Assignment 7: Curvature

For submission

1. Consider a geometry in which Euclid's 5th postulate is replaced by:

Through any point NO straight line can be drawn parallel to a given line.

Show that there is at least one triangle in this geometry whose angles sum to more than two right angles.
(Hint: On a line PQ, select two points A and B. Construct lines AC and BD perpendicular to PQ. What happens if AC and BD are extended in both directions?)

2. If you had before you a two dimensional surface of constant curvature, how could your determine whether the curvature was positive, negative or zero by measuring
(a) the sum of angles of a triangle;
(b) the circumference of a circle of known radius?

3. What is the difference between extrinsic and intrinsic curvature?

4. Imagine that you are a two dimensional being trapped in a flat two dimensional surface.

(a) How would you use geodesic deviation to confirm the flatness of your surface?

(b) Imagine that a three dimensional being picks up your surface and bends it into cylinder, without in any way stretching your surface. (This is just what happens when someone takes a piece of paper and rolls it into a cylinder.) You are still trapped in the surface. If you now use geodesic deviation to determine the curvature of your surface, would you get the same result as in (a)? Explain why.

For discussion in the recitation.

A. In a space with non-zero curvature, a geodesic is the analog of the straight line of ordinary Euclidean geometry. Why is it appropriate to take geodesics as their analogs? If you were in such a space, how would identify the geodesics?

B. Does it make sense to say that a space has a curved geometry if there is no higher dimensioned space into which the space can curve?

C. In a space with three or more dimensions, the curvature need not be the same in every two dimensional sheet that passes though some point in the space. Of course sometimes things are simple and the curvature does work out the same. Here's an example. Imagine that you are in an ordinary, three dimensional Euclidean space. You slice the space up into the flattest two dimensional sheets you can find, all built out of intersecting straight lines. The first set of sheets run left-right and up-down. The second set of sheets run left-right and front-back. The third set of sheets run up-down and front-back. You use geodesic deviation to determine the curvature of the sheets in each set. What is the curvature of:

(a) The left-right and up-down sheets?
(b) The left-right and front-back sheets?
(c) The up-down and front-back?

(d) Things need not work out so simply. In what space discussed in the chapter would the results be different?

D. The discovery of non-Euclidean geometries eventually precipitated a crisis in our understanding of what has to be and what just might be the case. At one extreme are necessities, such the truths of logic; they have to be true. At the other extreme are mundane factual matters--contingent statements that may or may not be true. Somewhere in between is a transition. Locating that transition has traditionally been of great importance in philosophy and philosophy of science. For if something is necessarily true, we need harbor no doubt over it. If something is contingent, the mainstream empiricist philosophy says we can only learn it from experience. Sometimes the contingent proposition is very broad. For example, consider the proposition that there never has been and never will be a magnet with only one pole. We may come to believe this proposition with ever greater confidence. But we can never be absolutely certain of it. We never know whether tomorrow will bring the counterexample.

Just where should the transition between necessity and contingency come?

Here is a list of propositions that begins with logical truths and bleeds off into ordinary contingent propositions. Sort them into necessary truths and contingent propositions. How are you deciding which is which?

If A and B are both true, then A is true.
If one of A or B is true and A is false, then B is true.
For any proposition A, either A is true or A is not true.
1 + 1 = 2
7 + 5 = 12
There are an infinity of prime numbers.
Every circle has one center.
The sum of the angles of a triangle is two right angles.
Only the fittest survive.
Every effect has a cause.
Every occurrence has a cause.
No effect comes before its cause.
Improbable events are rare.
Energy is always conserved.
Force equals mass times acceleration.
The earth has one moon.