idea of proper time, and how it relates to invariance,
• Justify calling the dot product an invariant, by considering spatial rotations and boosts, and their effect on the dot product
• calculate proper velocity, and see how the "gamma" factor falls out naturally, just by virtue of the dot products of the standard basis elements
• calculate the time-like and space-like components of proper momentum,
• and establish the correspondence between the unit spacetime bivectors and our Euclidean spatial basis.

We will use this spacetime bivector representation of the spatial basis in a later video, where we show how the STA form of Maxwell's equation can be derived from the Euclidean geometric algebra representation of the same.