My first real work [bib:1], although unpublished, was written with the enthusiasm of a young graduate student in the several year period following the weeklong conference in Erice (Italy) on Cosmology in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] under the supervision of Remo Ruffini, upgraded with more graduate student knowledge in my early years at UC Berkeley [1974-1978] where my PhD advisor was the legendary Abramham Taub, nearing retirement. 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 at Villanova University in the Department of Mathematical Sciences 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 the Max Planck Institute for Astrophysics in Munich where Cornelius Hoenselaers and Martin Walker were helpful [bib:8, 13] , and some interaction with Metin Gurses led to a correction of harmonic mapping solution generation work by Basilis Xanthopoulos [bib:20], and a short investigation of the geodesics crossings in Einstein-Strauss universes with Ruffini and Fabbri [bib:10-11], I moved on to the Harvard-Smithsonian Center for Astrophysics (CfA) and returned to the Bianchi cosmology problem (with the unpublished work appendix B on Bianchi quaternions growing up [bib:14]) which led to a closer examination of spinor field sources [bib:12], 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.
In 1983 when I arrived at Villanova University, the work of Appendix D of the unpublished work on spatially homogeneous electromagnetic fields grew into a more mature investigation including scalar fields and spatially self-similar symmetry [bib:21,27], influenced by Douglas Eardley at the CfA who had systematized this cosmological symmetry, and a mistaken application by Tomita [bib:19]. Meanwhile in 1983 passing through London, Malcolm MacCallum introduced me to his then postdoc Kjell Rosquist thinking we might begin a meaningful collaboration, and he was right. We started with several articles about some particular exact power law metrics [bib:22] and their general properties[ bib:23], important since they are important in the qualitative analysis of more general cosmological solutions. These turn out to be timelike self-similar models [bib:24,25,26,29], connecting up with my previous spatially self-similar investigation.
Extra dimensions began flowing into the exact solution industry, so a systematic analysis for the Hamiltonian dynamics of higher-dimensional cosmological models was called for [bib:28,31,33] and another correction, this time on the idea of "rotating" spacetimes [bib:30].
Meanwhile Kjell and I together established a general framework for analyzing the dynamics of spatially homogeneous cosmological models using qualitative methods [bib:34], and explored the power law lapse time gauges needed to express the simpler exact solutions which play a key role in qualitative analysis [bib:35,36]. In 1989 we were then joined by then student Claes Uggla in geometrizing this scenario using the Jacobi time gauge to incorporate the potential into the metric on the space of metrics to find hidden symmetries of the dynamics [bib: 38,39,40,41], and clarify late time dynamics [bib:43]. Implications of this understanding enable a unified derivation of most of the exact Friedmann solutions as well as the Schwarzschild solution in terms of a simple classical mechanical potential problem [bib:45], and the role played by Killing vector and tensor symmetries of the Jacobi metric in making exact solutions possible [bib:49,50]. The implications for exact solutions were then explored in a series of articles in the early 1990s [bib:53,55,56,58]. This launched Claes on his own journey into qualitative analysis of cosmological models, joining forces in a leadership position with the competing groups approaching this problem using orthonormal frame generated differential equations without the benefit of the underlying variational geometry intuition provided by the Lagrangian/Hamiltonian approach. We finished a last detail about the relationship of their adapted orthonormal frame choice to the automorphism symmetry of the dynamics in 1999, establishing a relation with the true Smarr-York minimal strain and minimal distortion shifts [bib:66].
In 1988 Remo Ruffini had became interested in the clarification of the definition of electric and magnetic fields and field lines in curved spacetimes, aware of work first by Lifshitz and later by Cattaneo and Ferrarese of the University of Rome Mathematics Department on splitting spacetime into space plus time in an alternative way to the then dominant slicing approach popularized by Misner, Thorne and Wheeler. A university student Paolo Carini and I, later joined by student Donato Bini in 1989, began systematically building a common notation and framework within which we could then fit any of the many different ways of approaching this problem, a framework we loosely referred to as gravitoelectromagnetism [bib:37]. Our first lengthy work was presented at a workshop on Elba organized by Ferrarese [bib:42], with an analysis of gyroscope motion first presented at the First William Fairbank Meeting [bib:44] in order to clarify exactly how the Stanford GP-B gyro experiment fits into the fully nonlinear theory of general relativity as opposed to the linearized theory in which it is always described. Paolo was then a PhD student at Stanford. We tried to summarize all the aspects of our unified framework for spacetime splittings in 1992 with a long article "The Many Faces of Gravitoelectromagnetism" which stands as our fundamental contribution to general theory [bib:46], briefly reported at MG6 [bib:47], together with a more specific analysis of the Minkowski spacetime, the Kerr black hole spacetime and the Godel rotating spacetime [bib:48,54]. With Daniel Wilkins we clarified the role of the Thomas precession in the transition from the nonlinear to the linear picture of gyroscope precession [bib:57]. We followed that with a systematic look at the various projection and boost maps between the local rest spaces of observers in relative motion [bib:59]. I was able to bring attention to our work in a plenary talk at MG7 at Stanford [bib:60], with additional session presentations [bib:61,62]. Because the Many Faces description of general spacetimes was still somewhat challenging to digest in practical applications, we gave a detailed accounting of stationary axisymmetric spacetimes in general and black hole spacetimes in particular, compared to the Minkowski and Godel spacetimes in a pair of articles in 1995, and clarifying the Abramowicz conformally redefined inertial force approach that was ambiguously presented in the literature [bib:64,65], reviewed in context for MG8 [bib:68,69]. Paolo left our collaboration for a high school teaching position after this. Donato and I continued working together (the "we" below), with occasional students joining our work.
