Teleparallel Equivalent of GR; Lie’s Third Theorem (2)

Below: A few notations to indicate topics covered. Author’s page numbering is ignored, page numbers referenced below are from the scanned PDF documents.

Notebook June 2012 – September 2012: Notes on the Teleparallel Equivalent of General Relativity; Lie’s Third Theorem

  • (p. 8) From a paper by Pereira et. al. Whether gravitation is a curved or torsioned spacetime — or equivalently a Riemannian or a Weitzenbock spacetime structure — turns out to be, at least classically, a matter of convention.
    • This was exactly Poincare’s point of view, but given the amazing success of GR circa 1920’s and the fact that Poincare did not have the precise example of teleparallel theory in hand, it is unfortunate but understandable that his very deep insight lost traction.
    • Even in the 1980’s, as evidenced by the arguments in the wonderful book of Torretti, the view that the choice of geometry cannot be conventional as Poincare had insisted was still dominant.
    • At this point (2012) the force of Poincare’s insight seems to be apparent. GR could have been discovered through teleparallel gravity — it is not at all an awkward reformulation that could only have been devised once one already knew the answers from ordinary GR, which is basically the charge that advocates of Poincare’s conventionalism have had to face.

(p. 9)…Βμ is, in Pereira’s conception, the gauge potential of the gravitational field. Its values lie in the Lie Algebra of the translation group. In my conception the values would lie in the Lie Algebroid of the translation groupoid.

(p. 11) Pereira makes a point of stating that the translation generators are able to act on the argument of any source field becausebut these relations look extremely suspicious to me. Pereira seems to be glossing over some essentially critical points in the definition of the alternative tangent space parametrized by the xα ‘s. This leads him to…a strange use of the functional derivative symbol

(p. 15) Thinking Again About the Big Picture

  • GR with an asymptotic group allows the definition of a Hamiltonian especially when there is SUSY.
    • But if we break up GR into pieces that have certain asymptotic symmetries we lose some very important insights about the theory of gravitation. The typical solution of GR (M.g) does not possess any symmetry!
    • The absence of a global time-like killing vector ⇒ no Hamiltonian can be written. To my mind this is the fundamental reason for the inability to quantize. With no group around what do we do?
    • Answer– Groupoids — Algebroids…

(p. 24) What is the Nature of a Groupoid?

  • There is a traditional and wonderfully beautiful argument that the mathematical theory of groups captures, in a parsimonious way, the conceptual structure of symmetry. The concept of symmetry that it codifies is the following:
    • Consider a system S and a set of operations O that may be performed on the set S, an element…O is called a symmetry of S if observations of S done just before and just after the performance ofare indistinguishable.
  • With this philosophical conception of system and symmetry operations, one is led parsimoniously to the abstract notion of group as it was defined by Galois.
  • I am looking for the analogous philosophic construction that would lead to the notion of groupoids. I have always felt uncomfortable with things like:
    • Hopf algebra symmetries, because of the beauty and generality of the above argument for the parsimonious connections between symmetry and the abstract theory of groups.
  • As of right now (Tuesday, September 4th 2012) it seems to me that, if we simply assume that the system S has an internal structure we may be led directly to the groupoid structure in a parsimonious fashion.

(Final) Notebook:

(p. 7) Some Notes on the Principle of Equivalence and Diffeomorphisms

  • It is argued by Stephen Weinberg in his book Gravitation and Cosmology that the principle of equivalence implies that all theories must be diffeomorphism invariantThus the curvature at point p is not gauge invariant.
  • The lesson is that we must in fact discuss the curvature at the location of a gauge invariant event. This is invariant under an active diffeomorphism.
  • (p. 8-9) Remark: This is actually quite similar to an example in electrodynamic theory. When a photon exceeds a certain critical energy, namely energy considerations allow for the formation of an electron proton pair in exchange for the sufficiently excited photon. There is however an apparent paradox because by changing reference frames, say boosting in the direction of the photon.
    • This is a kind of Einstein Hole argument in the case of Special Relativistic space. It’s not clear whether or not it is a close analogy, but it does seem to indicate something similarIn typical texts some handwaving is done and it is said that such processes occur only near an atom. It is never stated exactly how close the photon must be to the atom in order for this to work. This argument has so many problems, it’s not worth going through them now. End Remark