# Special relativity and accelerated frames

pervect
Staff Emeritus
Thanks, I think I can follow that chapter, but it will take some work. The first chapter is a little over my head, the notation is unfamiliar, although I can guess what it means.

But before digesting foundations for GR, I'd like to get SR straight.
A good basic plan, I'd recommend understanding the situation in non-wiggly coordinates first, the way it is presented in SR. This implies, though, not too much focus on acceleration, because the natural coordinate system of an accelerated observer is "wiggly". These coordinates not really wiggly in the spatial dimensions, but rather "wiggle" the time dimension. To thouroghly justify that statement I'd have to talk about the metric, which is one of the things I think you're trying to avoid at this point, so I won't clarify the vague general statement and hope it enlightens more than it confuses.

What you can do with the SR approach easily is to describe the coordinates of an accelerating object in an inertial frame, and calculate some basic coordinate independent observations such as proper time and proper acceleration on the accelerated clock, maybe even calculate the proper time of emission and reception of a few radar signals. What you'll be lacking is the "viewpoint" of the accelerated observer, the most direct route to that viewpoint is to learn about metrics and some of the (non-tensor) basics of how they can be used to deal with "wiggly" coordinate systems, but that can wait until after you learn about the non-wiggly coordinate approach in SR.

Yes. I would prefer to master the flat cases before think about others. My basic questions about acceleration in SR have been answered. After acquiring a sound understander of SR. It would be interesting to attempt to analyze the case of acceleration and see why these metrics are needed.

Dale,

This is tangential but I'm curious. Manifolds can be thought of as surfaces in a higher dimensional space. Can all manifolds be represented in a higher dimensional flat space? I saw a comment in a wiki article that suggested that a Klein bottle (a 2D manifold) cannot be represented in a 3D space but a 4D space does the trick. I suppose this is because the surface intersection in a Klein bottle in 3D leads to an ambiguous tangent plane.

Dale
Mentor
Manifolds can be thought of as surfaces in a higher dimensional space. Can all manifolds be represented in a higher dimensional flat space?
Yes, there are several "embedding theorems" about how many dimensions are needed.