A visual proof of the the theorem egregium of Gauss

Nothing better illustrates the difference between the intuitive understanding that something is true and being convinced by logic that something is true  than the visual proof of the theorem egregium of Gauss found in “Visual Differential Geometry and Forms” by Tristan Needham and  the 9 step algebraic proof in  “The Geometry of Spacetime” by Jim Callahan.

Mathematicians attempt to tie down the Gulliver of our powerful appreciation of space with Lilliputian strands of logic.

First: some background on the neurology of vision and our perception of space and why it is so compelling to us.

In the old days, we neurologists figured out what the brain was doing by studying what was lost when parts of the brain were destroyed (usually by strokes, but sometimes by tumors or trauma).  This wasn’t terribly logical, as pulling the plug on a lamp plunges you in darkness, but the plug has nothing to do with how the lightbulb or LED produces light.  Even so,  it was clear that the occipital lobe was important — destroy it on both sides and you are blind — https://en.wikipedia.org/wiki/Occipital_lobe but the occipital lobe accounts for only 10% of the gray matter of the cerebral cortex.

The information flowing into your brain from your eyes is enormous.  The optic nerve connecting the eyeball to the brain has a million fibers, and they can fire ‘up to 500 times a second.  If each firing (nerve impulse) is a bit, then that’s an information flow into your brain of a gigaBit/second.   This information is highly processed by the neurons and receptors in the 10 layers of the retina. Over 30 retinal cell types in our retinas are known, each responding to a different aspect of the visual stimulus.  For instance, there are cells responding to color, to movement in one direction, to a light stimulus turning on, to a light stimulus turning off, etc. etc.

So how does the relatively small occipital lobe deal with this? It doesn’t.  At least half of your the brain responds to visual stimuli.  How do we know?   It’s complicated, but something called functional Magnetic Resonance Imaging (fMRI) is able to show us increased neuronal activity primarily by the increase in blood flow it causes.

Given that half of your brain is processing what you see, it makes sense to use it to ‘see’ what’s going on in Mathematics involving space.  This is where Tristan Needham’s books come in.

I’ve written several posts about them.

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

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

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

 

 

OK, so what is the theorem egregium?  Look at any object (say a banana). You can see how curved it is by just looking at its surface (e.g. how it looks in the 3 dimensional space of our existence).  Gauss showed that you don’t

have to even look at an object in 3 space,  just perform local measurements (using the distance between surface points, e.g. the metric e.g.  the metric tensor) .  Curvature is intrinsic to the surface itself, and you don’t have to get outside of the surface (as we are) to find it.

 

 

The idea (and mathematical machinery) has been extended to the 3 dimensional space we live in (something we can’t get outside of).  Is our  universe curved or not? To study the question is to determine its intrinsic curvature by extrapolating the tools Gauss gave us to higher dimensions and comparing the mathematical results with experimental observation. The elephant in the room is general relativity which would be impossible without this (which is why I’m studying the theorem egregium in the first place).

 

So how does Callahan phrase and prove the theorem egregium? He defines curvature as the ratio of the area on a (small) patch on the surface to the area of another patch on the unit sphere. If you took some vector calculus, you’ll know that the area spanned by two nonCollinear vectors is the numeric value of their cross product.

 

 

The vectors Callahan needs for the cross product are the normal vectors to the surface.  Herein beginneth the algebra. Callahan parameterizes the surface in 3 space from a region in the plane, uses the metric of the surface to determine a formula for the normal vector to the surface  at a point (which has 3 components  x , y and z,  each of which is the sum of 4 elements, each of which is the product of a second order derivative with a first order derivative of the metric). Forming the cross product of the normal vectors and writing it out is an algebraic nightmare.  At this point you know you are describing something called curvature, but you have no clear conception of what curvature is.  But you have a clear definition in terms of the ratio of areas, which soon disappears in a massive (but necessary) algebraic fandango.

 

 

On pages 258 – 262 Callahan breaks down the proof into 9 steps involving various mathematical functions of the metric and its derivatives such as  Christoffel symbols,  the Riemann curvature tensors etc. etc.  It is logically complete, logically convincing, and shows that all this mathematical machinery arises from the metric (intrinsic to the surface) and its derivatives (some as high as third order).

 

 

For this we all owe Callahan a great debt.  But unfortunately, although I believe it,  I don’t see it.  This certainly isn’t to denigrate Callahan, who has helped me through his book, and a guy who I consider a friend as I’ve drunk beer with him and his wife while  listening to Irish music in a dive bar north of Amherst.

