Sean and Flavio,
I found your essay to be interesting. It takes me a while to digest the more interesting and solid works here, so it has taken me a bit of time to get to your paper. You make a number of very interesting points. I can only this morning discuss a few of these. I will try to follow up later today or tomorrow.
The relationship between shape dynamics (SD) and the extended configuration space (ECS) is one of conformal symmetry. SD has no explicit reference to scale, and scale independence is one aspect of conformal symmetry. The relationship between the two, or a decomposition SD --- > ECS is one where conformal symmetry is broken. General relativity has the group structure SO(3,1) ~ SL(2,C), and the extension of general relativity to conformal spacetime is to SO(4,2).
In my essay I work with the Britto, Cachazo, Feng, Witten (BCFW) recursion relationship. This is a Feynman diagram procedure in twostor space. Twistor space is the symmetry group SU(2,2) ~ SO(4,2), and so is a quantum or prequantum description of conformal relativity. The recursion relationship is a scaling principle which is conformal. In fact it leads to Yangian symmetries, which are a form of enveloping Lie algebras for gauge theory. Yangians consist of a gauge (or gauge-like) theory plus a dual theory related by a conformal symmetry, or Mobius symmetry and T-duality.
The BCFW recursion formula is a twistor theory, which has been used in the HopHat algorithm for computing gluon amplitudes at the LHC. Gravitation is in one sense the square of QCD gauge theory. After reading Giovanni Amelino-Camelia , which has regrettably and I think wrongly fallen down the community ranking, I suggested a connection between the boosts employed in κ-Minkowski and twistor theory. This might be a connection between string theory and the more loopy or triangulated theories like LQG. The Wheeler DeWitt equation has not time variables. Physically this means there is no Gaussian surface one may arrange in spacetime to localize energy. So HΨ[g] = i∂Ψ[g]/∂t = 0. The time variables is a coordinate time, used in QFT equations, which is not a proper variable in general relativity. Hence this equation is a constraint equation, classically NH = 0 and N^iH_i = 0. String theory on the other hand requires some external background field from which gravitons as closed strings are represented. This has been a problem the LQG folks like to point out --- never mind LQG has failed to produce even a first order renormalizable calculation. I speculate that somehow the two views of quantum gravity might connect, where LQG provides the background or constraint for string theory, and the field calculations of strings makes LQG more tractable.
Gravity as a dynamic force is conservative. The force in the Newtonian limit is given by F = -∇Φ(r), which is conservative. This means the force evaluated around a closed loop, such as an orbit, is zero. Thermodynamics gives nonzero evaluations for such forces. This is related to the matter in differential geometry that a p-form ω is closed if dω = 0, but a subset of them are exact when ω = dσ, or d^2 = 0. There is some cohomology behind this. The force is determined by the coboundary operator on a 0-form and we have by Stokes law
∫F•dr = ∫∫∇xF•da. da evaluated in the region enclosed by the closed loop.
Yet we know that ∇x∇Φ(r) = 0 (curl-grad = 0 or d^2 = 0) and so the force is conservative.
Verlinde's entropic gravity does not involve the dynamics of a particle in a gravity field. It involves the dynamics of an event horizon or holographic screen. The main idea is that the force on the screen over some unit distance is equal to the work
∫F•dr =W,
and this work is equal to the increase entropy of an event horizon. This by the Bekenstein theorem is S = k A/4L_p^2, for L_p = sqrt{Għ/c^3} --- the Planck length. So the entropy is a measure of how many Planck units of area there on the horizon. So the Verlinde hypothesis is
∫F•dr =TS,
or a force that displaces the horizon some increment gives
F•δr = TδS.
As a result some input of mass-energy into a black hole increases entropy, and this force is what evolves the event horizon, or equivalently the holographic screen.
Event horizons and screens have units of area, and in naturalized units with c = ħ = 1 the gravitation constant G is an area. So this measures the amount of information entangled with the black hole, or the entanglement entropy. As a result the theory you have built up with moments of inertia most likely applies on a holographic surface.
More later & cheers,
LC
