Vectors, One Forms, Metric in .NET

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or we can use it with higher rank tensors. Some examples are
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a Sb = g ac Scb
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T ab = g ac Tcb = g ac g bd Tcd e Rabcd = gae Rbcd In at spacetime, the metric ab is used to raise and lower indices. When we begin to prove results involving tensors, this technique will be used frequently.
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The Dot Product
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Earlier we brie y mentioned the scalar or dot product. The metric also tells us how to compute the dot or scalar product in a given geometry. In particular, the dot product is written as V W = Va W a Now we can use index raising and lowering to write the scalar product in a different way: V W = Va W a = gab V b W a = g ab Va Wb EXAMPLE 2-4 Consider the metric in plane polar coordinates with components given by gab = 1 0 0 r2 and g ab = 1 0 0
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1 r2
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Let V a = (1, 1) and Wa = (0, 1). Find Va , W a , and V W. SOLUTION 2-4 Proceeding in the usual manner, we nd Va = gab V b Vr = grr V r = (1)(1) = 1 V = g V = r 2 (1) = r 2 Va = (1, r 2 )
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In an analogous manner, we obtain W a = g ab Wb W r = grr Wr = (1)(0) = 0 1 1 W = g W = (1) = 2 2 r r 1 W a = 0, 2 r For the dot product, we nd V W = gab V a W b = grr V r W r + g V W 1 =0+1=1 = (1)(0) + r 2 r2 As a check, we compute V W = V a Wa = V r Wr + V W = (1)(0) + (1)(1) = 0 + 1 = 1
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Passing Arguments to the Metric
Thinking of a tensor as a map from vectors and one forms to the real numbers, we can think of the metric in different terms. Speci cally, we can view the metric as a second rank tensor that accepts two vector arguments. The output is a real number, the dot product between the vectors g (V, W ) = V W Looking at the metric tensor in this way, we see that the components of the metric tensor are found by passing the basis vectors as arguments. That is g (ea , eb ) = ea eb = gab In at space, we have ea eb = ab . EXAMPLE 2-5 Given that the basis vectors in cartesian coordinates are orthnormal, i.e., x x = y y = z z = 1 (2.19)
Vectors, One Forms, Metric
with all other dot products vanishing, show that the dot products of the basis vectors in spherical polar coordinates give the components of the metric. SOLUTION 2-5 The basis vectors in spherical coordinates are written in terms of those of cartesian coordinates using the basis vector transformation law ea =
b a eb
where the elements of the transformation matrix are given by
xb = a x
The coordinates are related in the familiar way: x = r sin cos , We have r = r r r x + y + z x y z = sin cos x + sin sin y + cos z y = r sin sin , z = r cos
Therefore, the dot product is grr = r r = sin2 cos2 + sin2 sin2 + cos2 = sin2 cos2 + sin2 + cos2 = sin2 + cos2 = 1 The basis vector is given by = r cos cos x + r cos sin y r sin z and the dot product is g = = r 2 cos2 cos2 + r 2 cos2 sin2 + r 2 sin2 = r 2 cos2 cos2 + sin2 + sin2 = r 2
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