Warped Passages (82 page)
and symmetry 197, 201, 213, 234
treatment of left/right-handed particles 164–5, 292
violates parity symmetry 163–4, 165
and weak gauge bosons 170, 200
weak gauge bosons 163, 165, 168, 169, 170, 182, 190–91, 200–201, 209, 213–14, 216–20, 241, 242, 245, 247, 249, 251, 254, 394
weak scale energy 144, 149, 222, 249, 255,
255
weak scale length 144–5, 149
weak scale mass 144, 219, 220, 242, 244, 250, 252, 254, 258, 268, 368, 401n, 403, 406
Weakbrane
392
, 393, 394, 395, 397–401,
399
, 405–8, 410–13, 415–17, 422, 430, 438
Weinberg, Stephen 78, 163, 169, 217, 235
Wess, Julius 259n, 261
Wess-Zumino model 261
Weyl, Hermann 196
Wieman, Carl 148
Wilczek, Frank 232
Wilson, Kenneth 224
winos 269, 271
Wise, Mark B. 402, 403, 404
Wiseman, Toby 432n
Witten, Edward 290, 293, 296, 305, 313, 315, 316, 318–19, 330–31, 332, 369, 452
Wood, Darien 183
Wu, C.S. 164
X-rays 27
Yang, C.N. 164
Yau, Shing-Tung 42
Young, Thomas 134
zero mass 43n, 191
zinos 269
Zumino, Bruno 259n, 261
Zweig, George 171n
LISA RANDALL
is an expert on particle physics, string theory, and cosmology. A member of the American Academy of Arts and Sciences and a winner of the Alfred P. Sloan Foundation Research Fellowship, she has been a tenured professor at Princeton, MIT, and Harvard, and she is one of the most highly cited physicists in her field. She lives in Cambridge, Massachusetts.
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WARPED PASSAGES
. Copyright © 2005 by Lisa Randall. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.
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*
We’ll discuss the Standard Model further in Chapter 7.
*
For British readers, a child’s climbing frame.
*
Questions I’ve actually been asked.
*
If you’re picky, you’ll object that Sam too has an age and therefore another dimension. However, I’ve assumed that Sam has been the same way for years so his age isn’t relevant.
*
This and other superscript numbers (1, 2,…) refer to the Math Notes at the end of the book.
*
Roald Dahl,
Charlie and the Chocolate Factory
(London: Puffin Books, 1998).
†
The full title is
Flatland: A Romance of Many Dimensions
.
*
This animated film, directed by Eric Martin, featured the voices of Dudley Moore and other members of the British theatrical comedy group
Beyond the Fringe
. It was very entertaining.
*
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 10
33
, 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 der Radioaktivität und Electronik
, vol. 4, pp. 411–62 (1907); see also Abraham Pais,
Subtle is the Lord
(Philadelphia: American Philological Association, 1982).
*
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.