Recent content by Orodruin

… shortly put: if it is infinite today, it was infinite then. But with a smaller scale factor.

Not only is a Master’s degree helpful, it is often a prerequisite to be eligible for a European PhD program. (Local variations apply depending on country)

Planes, I agree, but not with a single ball, which was my understanding from the discussion.

I believe that is intrinsically applying the Levi-Civita connection as the geodesics of the Levi-Civita connection are the paths that extremise path lengths. More generally, for an arbitrary connection you could define the surface of the ball as the set of points reached from the center after...

Not only why, but ”you cannot”. It is impossible to do it without parallel transport - or at least without parallel transport being defined - as curvature is a property of the affine connection that defines what parallel means. OP’s misunderstanding that parallel transport requires embedding is...

Free-streaming means that the (average) density of matter is so small that the mean-free path exceeds the size of the Universe. The Universe is pretty low density after all.

I mean, it is a two-dimensional gyroscope, i.e., one where the gyroscope is only free to rotate around one axis ... :wink: I'd like to add some additional structure to this: In a Euclidean space - such as the flat plane - it is clear what "the same direction" means between points (precisely...

The FW transport in GR is the correct thing to use for gyroscopes precisely because we are talking about its frame describing spacelike directions orthogonal to the timelike tangent of the world line. In the Riemannian setting, consider for example a gyroscope in regular Euclidean space, which...

Compare to the density of the Sun's photosphere. The number density of hydrogen there is about ##10^{24}## per cubic meter. But of course the Sun is rather small in comparison to the Universe and we must remember what we mean by the light in the early Universe not being free-streaming - it means...

The entire point of FW transport is to keep a spacelike vector, such as a polarization vector, orthogonal to the timelike tangent of the world line (and the timelike tangent is also FW transported). The corresponding definition in a Riemannian manifold would do the same, it would keep the...

The Levi-Civita connection is the unique metric compatible and torsion free connection on a Riemannian manifold (or pseudo-Riemannian). In general a connection has nothing to do with any basis, although its connection coefficients can be expressed in it. The Levi-Civita connection is...

Parallel transport is perfectly well defined along non-geodesic curves. The manifold described in the video is a Riemannian manifold so there is no need to define (or use for) a Fermi-Walker transport.

No, definitely not. There should be no outside reference. If he walks straight forward, he keeps the arrow in the same direction relative to himself. If he walks in a curved path, he needs to adjust the arrow accordingly when he turns. No, see above. No.

This is only correct if you view ##\Delta L## and ##\Delta B## as fixed numbers. In a setting where you have estimated errors, it is more appropriate to view them as statistical distributions, normally Gaussian distributions. If these are uncorrelated then you would obtain the result with the...

They are the same to the ordered considered in the ##\simeq## relation.