Warped Passages - Lisa Randall [248]
* Slices of ham do have some thickness, so they are in reality thin, but three-dimensional. Their size in this extra dimension is so small that it is a good approximation to think of them as two-dimensional. However, even with arbitrarily thin two-dimensional slices, we can imagine putting them together to make a three-dimensional object in this way.
† Again, for the pages to be truly two-dimensional they would have to be infinitely thin slices with no thickness at all in the third dimension. For now, though, two dimensions is a fine approximation for pages as thin as these.
* Or perhaps this story is a result of my having begun my education at the perhaps questionably named Lewis Carroll School, P.S. 179, in Queens.
* We will specify spatial dimensions in this and the following chapter. After introducing relativity, we will switch to spacetime, and consider time as an additional dimension.
† I will sometimes use scientific notation for very large or very small numbers. When a power of ten has a negative exponent, as in 10-33, it indicates a decimal number; for example, 10-33 is the number 0.000,000,000,000,000,000,000,000,000,000,001. This is an extremely tiny number and would be too cumbersome to write in full each time it occurs. A number with a positive exponent, such as 1033, has 33 zeroes after a 1,1,000,000,000,000,000,000,000,000,000,000,000, which is an enormous number that would also be difficult to write in full each time. I will often give a number in both scientific notation and in words the first time I use it.
† An order of magnitude is a factor of ten. Twenty-four orders of magnitude is 1,000,000,000,000,000,000,000,000, or one trillion trillion.
* The garden hose has always been a popular analogy to illustrate rolled-up dimensions. I learned it at math camp and it has most recently been described in Brian Greene’s Elegant Universe (Norton, 1999; Vintage, 2000). I’ll use this same analogy since it’s so good and because I want to expand on it in the following section (and in later chapters), in which I’ll also include sprinklers to explain extra-dimensional gravity.
* In this book a “massive” object means an object with mass. A massive object is to be distinguished from a “massless” object, which has zero mass (and travels at the speed of light).
* Only a year after the last time before 2004 that the Red Sox won the World Series—quite a while ago.
* Catwalks in the UK.
* Quoted in Anne Midgette, “At 3 score and 10, the music deepens,” New York Times, 28 January 2005.
* An address to a group of physicists at the British Association for the Advancement of Science in 1900.
† Presidential Address to British Association, 1871.
* The story might be apocryphal, but the reasoning is not.
* Letter from Isaac Newton to Robert Hooke, 5 February 1675.
* Gerald Holton, Einstein, History, and Other Passions (Cambridge, MA: Harvard University Press, 2000).
† Letter to E. Zschimmer, 30 September 1921.
* Velocity gives both speed and direction.
† Peter Galison, Einstein’s Clocks, Poincaré’s Maps: Empires of Time (New York: W.W. Norton, 2003).
‡ Don’t get me wrong—I like trains. But I wish they were better supported in the U.S.
* Although American trains don’t always coordinate time very well, Amtrak does appear to acknowledge special relativity when they say, “time and the space to use it” in their advertising slogan for the Acela, the high-speed train that travels the Northeast corridor. However, “time” and “space” are not precisely interchangeable. Although the slogan “space and the time to use it” does describe my more heavily delayed train rides, the phrase wouldn’t be a very compelling advertisement for a high-speed train.
* He did the experiment by timing objects rolling down an inclined plane.
* Albert Einstein, “Über das Relativitätsprinzip und die aus demselben gezogene Folgerungen” [“On the relativity principle and the conclusions drawn from it”], Jahrbuch