Consider three paths that begin at and end at enclosing the spacetime region depicted below
Staring at the above picture, we want to understand the analogy:
For those who better understand the mathematics of the first line, this analogy is likely to help motivate the physics as we go down the ladder. For those who better understand the physics of the bottom line, it is hoped that this analogy will help to motivate the mathematics as we move up the ladder.
Formulation
First, due to the symmetry of the diagram above, we can consider the simplified region
This diagram represents a very simple strict 2-category we will denote by with
Objects
Morphisms
2-Morphism
We will explicitly construct a 2-transport functor
where
Since both paths end up at the same point, the velocities must satisfy the relation
which implies that we can write
and
for some constant .
Let denote the 4-velocity of a charged particle and denote the corresponding dual 1-form, the acceleration 1-form may be written as
where is the curvature 2-form and the “acceleration 2-form” (corresponding to the acceleration bivector) is
I guess this is equal to . To see this, note that
We have
and
so that
I believe this is the clue that Urs mentioned, but made a bit more explicit. On the left is curvature and the right is the acceleration 2-form.
In my Galilean spacetime example, I think it is the “acceleration 2-form” that specifies the 2-transport even in the absence of an electromagentic field.
Consider two paths on a directed graph that both begin at the same point and end at the point as depicted in the figure below
The edges represent velocities and the area enclosed represents the acceleration.
If the acceleration is zero, then the area collapses and any two paths that connect the same two points must correspond to the same straight line, i.e. velocity does not change.
Given an interval and a smooth manifold , consider a smooth curve