Recent content by PAllen

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...

So, the easy part is as follows: my traveler has reached the left edge of the right bubble wall at t (coord time) of 4.95, z of 16.295, ##\tau=4.95\sqrt{.99}##, odometer reading of ##.495\sqrt{.99}##. The next part involves a slightly messy differential equation to track traveler through wall...

So, my choice for traveler world line is very convenient. It turns out it produces a constant integrand for the odometer integral of .1 (everywhere!). So “all” that remains is determining the proper time bounds along the traveler worldline. That might take some time (proper time since I am my...

So it is quite easy to analyze that light gets trapped in the wall in the original coordinates. Using my proposed diff.eq. for traveler T, I can similarly show that it gets trapped in the wall until v falls below .1. Still, at all times it is progressing against Eulerian observers, that also get...

Note that this section is using v constant and superluminal, which is not what we want. This basically says that if the bubble never slows down, you can never escape it.