Monday, January 03, 2005

Circular Reasoning
First Paul if you disinguish "Cartesian CS" from "Riemann normal CS" and other terms you have used without definition in context you need to give concrete examples of each in math notation, otherwise what you write is unintelligible vague and useless.

I do admit Paul that this subject gets confusing because there are several different connections each with a different covariant derivative. Visser even gets confused on that in his notes. His key equation for

A = (LC) + S + Q

is wrong.

When I asked you what Del means in the definition of nonmetricity S you said |w and that is wrong as I will show in a simple case directly. You were, however, citing Visser correctly. Visser got sloppy easy to do. In 1916 GR this is not a problem because S = Q = 0 by hypothesis.

The point is if you write

A = (LC) + S + Q

Then S and Q must be GCT tensors, i.e. defined not with respect to the A covariant derivative |w but, rather, with respect to the (LC) covariant derivative ;w.

In general if you have any connection C and wish to add a C-tensor field to it like B so that

C' = C + B

Then you must define B in terms of |c not in terms of |c'. This is an ITERATIVE PROCEDURE to avoid circular reasoning here equivalent to dividing by zero to get 1 = 0 or anything you like without logic. It's not unlike the Russell Paradox.

Example 1 1916 GR

For example, 1916 GR itself where we use the ordinary flat covariant ,w to define the (LC) curved covariant ;w.

(LC)^abc = (1/2)g^a^d(gdc,b + gbd,c - gbc,d)

Note that ,b are ordinary flat spacetime partial derivatives.

Essentially the ; curved spacetime covariant derivative is of the form

; = , + (LC)

Like in EM where

p = mv - (e/c)A

The EM vector potential is in internal fiber space with S1 circle fiber what (LC) is in the curved spacetime of 1916 GR. The EM magnetic field curlA is like the vacuum conformal curvature (LC) curl of itself. The Bohm-Aharonov effect is like the Vilenken-Taub "curvature without curvature" from a closed non-exact connection 1-form (R. Kiehn).

Now, if you use what Visser wrote in his notes literally in the case Q = S = 0 for 1916 GR then one gets the NONSENSE result

(LC)^abc = (1/2)g^a^d(gdc;b + gbd;c - gbc;d) = 0

Since S is from gdc;b = 0 in 1916 GR.

Therefore, ignoring the iterative emergence of new orders results in the absurd result (LC) = 0 if you use Visser's erroneous formula in this simplest of all cases.

Example 2 Alex Poltorak's theory, torsion Q = 0 but non-metricity =/= 0.

A = (LC) + S

But what is S?

Sabc = (1/2)(Dagbc + Dbgac - Dcgab)

The problem is what is D?

We have two choices

1. D = ; = , + (LC)

2. D = | = ; + S

Visser says use 2, but as shown in Example 1 that leads to nonsense. One must use 1


Sabc = (1/2)(gbc;a + gac;b - gab;c)

Sabc is a GCT tensor. It is not consistent with 1916 GR unless Sabc = 0, but you can consider a new larger theory where Sabc is a new dynamical field from HYPERSPACE for example. It is not RIGID. It is not NONDYNAMICAL. It is not "just differential geometry". So I do not understand Alex's "physics" here.

Suppose we use Visser's wrong formula literally, so that

S'abc = (1/2)(gbc|a + gac|b - gab|c)

where | = ; + S

Symbolically (forgetting sums and indices) this is

S' = g| = g; + S'g

Putting in the indices, you will get a God-Awful MESS that I bet has no consistent solution other than S' = 0.


S' = g| = g; + (g; + S'g)g = g; + g;g + S'gg

Indeed this is an infinite series. I suspect its only consistent solution is S' = 0.

Also there is no reason to think that this S' even if you could get a consistent solution S' =/= 0 is a GCT tensor, which you need for your problem!

Indeed, this is close to a rigorous proof that you are barking up the wrong tree.

Of course none of this happens in Einstein's 1916 GR, which is why it is unfamiliar.

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