I wrote: >In article <4486EF88.firstname.lastname@example.org>, Ralph Hartley ><email@example.com> wrote: [snip] >>[T]here 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. >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. One tricky way around this would be to allow some particles of "negative mass", i.e. with negative deficit angles. That way the total deficit angle in a spacelike slice could be limited to 2 pi, but you could still analyse a group of particles whose collective deficit angle would exceed 2 pi. For example: to infinity ^ ^ A | | B | | | | | | . 3 4 . / \ / \ / \ 1 . . 2 \ / \ / \ / . 3 4 . | | | | | | A | | B v v to infinity Take the interior of this diagram as flat space, and identify the pairs of lines that run between 1-3, 3-A, 2-4, and 4-B. The points marked 1,2,3 and 4 are singularities, while A and B are just marked to clarify the identification of the edges. The angular deficit around the points 1 and 2 individually both exceed pi, and as a group their total deficit exceeds 2pi. The negative deficits around 3 and 4 mean that the total angular deficit of this connected spacelike slice does not exceed 2pi. Of course, there might be good reasons to rule out these negative mass particles as unphysical.