## 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://www2.mathematik.tu-darmstadt.de/~bruhn/CommentaryApp01P89.html

where I pointed to a calculation in Sect.2 of

http://www2.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,
∂_{α}∂_{β}
=
∂_{β}∂_{α} . . . .

### References

[1] S.M. Carroll, *Lecture Notes on General Relativity*,

http://xxx.lanl.gov/PS_cache/gr-qc/pdf/9712/9712019v1.pdf