Tag Archives: relativity

Book Review: Hawking Hawking

To this neurologist, Stephen Hawking’s greatest contribution wasn’t in physics. I ran a muscular dystrophy clinic for 15 years in the 70s and 80s. Few of my ALS patients had heard of Hawking back then. I made sure they did. Hawking did something for them, that I could never do as a physician — he gave them hope.

Which brings me to an excellent biography of Hawking by Charles Seife “Hawking Hawking” which tries to strip away the aura and myths that Hawking assiduously constructed and show the man underneath.

Even better, Seife is an excellent writer and has the mathematical and scientific  chops (Princeton math major, Yale masters in math) to explain the problems Hawking was wrestling with.

Hawking was smart.  One story tells it all (p. 328).  Apparently there were only 3 other physics majors at Oxford that year.  They were all given a set of 13 problems on electromagnetism and a week to do them.    One of the others (Derek Powney) tells the tale. “I discovered very rapidly that I couldn’t do any of them”.  So he teamed up with one of the others, and by the end of the week they’d done 1.5 problems.  The thrd student (working alone) solved one. 

At the end of the week “Stephen as always hadn’t even started”. He went to his room and came out 3 hours later. “Well, I’ve only had the time to do the first ten.”  “I think at that point we realized that it’s not just that we weren’t on the same street, we weren’t on the same planet.”

Have you ever had an experience like that?  I’ve had two.  The first occurred in grade school. I was a pretty good piano player, better than the rest of Dr, Rudnytsky’s students.  Then, someone told me that at age 3 his son would tell him what notes passing trains were whistling on, and that later on he’d sit behind a door listening to his father give lessons, and then come in afterwards and play by ear what the students had been playing.  The second occurred a within day or so of starting my freshman year in college. My roommate told me about a guy who thought he ought to know everybody in our class of 700+.  So he got out the freshman herald which had our pictures and names and a day later knew everyone in the class by name. 

The reason people of a scientific bent should read the book, is not the sociology, or the complicated sexuality of Hawking and his two wives, and god knows what else.  It is the excellent explanations of the problems in math and physics that Hawking faced and solved.  Even better, Seife puts them in context of the work done before Hawking was born.  

Two  examples

1. pp. 14 – 18 — a superb explanation of what Einstein did to create special relativity. 

2. pp. 240 – 245 an excellent description of the horizon problem, the flatness problem and how inflation solved it. 

Any really good book will teach you something.  People in physics, math and biology are consumed with the idea of information.  The book (pp. 131 – 134) explains why Hawking was so focused on the black hole information paradox.  It always seemed pretty arcane and superficial to me (on the order of how many angels could dance on the head of a pin).  

Wrong ! Wrong !

The black hole information paradox is at the coalface of ignorance in modern physics.  Why?  Because the two great theories we have in  (quantum mechanics and general relativity) disagree with what happens to the information contained in an object (such as an astronaut) swallowed by a black hole.  Relativity says it’s destroyed, while quantum mechanics says that’s impossible. 

So reconciling the two descriptions would lead to a deeper theory, and showing that one was wrong, would discredit a powerful theory. 

So even if you’re not interested in the sociology of the circles Hawking moved in or his sex life, there is a lot of well-explained physics and math to be learned for the general reader.  

The black hole information paradox resembles a similarly unresolved pair of phenomena in the world we live in, the Cartesian dualism between flesh and spirit.  It is writ large in biology.

Chemistry is great and can provide mechanistic explanations what we see, such as the example from the following old post, produced after the ***

It’s quite technical, but is an elegant explanation of how different cells make different amounts of two different forms of a muscle protein (beta actin and gamma actin ).  I never thought we’d have an explanation this good, but we do.  Well that’s the flesh and the physicality of the explanation.  Asking why different cells would want this, or what the function of all is puts you immediately in the world of spirit (ideas, which are inherently noncorporeal).  Physical chemistry and biochemistry are silent, and all the abstract explanations science gives us (the function, the why, the reason) is essentially teleological. 


