Understanding the Riemann curvature tensor is like doing a spinal tap

Back in the day when I was doing spinal taps, I spent far more time setting them up (positioning the patient so that the long axis of the spinal column was parallel to the floor and the vertical axis of the recumbent patient was perpendicular to the floor) than actually doing the tap.  Why? because then, all I had to do was have the needle parallel to the floor, with no guessing about how to angle it when the patient had rolled (usually forward into the less than firm mattress of the hospital bed).

So it is with finally seeing what the Riemann curvature tensor actually is, why it is the way it is, and why the notation describing it is such a mess.  Finally on p. 290 of Needham’s marvelous book “Visual Differential Geometry and Forms” the Riemann curvature tensor is revealed in all its glory.  Understanding it takes far less time than understanding the mathematical (and geometric) scaffolding required to describe it, a la spinal taps.

Over the years while studying relativity, I’ve seen it in one form or other (always algebraic) without understanding what the algebra was really describing.

Needham will get you there, but you have to understand a lot of other things first. Fortunately almost all of them are described visually, so you see what the algebra is trying to describe.  Along the way you will see what the Lie bracket of two vector fields is all about along with holonomy.  And you will really understand what curvature is.  And Needham will give you 3 ways to understand parallel transport (which underlies everything — thanks Ashutosh)

Needham starts off with Gauss’s definition of curvature of a surface — the angular excess of a triangle, divided by its area.

Here is why this definition is enough to show you why the surface of a sphere is curved.   Go to the equator.  Mark point one, then point two 1/4 of the way around the sphere.  Form longitudes (perpendiculars) there and extend them as great circles toward the North pole. You now have a triangle containing 3 right angles, (clearly an angular excess from Euclid who states that the sum the angles of a triangle is two right angles).  The reason, of course, is because the sphere is curved.

Ever since I met a classmate 12 years ago at a college reunion who was a relativist working with Hawking, I decided to try to learn relativity so I’d have something intelligent to say to him if we ever met again (COVID19 stopped all that although we’re still both alive).

Now that I understand what the math of relativity is trying to describe, I may be able to understand general relativity.

Be prepared for a lot of work, but do start with Needham’s book.  Here are some links to other things I’ve written about it.  It will take some getting used to as it is very different from any math book you’ve ever read (except Needham’s other book).

12 July 21 — https://luysii.wordpress.com/2021/07/12/a-premature-book-review-and-a-60-year-history-with-complex-variables-in-4-acts/

4 Dec 21 — https://luysii.wordpress.com/2021/12/04/a-book-worth-buying-for-the-preface-alone-or-how-to-review-a-book-you-havent-read/

7 Mar 22 — https://luysii.wordpress.com/2022/03/07/visual-differential-geometry-and-forms-q-take-3/

27 June 22 — https://luysii.wordpress.com/2022/06/27/the-chinese-room-argument-understanding-math-and-the-imposter-syndrome/

17 July 22 — https://luysii.wordpress.com/2022/07/17/a-visual-proof-of-the-the-theorem-egregium-of-gauss/

What Cassava Sciences should do now

Apparently someone important didn’t like the way Cassava Sciences analyzed their data and their stock tanked again today..  Unfortunately all of this seems to be behind a paywall, and the someone important isn’t named.  I’d love a link if any reader knows of one — just put it in as a  comment below.

I’m not important, but I thought Cassava’s results were quite impressive.  They had enough cases and enough time for the results to be statistically significant

For one thing,  Cassava dealt with severely impaired people (see below) who would be expected to show greater neuronal dropout, senile plaques and neurofibrillary tangles, than recently diagnosed patients.   Neuronal loss is not reversible in man, despite hoards of papers showing the opposite in animals.

Since everything turns on ADAS-CoG, here is a link to a complete description along with some discussion — https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5929311/

On a slide from Cassava’s presentation yesterday the ADAS-CoG average of the 50 patients on entry 9 months ago was 16.6.  With a perfect score of 70, it’s clear that these people were significantly impaired (please look at the test items to see how simple the tasks in ADAS-CoG actually are).    So an improvement of 3 points at 9 months  is significant, particularly since a drop of 5 points is expected each year — yes I’ve seen plenty of Alzheimer patients with ADAS-CoG scores of zero or close to it.

So an increase of 3 points in this group is about a16% improvement.

Here’s what Cassava should do now.  Their data should be re-examined as follows.  Split the ADAS-CoG scores into 3 groups: highest middle and lowest. Quartiles are usually used, but I don’t think 50 patients is enough to do this.  Then examine the median improvement in each of the three.  I’d use median rather than average as with small numbers in each group, a single outlier can seriously distort things — think of the survival of Stephen Hawking in a group of 12 ALS patients.

If the patients with the highest ADAS-CoG scores have the highest median improvement, there is no reason mildly impaired individuals should have a less than 16% improvement in their scores.  This means that a person with ADAS-CoG of 60 should achieve a perfect score of 70,  e.g. return to normal.

This would be incredibly useful for early Alzheimer’s disease.

There is a precedent for this.  Again it’s Parkinson’s disease.

As I mentioned in an earlier post, I was one of the first neurologists in the USA to use L-DOPA for Parkinsonism.  All patients improved, and I actually saw one or two wheelchair bound Parkinsonians walk again (without going to Lourdes).  They were far from normal, but ever so much better.

However,  treated mildly impaired Parkinsonians became indistinguishable from normal, to the extent that I wondered if I’d misdiagnosed them. These results were typical.   For a time, in the early 70s neurologists thought that we’d actually cured the disease.  It was a very heady time.  We were masters of the neurologic universe — schizophrenia was too much dopamine, Parkinsonism not enough. Bring on the next neurotransmitter, bring on the next disease.

