Talk:Aichelburg-Sexl ultraboost

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[edit] Executive summary

A summary for a layman of exactly what forces the passing observer experiences, and what's special about them in this class of scenario, would be handy. --Christopher Thomas 06:10, 19 February 2006 (UTC)

Sorry, Christopher, I can't tell from this exactly what puzzled you. Did you understand that the A-S is a gravitational wave which has the nature of a sudden impulse? That it is a limiting case of gravitational waves (exact vacuum solutions) which approximately describe the physical experience of an observer who whizzes by a nonrotating massive object (spherically symmetric in its rest frame) at some ultrarelativistic speed?

Or are you asking about my comment (someplace, here or there) that near time zero (the moment of closest approach by the hyper-relativistic observer to the massive object), the tidal forces fall off more slowly than one might expect from the m/r^3 falloff in the Schwarzschild vacuum? Or that there are no "force beams" anywhere in sight?

Not sure I understand what you mean by what's special about them in this class of scenario. Do you mean: what's special about certain observers? (Which ones?) Or what's special about certain spacetimes? (A-S ultraboost of the Schwarzschild vacuum, versus other ultraboosts, e.g. competing ultraboosts of the Kerr vacuum?)

I never did regard this article as complete, and when I have time intend to add more about the Bel decomposition. At present it is probably more important to write about background such as, well, the Bel decomposition! ---CH 06:21, 19 February 2006 (UTC)

Pretty much all of the above clarifications helped, and would be useful to fold in the article. I'm in engineering, so my math isn't as good as a Real Physicist's, which means I'll have a harder time seeing implicitly what's going on from the equations. The question of "what's special?" is intended to be along the lines of, "what is it that makes this class of solution to this particular problem noteworthy?". From what I can gather, it's that it's a limiting case (v -> c), and that the solution _is_ exact, but it's quite possible there's more I'm missing. Stating the answer to this question outright would be handy too. The idea is to have the article make sense if read by a first-year student (I understand there are formal guidelines about this somewhere, but that it's still very much a grey area). --Christopher Thomas 03:54, 20 February 2006 (UTC)

It's exact if you allow distribution valued curvature tensors. That is very convenient for making simple models but also leads to undesirable technical problems. OTH, the ultraboost solutions are obtained via certain limits of unobjectionable exact solutions (in the article I exhibited some gravitational plane waves with Gaussian pulse profiles which converge to the A-S ultraboost. But taking limits like this is tricky. This is a point which I need to make in a separate article when time permits. First, all Lorentzian manifolds admit Penrose limits, which essentially examine the geometry very near an arbitrary null geodesic. Interestingly enough, even in simple exact solutions like Schwarzschild, some null geodesics are distinguished by having a different Penrose limit from the generic null geodesic. Second, there are usually a variety of other spacetimes which can be obtained as limiting cases by changing the coordinates and then letting the parameters of the solution approach zero or infinity (typically). In the case of the Schwarzschild vacuum, for example, MacCallum et al. have shown that there are precisely five possibilities:

the Minkowski vacuum (flat spacetime, Petrov type N) is obviously the $m \rightarrow \infty$ limit, and less obviously, can also be obtained by changing to new coordinates and letting $m \rightarrow \infty$ ,
the unique Petrov type D Kasner vacuum, aka the only nontrivially axisymmetric case of the Kasner vacuum (a two parameter anisotropic exact vacuum solution which is itself the mass -> 0 limiting case of the Kasner dust), arises as a mass -> infinity limit,
a certain family of EK9 gravitational plane waves, i.e. gravitational waves without any extra symmetries (inhomengenous Petrov type N spacetimes with a five dimensional isometry group),
linearly polarized EK10 gravitational plane waves (homogeneous Petrov type N spacetime with six dimensional isometry group),
circularly polarized EK11 gravitational plane waves (homogeneous Petrov type N spacetime with six dimensional isometry group).

I know this Petrov stuff must be over your head (this reply is partly a note for future work), but see Petrov classification and gravitational plane wave to get (I hope) some idea of what the technical buzzwords mean.

Anyway, when I get a chance I certainly intend to improve this article, but I think I need more background articles first.---CH 23:54, 20 February 2006 (UTC)