Bianchi classification of 3-geometries: original papers in translation


the papers

translated by Robert T. Jantzen, Department of Mathematical Sciences, Villanova University, Villanova, PA 19085-1699 USA

with help from Remo Ruffini, International Center for Relativistic Astrophysics, University of Rome "la Sapienza", Piazzale Aldo Moro, 2, I-00185 Roma, Italy

publication in journal General Relativity and Gravitation

The translation of the first Bianchi paper together with editorial commentary was published in the Journal General Relativity and Gravitation as Golden Oldie #23:

The modern version of the Bianchi classification of Lie algebras due to Schücking and Behr was published as Golden Oldie #30:

translation history

I originally translated these papers from Italian to English in 1972 after my sophomore year as an undergraduate physics major at Princeton University with no knowledge of Italian or group theory as a project from a student initiated seminar by Remo Ruffini on Differential Geometry for General Relativity (initiated by junior Jim Isenberg). Three semesters of college Spanish together with an Italian-English dictionary and a beginning Italian text enabled me to struggle through 95 percent of it with Remo Ruffini consulting on the remaining 5 percent. The displayed formulas were photocopied and literally cut and pasted into the typewritten text (supported by NSF grant GP-30799X to Princeton University) but never distributed except for a few bound photocopies made by Jim Isenberg in 1977 as a graduate student at the University of Maryland in the Misner-Brill group.

In 1998 at the request of Andrzej Krasinski for the General Relativity and Gravitation journal series on old papers that played an important role in the development of relativity, armed with some fluency in Italian gained over nearly 20 years of regular visits to Remo Ruffini's Relativistic Astrophysics group at the University of Rome, I laboriously scanned in the poor quality photocopy in my possession with optical character recognition software, cut out the useless garbled formulas, and created a LaTeX document, improved the rough translation, and typed in the many tedious displayed formulas, although a year intervened until I could finish the formula insertion in 1999.

historical background

Born in Parma, Italy on January 18, 1856, Luigi Bianchi was a student of Ulisse Dini and Enrico Batti at the Scuola Normale Superiore of Pisa. He became of professor at the University of Pisa in 1886 and then the director of the Scuola Normale Superiore of Pisa in 1918, a position he held until his death in 1928.

His mathematical contributions, published in eleven volumes by the Italian Mathematical Union, cover a rather wide range of topics. In the field of Riemannian geometry, he is most well known for his discovery of the "Bianchi identities" (Rend. Accad. Naz. dei Lincei 11, 3 (1902)). In 1897 following the results of R. Lipshitz (J. für die reine und aug. Math. 2, 1 (1870)) and W. Killing (J. für die reine und aug. Math. 109, 121 (1892)), and using the theory of continous groups developed by S. Lie (S. Lie and F. Engel, Theorie der Transformationsgruppen, Vol 1 (1888) and Vol 3 (1893)), he gave the complete classification of the isometry classes of Riemannian 3-manifolds categorized by his famous nine types identified by Roman numerals: I - IX. Neither special nor general relativity existed at the time, but they were soon to follow.

In 1951 this work of Bianchi was introduced into relativistic cosmology by Abraham Taub (a graduate student at Princeton in the 30's with O. Veblen as a thesis advisor and for a short time Instructor in Mathematics at Princeton University and Assistant at the Institute for Advanced Studies in Princeton) in his article Empty Spacetimes Admitting a Three-Parameter Group of Motions, Annals of Mathematics, vol. 53, 1951, pp.472-490, which refers to Bianchi's classification of the Lie algebras of the 3-dimensional homogeneity groups found in his Lectures on the theory of finite continuous transformation groups article. This was not long after Kurt Gödel at the Institute for Advanced Studies published his famous paper on stationary rotating cosmologies that lead to interest in spatially homogeneous spacetimes (written while Taub was a member of the Institute), and a followup article on expanding rotating cosmologies contemporary to Taub's Bianchi article:

Spacetimes which are spatially homogeneous have a time-dependent spatial geometry which is a homogeneous 3-geometry. Therefore the spacetime has an r-dimensional isometry group acting on a family of hypersurfaces with r = 3 (simply transitive action), r = 4 (locally rotationally symmetric), or r = 6 (isotropic). The case r = 3 became known as Bianchi cosmologies after Taub's article. Luther P. Eisenhart, a professor of Mathematics at Princeton University, served as a principle source of English language discussion of much of the early work in differential geometry and Lie group theory through his two books (later reprinted in paperback by Dover Publications):

The Bianchi cosmological models sat for a decade before the renaissance of general relativity that occurred in the early 60's. O. Heckmann and E. Schücking revived this work in 1958 (later summarized in their chapter Relativistic Cosmology, in Gravitation, an Introduction to Current Research, Wiley, 1962, edited by Louis Witten, father of Edward, now at Princeton). An investigation into the nature of the initial singularity of our universe by the Russian school of Lifshitz and Khalatnikov, later joined by Belinsky, independently led to the Bianchi models in describing in some approximation that was controversial at the time how the spacetime metric behaved along timelike curves approaching the "generic" spacelike singularity. This chaotic behavior inspired Misner's (Bianchi type IX) Mixmaster Universe in the USA and later complementary singularity work by Hawking and Ellis in the UK. István Ozsváth, inspired by Misner, returned to Gödel's second article in 1969 in his revival of the Bianchi Cosmology work (Spatially Homogeneous World Models, J. Math. Phys. 11, pp.2860-2870 (1970)). The Bianchi classification itself had just been updated by C.G. Behr in unpublished work reported in a 1968 article (Dyadic Analysis of Spatially Homogeneous World Models, F.B. Estabrook, W.D. Wahlquist, and C.G. Behr, J. Math. Phys. 9, pp.497-504).

In 1972 (during the "golden age of relativity" at Princeton) when John Wheeler was bringing in proofs of his new book Gravitation with Charles Misner and Kip Thorne (Freeman, SanFrancisco, 1973) to our sophomore Modern Physics class at Princeton University, Jim Isenberg, then a junior, was recruiting students to fill the quota for a student initiated seminar on Differential Geometry for General Relativity to be offered by Wheeler's collaborator Remo Ruffini. Following Wheeler's teaching style, Remo wanted to get the students more involved by doing special projects. The Landau Lifshitz text The Classical Theory of Fields (L.D. Landau and E.M. Lifshitz, Permagon Press, latest edition from Butterworth-Heinemann) had a section on the Bianchi cosmologies and Remo wanted a student to help him translate the original papers of Bianchi on this topic. Somehow I volunteered, but it immediately became clear that this was very inefficient so I boned up on some elementary Italian based on my 3 semesters of college Spanish and tackled the job during the summer while working as a carpenter with my dad during the day.

This was followed by a junior paper on Bianchi cosmology and a later a senior thesis begun in 1973. My advisor, Remo, excited in this period from investigating the orbits of particles in rotating black hole spacetimes with another undergraduate from his seminar (Mark Johnston, whose graphics led to the famous Marcel Grossmann Meeting logo), was curious about rotating cosmologies and wondered about talking to Gödel himself on this topic. Looking him up in the phone book (still naive times for celebrities), Remo found him, called him up and arranged for me to meet him at his office at the Institute, where he informed me about recent work by Michael P . Ryan, Jr. that I had not been aware of, initiating my own work in the dynamics of Bianchi cosmology. Ryan, a student of Misner, had taken off where Misner had left his Mixmaster universe and began studying more general Bianchi type spatially homogeneous spacetimes, later summarized in the book Homogenous Relativistic Cosmologies with Lawrence Shepley (Princeton University Press, 1975, out of print). Gödel, though his only published work in relativity was 20 years old at the time, had still been following current developments related to it.

Remo channeled me toward grad school at UC Berkeley to work with Abe Taub just before his retirement in 1978, during which time I learned more about Lie groups and their role in the dynamics of these cosmological models. However, the Bianchi translation, though typed up by a secretary, never found a wider audience, and sat for 27 years until GRG asked me if I might translate the long article, not knowing that it had essentially already been done.

