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Generator Quick Response Code in .NET Fig. 2-4. An admittedly crude representation.The blob represents a manifold (basically a

Fig. 2-4. An admittedly crude representation.The blob represents a manifold (basically a
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space of points). Tp is the tangent space at a point p. The tangent vectors live here.
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maps them to the real numbers. We can write a general tensor in the following way: T = Tabc lmn a b c el em en We will have more to say about one forms, basis vectors, and tensors later.
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In relativity it is often necessary to change from one coordinate system to another, or from one frame to another. A transformation of this kind is implemented with a transformation matrix that we denote by a b . The placement of the indices and where we put the prime notation will depend on the particular transformation. In a coordinate transformation, the components of a b are formed by taking the partial derivative of one coordinate with respect to the other. More speci cally,
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xa xb
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(2.3)
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The easiest way to get a handle on how to apply this is to simply write the formulas down and apply them in a few cases. Basis vectors transform as ea =
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(2.4)
We can do this because, as you know, it is possible to expand any vector in terms of some basis. What this relation gives us is an expansion of the basis vector ea in terms of the old basis eb . The components of ea in the eb basis are given by a b . Note that we are denoting the new coordinates by primes and the old coordinates are unprimed indices. EXAMPLE 2-1 Plane polar coordinates are related to cartesian coordinates by x = r cos and y = r sin
Describe the transformation matrix that maps cartesian coordinates to polar coordinates, and write down the polar coordinate basis vectors in terms of the basis vectors of cartesian coordinates.
Vectors, One Forms, Metric
SOLUTION 2-1a x Using a b = x b , the components of the transformation matrix are
x x r
x = cos r x = = r sin
and and
y = sin r y = = r cos
Using (2.4), we can write down the basis vectors in polar coordinates. We obtain er = e =
+ x ex +
r ex
r e y = cos ex + sin e y y e y = r sin ex + r cos e y
The components of a vector transform in the opposite manner to that of a basis vector (this is why, an ordinary vector is sometimes called contravariant; it transforms contrary to the basis vectors). This isn t so surprising given the placement of the indices. In particular, Va =
a b bV =
xa b V xb
(2.5)
The components of a one form transform as a = Basis one forms transform as a = dx a =
a b dx b b a
(2.6)
(2.7)
To nd the way an arbitrary tensor transforms, you just use the basic rules for vectors and one forms to transform each index (OK we aren t transforming the indices, but you get the drift). Basically, you add an appropriate for each index. For example, the metric tensor, which we cover in the next section, transforms as ga b =
c a d b
The Metric
At the most fundamental level, one could say that geometry is described by the pythagorean theorem, which gives the distance between two points (see
Vectors, One Forms, Metric
Fig. 2-5. The pythagorean theorem tells us that the lengths of a, b, c are related by
a2 + b 2 .
Fig. 2-5). If we call P1 = (x1 , y1 ) and P2 = (x2 , y2 ), then the distance d is given by d= (x1 x2 )2 + (y1 y2 )2
Graphically, of course, the pythagorean theorem gives the length of one side of a triangle in terms of the other two sides, as shown in Fig. 2-3. As we have seen, this notion can be readily generalized to the at spacetime of special relativity, where we must consider differences between spacetime events. If we label two events by (t1 , x1 , y1 , z 1 ) and (t2 , x2 , y2 , z 2 ), then we de ne the spacetime interval between the two events to be ( s)2 = (t1 t2 )2 (x1 x2 )2 (y1 y2 )2 (z 1 z 2 )2 Now imagine that the distance between the two events is in nitesimal. That is, if the rst event is simply given by the coordinates (t, x, y, z), then the second event is given by (t + dt, x + dx, y + dy, z + dz). In this case, it is clear that the differences between each term will give us (dt, dx, dy, dz). We write this in nitesimal interval as ds 2 = dt 2 dx 2 dy 2 dz 2 As we shall see, the form that the spacetime interval takes, which describes the geometry, is closely related to the gravitational eld. Therefore it s going to become quite important to familiarize ourselves with the metric. In short, the interval ds 2 , which often goes by the name the metric, contains information about how the given space (or spacetime) deviates from a at space (or spacetime). You are already somewhat familiar with the notion of a metric if you ve studied calculus. In that kind of context, the quantity ds 2 is often called a line element. Let s quickly review some familiar line elements. The most familiar is
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