Tag Archives: The geometry of spacetime

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.

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.

Two math tips

Two of the most important theorems in differential geometry are Gauss’s Theorem egregium and the Inverse function theorem. Basically the theorem egregium says that you don’t need to look at the shape of a two dimensional surface (say the surface of a walnut) from outside (e.g. from the way it sits in 3 dimensional space) to understand its shape. All the information is contained in the surface itself.

The inverse function theorem (InFT) is used over and over. If you have a continuous function from Euclidean space U of finite dimension n to Euclidean space V of the same dimension, and certain properties of its derivative are present at a point x of U, then there exists a another function to get you back from space V to U.

Even better, once you’ve proved the inverse function theorem, proof of another important theorem (the implicit function theorem aka the ImFT) is quite simple. The ImFT lets you know if given f(x, y, .. .) –> R (e.g. a real valued function) if you can express one variable (say x) in terms of the others. Again sometimes it’s difficult to solve such an equation for x in terms of y — consider arctan(e^(x + y^2) * sin(xy) + ln x). What is important to know in this case, is whether it’s even possible.

The proofs of both are tricky. In particular, the proof of the inverse function theorem is an existence proof. You may not be able to write down the function from V to U even though you’ve just proved that it exists. So using the InFT to prove the implicit function theory is also nonconstructive.

At some point in your mathematical adolescence, you should sit down and follow these proofs. They aren’t easy and they aren’t short.

Here’s where to go. Both can be found in books by James J. Callahan, emeritus professor of Mathematics at Smith College in Northampton Mass. The proof of the InVT is to be found on pages 169 – 174 of his “Advanced Calculus, A Geometric View”, which is geometric, with lots of pictures. What’s good about this proof is that it’s broken down into some 13 steps. Be prepared to meet a lot of functions and variables.

Just the statement of InVT involves functions f, f^-1, df, df^-1, spaces U^n, R^n, variables a, q, B

The proof of InVT involves functions g, phi, dphi, h, dh, N, most of which are vector valued (N is real valued)

Then there are the geometric objects U^n, R^n, Wa, Wfa, Br, Br/2

Vectors a, x, u, delta x, delta u, delta v, delta w

Real number r

That’s just to get you through step 8 of the 13 step proof, which proves the existence of the inverse function (aka f^-1). The rest involves proving properties of f^-1 such as continuity and differentiability. I must confess that just proving existence of f^-1 was enough for me.

The proof of the implicit function theorem for two variables — e.g. f(x, y) = k takes less than a page (190).

The proof of the Theorem Egregium is to be found in his book “The Geometry of Spacetime” pp. 258 – 262 in 9 steps. Be prepared for fewer functions, but many more symbols.

As to why I’m doing this please see https://luysii.wordpress.com/2011/12/31/some-new-years-resolutions/

An old year’s resolution

One of the things I thought I was going to do in 2012 was learn about relativity.   For why see https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/.  Also my cousin’s new husband wrote a paper on a new way of looking at it.  I’ve been putting him off as I thought I should know the old way first.

I knew that general relativity involved lots of math such as manifolds and the curvature of space-time.  So rather than read verbal explanations, I thought I’d learn the math first.  I started reading John M. Lee’s two books on manifolds.  The first involves topological manifolds, the second involves manifolds with extra structure (smoothness) permitting calculus to be done on them.  Distance is not a topological concept, but is absolutely required for calculus — that’s what the smoothness is about.

I started with “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. Eventually I got through a third of its 380 pages of text.  I thought that might be enough to help me read his second book “Introduction to Smooth Manifolds” but I only got through 100 of its 600 pages before I could see that I really needed to go back and completely go through the first book.

This seemed endless, and would probably take 2 more years.  This shouldn’t be taken as a criticism of Lee — his writing is clear as a bell.  One of the few criticisms of his books is that they are so clear, you think you understand what you are reading when you don’t.

So what to do?  A prof at one of the local colleges, James J. Callahan, wrote a book called “The Geometry of Spacetime” which concerns special and general relativity.  I asked if I could audit the course on it he’d been teaching there for decades.  Unfortunately he said “been there, done that” and had no plans ever to teach the course again.

Well, for the last month or so, I’ve been going through his book.  It’s excellent, with lots of diagrams and pictures, and wide margins for taking notes.  A symbol table would have been helpful, as would answers to the excellent (and fairly difficult) problems.

This also explains why there have been no posts in the past month.

The good news is that the only math you need for special relativity is calculus and linear algebra.  Really nothing more.  No manifolds.  At the end of the first third of the book (about 145 pages) you will have a clear understanding of

l. time dilation — why time slows down for moving objects

2. length contraction — why moving objects shrink

3. why two observers looking at the same event can see it happening at different times.

4. the Michelson Morley experiment — but the explanation of it in the Feynman lectures on physics 15-3, 15-4 is much better

5. The Kludge Lorentz used to make Maxwell’s equations obey the Galilean principle of relativity (e.g. Newton’s first law)

6. How Einstein derived Lorentz’s kludge purely by assuming the velocity of light was constant for all observers, never mind how they were moving relative to each other.  Reading how he did it, is like watching a master sculptor at work.

Well, I’ll never get through the rest of Callahan by the end of 2012, but I can see doing it in a few more months.  You could conceivably learn linear algebra by reading his book, but it would be tough.  I’ve written some fairly simplistic background linear algebra for quantum mechanics posts — you might find them useful. https://luysii.wordpress.com/category/linear-algebra-survival-guide-for-quantum-mechanics/

One of the nicest things was seeing clearly what it means for different matrices to represent the same transformation, and why you should care.  I’d seen this many times in linear algebra, but seeing how simple reflection through an arbitrary line through the origin can be when you (1) rotate the line to the x axis by tan(y/x) radians (2) change the y coordinate to – y  — by an incredibly simple matrix  (3) rotate it back to the original angle .

That’s why any two n x n matrices X and Y represent the same linear transformation if they are related by the invertible matrix Z in the following way  X = Z^-1 * Y * Z

Merry Christmas and Happy New Year (none of that Happy Holidays crap for me)