The last post “The death of the synonymous codon – II” puts you exactly at the nidus of the failure of chemical reductionism to bag the biggest prey of all, an understanding of the living cell and with it of life itself.  We know the chemistry of nucleotides, Watson-Crick base pairing, and enzyme kinetics quite well.  We understand why less transfer RNA for a particular codon would mean slower protein synthesis.  Chemists understand what a protein conformation is, although we can’t predict it 100% of the time from the amino acid sequence.  

Addendum 30 April ’21:  Called to task on the above  by a reader.  This statement is no longer true.  The material below the *** was bodily lifted from something I wrote 10 years ago.  Time and AI have marched on since then.

So we do understand exactly why the same amino acid sequence using different codons would result in slower synthesis of gamma actin than beta actin, and why the slower synthesis would allow a more leisurely exploration of conformational space allowing gamma actin to find a conformation which would be modified by linking it to another protein (ubiquitin) leading to its destruction.  Not bad.  Not bad at all.

Now ask yourself, why the cell would want to have less gamma actin around than beta actin.  There is no conceivable explanation for this in terms of chemistry.  A better understanding of protein structure won’t give it to you.  Certainly, beta and gamma actin differ slightly in amino acid sequence (4/375) so their structure won’t be exactly the same.  Studying this till the cows come home won’t answer the question, as it’s on an entirely different level than chemistry.


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. “

A Mathematical Near Death Experience

As I’ve alluded to from time to time, I’m trying to learn relativity — not the popularizations, of which there are many, but the full Monty as it were, with all the math required. I’ve been at it a while as the following New Year’s Resolution of a few years ago will show.

“Why relativity? It’s something I’ve always wanted to understand at a deeper level than the popularizations of it (reading the sacred texts in the original so to speak). I may have enough background in math, to understand how to study it. Topology is something I started looking at years ago as a chief neurology resident, to get my mind off the ghastly cases I was seeing.

I’d forgotten about it, but a fellow ancient alum, mentioned our college president’s speech to us on opening day some 55 years ago. All the high school guys were nervously looking at our neighbors and wondering if we really belonged there. The prez told us that if they accepted us that they were sure we could do the work, and that although there were a few geniuses in the entering class, there were many more people in the class who thought they were.

Which brings me to our class relativist. I knew a lot of the physics majors as an undergrad, but not this guy. The index of the new book on Hawking by Ferguson has multiple entries about his work with Hawking (which is ongoing). Another physicist (now a semi-famous historian) felt validated when the guy asked him for help with a problem. He never tooted his own horn, and seemed quite modest at the 50th reunion. As far as I know, one physics self-proclaimed genius (and class valedictorian) has done little work of any significance. Maybe at the end of the year I’ll be able to read the relativist’s textbook on the subject. Who knows? It’s certainly a personal reason for studying relativity. Maybe at the end of the year I’ll be able to ask him a sensible question.”

Well that year has come and gone, but I’m making progress, going through a book with a mathematical approach to the subject written by a local retired math prof (who shall remain nameless). The only way to learn any math or physics is to do the problems, and he was kind enough to send me the answer sheet to all the problems in his book (which he worked out himself).

I am able to do most of the problems, and usually get the right answer, but his answers are far more elegant than mine. It is fascinating to see the way a professional mathematician thinks about these things.

The process of trying to learn something which everyone says is hard, is actually quite existential for someone now 76. Do I have the mental horsepower to get the stuff? Did I ever? etc. etc.

So when I got to one problem and the profs answer I was really quite upset. My answer appeared fairly straightforward and simple, yet his answer required a long derivation. Even though we both came out with the same thing, I was certain that I’d missed something really basic which required all the work he put in.

One of the joys of reading math these days (at least math books written by someone who is still alive) is that you can correspond with them. Mathematicians are so used to being dumped on by presumably intellectual people, that they’re happy to see some love. Response time is usually under a day. So I wrote him the following

“Along those lines, you do a lot of heavy lifting in your answer to 3a in section 4.3. Why not just say the point you are trying to find in R’s world is the image under M of the point (h.h) in G’s world and apply M to get t and z.”

Now usually any mathematician I EMail about their books gets back quickly — my sardonic wife says that it’s because they don’t have much to do.

