Eric Forgy Position, Velocity, and Acceleration

See the corresponding nLab page

Warning: Under Construction

Idea

Consider three paths that begin at $x_0$ and end at $x_{2\tau}$ 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 $P^2(\mathbb{D}^2)$ with

• Objects $\{x_0,x_\tau,x_{2\tau}\}$
• Morphisms $\{x_0\stackrel{v_0}{\to}x_{\tau},x_\tau\stackrel{v_\tau}{\to}x_{2\tau},x_0\stackrel{v}{\to}x_{2\tau}\}$
• 2-Morphism $\{v_0\stackrel{F}{\Rightarrow}v_\tau\}$

We will explicitly construct a 2-transport functor

where

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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 $a$.

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Let $V$ denote the 4-velocity of a charged particle and $v$ denote the corresponding dual 1-form, the acceleration 1-form may be written as

where $F$ is the curvature 2-form and the “acceleration 2-form” (corresponding to the acceleration bivector) is

I guess this is equal to $F$. 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” $v\wedge a$ that specifies the 2-transport even in the absence of an electromagentic field.

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Relation to Discrete Calculus

See page 68.

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Consider two paths $\gamma^+,\gamma^-$ on a directed graph $G$ 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.

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Given an interval $I$ and a smooth manifold $M$, consider a smooth curve

to be a section of the trivial bundle

treating $I$ as the base space and denote $\gamma_t = \gamma(t)$ and $M_t = \pi^{-1}(t)$.

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