Vectors, One Forms, Metric in .NET

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that of ordinary cartesian coordinates. That one is given by ds 2 = d x 2 + d y 2 + d z 2 For spherical coordinates, we have ds 2 = dr 2 + r 2 d 2 + r 2 sin d 2 Meanwhile, the line element for cylindrical coordinates is ds 2 = dr 2 + r 2 d 2 + dz 2 (2.10) (2.9) (2.8)
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We can write these and other line elements in a succinct way by writing the coordinates with indices and summing. Generally, the line element is written as ds 2 = gab (x) dx a dx b (2.11)
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where gab (x) are the components of a second rank tensor (note that we can write these components as a matrix) called the metric. You can remember what this thing is by recalling that the components of the metric are given by the coef cient functions that multiply the differentials in the line element. For a metric describing ordinary three-dimensional space, these components are arranged into a matrix as follows: g11 g12 g13 gij = g21 g22 g23 g31 g32 g33 For example, looking at (2.8), we see that for cartesian coordinates we can write 1 0 0 gij = 0 1 0 0 0 1 When dealing with spacetime, we take the convention that the time coordinate is labeled by x 0 and write the matrix representation of the metric as g00 g10 = g20 g30 g01 g11 g21 g31 g02 g12 g22 g32 g03 g13 g23 g33
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For spherical coordinates, we make the identi cation (r, , ) x 1 , x 2 , x 3 and using (2.9) write 1 0 0 r2 gij = 0 0 0 0 2 2 r sin
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(2.12)
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For cylindrical coordinates, the matrix takes the form 1 0 gij = 0 r 2 0 0 0 0 1
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(2.13)
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In many cases, like the ones we have considered so far, the metric has components only along the diagonal. However, be aware that this is not always the case. For example, a metric can arise in the study of gravitational radiation that is called the Bondi metric. The coordinates used are (u, r, , ) and the line element can be written as ds 2 = f 2 e g 2r 2 e2 du 2 + 2e2 dudr + 2gr 2 e2 dud r r 2 e2 d 2 + e 2 sin2 d 2
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(2.14)
Here f, g, , are functions of the coordinates (u, r, , ). With these coordinates, we can write the matrix representation of the metric as guu gr u = g u g u gur grr g r g r gu gr g g gu gr g g
A good piece of information to keep in the back of your mind is that the metric is symmetric; i.e., gab = g ba . This information is useful when writing down components of the metric associated with the mixed terms in the line element. For example, in this case we have 2e2 dudr = e2 dudr + e2 dr du = gur dudr + gr u dr du 2gr 2 e2 dud = gr 2 e2 dud + gr 2 e2 d du = gu dud + g u d du
With this in mind, we write f 2 e g 2r 2 e2 r e2 gab = gr 2 e2 0
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The metric is a coordinate-dependent function, as can be seen from the examples discussed so far. Furthermore, recall that different sign conventions are used for space and time components. As an example, consider at Minkowski space written with spherical coordinates. It is entirely appropriate to use ds 2 = dt 2 dr 2 r 2 d 2 r 2 sin2 d 2 and it is equally appropriate to use ds 2 = dt 2 + dr 2 + r 2 d 2 + r 2 sin2 d 2 The important thing is to make a choice at the beginning and stick with it for the problem being solved. When reading textbooks or research papers, be aware of the convention that the author is using. We will use both conventions from time to time so that you can get used to seeing both conventions.
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