Saturday, July 14, 2018

Friday Science: General Relativity 1

A little over four years ago, I found a great book on relativity. Peter Collier's A Most Incomprehensible Thing: Notes Toward a Very Gentle Introduction to the Mathematics of Relativity. I've gone through a little less than half of it.

His concept is to introduce all the math needed for special and general relativity in the first 100 pages or so. He tries not to assume that you've had any of the math beyond algebra. He does pretty well although I think you probably need to have done some of it before to get it.

BTW, I first saw an approach something like this in the summer of 1983 at Rose-Hulman. I was given a physics textbook by Marion and Hornyak that interspersed calculus lessons with the physics. I thought it was brilliant--introduce the background math as you need it. It's something like problem-based learning.

I've basically finished Hawking. So on Fridays I hope more or less to alternate between Susskind's book on quantum mechanics and Collier. I don't entirely have down everything from his 58 pages on special relativity. Maybe I'll go back at some point. But I want to move forward through his 180 or so pages on general relativity.

4.1 Introducing the Manifold
a. Special relativity functions on the basis of what is called "Minkowski" space, which is flat.
  • 3.2.2 Time for a flashback. In chapter 3, he introduces Minkowski space or spacetime. In Newtonian mechanics, we talk about three-dimensional space, Euclidean space. 
  • For special relativity, Einstein drew on the idea of four-dimensional space, with time as the fourth dimension (spacetime). 
  • This is named for the German mathematician, Hermann Minkowski (1864-1909).
  • In Minkowski space, parallel lines never meet, so it is still flat space.
In general relativity, space curves, so we need some new math. Einstein, with the help of David Hilbert, found this math in the work of the German Bernhard Reimann (1826-66).
b. In general relativity, matter and energy curve spacetime. Gravity is not considered a force but a property of the curvature of spacetime. The idea of a "Riemannian manifold" is used to model this. A manifold is a smoothly curved space that is locally flat.

It would be like an ant walking on an apple. The ant thinks it is walking straight, but it is curving around the apple. Such a path on a sphere or curved surface is called a geodesic.

A circle is a one-dimensional manifold. If you walk on the perimeter and the circle is large enough, it just seems like you are walking straight. A sphere is a two-dimensional manifold. We can speak of a manifold as n-dimensional when locally it can be described by n dimensions.

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