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The last basis vector is = r sin sin x + r sin cos y Therefore, the last component of the metric is g = = r 2 sin2 sin2 + r 2 sin2 cos2 = r 2 sin2 sin2 + cos2 = r 2 sin2 As an exercise, you can check to verify that all the other dot products vanish.
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Null Vectors
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A null vector V a is one that satis es gab V a V b = 0 (2.20)
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The Metric Determinant
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The determinant of the metric is used often. We write it as g = det (gab ) (2.21)
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1. The following is a valid expression involving tensors: (a) S a Tab = S c Tab (b) S a Tab = S a Tac (c) S a Tab = S c Tcb Cylindrical coordinates are related to cartesian coordinates via x = r cos , y = r sin , and z = z. This means that zz is given by (a) 1 (b) 1 (c) 0
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Vectors, One Forms, Metric
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If ds 2 = dr 2 + r 2 d 2 + dz 2 then (a) grr = dr, g = r d , and gzz = dz (b) grr = 1, g = r 2 , and gzz = 1 (c) grr = 1, g = r, and gzz = 1 (d) grr = dr 2 , g = r 2 d 2 , and gzz = dz 2 The signature of 1 0 = 0 0 0 1 0 0 0 0 1 0 0 0 0 1
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is (a) (b) (c) (d) 5.
2 2 0 1
In spherical coordinates, a vector has the following components: X a = 1 1 r, r sin , cos2 . The component X is given by (a) (b) (c) (d) 1/ cos2 cos2 r 2 tan2 r 2 / cos2
CHAPTER
More on Tensors
In this chapter we continue to lay down the mathematical framework of relativity. We begin with a discussion of manifolds. We are going to loosely de ne only what a manifold is so that the readers will have a general idea of what this concept means in the context of relativity. Next we will review and add to our knowledge of vectors and one forms, and then learn some new tensor properties and operations.
Manifolds
To describe curved spacetime mathematically, we will use a mathematical concept known as a manifold. Basically speaking, a manifold is nothing more than a continuous space of points that may be curved (and complicated in other ways) globally, but locally it looks like plain old at space. So in a small enough neighborhood Euclidean geometry applies. Think of the surface of a sphere or the surface of the earth as an example. Globally, of course, the earth is a curved surface. Imagine drawing a triangle with sides that went from the equator to
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More on Tensors
Fig. 3-1. The surface of a sphere is an example of a manifold. Pick a small enough
patch, and space in the patch is Euclidean ( at).
the North Pole. For that kind of triangle, the familiar formulas of Euclidean geometry are not going to apply. But locally it is at, and good old Euclidean geometry applies. Other examples of manifolds include a torus (Fig. 3-2), or even a more abstract example like the set of rotations in cartesian coordinates. A differentiable manifold is a space that is continuous and differentiable. It is intuitively obvious that we must describe spacetime by a differentiable manifold, because to do physics we need to be able to do calculus. Generally speaking, a manifold cannot be completely covered by a single coordinate system. But we can cover the manifold with a set of open sets Ui called coordinate patches. Each coordinate patch can be mapped to at Euclidean space.
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