Manifold / Atlas / Chart (Building Simple Example)

[ldechent]

I’m studying GR and am curious about manifold, atlas and charts. I have an idea for building a simple example, in one dimension, and wanted to ask if what I’m doing below is “legal”/correct. Imagine a space flight that can be divided into three segments:

• A-B: velocity starts at zero and it increases at a constant rate to the cruising velocity
• B-C: velocity is constant at the cruising velocity
• C-D: velocity decreases at a constant rate from the cruising velocity to zero

Can I say that the above three scenarios can correspond to three charts which we would use in an atlas for a manifold?

The metrics for the first and last chart vary with position to offset or counter (probably could find a better word) the change to velocity. This is done in a way that points on the travel line that are equally spaced chronologically will appear equally spaced. We might say that A-B and C-D are sort of “cousins” to semi-log graph paper. Comment and suggestions are appreciated.

 

[bcrowell]

Hi, Idechent,

Welcome to PF!

Manifolds don't relate to motion. In fact, manifolds are even more basic than measurement. A manifold is a purely topological object. For example, a coffee cup is the same manifold as a doughnut.

When we add a metric onto a manifold so that we can have a system of measurement, that's extra machinery, like the air conditioner in a car.

-Ben

 

[WannabeNewton]

A manifold is an entity that is equipped with a maximal atlas which is composed of smoothly sewn charts which are homeomorphisms from an open subset of the manifold to an open subset of R^n. A manifold doesn't have to be geometric per say; there are sets of rotations that qualify as manifolds. However, the scenario which you have described is not really a manifold.