Recent content by PAllen

So, FW transport along any curve will preserve all of these properties. Parallel transport will only preserve them along a geodesic. For an arbitrary curve, parallel transport will preserve mutual orthogonality, but the timelike vector will no longer be tangent to the curve.

But that operation is typically called the exponential map (and is a completely rigorous construct): https://en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry) I wonder if this is what @cianfa72 is thinking of.

That is an exceedingly narrow statement of the equivalence principle. The most commonly accepted meaning of strong equivalence principle is given in (especially the end) of section 3.1.2 of the following: https://link.springer.com/article/10.12942/lrr-2014-4

That's not an update. It shares exactly one author, and the topic is not the same. Obviously, there is some relationship, but that's all. Here is how they refer to the paper I gave: "Regardless of what model cosmology is to be the standard in future, exploring more than one model is important...

I wondered if anyone had ever treated voids using numerical relativity rather than more indirect approximations. I came across the following, which does exactly this: https://academic.oup.com/mnras/article/536/3/2645/7923505 I note the conclusions are generally in agreement with other methods...

While the FLRW metric, when used for a realistic universe, is purely an emergent approximation at large scales, there is one aspect of if that applies locally. That is, if it suggests that there is a cosmological constant, this applies everywhere and has effects (too small to be observed)...

Yeah, but if you are talking about galaxies, you are outside the FLRW idealized model - that has only perfect fluid with different equations of state. So, to the same degree you ignore clumping to use an ideally homogenous model with cosmological constant, you ignore it when modeling a...

As far as I know, the common models of quintessence still use the FLRW equation metric, just with particular forms of ##a(t)##. In these, the cosmological principle is obeyed, and there is no spatial variation of the quintessence within a slice of constant cosmological time.

I don’t think the difficulty is insurmountable. I have a counterexample from my own experience. I wanted to learn calculus before the age it was available in my high school. So I went to a library, looked over different textbooks checking how readable I found them and whether it looked like...

I like it and will probably play with it some. I'll ask if I have any questions. I appreciate your effort. I disagree with the decision not to use https, but that horse is already dead. My browser offers no objection to displaying the site (Firefox on Windows with two security plugins).

No, because its total surface area is finite. It is actually just a small piece of the hyperbolic plane, it just 'looks' big in cartesian coordinates in ##E^3##.

I can’t seem to let this go, partly because some have asked for how to make the embedding result (part of Euclidean plane in 3-sphere) more comprehensible as well as using only intrinsic metric tools. This post will attempt to address both of these goals head on. I will note, in passing, that...

I don't know if I can track it down from a long ago posts of mine, but I discussed a paper by a mathematician specializing GR that emphasized @PeterDonis point that Birkhoff' theorem is a local theorem. Then it proceeded to demonstrate literally a dozen or more wild topologies that could...

Radius of curvature = 1

When discussing isometric embeddings, there is a little more to it. The metric of the submanifold is taken to be induced from the manifold metric. This means that, e.g. the distance along a curve on the 2-sphere embedded in E3 is, in fact required to be that same whether you use the manifold...