Recent content by Pencilvester

Ooooh, okay, gotcha. I understood one of the OP’s questions to essentially be “Will two different inertial frames ever disagree on whether an object’s velocity is changing or not?” and I had assumed that’s the question you were answering. I did not read carefully enough!

Could you enlighten me on this? I was under the impression that with the unique class of cartesian coordinate systems associated with each inertial frame, whether or not an object’s velocity is changing is invariant (among those coordinate systems). But are you simply saying that non-inertial...

There will always be agreement between inertial frames on which objects are accelerating and which are not.

If you have two objects at rest relative to each other* in flat spacetime, both equipped with accelerometers, and both accelerometers always read zero, then, as measured in an inertial frame (and each object’s reference frame is inertial in this scenario), neither object’s velocity will change...

I’m always surprised at the time difference, (and I know this isn’t exactly the same scenario that OP described), but if the spaceship is providing just 1g of acceleration in the direction of travel for the first half of the journey and in the opposite direction for the second half (starting and...

I think you’re confusing yourself by unnecessarily introducing a “target”. Try reformulating your scenario simply with a laser that leaves a scorch mark on a blank piece of paper that is at rest relative to the laser (and presumably the ground). You are thinking that if you fire the laser...

Fantastic, thank you!

Thanks! I will definitely read through this, but did you mean to link to something else for the second one? The URLs appear to be identical.

I have been thinking about rotation number of regular smooth curves in different surfaces. Here is how I’ve been defining these things: a regular smooth curve is a map from ##S^1 \rightarrow \mathbb{R}^2## whose derivative is non-vanishing. If we have a regular smooth curve ##\gamma## as well...

I suppose this is a semantics issue that depends on your definition of “tensorial”— if you want this property to be a part of your definition, which, given tensors’ linearity, doesn’t seem unreasonable, then I see your point. But if by “tensorial” you simply mean that it transforms like a...

I believe this directly answers my question, so thank you! Yes, though I did not express this fact in my original post, you correctly inferred my thinking, so thanks again!

I'm building a mental framework for the Levi-Civita connection that is intuitive to me. I start by imagining an arbitrary manifold with arbitrary coordinates embedded in a higher dimensional Euclidean space, then if I take the derivative of an arbitrary coordinate basis vector with respect to...

Oooookay, ##\theta## and ##\phi## are standard angular coordinates on a 2-sphere; they are well defined. We explicitly defined ##r## to be the circumference of any given 2-sphere divided by ##2\pi##; it is also well defined. We made a few demands of orthogonality on the coordinates, but...

Ok, ok, I think I get what you’re saying, so let me attempt to paraphrase again, and you can tell me if I’m getting there. The line element that includes ##b## as an unspecified function is expressed the way it is because you’re using a specific coordinate system, namely a specific coordinate...

I think I see what you're saying. I was originally thinking that by choosing, for example, ##b(t) = t^2 + 1##, then performing the coordinate transformation where ##dt' = \sqrt{b} ~ dt##, this would mean ##b(t) \neq 1## but we're still able to make it go away with the change of coordinates...