Fo days, I heard nothing. I figured that he was trying to figure out a nice way to tell me to take up watching sports or golf, and that relativity was a mountain my intellect couldn’t climb. True existential gloom set in. Then I go the following back.

“You are absolutely right about the question; what you propose is elegant and incisive. I can’t figure out why I didn’t make the simple direct connection in the text itself, because I went to some pains to structure everything around the map M. But all that was fifteen or more years ago, and I have no notes about my thinking as I was writing.”

A true mathematical (and existential) near death experience.

Old and New Year’s Resolutions

I haven’t posted in a while because I was preparing for and recovering from some ‘minor’ surgery. As a practicing clinical neurologist, I was called in multiple times after people didn’t wake up, or stroked out from what was thought by all to be trouble free surgery. I came to the conclusion that there is no such thing as minor surgery. Although mine has gone well, I knew all the things which could go wrong, having seen nearly all of them. This sort of thing has a way of concentrating the mind, leaving room for little else.

So the old year’s resolutions are to get a few of the posts I’ve been sitting on out before the end of the year (probably not in this exact order)

Post #1 — further death of the synonymous codon

Post #2 — Heraclitus was right (about the nervous system) — you can’t step into the same brain twice

Post #3 — Book Review of Duncan Watts book

Post #5 — Unhappy 50th birthday for the War on Poverty

Post #6 — Gloating about the minimal hurricane season despite dire predictions about it

Post #7 — What sleep does and why babies sleep so much

Post #8 — The mating dance of ligand and receptor

The new year resolution — to go through the text on relativity by a classmate I hadn’t met until my 50th reunion. I’ve been through most of the math behind it (I hope). It’s the only time (I think) I’ve used the old school tie gambit to get something I wanted. He’s incredibly busy, still writing papers with Hawking etc. etc. but will answer at least a few questions when I get stuck (as I’m certain to do) purely because we were classmates. I doubt that he’d do this for any other 75 year old retired non physicist.

As my kids say, knowing someone can get you in the door, but you have to perform once you’re inside.

The old Ivy League school tie ain’t what it used to be. A cousin’s kid couldn’t get into a grad school in a subtype of English Lit despite a recent degree from one. Back in the day, it did mean a lot. If you were a premed at my institution and the premed advisor put his hand on your shoulder to say you were okay, you got into Columbia Med School. He was already famous and an operation named for him is still in use. A classmate, a smart guy, majored in Near East studies just because he was interested in it. That was enough for Chase which hired him as a banker. He never went near the mideast in his career.

Urysohn’s Lemma

“Now we come to the first deep theorem of the book,. a theorem that is commonly called the “Urysohn lemma”.  . . .  It is the crucial tool used in proving a number of important theorems. . . .  Why do we call the Urysohn lemma a ‘deep’ theorem?  Because its proof involves a really original idea, which the previous proofs did not.  Perhaps we can explain what we mean this way:  By and large, one would expect that if one went through this book and deleted all the proofs we have given up to now and then handed the book to a bright student who had not studied topology, that student ought to be able to go through the book and work out the proofs independently.  (It would take a good deal of time and effort, of course, and one would not expect the student to handle the trickier examples.)  But the Uyrsohn lemma is on a different level.  It would take considerably more originality than most of us possess to prove this lemma.”

The above quote is  from  one of the standard topology texts for undergraduates (or perhaps the standard text) by James R. Munkres of MIT. It appears on  page 207 of 514 pages of text.  Lee’s text book on Topological Manifolds gets to it on p. 112 (of 405).  For why I’m reading Lee see https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/.

Well it is a great theorem, and the proof is ingenious, and understanding it gives you a sense of triumph that you actually did it, and a sense of awe about Urysohn, a Russian mathematician who died at 26.   Understanding Urysohn is an esthetic experience, like a Dvorak trio or a clever organic synthesis [ Nature vol. 489 pp. 278 – 281 ’12 ].

Clearly, you have to have a fair amount of topology under your belt before you can even tackle it, but I’m not even going to state or prove the theorem.  It does bring up some general philosophical points about math and its relation to reality (e.g. the physical world we live in and what we currently know about it).