With student Andrea Merloni we revisited the Cattaneo-Ferrarese quasi-natural frames in the context of gravitelectromagnetism [bib:70], and joined by Fernando de Felice we introduced relative Frenet-Serret frames appropriate for observer interpretation of kinematics and the introduction of a generalized centrifugal force, and gave the geometric interpretation of the spacetime Frenet-Serret description of stationary spacetimes, introducing the minimal intrinsic rotation observers which are complementary to the extremely accelerated observers [bib:71,72,73,76]. As the new century opened in 2000, Donato and I extended the embedding diagram tool for visualizing the equatorial plane geometry from the slicing point of view to an arbitrary observer family [bib:74], and then expressed Maxwell's equations in space plus time form for an arbitrary observer family with the student Cristiano Germani [bib:75,83]. We also began collaborating with Bahram Mashhoon on a number of issues, first studying the clock effect and Sagnac effect in circular orbits in stationary axisymmetric spacetimes [bib:77] and then extending these ideas to holonomy (parallel transport) around circular orbits and to spin transport (Fermi-Walker transport) relevant to orbiting gyroscopes [bib:91]. Donato reported on our activities at the 2000 Spanish Relativity conference for which we assembled a lengthy bibliography of literature citations for gravitoelectromagnetism and related areas [bib:87,88], and on inertial forces at MG10 [bib:106].
In the meantime, my Princeton undergraduate translation of the Bianchi treatise on 3-dimentional isometry groups and their classification was updated for publication as a Golden Oldie in the journal GRG [bib:78] and triggered by the death of Abe Taub, I stumbled onto the Princeton Mathematics Community in the 1930s Oral History Project and converted it to a website for Princeton University with a wealth of supporting materials about the background and later history [bib:79,80]. Not too long after an unpublished manuscript on the Bianchi-Behr classification of spatially homogeneous cosmological models was published as another Golden Oldie with commentary by a number of interested parties, including myself [bib:99]. Then Remo turned 65 so I remembered our 30 years together [bib:101].
Inspired by Remo, student Christian Cherubini brought a new way of looking at the Teukolsky master equation for perturbations of type D spacetimes coupled with my expertise on the de Rham tensor Laplacian and Donato's perturbation background [bib:82,93,94,96]. Joined by Bahram, we used this tool to investigate the effects of the gravitomagnetic monopole in the Kerr-Taub-NUT spacetime on perturbations [bib:97,98].
Independently we worked with student Giovanni Miniutti on the Cotton-York and Simon tensor in stationary spacetimes and their relationship to the Papapetrou field and their special connection to the principal null directions of the Weyl tensor for black hole spacetimes [bib:90], which led to examining boost transformations of the Weyl tensor and discovering that the Carter observers minimize the gravitational super-energy [bib:94], then extending our results to Einstein-Maxwell spacetimes [bib:105,111]. With student Marco de Mattia we investigated the comparison of the effects of the gravitomagnetic monopole in the Taub-NUT solution on equatorial plane orbits compared to rotation [bib:103]. With former student Gianluca Cruciani, we evaluated completely parallel transport around circular orbits in stationary axisymmetric spacetimes [bib:108].
In 2004 at GR17 and then immediately afterwards at another Elba conference organized by Ferrarese, I got Jim York's attention revisiting the minimal distortion shift vector field with his then recent rethinking of the role of the desensitized lapse [bib:109,110] and its relation to the Taub time gauge, which independently reared its head in a mistaken exact solutions paper which needed correction [bib:112]. This was an idea I had but never pursued way back in Chapel Hill decades before [bib:5]. This resulted in a joint paper finally, a welcome outcome considering we had never had a chance to work together on a project while I was his postdoc [bib:115].
In 2005 student Andrea Geralico with strong computer algebra system skills joined us to study the equations of motion of test bodies with internal structure, first in the Schwarzshild spacetime and then more generally in algebraically special spacetimes and then in the Kerr black hole spacetimes [bib:113,117,118]. In parallel we were able to make some progress in expressing the coordinate transformation to Fermi coordinates near a uniformly rotating orbit in a rotating black hole spacetime with Andrea's strong computer algebra skills [bib:114], and then update old work on Frenet-Serret frames along null world lines [bib:116]. He also helped us complete our earlier work with Christian on parallel and Fermi-Walker transport around circular orbits [bib:119] and to investigate the effects of uniform acceleration on particle motion in the C-metric [bib:120].