 

 

Callahan’s proof is the way Gauss himself did it and Callahan told me that Gauss didn’t have the notational tools we have today making the theorem even more outstanding (egregious).

 

Well now,  onto Needham’s geometrical proof.  Disabuse yourself of the notion that it won’t involve much intellectual work on your part even though it uses the geometric intuition you were born with (the green glasses of Immanuel Kant — http://onemillionpoints.blogspot.com/2009/07/kant-for-dummies-green-sunglasses.html)

 

Needham’s definition of curvature uses angular excess of a triangle.  Angles are measured in radians, which is the ratio of the arc subtended by the angle to the radius of the circle (not the circumference as I thought I remembered).  Since the circumference of a circle is 2*pi*radius, radian measure varies from 0 to 2*pi.   So a right angle is pi/2 radians.

 

Here is a triangle with angular excess.  Start with a sphere of radius R.  Go to the north pol and drop a longitude down to the equator.  It meets the equator at a right angle (pi/2).  Go back to the north pole, form an angle of pi/2 with the first longitude, and drop another longitude at that angle which meets the equator at an angle of pi/2.   The two points on the equator and the north pole form a triangle, with total internal angles of 3*(pi/2).  In plane geometry we know that the total angles of a triangle is 2 (pi/2).  (Interestingly this depends on the parallel postulate. See if you can figure out why).  So the angular excess of our triangle is pi/2.  Nothing complicated to understand (or visualize) here.

 

Needham defines the curvature of the triangle (and any closed area) as the ratio between the angular excess of the triangle to its area

 

 

What is the area of the triangle?  Well, the volume of a sphere is (4/3) pi * r^3, and its area is the integral (4 pi * r^2).  The area of the north hemisphere, is 2 pi *r^2, and the area of the triangle just made is 1/2 * Pi * r^2.

 

 

So the curvature of the triangle is (pi/2) / (1/2 * pi * r^2) = 1 / r^2.   More to the point, this is the curvature of a sphere of radius r.

 

 

At this point you should have a geometric intuition of just what curvature is, and how to find it.  So when you are embroiled in the algebra in higher dimensions trying to describe curvature there, you will have a mental image of what the algebra is attempting to describe, rather than just the symbols and machinations of the algebra itself (the Lilliputian strands of logic tying down the Gulliver of curvature).

 

The road from here to the Einstein gravitational field equations (p. 326 of Needham) and one I haven’t so far traversed,  presently is about 50 pages.Just to get to this point however,  you have been exposed to comprehensible geometrical expositions, of geodesics, holonomy,  parallel transport and vector fields, and you should have mental images of them all.Interested?  Be prepared to work, and to reorient how you think about these things if you’ve met them before.  The 3 links mentioned about will give you a glimpse of Needham’s style.  You probably should read them next.

The Chinese Room Argument, Understanding Math and the imposter syndrome

The Chinese Room Argument

 was first published in a 1980 article by American philosopher John Searle. He imagines himself alone in a room following a computer program for responding to Chinese characters slipped under the door. Searle understands nothing of Chinese, and yet, by following the program for manipulating symbols and numerals just as a computer does, he sends appropriate strings of Chinese characters back out under the door, and this leads those outside to mistakenly suppose there is a Chinese speaker in the room.

 

So it was with me and math as an undergraduate due to a history dating back to age 10.  I hit college being very good at manipulating symbols whose meaning I was never given to understand.  I grew up 45 miles from the nearest synagogue.  My fanatically religious grandfather thought it was better not to attend services at all than to drive up there on the Sabbath.  My father was a young lawyer building a practice, and couldn’t close his office on Friday.   So my he taught me how to read Hebrew letters and reproduce how they sound, so I could read from the Torah at my Bar Mitzvah (which I did comprehending nothing).  Since I’m musical, learning the cantillations under the letters wasn’t a problem.

 

I’ve always loved math and solving problems of the plug and chug variety was no problem.  I’d become adept years earlier at this type of thing thanks to my religiously rigid grandfather.   It was the imposter syndrome writ large.  I’ve never felt like this about organic chemistry and it made a good deal of intuitive sense the first time I ran into it.  For why have a look at — https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/

 

If there is anything in math full of arcane symbols calling for lots of mechanical manipulation, it is the differential geometry and tensors needed to understand General relativity.   So I’ve plowed through a lot of it, but still don’t see what’s really going on.