I’ve talked about the large number of extremely precise definitions to be found in math (particularly topology).  Actually what topology is about, is space, and what it means for objects to be near each other in space.  Well, physics does that too, but it uses numbers — topology tries to get beyond numbers, and although precise, the 202 definitions I’ve written down as I’ve gone through Lee to this point don’t mention them for the most part.

Essentially topology reasons about our concept of space qualitatively, rather than quantitatively.  In this, it resembles philosophy which uses a similar sort of qualitative reasoning to get at what are basically rather nebulous concepts — knowledge, truth, reality.   As a neurologist, I can tell you that half the cranial nerves, and probably half our brains are involved with vision, so we automatically have a concept of space (and a very sophisticated one at that).  Topologists are mental Lilliputians trying to tack down the giant Gulliver which is our conception of space with definitions, theorems, lemmas etc. etc.

Well one form of space anyway.  Urysohn talks about normal spaces.  Just think of a closed set as a Russian Doll with a bright shiny surface.  Remove the surface, and you have a rather beat up Russian doll — this is an open set.  When you open a Russian doll, there’s another one inside (smaller but still a Russian doll).  What a normal space permits you to do (by its very definition), is insert a complete Russian doll of intermediate size, between any two Dolls.

This all sounds quite innocent until you realize that between any two Russian dolls an infinite number of concentric Russian dolls can be inserted.  Where did they get a weird idea like this?  From the number system of course.  Between any two distinct rational numbers p/q and r/s where p, q, r and s are whole numbers, you can  always insert a new one halfway between.  This is where the infinite regress comes from.

For mathematics (and particularly for calculus) even this isn’t enough.  The square root of two isn’t a rational number (one of the great Euclid proofs), but you can get as close to it as you wish using rational numbers.  So there are an infinite number of non-rational numbers between any two rational numbers.  In fact that’s how non-rational numbers (aka real numbers) are defined — essentially by fiat, that any series of real numbers bounded above has a greatest number (think 1, 1.4, 1.41, 1.414, defining the square root of 2).

What does this skullduggery have to do with space?  It says essentially that space is infinitely divisible, and that you can always slice and dice it as finely as you wish.  This is the calculus of Newton and the relativity of Einstein.  It clearly is right, or we wouldn’t have GPS systems (which actually require a relativistic correction).

But it’s clearly wrong as any chemist knows. Matter isn’t infinitely divisible, Just go down 10 orders of magnitude from the visible and you get the hydrogen atom, which can’t be split into smaller and smaller hydrogen atoms (although it can be split).

It’s also clearly wrong as far as quantum mechanics goes — while space might not be quantized, there is no reasonable way to keep chopping it up once you get down to the elementary particle level.  You can’t know where they are and where they are going exactly at the same time.

This is exactly one of the great unsolved problems of physics — bringing relativity, with it’s infinitely divisible space together with quantum mechanics, where the very meaning of space becomes somewhat blurry (if you can’t know exactly where anything is).

Interesting isn’t it?

Why math is hard (for me) and organic chemistry is easy

I’ve been reading a lot of hard core math lately (I’ll explain why at the end), along with Clayden et al’s new edition of their fabulous Organic Chemistry text.  The level of sophistication takes a quantum jump about 2/3 of the way through (around pp. 796) and is probably near to the graduate level.  The exercise is great fun, but math and orgo require quite distinct ways of thinking.  Intermixing both on a daily basis brought home just how very different they are.

First off, the concepts in organic chemistry are fuzzy.  On p. 796 the graph of the Karplus relationship between J splitting in NMR and the dihedral angle of the hydrogens being split is shown.  It’s a continuous curve as the splitting is maximal at 180, zero at 90 and somewhat less than maximal at 0 degrees.

There is nothing like this in math.  Terms are defined exactly and the logic is that of true, false and the excluded middle (e.g. things are either true or false).   Remember the way that the square root of 2 was proved not to be the ratio of two whole numbers.  It was assumed that it could be done, and than it was shown no matter how you sliced it, a contradiction was reached.   The contradiction then implied that the opposite was true — if the negative of a proposition leads to a contradiction (it’s false) than the proposition must be true.  Math is full of proofs like this.Or if you are trying to prove A implies B, proving the contrapositive ( not B implies not A) will do just as well.  You never see stuff like this in orgo.