We hadn’t cured anything of course, and the underlying loss of dopamine neurons in the substantia nigra continued.  Reality intruded for me with one such extremely normal appearing individual I’d diagnosed with Parkinsonism a few years earlier. He needed surgery, meaning that he couldn’t take anything by mouth for a while.  L-DOPA could only be given orally, and he looked quite Parkinsonian in a day or two.

If reanalysis of the existing data shows what I hope, Cassava Sciences should start another study in Alzheimer patients with ADAS-CoG scores of over 50.  If I’m right the results should be spectacular (and lead to immediate approval of the drug).

A little blue sky.  Sumafilam will then come to be known as intellectual Viagra, as all sorts of oldsters (such as yrs trly) will try to get it Alzheimer’s or no Alzheimer’s.

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.

 

General relativity at last

I’ve finally arrived at the relativistic gravitational field equation which includes mass, doing ALL the math and understanding the huge amount of mathematical work it took to get there:  Chistoffel symbols (first and second kind), tensors, Fermi coordinates, the Minkowski metric, the Riemann curvature tensor (https://luysii.wordpress.com/2020/02/03/the-reimann-curvature-tensor/) geodesics, matrices, transformation laws, divergence of tensors, the list goes on.  It’s all covered in a tidy 379 pages of a wonderful book I used — “The Geometry of Spacetime” by James J. Callahan, professor emeritus of mathematics at Smith college.  Even better I got to ask him questions by eMail when I got stuck, and a few times we drank beer and listened to Irish music at a dive bar north of Amherst.

Why relativity? The following was written 8 years ago.  Relativity is 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 (Jim Hartle).  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 took 6 years or so.

Well as the years passed, Hartle was close enough to Hawking that he was chosen to speak at Hawking’s funeral.

We really don’t know why we like things and I’ve always like math.  As I went on in medicine, I liked math more and more because it could be completely understood (unlike medicine) –Why is the appendix on the right and the spleen on the left — dunno but you’d best remember it.

Coming to medicine from organic chemistry, the contrast was striking.  Experiments just refined our understanding, and one can look at organic synthesis as proving a theorem with the target compound as statement and the synthesis as proof.

Even now, wrestling with the final few pages of Callahan today took my mind off the Wuhan flu and my kids in Hong Kong just as topology took my mind off various neurologic disasters 50 years ago.

What’s next?  Well I’m just beginning to study the implications of the relativistic field equation, so it’s time to read other books about black holes, and gravity.  I’ve browsed in a few — Zee, Wheeler in particular are written in an extremely nonstuffy manner, unlike medical and molecular biological writing today (except the blogs). Hopefully the flu will blow over, and Jim and I will be at our 60th Princeton reunion at the end of May.  I better get started on his book “Gravity”

One point not clear presently.  If mass bends space which tells mass how to move, when mass moves it bends space — so it’s chicken and the egg.  Are the equations even soluble.

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. “

Stephen Hawking R. I. P.

Stephen Hawking, brilliant mathematician and physicist has died.  Forget all that. He did something for my patients with motor neuron disease that I, as a neurologist, could not do.  He gave them hope.

What has chemistry done for them?  Quite a bit, but there’s so much left.

Chemistry, when successful, just becomes part of the wallpaper and ignored. All genome sequencing depends on what some chemist did.

For one spectacular example of what, without chemistry, would be impossible is Infantile Spinal Muscular Atrophy (Werdnig Hoffmann disease).  For the actual molecular biology behind it — please see — https://luysii.wordpress.com/2016/12/25/tidings-of-great-joy/.   Knowing the cause has led to not one but two specific therapies — an antisense oligonucleotide and a virus which infects neurons and actually changes the gene.

So knowing what the cause of a disease is should lead to a treatment, shouldn’t it?  Hold that thought.  Sometimes one form of motor neuron disease (amyotrophic lateral sclerosis or ALS) can be hereditary.  Find out what is being inherited to find how ALS is caused.

Well, the first protein in which a mutation is associated with familial ALS (FALS) was found exactly 25 years ago.  It is called superoxide dismutase (SOD1).  Over 150 mutations have been found in the protein associated with FALS, and yet despite literally thousands of papers on the subject we don’t know if the mutations cause a loss of function, a gain of function (and if so what that function is), an increased tendency to fold incorrectly, and on and on and on.  It’s a fascinating puzzle for the protein chemist and over the years my notes on the papers I’ve read about SOD1 have ballooned to some 25,000 words.

If you’re tired of working on SOD1, try a few of the other proteins in which mutations have been associated with FALS — Alsin, TAF15, Ubiquilin, Optineurin, TBK1 etc. etc.  The list is long.

Now it’s biology’s turn.  Motor neurons go from the spinal cord (mostly) and brain to produce muscle contraction.  Why should only this tiny (but crucial) minority of cells be affected.  The nerve fibers leave the spinal cord and travel to muscle in nerves which contain sensory nerve fibers making the same long trip, yet somehow these nerves are spared.

More than that, why should these mutations affect only these neurons, and that often after decades.  Also why should great athletes (Lou Gehrig, Ezzard Charles, etc. etc. ) get the disease.

One closing point.  Hawking shows why, in any disease median survival (when 50% of those afflicted die) is much a more meaningful statistic than average duration of survival.  Although he gave my patients great hope, they all died within a few years even as he mightily extended average survival.