Vectors as axes.

[Ken S. Tucker]

Since I was introduced to Vector Analysis, I have had
reservations (ambiguity) about a separate meaning of
axes and equations defining vectors and lines on/in a
selected Coordinate System (CS).
The thought becomes more acute in generalized CS,
such as in curvilinear, n-dimensional, CS's, inclusive
of those where the Curvature tensor is nonzero.

So consideration was given to treating axes as vectors,
and is much more successful than I expected.
The procedure, to start regards the Kronecker Delta
transformation.

Defining axes as vectors, (rank 1 contravariant tensors
herein), and then subjecting them to the Kronecker Delta
requirement produces a generalized non-symmetrical metric
tensor in Eq.(4) far below.
(Some overview of the math is below ------------).
It has been lab tested and verified using GR & Unitivity.

g_uv = x_u;v - [uw,v]x^w Eq.(4).

I intend to post some abbreviated results, unless the
assumptions I've made are erroneous.
Regards
Ken S. Tucker


----------------------------------------------------------
NOTATIONAL NOTE: & is partial diff, a,b are indices....

&x^a / &x^b = Kronecker Delta = 1or 0 if a=b or a=/=b respectively.

We'll write that in ascii as, &(a/b).
----------------------------------------------------------
COVARIANT FORM of KRONECKER DELTA:

We next consider the covariant derivative of x^a, w.r.t to x^b,

x^a;b = &(a/b) + {a,bc} x^c Eq.(1)

( {a,bc} is a Christoffel of the 1st kind).

The difference between tensors is a tensor therefore

{a,bc} x^c Eq.(2)

is a tensor.

This transformation proof follows,

{u,vw}' x'^w = &( u'/a , b/v' ) {a,bc} x^c .

where x'^u = &(u'/a) x^a.

When the Curvature tensor R_abcd = 0 then Eq.(2)=0.
If R_abcd =/=0 then Eq.(2) =/=0.

------------------------------------------------------
OUTER PRODUCT of metric tensor g_ae, ON EQ.(1).

g_ae ( &(a/b) + {a,bc} x^c = &(a/b) + g^ad [bc,d} x^c )

=> g_eb + [ec,b] x^c = x_e;b

and a generalized metric,

g_eb = x_e;b - [ec,b] x^c Eq.(3) .
----------------------------------------------------------------------------
Tidying up Eq.(3) for theoretics applications renders,

g_uv = x_u;v - [uw,v]x^w Eq.(4) .

-------------------------------------------------------------------------

[Ken S. Tucker]

Hello again.
From my 1st post, a Generalized metric tensor was
derived, (the reasoning provided therein),

----------------------------------------------------------------------------

g_uv = x_u;v - [uw,v]x^w Eq.(4) .

----------------------------------------------------------------------------

Let's subject that to a theoretical experiment, using
the axis of time (x^0 =ct) only, to produce,

g_00 = x_0 ; 0 - [00,0} x^0 Eq.(4.0)

I'll try using the symbol ' Φ ' for gravitational potential,
and follow GR's

g_00 = 1 - 2Φ

as a test.

The Christoffel reduces to,

[00,0] = 1/2 g_00,0

wherein we'll presume the x^0 = r in Φ = (GM/rc^2),
to give us,

1/2 g_00,0 = -Φ/x^0

that subs into Eq.(4.0) to provide,

g_00 = x_0 ; 0 - Φ Eq.(4.1).

That in turn yields,

x_0 ; 0 = 1 - Φ .

and everything balances, quite nicely.
From that we can conclude the Generalized metric tensor
is in accord with GR at 1st glance.
Regards
Ken S. Tucker

ΔΘΛΞΠΣΦΨΩαβγδεζηθκλμνξπρφψ∂∆∑√∫≈≠≡⌠⌡⌡

[Ken S. Tucker]

TYPO

> Let's subject that to a theoretical experiment, using
> the axis of time (x^0 =ct) only, to produce,
>
> g_00 = x_0 ; 0 - [00,0} x^0 Eq.(4.0)
>
> I'll try using the symbol ' Φ ' for gravitational potential,
> and follow GR's
>
> g_00 = 1 - 2Φ
>
> as a test.
>
> The Christoffel reduces to,
>
> [00,0] = 1/2 g_00,0
>
> wherein we'll presume the x^0 = r in Φ = (GM/rc^2),
> to give us,
>

1/2 g_00,0 = -Φ/x^0 ***** TYPO

Should be + Φ/x^0
(Material below is unaffected).