In 2007 with Christian we were led to examine the Kasner solution as an example of the speciality index whose values help classify the Weyl tensor Petrov type, and discovered a nice connection with the Lifshitz-Khalatnikov parametrization of the Kasner indices and the Kasner map which is associated with gravitational chaos [bib:121, 122].
In 2008 my old flatmate from a year at the Harvard-Smithsonian CFA days Luigi Stella began pushing us to think about accelerated orbits around compact objects based on the Poynting-Robertson effect, which had never received a proper general relativistic treatment. This led to a series of articles [bib:123, 125, 129] over a number of years investigating this mathematically interesting problem that may have physical implications. A new friend Oldrich Semerak joined us in this study after our first article.
In 2009 with both Christian and Andrea we introduced a novel interpretation of the gauge-invariant description of the dynamics of the notorious Bianchi type IX Mixmaster universe: the electrocardiogram of the universe. The real and imaginary parts of the speciality index, the scale-invariant part of the Weyl tensor, each trace out an electrocardiogram-like pulse graph versus time for a spatially homogeneous spacetime, and for each value of the single spatial coordinate of a plane symmetric Gowdy spacetime, which we investigated for the exact Lim solutions describing spikes in the BKL oscillatory approach to an initial cosmological singularity [bib:124]. For a Mixmaster universe or more general oscillatory approach to an initial singularity, there are an infinite number of such pulses corresponding to curvature wall collisions in the Hamilitonian picture of the dynamical evolution of the spatial geometry.
In 2011-2012 with Andrea we considered a number of calculations related to black hole spacetimes and the larger class of stationary spacetimes to which they belong. First we studied the deviation from geodesic behavior caused by the spin of a test particle in the Schwarzschild spacetime [bib: 127], then got as close as possible to closed form expressions for Fermi coordinates in that spacetime [bib: 128] and finally understood the special nature of the time slicing in black hole spacetimes connected to the separability of the geodesic equations of motion [bib: 130].
With Remo Ruffini we resurrected the first papers by Enrico Fermi on the general topic of the classical electrodynamics of a classical extended electron which had accompanied the formation of special relativity as a pedagogical topic of great interest at the time. From those papers came the well-known Fermi-Walker transport developed to study this problem, but the original motivation was largely forgotten. We revisited Fermi's calculation completing its natural consequences for the total 4-momentum observed by an inertial observer, a question which arose after Fermi had lost interest in the problem [bib: 131, 132]. It turns out only the sequence of 4-momenta of the electromagnetic field in which the electron is moving observed by a sequence of inertial observers momentarily at rest with respect to that accelerated electron satisfies a conservation law as in the source-free Maxwell equation case.
This forced duty on Fermi after our own independent interest in his coordinates and transport coincided with the need to give a public talk to a general audience at Villanova about what I do, so I needed some catchy way to explain general relativity, weaving in my Italian slant on the subject. Serendipitously the answer appeared at just the right moment through a newly published book Pasta by Design that described pasta surfaces mathematically, and one helical pasta shape in particular was a natural candidate for illustrating the spacetime geometry of planets orbiting the sun in helical orbits in spacetime: cavatappo (singular)/ cavatappi (plural). "Corkscrew pasta." bob had already done the groundwork exploring the geodesics on the torus (donut) [bib: 137], an analysis which could also be used to understand circular orbits in Newtonian and relativistic gravity, and the cavatappo surface just required minor surgery, a cut and stretch of the torus [bib: 138]. With a little more inspiration, this resulted in the orthogonally tilted cavatappo surface whose Lorentizan version exactly describes the classical electron model [bib: 139] which led Fermi to introduce his famous coordinates and corresponding transport, as well as providing a nice toy example of the slicing and threading splittings of spacetime.
Meanwhile joint work with my Villanova Math Dept colleague and friend Klaus Volpert resulted in an article [bib: 135] on the self-similarity of the income distribution on the low and high ends which allowed us to "split the Gini index in two." In an amazing coincidence the increasing income inequality in the US rose to prominence in the national political discussion just as our article appeared in the academic year 2011-2012, which won us a small award from the MAA in 2013. Klaus had actually inspired me to begin my torus geodesic study.13-feb-2014: robert t jantzen in progress
Much of the work since 1992 after my spatially homogeneous cosmology period wound down has been done with my collaborator Donato Bini, CNR Applied Mathematics, Rome, Italy, and sponsored by ICRA (since 1985) as well as ICRANet (since 2005). My mathematical article on the Gini index is coauthored with my Villanova Math Dept colleague and friend Klaus Volpert.
Student and former student coauthors: