New Math and Folklore

July 25, 2008

see also the sequel


Evans is complaining that I would always misrepresent his work. So what is this?

This tops all:

From Evans' todays blog
http://www.atomicprecision.com/blog/2008/07/24/a-short-history-of-cyberstalking-2/

. . . The incident which catalysed the barrister’s letter was a typical one
in which Bruhn asserts that d ^ (d ^ omega) is zero.

Evans should have a look at
http://en.wikipedia.org/wiki/Exact_form

He will find this:

... a closed form is a differential form a whose differential is zero (da = 0),
and an exact form is a differential form that is the differential of another differential form
(a = db for some differential form b, known as a primitive for a).
Since d² = 0, to be exact is a sufficient condition to be closed.
In this text the symbol ^ (or Ù) is suppressed.

In simpler words:

α = dÙω is an exact form since being the differential of the form ω.

Due to

                dÙd = d² = 0

we may conclude for the exact form α

                dÙα = dÙ(dÙω) = (dÙd)Ùω = 0Ùω = 0.

This is folklore!

Some additional archaeology: Evans came across the equation dÙdÙω = 0 by my hint in
http://www.mathematik.tu-darmstadt.de/~bruhn/CommentaryApp01P89.html
where I pointed to a calculation in Sect.2 of
http://www.mathematik.tu-darmstadt.de/~bruhn/onMwesPaper100-2.html

There I had performed a calculation that was left to the reader in Evans' standard textbook, S.M. Carroll's Lecture Notes [1] between [1,(3.138)] and [1,(3.140)] And S.M.Carroll assumed that an attentive reader would have kept in mind what he said on p.22:

Another interesting fact about exterior differentiation is that, for any form A,

                d(dA) = 0 ,                                 (1.84)

which is often written d² = 0. This identity is a consequence of the definition of d and the fact that partial derivatives commute, ∂αβ = ∂βα . . . .


References

[1] S.M. Carroll, Lecture Notes on General Relativity,
      http://xxx.lanl.gov/PS_cache/gr-qc/pdf/9712/9712019v1.pdf



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