Tag Archives: John Wheeler

Relativity becomes less comprehensible

“To get Hawking radiation we have to give up on the idea that spacetime always had 3 space dimensions and one time dimension to get a quantum theory of the big bang.”  I’ve been studying relativity for some years now in the hopes of saying something intelligent to the author (Jim Hartle), if we’re both lucky enough to make it to our 60th college reunion in 2 years.  Hartle majored in physics under John Wheeler who essentially revived relativity from obscurity during the years when quantum mechanics was all the rage. Jim worked with Hawking for years, spoke at his funeral and wrote this in an appreciation of Hawking’s work [ Proc.Natl. Acad. Sci. vol. 115 pp. 5309 – 5310 ’18 ].

I find the above incomprehensible.  Could anyone out there enlighten me?  Just write a comment.  I’m not going to bother Hartle

Addendum 25 May

From a retired math professor friend —

I’ve never studied this stuff, but here is one way to get more actual dimensions without increasing the number of apparent dimensions:
Start with a 1-dimensional line, R^1 and now consider a 2-dimensional cylinder S^1 x R^1.  (S^1 is the circle, of course.)  If the radius of the circle is small, then the cylinder looks like a narrow tube.  Make the radius even smaller–lsay, ess than the radius of an atomic nucleus.  Then the actual 2-dimensional cylinder appears to be a 1-dimensional line.
The next step is to rethink S^1 as a line interval with ends identified (but not actually glued together.  Then S^1 x R^1 looks like a long ribbon with its two edges identified.  If the width of the ribbon–the length of the line interval–is less, say, than the radius of an atom, the actual 2-dimensional “ribbon with edges identified” appears to be just a 1-dimensional line.
Okay, now we can carry all these notions to R^2.  Take S^1 X R^2, and treat S^1 as a line interval with ends identified.  Then S^1 x R^2 looks like a (3-dimensional) stack of planes with the top plane identified, point by point, with the bottom plane.  (This is the analog of the ribbon.)  If the length of the line interval is less, say, than the radius of an atom, then the actual 3-dimensional s! x R^2 appears to be a 2-dimensional plane.
That’s it.  In general, the actual n+1-dimensional S^1 x R^n appears to be just n-space R^n when the radius of S^1 is sufficiently small.
All this can be done with a sphere S^2, S^3, … of any dimension, so that the actual k+n-dimensional manifold S^k x R^n appears to be just the n-space R^n when the radius of S^k is sufficiently small.  Moreover, if M^k is any compact manifold whose physical size is sufficiently small, then the actual k+n-dimensional manifold M^k x R^n appears to be just the n-plane R^n.
That’s one way to get “hidden” dimensions, I think. “
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High level mathematicians look like normal people

Have you ever had the pleasure of taking a course from someone who wrote the book? I did. I audited a course at Amherst from Prof. David Cox who was one of three authors of “Ideals, Varieties and Algorithms” It was uncanny to listen to him lecture (with any notes) as if he were reading from the book. It was also rather humbling to have a full professor correcting your homework. We had Dr. Cox for several hours each weak (all 11 or 12 of us). This is why Amherst is such an elite school. Ditto for Princeton back in the day, when Physics 103 was taught by John Wheeler 3 hours a week. Physics 103 wasn’t for the high powered among us who were going to be professional physicists (Heinz Pagels, Jim Hartle), it was for preMeds and engineers.

Dr. Cox had one very useful pedagogical device — everyone had to ask a question at the beginning of class, Cox being of the opinion that there is no such thing as a dumb question in math.

Well Dr. Cox and his co-authors (Little and O’Shea) got an award from the American Mathematical sociecty for their book. There’s an excerpt below. You should follow the link to the review to see what the three look like along with two other awardees. http://www.ams.org/publications/journals/notices/201604/rnoti-p417.pdf. Go to any midsize American city at lunchtime, and you’d be hard pressed to pick four of the five out of the crowd of middle aged men walking around. Well almost — one guy would be hard to pick out of the noonday crowd in Williamsburg Brooklyn or Tel Aviv. Four are extremely normal looking guys, not flamboyant or bizarre in any way. This is certainly true of the way Dr. Cox comports himself. The exception proving the rule however, is Raymond Smullyan who was my instructor in a complex variables course back in the day– quite an unusual and otherworldly individual — there’s now a book about him.

