Gauge Gravity: An Abridgment on the Algebroid Approach

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This webpage gives quick access to Karatheodoris’ theory by presenting a collection of excerpts, from various notebooks, and listing commentaries/conclusions/summaries/insights.

Excerpts from: An Algebraic Approach to Gauge Gravity

April 16, 2012

(p. 15) The study I have initiated on the measurability of the electromagnetic field, with the goal of generalizing to the torsion field of the teleparallel gravity is quite substantial.

May 28, 2012

(p. 24) Relationship between noncommutativity and gravity. Szabo et.al. have written a nice paper discussing the fact that Noncommutative u(1) gauge theory contains teleparallel gravity. Ref. + 01050492v2 25 May 2001. This is quite an old paper but it may be helpful for my concept of gauge gravity. “In my conception of gauge gravity I have a displacement algebra…and a gauge algebra…” (notes below, were followed up with significant work in the notebook )

In-My-Conception-for-2-7-2012

July 15, 2012

(p. 81-83) “So, to review what we have learned about gravity and quantum gravity over the last few years” (p. 1-2 below)

Summary-of-Conclusions-1

Excerpts From: Teleparallel Equivalent of GR; Lie’s Third Theorem

August 2012

(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. 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 of … are 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.