[The Schwartzchild metric] reveals another peculiarity in the interior of the horizon: the function [1/(1-2MG/r)] changes sign, implying that behind the horizon the coordinate t becomes space-like, wheras the radial coordinate r becomes time-like. This interchange of space-like and time-like means that the singularity is not a place–it is a time. One approaches the singularity, not as one would approach a location in space, but as one would approach the end of the world. (Susskind, Nature Physics 2 p. 665

This is a very nice way to explain the structure of conformal diagrams containing a singularity; kind of obvious, once you think about it, but I’ve never seen it expressed quite that way before.

But in the Riemannian geometry … there is contained a residual element of rigid geometry–with no good reason, as far as I can see; it is due only to the accidental development of Riemannian geometry from Euclidean geometry. The metric allows the magnitudes of two vectors to be compared, not only at the same point, but at any two arbitrarily seperated points. A true infinitesimal geometry should, however, recognize only a principle of transferring the magnitude of a vector to an infinitesimally close point and then … the integrability of the magnitude of a vector is no more to be expected than the integrability of its direction … only the ratios of the components g(i,k) are detemined by the metric at [a point] p. Physically also, only the ratios of the g(i,k) have a direct physial meaning. -H. Weyl, 1918

I have failed miserably to keep up my goal of reading a RMP article each week. I got stuck on a particular article about quantum mechanics, special relativity, and information theory- I wanted to keep at it rather than move onto something else. Then there was all the studying of differential geometry over the summer. I have let my reading of Gravitation slide the last several weeks. Poor sleep makes it difficult to concentrate on such things. Then there’s NHL center ice…

I’m considering a subscription to the band new Nature Physics. I’m not terribly impressed with the feature articles of issue 1 (available online).

A new round of job applications are around the corner. This is my third year of a three-year contract. My conversion to tenure-track has been approved by the Provost. But, from the start, I considered the commitment to go both ways. If they want to keep me, they’ll have to pay me more. While I’m not in the mood to try to sell a house, I could be lured away.

Finished Penrose. Mowed the lawn. Dug a ditch. Played 9 holes (7 over). Made roast lemon rosemary chicken with potatoes, carrots, parsnips, and a turnip. Worked through exercises from two sections of Gravitation.

Now that the Penrose book is finally, finally over, I can make more headway in Gravitation, and Zee’s QFT book is on the way.

Those last sections on loop variables and twister theory were just plain rediculous. Nice Epilogue, though.

Been reading: Penrose and Capote (In Cold Blood). Penrose has got me thinking in good directions. In a few weeks we’re having a visit by Anthony Leggett, recent Nobel Laureate in Physics, who works in the foundations of quantum mechanics. Some of the thoughts I’ve had coming from Penrose’s ideas will make good conversation fodder.

Capote is a master. The greatest feat of any writer is to be invisible. As we say in Guitar Craft, any idiot can play something complicated. Capote’s prose is not simple; but he makes it seem simple, natural, and inevitable. Brilliant.

In the mean time I’ve got to get PR for Leggett’s visit in order, keep two student research projects progressing, and re-format a paper to antiquated standards for re-submission.

It is my last few days of relative freedom before the sure-to-be hellish double-duty of the first summer term. I have been buried in the Hubbards’ Vector Calculus, Linear Algebra, and Differential Forms. This is a brilliant book. Every aspiring physicist should take a full year course from this tome their sophomore year. It’s sad I could get a PhD without having seen this contemporary approach to calculus. Penrose has shown me the light, and my Lipschitz ratio is tingling.. I am re-acquiring my penchant for formal mathematical physics. It’s sad I probably won’t get to teach much of this in Mathematical Physics next fall. Linear algebra will be high on my list of topics, but I will probably have to take a more traditional approach.

Still plowing through Penrose’s The Road to Reality. I find it amusing Penrose thinks of this as a popular book. I am barely, barely hanging on. I would be quite shocked if anyone without a PhD in physics or mathematics can make it though understanding 1/10th of the material.

Penrose:

These things, however, depend upon certain basic notions of the calculus, so, in order to convey something of this magic to the reader, it will be necessary first to say something about these basic notions. There is, of course, an additional reason for doing this. Calculus is absolutely essential for a proper understanding of physics!

Amen! I taught calc-based physics to high school students in a New Orleans suburb as Jesus glared at me from several well-selected vantage points around the room. At that level calculus simplifies, immenseley.

All you need is derivatives and integrals of polynomials and basic triginometric functions, and suddenly all these disarpate results make sense. The two courses have the same essential learning goals, why strip away their most important tool for extracting meaning.

Penrose, 3.3:

Mathematical ideas develop, and various types of problem seem to arise naturally. Some of these … can lead to an essential extension of the original mathematical concepts in terms of which the original problem had been formulated… accordingly, the development of mathematics may seem to diverge from what it had been set up to achieve, namely simply to reflect physical behavior. Yet, in many instances, this drive for mathematical consistency and elegance takes us to mathematical structures and concepts which turn out to mirror the physical world in a much deeper and more broad-ranging way than those that we started with.

Penrose certainly doesn’t have the deft, cutting prose of Dawkins, and he’s prone to loose use of physics-speak, but his observations are salient. This is a question I have spent a good deal of time pondering: just what parts of modern mathematics are inevitable extensions of arithmetic? Supposing we could run the history of mathematics over again, or that we’re able to see a parallel developed by intelligent life elsewhere. Surely, notations would be different, and in many cases, this would be a good thing. But what of the structure, the foundational concepts, of calculus, functional analysis, abstract algebras?

The obvious natural extension of mathematics, to me, ends at complex analysis. It brings closure to so many basic issues, and an elegance and beauty I am still in awe of. Beyond that? There is a theorem for universal computation… that any system capable of (some minimal set of capabilities) is computationally equivalent. Is there an equivalent for formal systems? Is it the same theorem? It seems like the minimal requirements in this case would be the same as that for Goedel’s theorem…

Penrose, 1.3:

What I mean by [Platonic] ‘existence’ is really just the objectivity of mathematical truth. Platonic existence, as I see it, refers to the existence of an objective external standard that is not dependent upon our individual opinions nor upon our particular culture.

He follows in an example of Fermat’s last theorem: was it not “true” before it was proved? I like Penrose’s approach here: it is a soft Platonism, accepting objectivity, which is pretty necessary to assign meaning to scientific explanations, but placing it in a different category of existence form physics things. Has he says:

[Mathematical forms] do not have spatial locations; nor do they exist in time. Objective mathematical notions must be thought of as timeless entities and are not to be regarded as being conjured into existence at the moment thet they were first humanely perceived.

This is a nice, functional approach to the nature of mathematical reality. It leaves room for the mystery without endorsing mysticism. As I discussed on WKU, this problem has really been the center of philosophy from the beginning: seeking truths as compelling as mathematical theorems in all inquiries.

The short answer: we are not endowed with a category for the ontological status of mathematical forms, so we apply the concept of existence as evolved. The long answer will require a book.

I must say I’m delighted to find such sitmulating thought in the first 20 pages, of more than 1000.