The essential beginner's guide to string theory The Little Book of String Theory offers a short, accessible, and entertaining introduction to one of the most talked-about areas of physics today. String theory has been called the "theory of everything." It seeks to describe all the fundamental forces of nature. It encompasses gravity and quantum mechanics in one unifying theory. But it is unproven and fraught with controversy. After reading this book, you'll be able to draw your own conclusions about string theory.Steve Gubser begins by explaining Einstein's famous equation E = mc2, quantum mechanics, and black holes. He then gives readers a crash course in string theory and the core ideas behind it. In plain English and with a minimum of mathematics, Gubser covers strings, branes, string dualities, extra dimensions, curved spacetime, quantum fluctuations, symmetry, and supersymmetry. He describes efforts to link string theory to experimental physics and uses analogies that nonscientists can understand. How does Chopin's Fantasie-Impromptu relate to quantum mechanics? What would it be like to fall into a black hole? Why is dancing a waltz similar to contemplating a string duality? Find out in the pages of this book. The Little Book of String Theory is the essential, most up-to-date beginner's guide to this elegant, multidimensional field of physics.
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THE AIM OF THIS CHAPTER IS TO PRESENT THE MOST FAMOUS equation of physics: E = mc2. This equation underlies nuclear power and the atom bomb. It says that if you convert one pound of matter entirely into energy, you could keep the lights on in a million American households for a year. E = mc2 also underlies much of string theory. In particular, as weâll discuss in chapter 4, the mass of a vibrating string receives contributions from its vibrational energy.
Whatâs strange about the equation E = mc2 is that it relates things you usually donât think of as related. E is for energy, like the kilowatt-hours you pay your electric company for each month; m is for mass, like a pound of flour; c is for the speed of light, which is 299,792,458 meters per second, or (approximately) 186,282 miles per second. So the first task is to understand what physicists call âdimensionful quantities,â like length, mass, time, and speed. Then weâll get back to E = mc2 itself. Along the way, Iâll introduce metric units, like meters and kilograms; scientific notation for big numbers; and a bit of nuclear physics. Although itâs not necessary to understand nuclear physics in order to grasp string theory, it provides a good context for discussing E = mc2. And in chapter 8, I will come back and explain efforts to use string theory to better understand aspects of modern nuclear physics.
Length, mass, time, and speed
Length is the easiest of all dimensionful quantities. Itâs what you measure with a ruler. Physicists generally insist on using the metric system, so Iâll start doing that now. A meter is about 39.37 inches. A kilometer is 1000 meters, which is about 0.6214 miles.
Time is regarded as an additional dimension by physicists. We perceive four dimensions total: three of space and one of time. Time is different from space. You can move any direction you want in space, but you canât move backward in time. In fact, you canât really âmoveâ in time at all. Seconds tick by no matter what you do. At least, thatâs our everyday experience. But itâs actually not that simple. If you run in a circle really fast while a friend stands still, time as you experience it will go by less quickly. If you and your friend both wear stopwatches, yours will show less time elapsed than your friendâs. This effect, called time dilation, is imperceptibly small unless the speed with which you run is comparable to the speed of light.
Mass measures an amount of matter. Weâre used to thinking of mass as the same as weight, but itâs not. Weight has to do with gravitational pull. If youâre in outer space, youâre weightless, but your mass hasnât changed. Most of the mass in everyday objects is in protons and neutrons, and a little bit more is in electrons. Quoting the mass of an everyday object basically comes down to saying how many nucleons are in it. A nucleon is either a proton or a neutron. My mass is about 75 kilograms. Rounding up a bit, thatâs about 50,000,000,000, 000,000,000,000,000,000 nucleons. Itâs hard to keep track of such big numbers. There are so many digits that you canât easily count them up. So people resort to whatâs called scientific notation: instead of writing out all the digits like I did before, you would say that I have about 5 Ă 1028 nucleons in me. The 28 means that there are 28 zeroes after the 5. Letâs practice a bit more. A million could be written as 1 Ă 106, or, more simply, as 106. The U.S. national debt, currently about $10,000,000,000,000, can be conveniently expressed as 1013 dollars. Now, if only I had a dime for every nucleon in me . . .
