General Relativity as a Gauge Theory
Another look at Maxwell's Electromagnetic Field theory as a gauge theory.
The global internal symmetry group U(1)em acting on a source field like the Dirac electron is locally gauged. This demands a compensating 1-form field "potential" A to keep the dynamical action invariant under the U(1) group with an arbitrary variable phase function conjugate to the single charge operator in the Lie algebra of U(1).
The "field" is the 2-form
F = dF
Maxwell's field equations are
dF = d^2A = 0
d*F = *J 3-form eq.
*d*F = J 1-form eq.
d^2*F = d*J = 0 is local electric current density conservation
Total charge Q is a constant of the motion.
Next 1915 General Relativity:
1905 Special Relativity has the global translation space-time symmetry group T4 that guarantees conservation of total energy and total linear momentum for every system with a dynamical action invariant under T4 (Noether's theorem).
Locally gauge T4 to Diff(4), the compensating potential 1-form is B.
The total tetrad 1-form field is
e = 1 + B
When B = 0 we have Special Relativity without any gravity and inertia, i.e. when B = 0 all rest masses must be zero in order to obey Einstein's Equivalence Principle.
We impose the constraint (zero torsion)
De = D(1 + B) = 0
Where we need a second 1-form W the "spin connection" i.e.
D = d + W/
Since d1 = 0
De = dB + W/\(1 + B) = 0
dB = - W/\(1 + B)
A formal solution is the 1-form spin-connection
- dB/\(1 + B)^-1 = *W 3-form equation
Where the inverse operation reduces the rank of the form on the LHS.
(1 + B)^-1 = 1 - B + B/\B + B/\B/\B + ...
However, now we need the tangent space indices raised and lowered with the flat Minkowski metric
Both 1 & B have only 1 index and we must have the result be a 1-form.
Therefore, it follows that
(1 + B)^-1 = 1 - B
Therefore, we get the 1-form solution
-*[dB/\(1 - B)] = W
i.e. with tangent indices
-*dB^a/\(1 - B)b = W^ab
The curvature 2-form is
R = DW = dW + W/\W
Since W is a 1-form with 2 indices
(W/\W)^b means W^ac/\W^cb with summation convention
The Bianchi identity is
DR = D^2W = 0
Einstein's field equation with sources is
D*W = *J a 2-form equation
Note that the electric currents are 1-forms, but the stress-energy currents are 2-forms.
Now this really is like EM when you realize it is B that is like A in EM not W!
W = -*[dB/\(1 - B)]
dB/\(1 - B) is a 3-form and its Hodge dual is the 1-form.
But W has a dB factor that is analogous to dA = F as in
d*F = d*dA = *J(electric current)
Hence, it is natural that
D*W = *J(momentum-energy source)
is the Cartan form version of
Guv = kTuv
D*J(momentum-energy translation source) = 0
as local stress-energy current density conservation.
On Nov 6, 2005, at 2:45 PM, Jack Sarfatti wrote:
Yes, Gennady I simply made a typo saying "R" when I should have said "W" that I did not see until you pointed out the objection of the 3rd derivative. Thanks! The basic idea I had is still correct, but here is how I should have said it
Locally gauge T4 to get Diff(4) with B as the compensating 1-form field.
e = 1 + B
De = 0
dB + W/\(1 + B) = 0
R = DW is the curvature 2-form
DR = 0 is Bianchi identity
D*W = *J(source)
is Einstein's eq.
D*J(source) = 0
is local conservation of current.
OK I will rewrite it all to fix that.