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Special Relativity
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Consider two frames in standard con guration. The phenomenon of length contraction can be described by saying that distances are shortened by a factor of (a) (b) (c) 1 + 2 1 2 1 + 2 c2
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Vectors, One Forms, and the Metric
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In this chapter we describe some of the basic objects that we will encounter in our study of relativity. While you are no doubt already familiar with vectors from studies of basic physics or calculus, we are going to be dealing with vectors in a slightly different light. We will also encounter some mysterious objects called one forms, which themselves form a vector space. Finally, we will learn how a geometry is described by the metric.
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A vector is a quantity that has both magnitude and direction. Graphically, a vector is drawn as a directed line segment with an arrow head. The length of the arrow is a graphic representation of its magnitude. (See Figure 2-1).
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Fig. 2-1. Your basic vector, a directed line segment drawn in the x y plane.
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The reader is no doubt familiar with the graphical methods of vector addition, scalar multiplication, and vector subtraction. We will not review these methods here because we will be looking at vectors in a more abstract kind of way. For our purposes, it is more convenient to examine vectors in terms of their components. In the plane or in ordinary three-dimensional space, the components of a vector are the projections of the vector onto the coordinate axes. In Fig. 2-2, we show a vector in the x y plane and its projections onto the x and y axes. The components of a vector are numbers. They can be arranged as a list. For example, in 3 dimensions, the components of a vector A can be written as A = A x , A y , A z . More often, one sees a vector written as an expansion in terms of a set of basis vectors. A basis vector has unit length and points along the direction of a coordinate axis. Elementary physics books typically denote the basis for cartesian coordinates by (i, j, k), and so in ordinary three-dimensional
x Wx
Fig. 2-2. A vector W in the x y plane, resolved into its components Wx and Wy . These
are the projections of W onto the x and y axes.
Vectors, One Forms, Metric
cartesian space, we can write the vector A as A = Ax i + A y j + Az k In more advanced texts a different notation is used: A = Ax x + A y y + Az z This has some advantages. First of all, it clearly indicates which basis vector points along which direction (the use of (i, j, k) may be somewhat mysterious to some readers). Furthermore, it provides a nice notation that allows us to de ne a vector in a different coordinate system. After all, we could write the same vector in spherical coordinates: A = Ar r + A + A There are two important things to note here. The rst is that the vector A is a geometric object that exists independent of coordinate system. To get its components we have to choose a coordinate system that we want to use to represent the vector. Second, the numbers that represent the vector in a given coordinate system, the components of the vector, are in general different depending on what coordinate system we use to represent the vector. So for the example we have been using so far A x , A y , A z = Ar , A , A .
New Notation
We are now going to use a different notation that will turn out to be a bit more convenient for calculation. First, we will identify the coordinates by a set of labeled indices. The letter x is going to be used to represent all coordinates, but we will write it with a superscript to indicate which particular coordinate we are referring to. For example, we will write y as x 2 . It is important to recognize that the 2 used here is just a label and is not an exponent. In other words, y2 = x 2
and so on. For the entire set of cartesian coordinates, we make the following identi cation: (x, y, z) x 1 , x 2 , x 3