Recent content by PAllen

One can discuss the topology of limiting surfaces approaching a singularity. In that sense, the topology of the initial big bang singularity for an open universe (either flat or hyperbolic) is R3 meaning ordinary 3-space. This is simply because every isotropic surface that exists in the manifold...

Also, it should be noted that spherically symmetric nonrotating collapse to a BH per general relativity does predict unbounded density for the original matter falling in. This is because, while the singularity can be thought of being the limit of a collapsing hypercylinder, all the original...

But that is part of the measuring device. In the continuum approximation, you can take the limit as charge approaches zero, voltage remaining constant.

One other example is simply that if you charge metal ball or plate with static electricity, then you measure a potential difference (equivalent to voltage) between different points of the space near it, even though there are no electrons existing or flowing at or between those points. This gets...

Yes, I agree with you.

I don’t think that works. I tried that. Remember f is a function (hidden within r) of all 4 coordinates. That means zeta must be as well. Then its total derivative could not look like that. One way to check a method is to try it on the Kerr metric. If it seems to be able to work on the Kerr...

My current view is back to the idea that you can have synchronous, but not diagonal. However, the trick used by Hiscock for diagonalization does not work in the full 3+1 context, so diagonalization really is not possible, even in a finite patch (in the wall, that is). But Gaussian normal form...

True, but if you switch to coordinates that manifest the orthogonality, then the Eulerian world lines do have constant spatial coordinates. Similar to going from Gullestrand-Panlieve to Lemaitre. So now I think there is something I want to capture here, but need better wording that what I said...

good catch, I'll fix.

I would like to summarize conclusions scattered across this thread. The following are key references: The main document that generalized Alcubierres original to a bubble that starts and stops, and derives many featrues: https://scipost.org/SciPostPhysLectNotes.10/pdf A paper @PeterDonis has...

A curve orthogonal to all Eulerian world lines it touches must have zero t component. But one that is orthogonal to just one need not; its tangent at that point has no t component. Note, besides my own work, which may have errors. I have seen no claim in any of the papers that the geodesics of...

This I don’t believe. One thing that is confusing about the notation in some of these papers is that f is a function of all 4 spacetime coordinates. That leads to different Euler Lagrange equations than if you don’t notice this.

There is always a geodesic orthogonal to any world line at any point. What happens is the spacetime geodesic and the slice geodesic are tangent at that point, but diverge elsewhere. This is easily possible. Consider a diameter line in a ball. At the 2 sphere surface, a great circle and a tangent...

Just to take up a detail discussed earlier: whether There is a difference between computing distance using a geodesic of a Euclidean slice, vs a spacetime geodesic orthogonal to a bubble rider. The claim was made that these are the same. I do not believe this is true. In particular, I think the...

Before the full solution, I can put an upper bound on the odometer reading. It is easy to compute, using the full metric, rigorously, that for my definition of T, at all time, ##d\tau/dt=\sqrt{.99}##. Thus, even if the traveler is assumed trapped at the beginning of the wall until the bubble...