The Road to Reality

Exercise [12.16]
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Author:  jbeckmann [ 05 Dec 2010, 09:13 ]
Post subject:  Exercise [12.16]

Here is my (WRONG!) proposal:
Exercise_12_16.pdf [31.75 KiB]
Downloaded 129 times


Better try this one:
Exercise_12_16amended.pdf [33.13 KiB]
Downloaded 149 times

Author:  Roberto [ 06 Dec 2010, 14:27 ]
Post subject:  Re: Exercise [12.16]

I think your proof of "a_[rs a_u]v implies that a_rs is simple " is not correct.

The proof is based on the assumption that there is a vector x^v satisfying condition
x^m x^n a_mn =1

This is never possible, because being a_mn antisymmetric and x^m x^n symmetric their contraction is always 0.
This can easily be seen in the simple 2 dimensional case where
a_11=a_22=0, a_12 = -a_21=a
x^m x^n a_mn= x_1 x_2 a - x_2 x_1 a = 0.

The formal proof for the general case is:
x^m x^n a_mn = x^n x^m a_nm = - x^n x^m a_mn = - x^m x^n a_mn
(where in the first step I just renamed the contracted indices and in second step antisymmetry of a was exploited) therefore x^m x^n a_mn =0.

Author:  jbeckmann [ 06 Dec 2010, 20:42 ]
Post subject:  Re: Exercise [12.16]

You are right, Roberto! Thanks for the hint and sorry for the error.
I hope this fixes the bug:
Exercise_12_16amended.pdf [33.13 KiB]
Downloaded 134 times

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