[Ken S. Tucker]

On Jan 3, 11:03 am, "Ken S. Tucker" <dynam...@vianet.on.ca> wrote:
> > Let's subject that to a theoretical experiment, using
> > the axis of time (x^0 =ct) only, to produce,
>
> > g_00 = x_0 ; 0 - [00,0} x^0 Eq.(4.0)
>
> > I'll try using the symbol ' Φ ' for gravitational potential,
> > and follow GR's
>
> > g_00 = 1 - 2Φ
>
> > as a test.
>
> > The Christoffel reduces to,
>
> > [00,0] = 1/2 g_00,0
>
> > wherein we'll presume the x^0 = r in Φ = (GM/rc^2),
> > to give us,
>

>> 1/2 g_00,0 = Φ/x^0

> > that subs into Eq.(4.0) to provide,
>
> > g_00 = x_0 ; 0 - Φ Eq.(4.1).
>
> > That in turn yields,
>
> > x_0 ; 0 = 1 - Φ .
>
> > and everything balances, quite nicely.
> > From that we can conclude the Generalized metric tensor
> > is in accord with GR at 1st glance.

--------------------------------------------------------------------------------------------
As a calculus student many names of derivatives are encountered.
A beginnners list is,
1) Ordinary uses d
2) Partial uses ∂
those together makes a
3) Total derivative by summing the
partials ( ∂ ) of independant variables.
4) GRAD is the maximum slope of a curve,
usually a Vector Analysis thing.

Now is the "covariant derivative" applied to any rank of tensor,
D (x^ijj.._abc...) / D parameter(s)
(usually a single parameter).

That in turn yields up the "Intrinsic, iow's absolute" derivative,

δ(tensor) =D (tensor) / D (parameters) x D (parameters)

I'm having a problem expressing "covariant" and "absolute"
derivatives in English.

Allow me to provide a physical example.
Let U^μ be a points relative velocity in 4D.

((In terms of relativity U_i = 0 , i=1,2,3, removes absolute
uniform velocity, U_i is a covariant 3 vector in space))

Then the absolute derivative of U^μ ,expressed as

δ ( U^μ ) = 0 (1)

as absolute acceleration cannot exist. That's naturally
common sense, since one may place the CS origin
attached to any point, and the Laws of Nature apply
in that Generally Relativistic.

Eq.(1) makes GR straight forward as does the explanation
provided.

I ride two ways, math & physics, here's a sample
that unifies them,

Let a flat line be X.

Let a minimum variation from a flat line be "h".

Allow the variation to be expressed by,
S^2 = X^2 + h

I enable S dS = X dX and S ∆S = X ∆X.

to enable the continuum in the 1st place and a quantized
variation in the incremental form in the later.
(( I'm unifying continuum and incremental derivarives ))

As a theoretician I wear two hats,
math LHS and physics RHS (as above) enabling both.
That way I circumvent the philosophy of a prejudicial
formation as to 'is the universe a continuum or quantized'
Regards
Ken S. Tucker
ΣΘΛΞΠΣΦΨΩαβγδεζηθκλμνξπρφψ∂∆∑√∫≈≠≡⌠⌡⌡

[brad]

On Feb 2, 6:33 pm, "Ken S. Tucker" <dynam...@vianet.on.ca> wrote:





> to enable the continuum in the 1st place and a quantized
> variation in the incremental form in the later.
> (( I'm unifying continuum and incremental derivarives ))

Have you investigated Exotic smoothness and quantum gravity?
Google Asselmayer-Maluga and also on the arxiv.

Brad

[Ken S. Tucker]

Hi Brad, nice to meet you.
I got 37K+ hit's from googling that, which one do you recommend?
(I'm on dial-up here in canutistan, very -s l o w- speed).

Yes, I've researched the problem, at this site ,

http://physics.trak4.com/

is a 2 pg brief,

http://physics.trak4.com/GR_Charge_Couple.pdf

The latter, provides an electronic solution to GR's Guv=Tuv.
Electronic means relating two charges, I find GR theory
produces a residual attactive force.in charge couples.
That's in the brief near Eq.(5).
That reduces to Newtonian like gravity, when summed over
many charge couples.

If you'll permit units of Action= charge^2 then Eq.(4)
becomes S^2 = X^2 + an action (ab).
So GR does provide for 'quantized gravity'.

The subject has gained importance.
It implies our current g-wave detectors will remain nulled.
When one takes the time based derivative of the metric
in Eq.(2), it creates ElectroMagnetic Radiation, (EMR),
but NOT g-waves.
We have/had high expectations that we might use the
g-wave spectrum as an astronomical tool, and I guess
we've spent ~$1billion globally using LIGO type units to
view that spectrum, but the LIGO's decade+ of silence is
deafening.
The negative result of the Micheson Morley experiment
actually enabled a serious consideration of Relativty,
upon which a new physics was born.
A negative result of LIGO is an unexcepted possibilty.
Regards
Ken S. Tucker