The Signature of a Metric in Visual Studio .NET

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The Signature of a Metric
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The sum of the diagonal elements in the metric is called the signature. If we have ds 2 = dt 2 + dx 2 + dy 2 + dz 2 then 1 0 = 0 0 0 1 0 0 0 0 1 0 0 0 0 1
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The signature is found to be 1 + 1 + 1 + 1 = 2
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Vectors, One Forms, Metric
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The Flat Space Metric
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By convention, the at metric of Minkowski spacetime is denoted by ab . Therefore 1 0 = 0 0 0 0 0 1 0 0 0 1 0 0 0 1
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provided that ds 2 = dt 2 + dx 2 + dy 2 + dz 2 .
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The Metric as a Tensor
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So far we have casually viewed the metric as a collection of the coef cients found in a given line element. But as we mentioned earlier, the metric is a symmetric second rank tensor. Let s begin to think about it more in this light. In fact the metric g, which we sometimes loosely call the line element, is written formally as a sum over tensor products of basis one forms g = gab dx a dx b First, note that the metric has an inverse, which is written with raised indices. The inverse is de ned via the relationship
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c gab g bc = a
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(2.15)
c where a is the familiar (hopefully) Kronecker delta function. When the metric is diagonal, this makes it easy to nd the inverse. For example, looking at the metric for spherical coordinates (2.12), it is clear that all components gab = 0 when a = b. So using (2.15), we arrive at the following:
grr grr = 1 grr = 1 g g = g r 2 = 1 g = g g 1 r2 2 2 = g r sin = 1 g = 1 r 2 sin2
Vectors, One Forms, Metric
These components can be arranged in matrix form as 1 0 = 0 0
1 r2
g ab
1 r 2 sin2
(2.16)
Index Raising and Lowering
In relativity it is often necessary to use the metric to manipulate expressions via index raising and lowering. That is, we can use the metric with lowered indices to lower an upper index present on another term in the expression, or use the metric with raised indices to raise a lower index present on another term. This probably sounds confusing if you have never done it before, so let s illustrate with an example. First, consider some vector V a . We can use the metric to obtain the covariant components by writing Va = gab V b (2.17)
Remember, the summation convention is in effect and so this expression is shorthand for Va = gab V b = ga0 V 0 + ga1 V 1 + ga2 V 2 + ga3 V 3 Often, but not always, the metric will be diagonal and so only one of the terms in the sum will contribute. Indices can be raised in an analogous manner: V a = g ab Vb Let s provide a simple illustration with an example. EXAMPLE 2-2 Suppose we are working in spherical coordinates where a contravariant vector X a = (1, r, 0) and a covariant vector Ya = (0, r 2 , cos2 ). Find X a and Y a . SOLUTION 2-2 Earlier we showed that grr = grr = 1, g = r 2 , g = 1 g = r 2 sin2 . Now X a = gab X b
1 , g r2
(2.18)
= r 2 sin2 ,
and so
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X r = grr X r = (1)(1) = 1 X = g X = r 2 (r ) = r 3 X = g X = r 2 sin2 (0) = 0 Therefore, we obtain X a = 1, r 3 , 0 . For the second case, we need to raise indices, so we write Y a = g ab Yb This gives Y r = grr Yr = (1)(0) = 0 Y = g Y = Y = g Y = and so Y a = 0, 1, cot2 . r
1 r2
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