Letâs get back to dimensionful quantities in physics. Speed is a conversion factor between length and time. Suppose you can run 10 meters per second. Thatâs fast for a personâreally fast. In 10 seconds you can go 100 meters. You wouldnât win an Olympic gold with that time, but youâd be close. Suppose you could keep up your speed of 10 meters per second over any distance. How long would it take to go one kilometer? Letâs work it out. One kilometer is ten times 100 meters. You can do the 100-meter dash in 10 seconds flat. So you can run a kilometer in 100 seconds. You could run a mile in 161 seconds, which is 2 minutes and 41 seconds. No one can do that, because no one can keep up a 10 m/s pace for that long.
Suppose you could, though. Would you be able to notice the time dilation effect I described earlier? Not even close. Time would run a little slower for you while you were pounding out your 2:41 mile, but slower only by one part in about 1015 (thatâs a part in 1,000,000,000,000,000, or a thousand million million). In order to get a big effect, you would have to be moving much, much faster. Particles whirling around modern accelerators experience tremendous time dilation. Time for them runs about 1000 times slower than for a proton at rest. The exact figure depends on the particle accelerator in question.
The speed of light is an awkward conversion factor for everyday use because itâs so big. Light can go all the way around the equator of the Earth in about 0.1 seconds. Thatâs part of why an American can hold a conversation by telephone with someone in India and not notice much time lag. Light is more useful when youâre thinking of really big distances. The distance to the moon is equivalent to about 1.3 seconds. You could say that the moon is 1.3 light-seconds away from us. The distance to the sun is about 500 light-seconds.
A light-year is an even bigger distance: itâs the distance that light travels in a year. The Milky Way is about 100,000 light-years across. The known universe is about 14 billion light-years across. Thatâs about 1.3 Ă 1026 meters.
E = mc2
The formula E = mc2 is a conversion between mass and energy. It works a lot like the conversion between time and distance that we just discussed. But just what is energy? The question is hard to answer because there are so many forms of energy. Motion is energy. Electricity is energy. Heat is energy. Light is energy. Any of these things can be converted into any other. For example, a lightbulb converts electricity into heat and light, and an electric generator converts motion into electricity. A fundamental principle of physics is that total energy is conserved, even as its form may change. In order to make this principle meaningful, one has to have ways of quantifying different forms of energy that can be converted into one another.
A good place to start is the energy of motion, also called kinetic energy. The conversion formula is K =
mv2, where K is the kinetic energy, m is the mass, and v is the speed. Imagine yourself again as an Olympic sprinter. Through a tremendous physical effort, you can get yourself going at v = 10 meters per second. But this is much slower than the speed of light. Consequently, your kinetic energy is much less than the energy E in E = mc2. What does this mean?
It helps to know that E = mc2 describes ârest energy.â Rest energy is the energy in matter when it is not moving. When you run, youâre converting a little bit of your rest energy into kinetic energy. A very little bit, actually: roughly one part in 1015. Itâs no accident that this same number, one part in 1015, characterizes the amount of time dilation you experience when you run. Special relativity includes a precise relation between time dilation and kinetic energy. It says, for example, that if something is moving fast enough to double its energy, then its time runs half as fast as if it werenât moving.
Itâs frustrating to think that you have all this rest energy in you, and all you can call up with your best efforts is a tiny fraction, one part in 1015. How might we call up a greater fraction of the rest energy in matter? The best answer we know of is nuclear energy.