 

Enter Tristan Needham’s book “Visual Differential Geometry and Forms”.  I’ve written about it several times

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

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

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

 

If you’ve studied any math, his approach will take getting used to as it’s purely visual and very UNalgebraic.  But what is curvature but a geometric concept.

 

So at present I’m about 80 pages away from completing Needham’s discussion of general relativity.  I now have an intuitive understanding of curvature, torsion, holonomy, geodesics and the Gauss map that I never had before.   It is very slow going, but very clear.  Hopefully I’ll make it to p. 333.  Wish me luck.

The pleasures of reading Feynman on Physics – V

Feynman finally gets around to discussing tensors 376 pages into volume II in “The Feynman Lectures on Physics” and a magnificent help it is (to me at least).  Tensors must be understood to have a prayer of following the math of General Relativity (a 10 year goal, since meeting classmate Jim Hartle who wrote a book “Gravity” on the subject).

There are so many ways to describe what a tensor is (particularly by mathematicians and physicists) that it isn’t obvious that they are talking about the same thing.   I’ve written many posts about tensors, as the best way to learn something it to try to explain it to someone else (a set of links to the posts will be found at the end).

So why is Feynman so helpful to me?  After plowing through 370 pages of Callahan’s excellent book we get to something called the ‘energy-momentum tensor’, aka the stress-energy tensor.  This floored me as it appeared to have little to do with gravity, talking about flows of energy and momentum. However it is only 5 pages away from the relativistic field equations so it must be understood.

Back in the day, I started reading books about tensors such as the tensor of inertia, the stress tensor etc.  These were usually presented as if you knew why they were developed, and just given in a mathematical form which left my intuition about them empty.

Tensors were developed years before Einstein came up with special relativity (1905) or general relativity (1915).

This is where Feynman is so good.  He starts with the problem of electrical polarizability (which is familiar if you’ve plowed this far through volume II) and shows exactly why a tensor is needed to describe it, e.g. he derives  the tensor from known facts about electromagnetism.  Then on to the tensor of inertia (another derivation).  This allows you to see where all that notation comes from. That’s all very nice, but you are dealing with just matrices.  Then on to tensors over 4 vector spaces (a rank 4 tensor) not the same thing as a 4 tensor which is over a 4 dimensional vector space.

Then finally we get to the 4 tensor (a tensor over a 4 dimensional vector space) of electromagnetic momentum.  Here are the 16 components of Callahan’s energy momentum tensor, clearly explained.  The circle is finally closed.

He briefly goes into the way tensors transform under a change of coordinates, which for many authors is the most important thing about them.   So his discussion doesn’t contain the usual blizzard of superscripts and subscript.  Covariant and contravariant are blessedly absent. Here the best explanation of how they transform is in Jeevanjee “An introduction to Tensors and Group Theory for Physicists”  chapter 3 pp. 51 – 74.

Here are a few of the posts I’ve written about tensors trying to explain them to myself (and hopefully you)

https://luysii.wordpress.com/2020/02/03/the-reimann-curvature-tensor/

https://luysii.wordpress.com/2017/01/04/tensors-yet-again/

https://luysii.wordpress.com/2015/06/15/the-many-ways-the-many-tensor-notations-can-confuse-you/

https://luysii.wordpress.com/2014/12/28/how-formal-tensor-mathematics-and-the-postulates-of-quantum-mechanics-give-rise-to-entanglement/

https://luysii.wordpress.com/2014/12/08/tensors/

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.

The Reimann curvature tensor

I have harpooned the great white whale of mathematics (for me at least) the Reimann curvature tensor.  Even better, I understand what curvature is, and how the Reimann curvature tensor expresses it.  Below you’ll see the nightmare of notation by which it is expressed.

Start with curvature.  We all know that a sphere (e.g. the earth) is curved.  But that’s when you look at it from space.  Gauss showed that you could prove a surface was curved just be performing measurements entirely within the surface itself, not looking at it from the outside (theorem egregium).

Start with the earth, assuming that it is a perfect sphere (it isn’t because its rotation fattens its middle).  We’ve got longitude running from pole to pole and the equator around the middle.  Perfect sphere means that all points are the same distance from the center — e.g. the radius.  Call the radius 1.

Now think of a line from the north pole to the plane formed by the equator (radius 1).  Take the midpoint of that line and inscribe a circle on the sphere, parallel to the plane of the equator.  Its radius is the half the square root of 3 (or 1.73). This comes from the right angle triangle just built with hypotenuse is 1 and  one side 1/2.   The circumference of the equator is 2*pi (remember the sphere’s radius is 1).  The circumference of the newly inscribed circle is 1.73 * pi.

