Warped Passages - Lisa Randall [249]
* János Bolyai was a genius, but although his father, Farkas Bolyai, wanted him to be a mathematician, János was poor and joined the military and not the academy. Others initially discouraged János about his work on non-Euclidean geometry, and he eventually published it only because his father insisted on putting it in a book he was writing. Farkas, who was friends with Gauss, sent him the appendix that János wrote. But once again, János was in for disappointment. Although Gauss recognized János Bolyai’s genius, he replied only, “To praise it would amount to praising myself. For the entire content of the work…coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years.” (Letter from Gauss to Fartas Bolyair, 1832.) So once again, János’s mathematical career was thwarted.
* Because the gravitational field carries energy, the energy of the field must be taken into account when using Einstein’s equations. This makes solving for the gravitational field more subtle than it would be in Newtonian gravity.
* He did this on the Russian front while serving with the German army during World War I.
† Neil Ashby, “Relativity and the Global Positioning System,” Physics Today, May 2002, p. 41.
* The name refers to the electron, not to the character in Greek mythology.
* Quoted in Gerald Holton and Stephen J. Brush, Physics, the Human Adventure, from Copernicus to Einstein and Beyond (Piscataway, NJ: Rutgers University Press, 2001).
* Gerald Holton, The Advancement of Science, and Its Burdens (Cambridge, MA: Harvard University Press, 1998).
* Quoted in Gerald Holton and Stephen J. Brush, Physics, the Human Adventure, from Copernicus to Einstein and Beyond (Piscataway, NJ: Rutgers University Press, 2001).
* “Ultraviolet” means “high-frequency.”
† A blackbody is actually an idealization; real objects like coal aren’t perfect black-bodies.
* “…at any cost, that is, except for the inviolability of the two laws of thermodynamics.” Quoted in David Cassidy, Einstein and Our World, 2nd edn (Atlantic Highlands, NJ: Humanities Press, 2004).
* Abraham Pais, Subtle Is the Lord: The Science and Life of Albert Einstein (Philadelphia: American Philological Association, 1982).
† Gerald Holton, Thematic Origins of Scientific Thought, revised edn (Cambridge, MA: Harvard University Press, 1988).
* Quoted in Abraham Pais, Inward Bound: Of Matter and Forces in the Physical World (Oxford: Oxford University Press, 1986).
* Integers are the familiar whole numbers: 0, 1, 2, 3, and so on.
† We are focusing here on discrete spectra. When a free electron is absorbed by an ion, a continuous—not a discrete—spectrum of light is emitted.
* Wavelength is equal to Planck’s constant, h, divided by momentum.
† Although we need three coordinates to specify a point in space, we sometimes simplify and pretend that the wavefunction depends only on a single coordinate. This makes it easier to draw pictures of wavefunctions on a piece of paper.
* He even helped to decipher the Rosetta Stone.
* People are actually capable of detecting individual photons, but only in carefully prepared experiments. Usually, you see more standard light composed of many photons.
* Werner Heisenberg, Physics and Beyond: Encounters and Conversations, translated by Arnold Pomerans (New York: Harper & Row, 1971).
† Ibid. Owing to his German nationalism, he also participated in the German atomic bomb project.
‡ Gerald Holton, The Advancement of Science, and Its Burdens (Cambridge, MA: Harvard University Press, 1998).
* The GeV is a unit of energy that I will soon explain.
* “Tap” for British readers. We are assuming in this example that the faucet drips nonuniformly, which is not always true of real faucets.
* I will not derive the precise number here.
* The above reasoning is not entirely sufficient to fully