Connections, Curvature, Torsion, Non-metricity
On Feb 27, 2007, at 10:56 PM, Paul Zielinski wrote:
OK Jack, fire away. Give me your best shot.
Here is a direct quote from Carroll:
"...any set of connections can be expressed as some fiducial connection plus a tensorial correction." (Notes on General Relativity, p 59).
Now explain to us why this doesn't apply to the LC connection?
I told you a jillion times. Read the rest of what he says!
Clearly, a zero "correction" is no correction at all. For Carroll's statement to be non-trivial, the tensorial correction must be non-vanishing.
Yet you are saying, in effect, that in the case of the LC connection the tensorial correction is always exactly zero. I think you have to prove
this mathematically, in the abstract, without any appeal to any particular physical theory.
Sean Carroll like many others points out that most generally
Affine = LC + Torsion + Nonmetricity
Torsion and non-metricity are GCT 3rd rank tensors.
LC is not a tensor.
GR is precisely that theory in the set of theories with different connections such that
Affine = LC
End of trivial story.
Physically LC parallel transports vectors along worldlines.
Going around a closed worldline the difference of orientation of the vector with itself is a measure of the average curvature over the area of the loop - shrunk to zero - i.e. crystal disclination defect.
Limit of the ratio of angle deficit to area of shrinking loop is the sectional curvature dimensions 1/Area.
If there is non-metricity the length of the vector will also change in the parallel transport and in a loop you will get a discrepancy in lengths of the vector unless you impose a kind of Bohr-Sommerfeld quantization on the closed loop.
If there is torsion starting with two sides of an infinitesimal parallelogram and parallel transporting one against the other both ways the loop does not close to second order - there is a torsion gap, i.e. crystal dislocation defect.
In my theory you have a set of 0-forms the Goldstone phases of the vacuum coherence ODLRO Higgs field.
From the 0-forms comes a set of 1-forms.
There are 10 1-forms.
4 tetrad 1-forms & 6 spin connection 1-forms.
Then you have GR + torsion trivially.
i.e. a theory with
A = LC + Torsion Tensor
beyond 1915 GR of course.
dO-form = 1-form
d^2)-form = 0
However that 1-form need not be exact if the 0-form has a singularity, i.e.
Integral of the 1-form over a non-bounding 1-cycle without boundary = winding number integer
Bohr-Sommerfeld condition from non-trivial 1-homotopy.
Then you can pretend Stoke's theorem works i.e. flux without flux - these are 1D string vortex defects.
The integral of the 1-form around the loop is the surface integral of a ghostly 2-form flux through the area of the loop. This is related to the Bohm-Aharonov effect.
Similarly given 2 0-forms A & B, I define the non-closed 1-form as
C = dA/\B - A/\dB 1-form
dC = 2dA/\dB =/= 0 2-form
I can now play the same game one dimension higher! These are 0D point monopole stable defects.
Now we have star gate wormholes!
The mouth of the wormhole is the non-bounding 2-cycle without boundary - the portal like they see on the Skinwalker Ranch!
The deRham integral of 2-form C over the closed 2-surface is now an integer wrapping number ~ Bekenstein BITS for the entropy of black hole event horizons, de Sitter dark energy future observer horizons, Unruh effect, Hawking radiation, you name it. I got it all topologically pre-metrically.
There is again the world hologram ghostly volume 3-form without volume from the actual hologram area 2-form dC.
OK everything I need comes from C as the template.
C^a^b = dA^a/\B^b - A^a/\dB^b
The 4 Einstein curvature tetrad 1-forms are the diagonal matrix elements
The 6 torsion field spin connection 1-forms are the anti-symmetrized off-diagonal matrix elements
C^[a,b] = - C^[b,a]
Einstein's 1915 GR is the limit
C^[a,b] = 0
D = d + W^ac/\
e^a = I^a + C^a^a
The torsion 1-form is
T^a = De^a = 0
This determines the non-dynamical zero torsion spin connections W^ac from
de^a + W^ac/\e^c = 0
The curvature 2-form is
R^a^b = DW^a^b
The Einstein-Hilbert action density is
and the rest is history.
See Rovelli Ch II for the details.
Plus it's obvious how to generalize for torsion.
So I have 8 0-form Goldstone phases from 9 real Higgs fields with vacuum manifold S^8.
This fits most naturally in terms of stable defects into a 9+1 D spacetime. Hey that's interesting. Where have we seen that before?
I want to compactify to 3 large space dimensions so I can use
A^2 = A^aAa
B^2 = B^aBa
take square roots.
And I am back to the Bekenstein world holography bits with point defects in the world crystal's spacelike slices. They are the "lattice points"!
"If we knew what it was we were doing, it would not be called research, would it?"
- Albert Einstein