## New Rules of Tensor Calculusto Prove the Evans Lemma

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:

qμa Raμ := R

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

qaμ qμa Raμ = qaμ 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

qaμ qμa = 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

Raμ =: R qaμ

This equation requires the proportionality of the matrices (Raμ) and (qaμ) and thus is invalid in general.

with a scalar function R. Then follows

Raμ qμa = 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

Raμ = R qaμ         =>         Raμ qμa = 4 R

so

Raμ = ¼ R qaμ         =>         Raμ qμa = R

as well. The last equation is correct due to the definition of the scalar curvature R := gμν Rμν. We obtain

R = Rμν gμν = Rμa qaσ gσν gμν = Rμa qaσ δσμ = Rμa qaμ .

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

R qλa = ∂μνμλqνa − ωaμbqλb)                                 (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.

### Conclusion

#### The only thing the Doctores E & E have proven above is their unability of using the elements of Mathematical Physics. I know that Evans has got his PhD in Chemistry, and I doubt that Dr Eckardt has a degree in (Mathematical) Physics as well.

On

Dr Eckardt remarks that Dr Evans has given 13 proofs of his Lemma. After all the reader will not miss the other 12 proofs.

## Links to further applications of Evans' New Math

(02.10.2007) Comments on Evans' Collection of "Rebuttals"

(01.10.2007) Comments on Evans' Note 2 on the ECE Lemma

(30.09.2007) Comments on Evans' Note 1 on the Lorentz transform

(25.06.2007) The consequences of the invalidity of the Evans Lemma

(27.05.2007) Commentary on Evans' recent remark on the ECE Lemma

(09.04.2007) Review of the Evans Lemma

(12.03.2007) Evans "proves" the Evans Lemma again