Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.



Tetrads and Spin Connections as YangMills Gauge Potentials
Posted:
Oct 2, 2007 12:52 AM


PS Consider
F = T + R = (I/\ + A/\ + S/\)(I + A + S)
OK so now it really looks like YangMills F with A & S appearing symmetrically!
Note that
(I/\ + A/\ + S/\)I = 0
therefore we have the 2form field from the ten compensating gauge potentials of Poincare group
F = T + R = d(A + S) + (A + S)/\(A + S)
F is peculiar with mixed Lorentz group indices
T^a + S^a^b = dA^a + dS^a^b + A^a/\A^b + S^ac/\A^c + S^ac/\S^c^b
which splits into
T^a = dA^a + S^acA^c torsion field
R^a^b = dS^a^b + A^a/\A^b + S^ac/\S^c^b curvature field
On Oct 1, 2007, at 3:11 PM, Jack Sarfatti wrote:
v2 (expanded & corrected too many "/\" in R formulae) On Sep 29, 2007, at 6:22 PM, Jack Sarfatti wrote:
Note also, the effect of the equivalence principle makes a difference in comparing gravity fields to YangMills fields.
Look at the exterior covariant derivative 1form
D = (d + S)/\
note that d is usually written without the /\ that is tacitly understood since
d(pform) = (p + 1)form
dual to boundary operator
&(p + 1) dim manifold = pdim manifold so that Stokes theorem generalizes to
Integral of d(pform) on p + 1 dim manifold = Integral of (pform) on the boundary &(p + 1)manifold
for example when p = 0 in 3D space i.e. fundamental definite integral formula of calculus
d(0form) = gradient of a function
p = 1 e.g. loop integral of vector (1form) field is interior flux (2form) integral
d(1form)  curl of a vector (2form)
p = 2 Gauss's divergence theorem  end of short digression
d(2form) = divergence of a 1form vector field in 3D space.
can generalize to Minkowski spacetime of 1905 SR
Stoke's theorem is key. A star gate traversable wormhole time machine to past (or equivalently in this regard a weightless zero gforce geodesic glider warp drive bubble in sense of Alcubierre's toy model metric) held open by universally antigravitating positive zero point dark energy density with equal and opposite negative pressure (w = 1 with GR source factor 1 + 3w) depends on multiplyconnected spacetime corresponding to topological defects in the vacuum ODLRO field whose phase modulation determines the local curved tetrad fields and the local torsion field spin connections from localizing the 10parameter Poincare symmetry group of special relativistic quantum field theory. The latter is backgrounddependent the former is not.
Multiple connectivity means, in our specific problem of metric engineering practical warp and wormhole, closed 2 surfaces (portals or wormhole mouths) without boundary that are not boundaries of the 3D space wormhole tunnel.
<pastedGraphic8.tiff> http://www.kahl.net/astro/graphics/wormhole.gif
the 2D mouth is a circle (1 space dimension removed in picture at a fixed "time"
the 3D space Dr. Who walks through is the 2D tube in the picture  you are flatlander bug constrained to surface in the picture.
"dark energy" = "exotic matter"
Since the vacuum ODLRO field is singlevalued that means that the geometrodynamic field area density flux through a nonbounding closed 2D surface is quantized and this explains the HawkingBekenstein formula corresponding to point gravity monopole defects in the fabric of dynamical 3D space.
Entropy/kB = Horizon Area/4 Quantum of Area Flux = N
i.e. Newton's Planck area hG/c^3 ~ 10^66 cm^2
with the world hologram formulae
Size of wavelet Quantum Foam Bubble ~ N^1/6(Quantum of Area)^1/2
i.e. &L ~ N^1/6Lp = (Lp^2L)^1/3
L ~ N^1/2Lp (hologram)
Horizon area ~ (10^28 cm)^2 in our pocket universe on the Cosmic Landscape with &L ~ 10^13 cm and observed dark energy density ~ hc/NLp^4 ~ 10^29 gm/cc
Back to main point
d in a sense is e
i.e. in a flat spacetime in a geodesic GIF coordinate basis
d(0form) is "4gradient" d/dx^a on a 0form function since ea^u = Kronecker delta
ea = ea^u(d/dx^u)
e^a = e^audx^u
That is we can think of d/\ as e/\
or
D/\ = (e + S)/\
e = I + A
i.e.
