Non-Generated Diffeomorphisms (7)

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 September 2011 – October 2011:

(p. 07) Some thoughts 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 invariant.

(p.14) Been thinking: Diffeomorphism group and its relation to local translations and Yang-Mills groups…The Diffeomorphism group is more complex in this regard, as there are elements arbitrarily close to the identity that are not on a 1- parameter subgroup. An example is

  • (p. 21) There are diffeomorphismswhich:
    • are uniformly close to the identity for N large and ∝ small
    • are therefore contained in the “identity component” of the diffeomorphism
    • do not lie on any 1-parameter subgroup of diffeomorphism
  • Some natural questions present themselves
  • (p. 32) So, we have:
    • very near the origin
      • circles β†’ N wrapped circles
    • unit circles and nearby
      • circles β†’ N wrapped ellipses
    • far from origin
      • circles β†’ circles (1-wrapped)

(p. 76) So it’s clear induction will play a part.

(p. 135) So all the fun happens when n=0. Generally the differential equations that arise are of the form.

  • (p. 140-145) (review of steps for calculations and conclusions)
    • #1. (p. 140) We want to examine diffeomorphisms that are not generated by a vector field. More specifically, we require that the non-generated diffeomorphism be arbitrarily close to the identity, thereby removing certain easy ‘reasons’ why the diffeomorphisms in question are non-generated. We also only consider small perturbations of a small rotation…
    • #2. We are inspired by a small perturbation on a small rotation
    • #8. (p. 142) The pattern of the computation is to repeat this calculation until a conjecture can be made and a proof by induction can be done. Even at this point we can conjecture that
    • #11. (p. 145) Now we can calculate the cubic term and compare it with the prediction

(p. 146) Beginning of the calculation for the cubic (m+n=3).

(p. 151) End of (m+n+3) calculation.

Notebook October 2011 – December 2011: Non-Generated Diffeomorphisms cont’d ( Plus A Few Thoughts on Generalized Gauge Theory)

(p. 15-16) So, since I’ve made the interpolating function, it would seem that a sufficient constraint onso that it lies on a 1-parameter group containing the identity is thatIf this constraint is relaxed we must deal with a more complicated computation. Already at the level of (m+n)=3, there is an addition to the differential equation that determinesIt does, however, seem that only theare non-zero. So, the solution is:

(p. 17) It seems that one must retain products of 𝒷’s. We can check whether theapproximation was sensible by checking the group law for the final function. In addition it is not too hard to find the exact solution, all we have in the RHS of the differential equation is sums of exponentials..

(p. 19) Now we set to zero any quantity quadratic in the 𝒷’s . That means that 𝓹 must be equal to 1.

(p. 24) We have discovered an interpolating function in the case that we can ignore all quadratic terms in the perturbationthe question is:does this give us the right to ignore the perturbation for all 𝓽?

(p. 28-30) Small perturbation of a small rotation on a 1-parameter subgroup.

A Few Thoughts on Generalized Gauge Theory, November-December 2011:

(p. 6) The principle novelty of Hawking’s argument is that “the Black Hole” is firmly placed in the interaction region and is therefore not accessible to experiment. Hawking assumes that all that can be measured about the Black Hole is available on the boundary. Suppose we have a shell of matter in-falling to create a Black Hole

(p. 7) If Hawking is correct, the observers at infinity have no measurement paradox. What about the observer in the bulk that can experimentally determine whether or not a Black Hole has actually formed. Such an observer may then report to the boundary observers.