Spatially Homogeneous Cosmology: Background and Dynamics

R.T. Jantzen
unpublished 200-page report, U.C. Berkeley, 1977 [215 with corrections]


file link section title pages page total
shc_bad1.pdf 0. Introduction 5+2 7
1.51Meg 1. A First Look at Lie Groups and Lie Algebras 12 19
22pages 2. Integral Curves and Diffeomorphism Groups 3 22
shc_bad2.pdf 3. Dragging: Lie and Lagrange Derivatives 8 30
1.15Meg 4. Lie Groups Revisited 7 37
21pages 5. The Adjoint Action 6 42
shc_bad3.pdf 6. Matrix Groups 15 57
1.17Meg 7. Actions Lifted to TM from TM* 4 61
19 pages 8. Isometry Groups and Homogeneous Geometries 7 68
shc_bad4.pdf 9. Bianchi Classification 4 72
22 pages
10. Bianchi Groups, Adjoint and Automorphism Groups,
      Three-Geometries and Maximal Isometry Groups
12 84
shc_bad5.pdf 11. The Metric Manifold M3 22 106
29 p, 1.4Meg 12. Group Kinematics 7 113
shc_bad6.pdf 13. Spatially Homogeneous Spacetime Models 14 127
27 pages 14. Spatial Homogenization of Lagrangian and Routhian Formulations 5 129
1.1Meg 15. A Spatially Homogeneous Perfect Fluid Source 8 137
46p, 1,7Meg
16. Exploration of the Dynamical System 46 183
shc_bad8.pdf 17. Concluding Remarks 8 191
26p, 1.18Meg A. Notation and Formulas 6 197
  B. Geodesics of M 3 200
  C. Bianchi Quaternions 6 206
  D. A Spatially Homogeneous Electromagnetic Field 3 209
9p, 0.31Meg
References 6 215

This was written with the enthusiasm of a young graduate student in the several year period following the weeklong conference in Erice (Italy) on Cosmology in November of 1975 organized by Remo Ruffini and attended by Juergen Ehlers, C. Barry Collins, I. Khalatnikov, Volodia Belinski (?), Alexei Starobinsky, Kenji Tomita, Mauro Francavilla, Michael P. Ryan, Jr., myself, among others. Unfortunately the 5 or 6 main contributions that were ready for publishing (all long review type articles) had to be either dropped or found a new home since the putting together of the book dragged on longer than the contributors had the patience to wait.

In my case, the first half of this article contained the distilled knowledge of Lie group theory and differential geometry that was relevant to Bianchi Cosmology that I had put together in various ways since my undergraduate beginnings at Princeton University [1970-1974], upgraded with more graduate student knowledge in my early years at UC Berkeley [1974-1978]. When Larry Smarr came by to give a seminar about work with James York, Jr., on geometric choices for lapse and shift in GR , the ideas clicked with what I had been beginning to understand about the spatially homogeneous case, leading to my second published article [bib:3] containing the key formulas from chapter 10 on the Bianchi groups and their adjoint and automorphism groups and the key results of chapters 12-16 on the dynamics of the Bianchi cosmological models. [My first published article on tensor harmonics [bib:2] was more related to the foundations of my Ph.D. work with Abe Taub on harmonic analysis of perturbations of some of these models, summarized in [bib:4].] I spent my first postgraduate year with York at UNC Chapel Hill, the second with Remo Ruffini at the University of Rome, the third with Juergen Ehlers at the Max Planck Institute for Astrophysics in Munich, and the remaining 2 years in the Press-Eardley group at the Harvard-Smithsonian Center for Relativistic Astrophysics, landing my present teaching position in a mathematics department in 1983 here in the western suburbs of Philadelphia.

During this time, my appreciation for how things worked grew. At Chapel Hill it was natural to explore conformal ideas [bib:5-6], while in Rome I returned to the variational principle ideas [bib:7] that I had begun in Chapel Hill summarized in my first Marcel Grossmann Meeting (MG2, 1979) proceedings blurb [bib:9]. After a detour on soliton solution generating techniques at Max Planck where C. Hoenselaers and M. Walker were helpful [bib:8] and Einstein-Strauss universes with Ruffini and Fabbri [bib:10-11], I returned to the Bianchi cosmology problem (with appendix B on Bianchi quaternions growing up [bib:14]) which led to a closer examination of perfect fluid sources [bib:15], gauge freedom [bib:16], and the initial singularity studies [bib:17], and as a result of the Lake Como Varenna, Italy 1982 summer school in Cosmology, I was able to tie together many of these ideas in a very long document [bib:18] on a unified picture of the whole family of Bianchi cosmological models.

It was unfortunate that its predecessor, the original long article Spatially Homogeneous Cosmology: Background and Dynamics, never was published, especially since it contained some work on digesting Lie group theory for applications to classical mechanics, homogeneous manifolds, and metric dynamics that was never put together in this way anywhere else. Jim Isenberg made a few bound photocopies when he was a graduate student at the U of Maryland in the Misner-Brill group, but very few people apart from some students (Giuseppe Pucacco, for example) in Rome ever saw this work.

Its appendix D on spatially homogeneous electromagnetic fields eventually grew into a more mature investigation [bib:21,27] when I arrived at Villanova.

10-aug-2000: robert t jantzen  [scanned June-2006]