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/

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.