Recent content by Pencilvester

I did not know this! This is very helpful, thank you!

So, to make sure I'm understanding: any and all statements that can be made about the behavior of the invariant ##r := C / 2 \pi## across the manifold in Birkhoff's theorem are derived and not assumed (either explicitly or implicitly). I think my problem was that Schutz's section 10.1...

I personally don’t doubt this either, but has this been proven anywhere? If so, I’d be interested to see the way or ways this can be proven. 🙂

Right, so doesn’t making ##r## a coordinate that ought to cover the spacetime restrict the number of possible spherically symmetric spacetimes that are covered in Birkhoff’s theorem? Yes, that would be one potential example of a spherically symmetric spacetime that appears to me to be excluded...

In Birkhoff’s theorem, doesn’t assuming we can use r (defined as circumference divided by ## 2 \pi ## for any given sphere) as a coordinate across the spacetime implicitly assume that the spheres must always be getting bigger in some specific direction? Is there a version of the proof that...

A matrix (specifically a square matrix) is a decent way to express all of the components of a (1, 1) tensor. The problem is that the traditional convention for index placement in linear algebra has, as you mentioned, the first index representing row, and second index representing column...

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...