[SOLVED] Re: This Week's Finds in Mathematical Physics (Week 232) In article <4486EF88.firstname.lastname@example.org>, Ralph Hartley <email@example.com> wrote: > Greg Egan wrote: > > However, if M exceeds the maximum particle mass, then there is cyclic > > behaviour in v, with the total holonomy being a rotation by an angle > > that cycles back and forth between 0 and 2pi ... and then only > > becomes a boost after going through several cycles. > > > > I've put some plots on this page: > > > > http://gregegan.customer.netspace.net.au/GR2plus1/GR2plus1.htm > >Two comments, minor and major: > >Minor: >Your graphs look suspiciously like they go through *infinitely* many >cycles, like sin(1/x) near x=0. Do you trust your program to be >numerically stable? With lower values of M it's clear that there is a finite number of cycles. Basically as you increase M the plots, which start out monotonic decreasing, smoothly concertina to develop cyclic sections with an increasing number of cycles before the final plunge. I've lowered the values of M plotted on the web page to make this clearer. >Major: >You seem to assume a simple topology. For small M that's fine, for each >particle you cut out a wedge and glue the sides together. > >Looking just at a spacelike slice, each particle is the tip of a cone. > >But there is a limit to how many wedges you can cut out of a plane, and >still have the topology of a plane. If the deficit angle is 2Pi the >plane closes up into a sphere. > >If the deficit angle is more than 2Pi then it will become disconnected. > >So at *least* we can say that if the total Mass is greater than 2Pi the >space cannot have any connected spacelike slices. > >Does the whole space become disconnected, or does it just not have >connected spatial slices? Good point. I did wonder about this, but I clearly haven't given it enough thought. I'll have to check the literature more carefully; I expect someone has analysed this issue. In fact, when there's relative motion of the particles the problem seems to become more acute, because for a spacelike slice in the centre-of-mass frame the angular deficits that need to be accommodated are all larger. In the limit of N->infinity, we need to have v < sqrt(1-(M/Pi)^2) to avoid this problem. However, that restriction doesn't seem to rule out a tachyonic total momentum. It's clear that for N=2 there is no limit on v, because the worst the relativistic effects can do is make each angular deficit in the c.o.m. frame tend towards pi. For large N, my numeric results show a tachyonic transition well before the total angular deficit in the c.o.m. frame reaches 2pi.