Our understanding of nuclear energy rests squarely on E = mc2. Here is a brief synopsis. Atomic nuclei are made up of protons and neutrons. A hydrogen nucleus is just a proton. A helium nucleus comprises two protons and two neutrons, bound tightly together. What I mean by tightly bound is that it takes a lot of energy to split a helium nucleus. Some nuclei are much easier to split. An example is uranium-235, which is made of 92 protons and 143 neutrons. It is quite easy to break a uranium-235 nucleus into several pieces. For instance, if you hit a uranium-235 nucleus with a neutron, it can split into a krypton nucleus, a barium nucleus, three neutrons, and energy. This is an example of fission. We could write the reaction briefly as
U + n â kr + Ba + 3n + Energy,
where we understand that U stands for uranium-235, Kr stands for krypton, Ba stands for barium, and n stands for neutron. (By the way, Iâm careful always to say uranium-235 because thereâs another type of uranium, made of 238 nucleons, that is far more common, and also harder to split.)
E = mc2 allows you to calculate the amount of energy that is released in terms of the masses of all the participants in the fission reaction. It turns out that the ingredients (one uranium-235 nucleus plus one neutron) outweigh the products (a krypton atom, a barium atom, and three neutrons) by about a fifth of the mass of a proton. It is this tiny increment of mass that we feed into E = mc2 to determine the amount of energy released. Tiny as it seems, a fifth of the mass of a proton is almost a tenth of a percent of the mass of a uranium-235 atom: one part in a thousand. So the energy released is about a thousandth of the rest energy in a uranium-235 nucleus. This still may not seem like much, but itâs roughly a trillion times bigger as a fraction of rest energy than the fraction that an Olympic sprinter can call up in the form of kinetic energy.
I still havenât explained where the energy released in nuclear fission comes from. The number of nucleons doesnât change: there are 236 of them before and after fission. And yet the ingredients have more mass than the products. So this is an important exception to the rule that mass is essentially a count of nucleons. The point is that the nucleons in the krypton and barium nuclei are bound more tightly than they were in the uranium-235 nucleus. Tighter binding means less mass. The loosely bound uranium-235 nucleus has a little extra mass, just waiting to be released as energy. To put it in a nutshell: Nuclear fission releases energy as protons and neutrons settle into slightly more compact arrangements.
One of the projects of modern nuclear physics is to figure out what happens when heavy nuclei like uranium-235 undergo far more violent reactions than the fission reaction I described. For reasons I wonât go into, experimentalists prefer to work with gold instead of uranium. When two gold nuclei are slammed into one another at nearly the speed of light, they are utterly destroyed. Almost all the nucleons break up. In chapter 8, I will tell you more about the dense, hot state of matter that forms in such a reaction.
In summary, E = mc2 says that the amount of rest energy in something depends only on its mass, because the speed of light is a known constant. Itâs easier to get some of that energy out of uranium-235 than most other forms of matter. But fundamentally, rest energy is in all forms of matter equally: rocks, air, water, trees, and people.
Before going on to quantum mechanics, letâs pause to put E = mc2 in a broader intellectual context. It is part of special relativity, which is the study of how motion affects measurements of time and space. Special relativity is subsumed in general relativity, which also encompasses gravity and curved spacetime. String theory subsumes both general relativity and quantum mechanics. In particular, string theory includes the relation E = mc2. Strings, branes, and black holes all obey this relation. For example, in chapter 5 Iâll discuss how the mass of a brane can receive contributions from thermal energy on the brane. It wouldnât be right to say that E = mc2 follows from string theory. But it fits, seemingly inextricably, with other aspects of string theoryâs mathematical framework.
Chapter TWO
QUANTUM MECHANICS
AFTER I GOT MY BACHELORâS DEGREE IN PHYSICS, I SPENT A year at Cambridge University studying math and physics. Cambridge is a place of green lawns and grey skies, with an immense, historical weight of genteel scholarship. I was a member of St. Johnâs College, which is about five hundred years old. I particularly remember playing a fine piano located in one of the upper floors of the first courtâone of the oldest bits of the college. Among the pieces I played was Chopinâs Fantasie-Impromptu. The main section has a persistent four-against-three cross rhythm. Both hands play in even tempo, but you play four notes with the right hand for every three notes in the left hand. The combination gives the composition an ethereal, liquid sound.