Now pick a point on the smaller circle and follow a longitude down to the equator.  Call this point down1.  Move in one direction by 1/4 of the circumference of the sphere (pi/2).  Call that point on the equator down then across

Now go back to the smaller circle at the first point you picked and move in the same direction as you did on the equator by absolute distance pi/2 (not by pi/2 radians).  Then follow the longitude down to the equator.  Call that point across then down.  The two will not be the same.  Across then down is farther from down 1 than down then across.

The difference occurs because the surface of the sphere is curved, and the difference in endpoints of the two paths is exactly what the Reimann curvature tensor measures.

Here is the way the Riemann curvature tensor is notated.  Hideous isn’t it?

If you’re going to have any hope of understanding general relativity (not special relativity) you need to understand curvature.

I used paths in the example, Riemann uses the slope of the paths (e.g derivatives) which makes things much more complicated.  Which is where triangles (dels), and the capital gammas (Γ) come in.

To really understand the actual notation, you need to understand what a covariant derivative actually is, which is much more complicated, but knowing what you know now, you’ll see where you are going when enmeshed in thickets of notation.

What the Riemann curvature tensor is actually saying is that the order of taking covariant derivatives (which is the same thing as the order of taking paths)  is NOT commutative.

The simplest functions we grow up with are commutative.  2 + 3 is the same as 3 + 2, and 5*3 = 3*5.  The order of the terms doesn’t matter.

Although we weren’t taught to think of it that way, subtraction is not.  5 – 3 is not the same as 3 – 5.  There is all sorts of nonCommutativity in math.  The Lie bracket is one such, the Poisson bracket  another, and most groups are nonCommutative.  But that’s enough.  I wish I’d known this when I started studying general relativity.

Why it is sometimes good to read the preface7

“the (gravitational) field equations are derived  . . .  from an analysis of tidal forces.”  Thus sayeth p. xii of the preface to “The Geometry of Spacetime” by James J. Callahan.  This kept me from passing over pp. 174 –> on tides, despite a deep dive back into differentiating complicated functions, Taylor series etc. etc. Hard thinking about tidal forces finally gave me a glimpse of what general relativity is all about.

Start with a lemma.  Given a large object (say the sun) and a single small object (say a satellite, or a spacecraft), the path traced out by the spacecraft will lie in a plane.  Why?

All gravitational force is directed toward the sun (which can be considered as a point mass – it is my recollection that it took Newton 20 years to prove this delaying the publication of the Principia , but I can’t find the source).  This makes the gravitational force radially symmetric.

Now consider an object orbiting the sun (falling toward the sun as it orbits, but not hitting the sun because its angular momentum carries away). Look at two close by points in the orbit and the sun, forming a triangle.  The long arms of the triangle point toward the sun.  Now imagine in the next instant that the object goes to a fourth point out of the plane formed by the first 3.  Such a shift in direction requires a force to produce it, but in the model there is only gravity, so this is impossible meaning that all points of the orbit lie in a plane containing the sun considered as a point mass.

You are weightless when falling, even though you are responding to the gravitational force (paradoxic but true).   An astronaut in a space capsule is weightless because he or she is falling but the conservation of angular momentum keeps them going around the sun.

Well if the sun can be considered a point mass, so can the space capsule.  Call the local coordinate of the point representing the capsule point mass x.  The orbit of x around the sun is in the x — sun plane.

Next put two objects 1 foot above and below the x — sun plane.

object1

x  ——————————sun point mass

object2

Objects 1 and 2 orbit in a plane containing the sun point mass as well.  They do not orbit parallel to x (but very close to parallel).

What happens with the passage of time?   The objects approach each other.  To an astronaut inside the capsule it looks like a force (similar to gravity) is pushing them together.  These are the tidal forces Callahan is talking about.

Essentially the tidal are produced by gravitational force of the sun even though everything in the capsule floats around feeling no gravity at all.

Consider a great circle on a sphere — a longitudinal circle on the earth will do.  Two different longitudinal great circles will eventually meet each other.  No force is producing this, but the curvature of the surface in which they are embedded.

I think that  what appear to be tidal forces in general relativity are really due to the intrinsic curvature of spacetime.  So gravity disappears into the curvature of spacetime produced by mass.   I’ll have to go through the rest of the book to find out.  But I think this is pretty close, and likely why Callahan put the above quote into the preface.