Here’s part of the citation. The link also contains bios of all.

“Even more impressive than its clarity of exposition is the impact it has had on mathematics. CLO, as it is fondly known, has not only introduced many to algebraic geometry, it has actually broadened how the subject could be taught and who could use it. One supporter of the nomination writes, “This book, more than any text in this field, has moved computational algebra and algebraic geometry into the mathematical mainstream. I, and others, have used it successfully as a text book for courses, an introductory text for summer programs, and a reference book.”
Another writer, who first met the book in an REU two years before it was published, says, “Without this grounding, I would have never survived my first graduate course in algebraic geometry.” This theme is echoed in many other accounts: “I first read CLO at the start of my second semester of graduate school…. Almost twenty years later I can still remember the relief after the first hour of reading. This was a math book you could actually read! It wasn’t just easy to read but the material also grabbed me.”
For those with a taste for statistics, we note that CLO has sold more than 20,000 copies, it has been cited more than 850 times in MathSciNet, and it has 5,000 citations recorded by Google Scholar. However, these numbers do not really tell the story. Ideals, Varieties, and Algorithms was chosen for the Leroy P. Steele Prize for Mathematical Exposition because it is a rare book that does it all. It is accessible to undergraduates. It has been a source of inspiration for thousands of students of all levels and backgrounds. Moreover, its presentation of the theory of Groebner bases has done more than any other book to popularize this topic, to show the powerful interaction of theory and computation in algebraic geometry, and to illustrate the utility of this theory as a tool in other sciences.”

A book recommendation, not a review

My first encounter with a topology textbook was not a happy one. I was in grad school knowing I’d leave in a few months to start med school and with plenty of time on my hands and enough money to do what I wanted. I’d always liked math and had taken calculus, including advanced and differential equations in college. Grad school and quantum mechanics meant more differential equations, series solutions of same, matrices, eigenvectors and eigenvalues, etc. etc. I liked the stuff. So I’d heard topology was cool — Mobius strips, Klein bottles, wormholes (from John Wheeler) etc. etc.

So I opened a topology book to find on page 1

A topology is a set with certain selected subsets called open sets satisfying two conditions
l. The union of any number of open sets is an open set
2. The intersection of a finite number of open sets is an open set

Say what?

In an effort to help, on page two the book provided another definition

A topology is a set with certain selected subsets called closed sets satisfying two conditions
l. The union of a finite number number of closed sets is a closed set
2. The intersection of any number of closed sets is a closed set

Ghastly. No motivation. No idea where the definitions came from or how they could be applied.

Which brings me to ‘An Introduction to Algebraic Topology” by Andrew H. Wallace. I recommend it highly, even though algebraic topology is just a branch of topology and fairly specialized at that.

Why?

Because in a wonderful, leisurely and discursive fashion, he starts out with the intuitive concept of nearness, applying it to to classic analytic geometry of the plane. He then moves on to continuous functions from one plane to another explaining why they must preserve nearness. Then he abstracts what nearness must mean in terms of the classic pythagorean distance function. Topological spaces are first defined in terms of nearness and neighborhoods, and only after 18 pages does he define open sets in terms of neighborhoods. It’s a wonderful exposition, explaining why open sets must have the properties they have. He doesn’t even get to algebraic topology until p. 62, explaining point set topological notions such as connectedness, compactness, homeomorphisms etc. etc. along the way.

This is a recommendation not a review because, I’ve not read the whole thing. But it’s a great explanation for why the definitions in topology must be the way they are.

It won’t set you back much — I paid. $12.95 for the Dover edition (not sure when).