Role in General Relativity

Exact solutions have played an important role in Einstein's general theory of relativity and other metric theories of gravity in studying various aspects of the theory itself that are difficult to establish in general because of the difficulty of the generic field equations. The spatially homogeneous cosmological models (the Bianchi models and later the Kantowski-Sachs models) were originally introduced by Gödel and Taub (1948-1950) to reveal previously unrecognized aspects of general relativity regarding the the notion of an absolute time and causality as affected by a global rotation of the universe and in exploring Mach's principle. At the end of the next decade they were taken up as the simplest generalization of the spatially isotropic and homogeneous Friedmann models as possible models of the actual universe or of its early evolution which would exhibit spatial anisotropy. They then served as a toy model for the investigation of possible isotropization mechanisms that might explain the high degree of spatial isotropy and homogeneity observed in the cosmic blackbody radiation first reported in 1965. At about the same time they provided spatially homogeneous "minisuperspaces" as toy models not only for the emerging field of quantum cosmology introduced to explore more concretely certain issues in the quantization of general relativity but also to study the singularity structure of classical general relativity.

The key reason these mathematical universe models are so attractive, apart from the fact that they may have some physical motivation in describing small "global" anisotropies in the observed universe, is that they reduce the partial differential equations of general relativity to ordinary differential equations where many of the ideas of classical mechanics and qualitative analysis apply, allowing an arsenal of mathematical techniques to be applied in studying the behavior of the field equations and properties of their solutions.

summary of content

LaTeX computer files

old18bian.tex [ol18bianwithtoc.pdf] [table of contents is useful!]
"On 3-d spaces...": Latex master file (GRG formatting) for Bianchi article [67 pages including separate title page] for General Relativity and Gravitation Golden Oldie 23,  Volume 33, 2171-2253, 2001.
ol18bian_int.tex [ol18bian_int.pdf]
LaTex file of General Relativity and Gravitation Editorial Material for Golden Oldie 18 [12 pages] for General Relativity and Gravitation Golden Oldie 18, Volume 33, 2157-2170, 2001.
bianchi5.tex [bianchi5.pdf]
LaTeX file (GRG formatting): Lie algebra classification sections excerpted from Bianchi book [6 pages]; this English translation not to be published

Final remark: Guido Fubini (1879-1943), a former student of Bianchi, extended his work to 4 dimensions in a long article:

"Sugli spazi a quattro dimensioni che ammettono un gruppo continuo di movimenti," Ann. Mat. pura appli. [3] 9, 33-90 (1904); reprinted with his preliminary commentary in his "Opere Scelte," a cura dell'Unione matematica italiana e col contributo del Consiglio nazionale delle ricerche, Roma Edizioni Cremonese, 1957-62 [Fine Hall Library SM 8102.387].

Bianchi identities

Apparently the differential Bianchi identities satisfied by the Riemann curvature tensor were first discovered by Ricci in 1889 (or perhaps Aurel Voss in 1880?) but apparently he forgot about them according to a footnote on page 182 of The Absolute Differential Calculus by Tullio Levi-Civita (1927: Dover paperback 1977), who explained that they were given in a small marginal footnote of a paper by Padova (referring to a footnote of a contemporary paper by Ricci) and so Bianchi missed seeing them. Bianchi gave the full derivation when he published his result in 1902.

Mauro Picone and the Istituto per le Applicazione del Calcolo of CNR Italy

A small world turns out that my collaborator of several decades Donato Bini is a researcher at the Italian Istituto per le Applicazioni del Calcolo of the Italian Research Council, an institute (IAC)  founded by Luigi Bianchi's student Mauro Picone, and somehow they had a photo of Bianchi from the Picone family in the Institute archives, but the request for a photo suitable for a history of the various cosmological models in GR by John Barrow (with whom bob overlapped slightly at UC Berkeley in 1978) eventually turned up an official photo of Bianchi as Director of The Scuola Normale Superiore di Pisa  1918-1928.

9-oct-2010: robert.jantzen