A Lecture on New Math
given by Dr Horst Eckardt and Dr Myron W. Evans
with commentary by G.W. Bruhn

Copied from Evans' Blog
http://www.atomicprecision.com/blog/2007/03/19/usage-of-dummy-indices/

Subject: Fwd: Question: Usage of dummy indices
Date: Mon, 19 Mar 2007 02:38:56 EDT

A lot of expressions of Cartan geometry contain summations over indices, called “dummy indices”. Consider the following example wich occurs in several proves of the Evans lemma:

                q^μ_a R^a_μ := R

This term represents a double sum. It is multiplied then by q^a_μ to give (with the normalization of the tetrad)

                q^a_μ q^μ_a R^a_μ = q^a_μ R

This multiplication is inadmissible, as can be seen by using the extended form of the former equation. Dr Eckardt has not understood the concept of "dummy indices".

The question is: why is it allowed here to execute in this equation the substitution

                q^a_μ q^μ_a = 1 ?

Not allowed in addition because this equation is wrong: ... = 4 would be correct.

There are further terms present depending on the dummy indices.

Now Dr Eckardt restarts:

I believe that the correct order of arguments is as follows: Define

                R^a_μ =: R q^a_μ

This equation requires the proportionality of the matrices (R^a_μ) and (q^a_μ) and thus is invalid in general.

with a scalar function R. Then follows

                R^a_μ q^μ_a = R.

This gives the Evans Lemma.

Horst Eckardt

Final remark
The above conclusion requires proportionality and therefore does not apply in general. More, we would obtain

                R^a_μ = R q^a_μ         =>         R^a_μ q^μ_a = 4 R

so

                R^a_μ = ¼ R q^a_μ         =>         R^a_μ q^μ_a = R

as well. The last equation is correct due to the definition of the scalar curvature R := g^μν R_μν. We obtain

                R = R_μν g^μν = R_μ^a q_a^σ g_σν g^μν = R_μ^a q_a^σ δ_σ^μ = R_μ^a q_a^μ .

But it is completely dubious what the equation R = R_μ^a q_a^μ could help to prove Evans' Lemma who claims another proportionality (see equ.(11) in Evans' akeyderivations1and2.pdf):

                                R q_λ^a = ∂^μ (Γ^ν_μλq_ν^a − ω^a_μbq_λ^b)                                 (11)

And Dr Myron W Evans replied:

This is exemplary procedure again by Dr Eckardt, consisting of checking the mathematics. Both methods are equivalent, it seems to me, but your method is another useful check. The rule on the dummy indices is to choose a given contravariant index and match it with a given covariant index, id there is another index with the same label it is not counted. Then the pair of dummy indices can be relabelled. The most extensive and SELF CHECKING use of dummy indices is given in the appendices of chapter 17 of volume one. The rules are given in much greater detail than usually available in a textbook. These appendices work out the exercises for graduate students given by Carroll in chapter three.
Of course, nowhere in Carroll's publications such nonsense can be found.
For the wider readership here, it is easily possible to download all of Carroll’s notes and put them on your computer as desktop icons. Then you can double check statements made here concerning Carroll’s notes. In his book he greatly extends the notes. Teh two main books are:
1) S. P. Carroll, “Space-time and Geometry: An Introduction to Geenral Relativity” (Addison Wesley, New York, 2004). His downloadable notes are found by googling “The Cartan Structure Equations”. These are very clear, and are based on notes to graduate students at Harvard, UCSB and Chicago.
2) M. W. Evans, “Generally Covariant Unified Field Theory: the Geometrization of Physics” (Abramis, 2005, 2006 onwards), volumes 1 - 3 available, volumes 4 and 5 in prep.