If you are studying on your own, a wonderful explanation of just what is going on under the algebraic sheets of the Taylor series is to be found pp. 255 –> of Fundamentals of Physics by R. Shankar.  In addition to being clear, he’s funny.  Example: Nowadays we worry about publish and perish, but in the old days it was publish and perish.

Bye bye stoichiometry

Until recently, developments in physics basically followed earlier work by mathematicians Think relativity following Riemannian geometry by 40 years.  However in the past few decades, physicists have developed mathematical concepts before the mathematicians — think mirror symmetry which came out of string theory — https://en.wikipedia.org/wiki/Mirror_symmetry_(string_theory). You may skip the following paragraph, but here is what it meant to mathematics — from a description of a 400+ page book by Amherst College’s own David A. Cox

Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. This book is the first completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. Subjects discussed include toric varieties, Hodge theory, Kahler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem. This title features: numerous examples worked out in detail; an appendix on mathematical physics; an exposition of the algebraic theory of Gromov-Witten invariants and quantum cohomology; and, a proof of the mirror theorem for the quintic threefold.

Similarly, advances in cellular biology have come from chemistry.  Think DNA and protein structure, enzyme analysis.  However, cell biology is now beginning to return the favor and instruct chemistry by giving it new objects to study. Think phase transitions in the cell, liquid liquid phase separation, liquid droplets, and many other names (the field is in flux) as chemists begin to explore them.  Unlike most chemical objects, they are big, or they wouldn’t have been visible microscopically, so they contain many, many more molecules than chemists are used to dealing with.

These objects do not have any sort of definite stiochiometry and are made of RNA and the proteins which bind them (and sometimes DNA).  They go by any number of names (processing bodies, stress granules, nuclear speckles, Cajal bodies, Promyelocytic leukemia bodies, germline P granules.  Recent work has shown that DNA may be compacted similarly using the linker histone [ PNAS vol.  115 pp.11964 – 11969 ’18 ]

The objects are defined essentially by looking at them.  By golly they look like liquid drops, and they fuse and separate just like drops of water.  Once this is done they are analyzed chemically to see what’s in them.  I don’t think theory can predict them now, and they were never predicted a priori as far as I know.

No chemist in their right mind would have made them to study.  For one thing they contain tens to hundreds of different molecules.  Imagine trying to get a grant to see what would happen if you threw that many different RNAs and proteins together in varying concentrations.  Physicists have worked for years on phase transitions (but usually with a single molecule — think water).  So have chemists — think crystallization.

Proteins move in and out of these bodies in seconds.  Proteins found in them do have low complexity of amino acids (mostly made of only a few of the 20), and unlike enzymes, their sequences are intrinsically disordered, so forget the key and lock and induced fit concepts for enzymes.

Are they a new form of matter?  Is there any limit to how big they can be?  Are the pathologic precipitates of neurologic disease (neurofibrillary tangles, senile plaques, Lewy bodies) similar.  There certainly are plenty of distinct proteins in the senile plaque, but they don’t look like liquid droplets.

It’s a fascinating field to study.  Although made of organic molecules, there seems to be little for the organic chemist to say, since the interactions aren’t covalent.  Time for physical chemists and polymer chemists to step up to the plate.

Time to get busy

Well I asked for it (the answer sheets to my classmate’s book on general relativity). It came today all 347 pages of it + a small appendix “Light Orbits in the Schwarzschild Geometry”. It’s one of the few times the old school tie has actually been of some use. The real advantages of going to an elite school are (1) the education you can get if you want (2) the people you meet back then or subsequently. WRT #1 — the late 50s was the era of the “Gentleman’s C”.

It should be fun. The book is the exact opposite of the one I’d been working on which put the math front and center. This one puts the physics first and the math later on. I’m glad I’m reading it second because as an undergraduate and graduate student I became adept at mouthing mathematical incantations without really understanding what was going on. I think most of my math now is reasonably solid. I did make a lot of detours I probably didn’t need to make — manifold theory,some serious topology — but that was fun as well.

When you’re out there away from University studying on your own, you assume everything you don’t understand is due to your stupidity. This isn’t always the case (although it usually is), and I’ve found errors in just about every book I’ve studied hard, and my name features on errata web pages of most of them. For one example see https://luysii.wordpress.com/2014/05/01/a-mathematical-near-death-experience/

Read Einstein

Devoted readers of this blog (assuming there are any) know that I’ve been studying relativity for some time — for why see https://luysii.wordpress.com/2011/12/31/some-new-years-resolutions/.

