Fear of Physics

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Authors: Lawrence M. Krauss
Tags: General, science, Physics, energy, Mechanics
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dimensions of velocity are length/time. The dimensions of distance are length. Therefore, if the left-hand side has dimensions of length, and velocity has dimensions of length/time, clearly you must multiply velocity by time in order for the righthand side to have the dimensions of length.
    This kind of analysis can never guarantee you that you have the right answer, but it can let you know when you are wrong. And even though it doesn’t guarantee you’re right, when working with the unknown it is very handy to let dimensional arguments be your guide. They give you a framework for fitting the unknown into what you already know about the world.
    It is said that fortune favors the prepared mind. Nothing could be more true in the history of physics. And dimensional analysis can prepare our minds for the unexpected. In this regard, the ultimate results of simple dimensional analysis are often so powerful that they can seem magical. To demonstrate these ideas graphically, I want to jump to a modern example based on research at the forefront of physics—where the known and unknown mingle together. In this case, dimensional arguments helped lead to an understanding of one of the four known forces in nature: the “strong” interactions that bind “quarks” together to form protons and neutrons, which in turn make up the nuclei of all atoms. The arguments might seem a little elusive at first reading, but don’t worry. I present them because they give you the chance to see explicitly how pervasive and powerful dimensional arguments can be in guiding our physical intuition. The flavor of the arguments
is probably more important to carry away with you than any of the results.
    Physicists who study elementary-particle physics—that area of physics that deals with the ultimate constituents of matter and the nature of the forces among them—have devised a system of units that exploits dimensional analysis about as far as you can take it. In principle, all three dimensional quantities—length, time, and mass—are independent, but in practice nature gives us fundamental relations among them. For example, if there existed some universal constant that related length and time, then I could express any length in terms of a time by multiplying it by this constant. In fact, nature has been kind enough to provide us with such a constant, as Einstein first showed. The basis of his theory of relativity, which I will discuss later, is the principle that the speed of light, labeled c, is a universal constant, which all observers will measure to have the same value. Since velocity has the dimensions of length/time, if I multiply any time by c, I will arrive at something with the dimension of length—namely, the distance light would travel in this time. It is then possible to express all lengths unambiguously in terms of how long it takes light to travel from one point to another. For example, the distance from your shoulder to your elbow could be expressed as 10 –9 seconds, since this is approximately the time it takes a light ray to travel this distance. Any observer who measures how far light travels in this time will measure the same distance.
    The existence of a universal constant, the speed of light, provides a one-to-one correspondence between any length and time. This allows us to eliminate one of these dimensional quantities in favor of the other. Namely, we can choose if we wish to express all lengths as equivalent times or vice versa. If we want to do this, it is simplest to invent a system of units where the speed of light is
numerically equal to unity. Call the unit of length a “light-second” instead of a centimeter or an inch, for example. In this case, the speed of light becomes equal to 1 light-second/second. Now all lengths and their equivalent times will be numerically equal!
    We can go one step further. If the numerical values of light-lengths and light-times are equal in this system of units, why consider length and

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