I agree with Rovelli Ch 2 "Quantum Gravity" that the tetrad 1-forms
e^a = e^audx^u
are the most natural choice for the gravitational field with the most compact way of doing GR
ds^2 = guvdx^udx^v = (Minkowski)abe^ae^b = e^aea
Curvature 2-form is R^a^b = R^a^buvdx^u/\dx^v = DS^a^b = dS^a^b + S^ac/\S^cb
S^a^b = S^a^budx^u is spin-connection 1-form
e^a = I^a + B^a
I^a is for Minkowski spacetime
B^a is analogous to v = vudx^u = (h/m)dTheta = superflow 3-velocity 1-form
Since the B^a field is a spin 1 vector field under Lorentz group, it is obviously the natural starting point to understand gravity and torsion SIMPLY as the local gauging of the global 10-parameter Poincare group that IS 1905 special relativity analogous to internal symmetry Yang-Mills theory of electro-weak-strong forces. In fact The Pundits have not done so - inventing cumbersome formalisms at the 2nd rank GCT metric tensor, Levi-Civita connection level, that are not needed to get to the important physics quickly IMHO.
On Sep 16, 2007, at 9:06 AM, Paul Zielinski wrote:
Agreed, and this is the advantage of Einstein-Cartan, but my point here is that you don't need tetrads in order to separate the mathematical coordinate charts from the physical coordinate charts.
Maybe, but I never use mathematical coordinate charts in my conceptualization of the objective reality physics. I only think of relationships between locally coincident tiny detector observers "Alice" and "Bob" in strictly operational pragmatic terms. I agree with the Cornell Physics Department philosophy of the late 1950's (Feynman, Bethe, Salpeter, Morrison, Gold, Kinoshita ...) keep the math minimal. Fancy math is usually a cover for lack of a good physical idea as we see in string theory and loop quantum gravity - not always of course. Almost every theory paper on the gr-qc archive goes nowhere in terms of physical insight on the real problems, e.g. dark energy, dark matter - much fancy math, but very little physics. After looking at many of those papers I ask "So what?" "Why bother?" "What's your point?" Perhaps I am mistaken? ;-)
PS as a purely mathematical exercise fine - i.e. conceptual art. Let's see if LHC makes a difference if it ever goes online, similarly for LISA/LIGO.
Jack Sarfatti wrote:
On Sep 14, 2007, at 2:58 PM, Paul Zielinski wrote:
This is how the tetrad model accounts for the dual role of spacetime coordinates in 1915 GR. It is this tacit duality in Einstein's approach that has caused much confusion in the teaching of relativity theory IMHO. Of course one of my personal hobbyhorses is that you can also do all this without tetrads,
You need the tetrads to couple gravity to spinor fields - that's important since all the basic matter fields (leptons & quarks) are spinor fields. The spin 1 vector fields (including the curvature tetrads and torsion spin connections) are induced by localizing the different symmetry groups of the action of the spinor fields.