Probably some of you have looked at writings about relativity, and have seen equations containing terms like ( 1 – v^2/c^2)^1/2. You need a lot of math for general relativity (which is about gravity), but to my surprise not so much for special relativity.

Back in the early 50’s we were told not to study Calculus before reaching 18, as it was simply to hard for the young brain, and would harm it, the way lifting something too heavy could bring on a hernia. That all changed after Sputnik in ’58 (but too late for me).

I had similar temerity in approaching anything written by Einstein himself. But somehow I began looking at his book “Relativity” to clear up a few questions I had. The Routledge paperback edition (which I got in England) cost me all of 13 pounds. Routledge is a branch of a much larger publisher Taylor and Francis.

The book is extremely accessible. You need almost no math to read it. No linear algebra, no calculus, no topology, no manifolds, no differential geometry, just high school algebra.

You will see a great mind at work in terms you can understand.

Some background. Galileo had a theory of relativity, which basically said that there was no absolute position, and that motion was only meaningful relative to another object. Not much algebra was available to him, and later Galilean relativity came be taken to mean that the equations of physics should look the same to people in unaccelerated motion relative to each other.

Newton’s laws worked out quite well this way, but in the late 1800’s Maxwell’s equations for electromagnetism did not. This was recognized as a problem by physicists, so much so that some of them even wondered if the Maxwell equations were correct. In 1895 Lorentz figured out a way (purely by trying different equations out) to transform the Maxwell equations so they looked the same to two observers in relative motion to each other. It was a classic kludge (before there even were kludges).

The equation to transform the x coordinate of observer 1 to the x’ of observer 2 looks like this

x’ = ( x – v*t) / ( 1 – v^2/c^2)^1/2)

t = time, v = the constant velocity of the two observers relative to each other, c = velocity of light

Gruesome no ?

All Lorentz knew was that it made Maxwell’s equations transform properly from x to x’.

What you will see on pp. 117 – 123 of the book, is Einstein derive the Lorentz equation from
l. the constancy of the velocity of light to both observers regardless of whether they are moving relative to each other
2. the fact that as judged from observer1 the length of a rod at rest relative to observer2, is the same as the length of the same rod at rest relative to observer1 as judged from observer2. Tricky to state, but this just means that the rod is out there and has a length independent of who is measuring it.

To follow his derivation you need only high school algebra. That’s right — no linear algebra, no calculus, no topology, no manifolds, no differential geometry. Honest to God.

It’s a good idea to have figure 2 from p. 34 in front of you

The derivation isn’t particularly easy to follow, but the steps are quite clear, and you will have the experience of Einstein explaining relativity to you in terms you can understand. Like reading the Origin of Species, it’s fascinating to see a great mind at work.

Enjoy

The weirdness of gravity

We experience gravity every waking moment, so it’s hard to recognize just how strange the gravitational ‘force’ actually is. Push a toy sailboat, a rowboat, and a yacht with the same amount of force (effort). What happens?

The smaller the boat, the faster it moves. Physicists would say the acceleration (change in velocity over time e.g. from the boat not moving at all to moving somewhat) is inversely proportional to the mass of the boat. This is Newton’s famous second law force = mass * acceleration. This isn’t actually what he said which you’ll find at the end.

So in every force except gravity, the bigger the force the more the acceleration. In Galileo’s famous experiment (which Wikipedia says might actually not have occurred), he dropped 2 objects of different masses from the leaning tower of Pisa and found that they hit the ground at the same time, so the acceleration of both due to the ‘force’ of gravity is the for all objects regardless of their different masses.

This implies that gravity is a force that adjusts itself to the mass of the object it is pushing on to produce the same acceleration. Weird, but true.

General relativity says, that the motion must be considered not just in space and time, but in 4 dimensional space-time where space can become our conventional time and vice versa. Here all paths are as straight as possible — because the 4 dimensional space-time we inhabit has an intrinsic curvature, produced by the masses found within it.

What Newton said: “The change of motion is proportional to the motive force impressed and is made in the direction of the straight line in which that force is impressed” By motion Newton means what we call momentum — mass * velocity.

The change in momentum is of course a change in velocity — which is what acceleration actually is. Note that mass is assumed constant regardless of how fast the object is moving. This isn’t even true in special relativity (which doesn’t include gravity — that’s what general relativity is all about).