There just aren’t that many concepts in organic chemistry, even though the details of each and every reaction are beyond the strongest memory.  The crucial points are to have the orbitals of the various atoms firmly in mind and where they are in space.  This tells you how molecules will or won’t react, or how certain conformations will be stable (see anomeric effect).  Entropy in physics is a very subtle concept, but pretty obvious as used by organic chemists.  Two molecules are better than one etc. etc.  Also you see these concepts over and over.  Everything you study (just about) has carbon in it.  Chair and boat, cis and trans, exo and endo become part of you, without thinking much about them.

Contrast this with math.  I’m currently reading “Introduction to Topological Manifolds” (2nd. Edition) by John M. Lee.  I’ve got about 34 pages of notes on the first 95 pages (25% of the text), and made a list of the definitions I thought worth writing down — there are 170 of them.  Each is quite precise.  A topological embedding is (1) a continuous function (2) a surjective function (3) a homeomorphism.  No more no less.  Remove any one of the 3 (examples are given) and you no longer have an embedding.  The definitions are abstract for the most part, and far from intuitive.  That’s because  the correct definitions were far from obvious even to the mathematicians formulating them.  Hubbard’s excellent book on Vector Calculus says that it took some 200 years for the correct definition of continuity to be thrashed out.  People were arguing about what a function actually was 100 years ago.

As you read you are expected to remember exactly (or look up) the  170 or so defined concepts and use them in a proof.  So when you read a bit of Lee’s book, I’m always stopping and asking myself  ‘did I really understand what I just read’?  Clayden isn’t at all like that — Oh that’s just an intramolecular Sn2, helped because of the Thorpe Ingold effect, which is so obvious it shouldn’t be given a name.

Contrast this with:

After defining topological space, open set, closed set, compact, Hausdorff, continuous, closed map, you are asked to show that a continuous map from a compact topological space to a Hausdorff topological space is a closed map, and that such a map, if surjective as well is an embedding.   To get even close to the proof you must be able to hold all this in your head at once.  You should also remember that you proved that in a Hausdorff space compact sets are closed.

No matter how complicated the organic problem, you can always look at the molecule, and use the fabulous spatial processing capacity your brain gives you.   The interpretation of NMR spectra in terms of conformation certainly requires logical thinking — it’s sort of organic Sudoku.

I imagine a mathematician would have problems with the fuzzy concepts of organic chemistry.  Anslyn and Dougherty take great pains to show you why some reactions fall between Sn1 and Sn2, or E1cb.

So why am I doing this?  Of course there’s the why climb Everest explanation — because it’s there, big and hard, and maybe you can’t do it.  That’s part of it, but not all.  For reasons unknown, I’ve always like math, even though not terribly good at it.  Then there’s the surge for the ego should I be able to go through it all proving that I don’t have Alzheimer’s at 74.5 (at least not yet).  Then there is the solace (yes solace) that math provides.  Topology is far from new to me in 2011.  I started reading Hocking and Young back in ’70 when I was a neurology resident, seeing terrible disease, being unable to help most of those I saw, and ruminating about the unfairness of it all.  Thinking about math took me miles away (and still does), at least temporarily.  When I get that far away look, my wife asks me if I’m thinking about math again.  She’s particularly adamant about not doing this when I’m driving with her (or by myself).

The final reason, is that I went to college with a genius.  I met him at our 50th reunion after reading his bio in our 50th reunion book.  I knew several self-proclaimed geniuses back then, and a lot a physics majors, but he wasn’t one of them.  At any rate, he’s still pumping out papers on relativity with Stephen Hawking, and his entries in the index of the recent biography by Kitty Ferguson take up almost as many entries as Hawking himself.  He’s a very nice guy and agreed to answer questions from time to time.  But to understand the physics you need to really understand the math, and not just mouth it.

In particular, to understand gravity, a la relativity, you have to know how mass bends 4 dimensional space-time. This means you must understand curvature on manifolds, which means you must understand smooth manifolds, which means that you must understand topological manifolds which is why I’m reading Lee’s book.
So perhaps when the smoke clears, I might have something intelligent to say to my classmate.