There *is* no royal road to geometry

Archimerged has been studying. In particular, O’Neill’s 1995 book on The geometry of K___ B____ H___s, which title Ark is leaving out so that googling for that misleading keyword will not lead here. (This doesn’t mean he doesn’t believe in b____ h___s. He just feels that the metric has other applications). He bought it around 1996 and has read parts in great detail but skipped some things which he is now working through. He wants a very firm grasp of the Petrov classification (he skipped the beginning of chapter 4) and the precise meaning of principal null directions. Misled by the title into thinking that it dealt only with things he wasn’t interested in, he also skipped section 2.4, and so missed the observation that ∂t is spacelike in a small region just outside of the trapped surface at r = m. So now he knows that the metric is spinning even though outside of r = 2m, it is stationary, and the pretty picture in the previous post is misleading because the vertical dimension is not always timelike. He still hopes to find that there is a way to animate this metric so it looks like the word picture in his earlier musings.

He knows a lot of Wald’s General Relativity, but never became fluent in the language of wedge products xy (also known as bivectors), Λ2, Hodge star, etc. He did read Wald’s chapter on Spinor methods once, and realizes there is some connection between the multiplicity of roots of a certain fourth order polynomial and the principal null directions and Petrov classification, but it remains hazy. He used Mathematica on a 1-gig system to fully expand the Riemann and Weyl tensors for an arbitrary metric and figure some things out that way, never realizing that O’Neill (and others) have methods of finding the useful results with pencil and paper. He has lots of photocopies of papers, like Newman-Penrose, Goldberg-Sachs, but never became fluent in them either. When he tries to start at the beginning and work through, he has always gotten sidetracked on a tangent, for instance, reading topology books he got interested in whether you can or cannot exhibit a well-ordering of the reals and worked on that for a year. But right now, Ark needs to know exactly what the principal null directions are. (So why is he writing about section 1.4 of O’Neill? Well, probably it really is relevant…)

Ark is planning to make revisions to the following, which is partly wrong.

Some comments on the book. It has has lots of typos; perhaps Ark will publish a list (after he finishes that animation).

In Section 1.4, “Extending Manifolds,” subsection “Gluing topological spaces,” O’Neill considers two topological spaces M and N with subsets Μ ⊂ M and Ν ⊂ N that are to overlap in the final space Q. A matching map μ: Μ → Ν specifies how the subsets are to overlap and especially how the open sets of Q are to be determined. The topology of Q needs to be Hausdorff: for every pair of points p and qQ, there must be disjoint open sets Op and Oq with pOp, qOq, and OpOq = ∅.

O’Neill’s method of establishing a Hausdorff result involves a constraint on the matching map μ: Μ → Ν. Consider any sequence of points {pn} in Μ, together with {μ(pn)}, the image in Ν of the first under the matching map μ. Then space Q will be Hausdorff provided that there is no sequence which satisfies both

{pn} → pM \ Μ and {μ(pn)} → qN \ Ν.

In example 1.4.6, two planes M and N (each a copy of R2) are to be glued together along the open half-planes Μ ⊂ M and Ν ⊂ N (each {(r, t): r r = 0) axes (each infinitely long) and right half-planes. O’Neill specifies matching map μ(r,t) = (r, t/r), neglecting to note that when t is held constant at zero, t/r fails to diverge as r approaches zero, so the constraint on μ is not met. (Ark was proud of discovering this). When this approach is actually used in Chapter 4, O’Neill does not make this mistake, using something like (r, t + 1/r), which does diverge for all t as r → 0.

Ark proposes an equivalent method where the topology of the result and its Hausdorff nature is much easier to see. In the example, before even thinking of gluing the spaces together, he maps each plane onto an open subset of another plane, such that the parts to be glued overlap, while the parts which are to be distinct are mapped to disjoint regions. After gluing the two new planes together using the identity map, the resulting space Q consists of a plane less the positive r axis, Q = R2 \ {(r, t): r ≥ 0 & t = 0}. As a topological space, Q is the desired result, and as a manifold, it already contains the necessary coordinate systems. In particular, we have a set of (r, t) coordinates which cover the left half-plane and the top right quadrant, and another set which cover the left half-plane and the lower right quadrant, although they do not assign the same coordinates to a given point in the left half-plane.

map the planes onto the open sets

{(r, t): r r, t): t > 0} and {(r, t): r r, t): t 2 \ {(r, t): r ≥ 0 & t ≤ 0} and R2 \ {(r, t): r ≥ 0 & t ≥ 0},

via

ir(t) = (rt − et) / (r − 1) for r ir(t) = (0t − et) / (0 − 1) = et for r = 0

ir(t) = et for r > 0

and

jr(t) = (rt + et) / (r − 1).

Obviously the above is incomplete, and it turns out, partly wrong. But Figure 1.3 in O’Neill is very misleading, and Ark will put his version here. Meanwhile he wants to get this post out the door.

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