Nonholonomic Bases in Visual Studio .NET

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Nonholonomic Bases
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A nonholonomic basis is one such that the basis vectors are orthonormal with respect to the chosen metric. Another name for this type of basis is a noncoordinate basis and you will often hear the term orthonormal tetrad (more below). This type of basis is based on the fundamental ideas you are used to from freshman physics. A set of orthogonal vectors, each of unit length, are chosen for the basis. We indicate that we are working with an orthonormal basis by placing a hat or carat over the indices; i.e., basis vectors and basis one forms are written as
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An orthonormal basis is of interest physically and has use beyond mere mathematics. Such a basis is used by a physical observer and represents a basis with respect to the local Lorentz frame, while the coordinate basis represents
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Cartan s Structure Equations
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the global spacetime. As we move ahead in this chapter, we will learn how to transform between the two representations. We can expand any vector V in terms of a coordinate or a noncoordinate basis. Just like any expansion, in terms of basis vectors, these are just different representations of the same vector
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V = V a ea = V a ea
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Since this basis represents the frame of the local Lorentz observer, we can use the at space metric to raise and lower indices in that frame. As usual, the signs of the components can be read off the metric. For example, with a metric with the general form ds 2 = dt 2 dx 2 , we have a b = diag (1, 1, 1, 1). The basis vectors of a nonholonomic basis satisfy ea eb = a b (5.4)
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In a nutshell, the basic idea of creating a nonholonomic basis is to scale it by the coef cient multiplying each differential in the line element. Let s illustrate this with an example. In the case of spherical polar coordinates, a noncoordinate basis is given by the following: er = r , 1 e = , r e = 1 r sin
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An easy way to determine whether or not a given basis is holonomic is to calculate the commutation coef cients for the basis. We do this for the case of spherical polar coordinates in the next section.
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Commutation Coef cients
The commutator is de ned to be [A, B] = AB BA From calculus, we know that partial derivatives commute. Consider a function f (x, y). It is true that 2f 2f 2f 2f = =0 x y y x x y y x
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Let s rewrite this in terms of the commutator of the derivatives acting on the function f (x, y): 2f 2f = x y y x So we can write =0 , x y Looking at the holonomic basis for spherical polar coordinates, we had er = r = , r e = = , e = = x y y x f = , x y f
From the above arguments, it is clear that [er , e ] = er , e = e , e = 0 Now let s consider the nonholonomic basis we found for spherical polar coordinates. Let s compute the commutator er , e . Since we re new to this process we carry along a test function as a crutch. 1 er , e f = r , r = = f = r 1 f r 1 ( r f ) r
1 1 1 f + ( r f ) ( r f ) 2 r r r 1 f r2
Using the de nitions given for the nonholonomic basis vectors, the end result is er , e f = 1 f r2
1 = e f r
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The test function is just being carried along for the ride. Therefore, we can drop the test function and write 1 er , e = e r This example demonstrates that the commutators of a nonholonomic basis do not always vanish. We can formalize this in the following way: ei , e j = Ci j k ek
(5.5)
The Ci j k are called commutation coef cients. The commutation coef cients are antisymmetric in the rst two indices; i.e. Ci j k = C ji k If the following condition is met, then the basis set is holonomic. Ci j k = 0 i, j, k
We can also compute the commutation coef cients using the basis one forms, as we describe in the next section.
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