D/\ = I/\ + A/\ + S/\
I is when we have globally flat Minkowski spacetime and a Global Inertial Frame (GIF)
A & S are the compensating geometrodynamic field gauge potentials that first appear in a Global NonInertial Frame
in globally flat Minkowski spacetime where the curvature R^a^b = 0 and the torsion T^a = 0 but in 1916 GR R^a^b =/= 0 while still T^a = 0.
where
I/\ is in flat Minkowski spacetime of 1905 SR
A comes from using a GNIF and finally from localizing T4 to LIFs & LNIFs.
Note that A induces S that is not independent when the torsion T field vanishes globally.
In a GNIF
R = D/\S = 0
i.e.
R = (I/\ + A/\ + S/\)S = 0
and also
T = De = 0
i.e.
(I/\ + A/\ + S/\)(I + A) = 0
when rigid T4 is localized to T4(x) then we have possibility that
R = (I/\ + A/\ + S/\)S =/= 0
This notation makes the YangMills field structure more apparent.
Consider
F = T + R = (I/\ + A/\ + S/\)(I + A + S)
OK so now it really looks like YangMills F with A & S appearing symmetrically!
In 1916 GR S = S(A) is redundant as shown in Rovelli's eq. 2.89
S has and independent part when there is a torsion field i.e. full Poincare group is locally gauged.
That is 1916 GR is really a theory of the spin 1 YangMills curvature tetrad field A with a redundant S as given in Rovelli (2.89). Hence GR is renormalizable in t'Hooft's sense including vacuum ODLRO Higgs field that may give Salam's strong shortrange fgravity in addition to zero mass geometrodynamic field quanta. Indeed the composite quanta are spin 0, spin 1 and spin 2 from pairs of the fundamental spin 1 geometrodynamic quanta i.e. zero point fluctuations of the uncondensed part of the postinflation vacuum ODLRO field. Think of the geometrodynamic field random "dark energy" quanta like the zero point motions of helium 4 atoms in the T = 0 ground state that is only 10% coherent condensate even though the effective superfluid density is 100%.
typo corrected draft 2 On Sep 29, 2007, at 5:37 PM, Jack Sarfatti wrote:
The key equation is Rovelli's (2.89) for only the torsionfree curvatureonly spin connection in terms of the tetrads. It has quadratic and quartic parts. The quartic part can be put into the desired form but the quadratic part cannot. Also both parts depend on gradients in the tetrad component fields. It may be that only the torsion part of the spin connection can be put into the YangMills covariant derivative form. I have not yet confirmed that. However, this is really a side issue, as in general we need to treat the 6 spin connection 1forms S^a^b and the 4 tetrad 1forms e^a as independent YangMills type compensating local gauge field potentials in which we define the exterior covariant derivative as
D = d + S/\
Suppressing indices for simplicity. This is analogous to a YangMills theory where the curvature two form field is
R = DS
i.e. curvature field 2form = exterior covariant derivative of the spin connection YangMills potential with itself, i.e. in 1916 GR
R = dS + S/\S
This is completely analogous to the YangMills theory where
F = DA
= dA + A/\A
DF = 0
D*F = J*
DJ* = 0
In 1916 GR
DR = 0
D*R = *J
must translate in ordinary tensor notation to
Guv = kTuv
D*J = 0
corresponds to
Tuv^;v = 0 i.e. local energymomentum stress current densities conserved  all bets off on global integrals over spacelike surfaces.
All of the above is for zero torsion fields
T = De = 0
This is an auxiliary equation not found in the internal YangMills theories. The theory is more complex of course when T =/= 0 i.e. locally gauging the full 10parameter Poincare spacetime symmetry group. One must be careful on how to make the analogy of GR with YangMills theories. The analogy is perfect in Utiyama 1956 where there is only S and no e in the sense of the compensating field A where e = I + A because T4 is not locally gauged there. GCTs are put in adhoc  not pretty.
On Sep 28, 2007, at 4:25 PM, Jack Sarfatti wrote:
In trying to make gravity tetrad GR into a formal analog of YangMills I have posited
S^ac = w^acc'e^c'
e^c' are the Einstein tetrad 1forms
S^ac are the spinconnection 1forms (involving gradients of the tetrads in 2.88)
Rovelli has (2.88) for example. Now I had thought I had seen the equivalent of S^ac = w^acc'e^c' in Rovelli's book, but now I cannot find it.
Using it, the torsion field 2form is
T^a = de^a + S^ac/\e^c
= de^a + w^acc'e^c'/\e^c
which is like the YangMills field 2form
F^a = dA^a + w^acc'A^a/\A^c'
It is not clear that S^ac = w^acc'e^c' is consistent with (2.88)



