1

The Dyson quotations are to be found in his book From Eros to Gaia (Pantheon Books, New York, 1992), pages 325 and 314, respectively.

2

The Minkowski quote is from the Dover reprint of seminal publications on relativity The Principle of Relativity by Einstein, Lorentz, Weyl, and Minkowski (originally Methuen and Co., 1923).

3

In type, vectors are represented by boldface; in handwritten form an arrow is used: .

4

of The Feynman Lectures on Physics, vol. I.

5

of The Feynman Lectures on Physics, vol. I.

6

of The Feynman Lectures on Physics, vol. I.

7

of The Feynman Lectures on Physics, vol. I.

8

The electrons would actually win the race versus visible light because of the index of refraction of air. A gamma ray would make out better.

9

of The Feynman Lectures on Physics, vol. I.

10

Except for the one point at infinity.

11

We should mention one additional point for completeness. If you want to carry the hot-plate model of curved space over into three dimensions you must imagine that the length of the ruler depends not only on where you put it, but also on which orientation the ruler has when it is laid down. It is a generalization of the simple case in which the length of the ruler depends on where it is, but is the same if set north-south, or east-west, or up-down. This generalization is needed if you want to represent a three-dimensional space with any arbitrary geometry with such a model, although it happens not to be necessary for two dimensions.

12

Nobody—not even Einstein—knows how to do it if mass comes concentrated at points.

13

Approximately, because the density is not independent of radius as we are assuming.

14

of The Feynman Lectures on Physics.

15

R. V. Pound and G. A. Rebka, Jr., Physical Review Letters, vol. 4, p. 337 (1960).

16

of The Feynman Lectures on Physics.

17

Strictly speaking it is only a local maximum. We should have said that the proper time is larger than for any nearby path. For example, the proper time on an elliptical orbit around the earth need not be longer than on a ballistic path of an object which is shot to a great height and falls back down.