Daniel Finley (Professor)
Ph.D. 1968, University of California at Berkeley
Brief Description of Research
My current work involves exact descriptions of solutions of the
vacuum Einstein field equations, for
algebraically-degenerate Petrov types, especially those related to
non-trivial (twisting)
gravitational waves.
The complexity of the gravitational
equations has led me to interests in other sorts of nonlinear physics
and in general methods for studying them, the most common being
infinite-dimensional Lie algebras of local and/or non-local symmetries
of the associated nonlinear PDE's.
The main interest is in solutions to Einstein's field equations, and in
perturbations of both the solutions themselves and massless field equations
defined over the corresponding curved spaces. The approach to solutions is
physical in motivation but mathematical in content. I often use the methods of
complex differential manifolds, leading to the study of the structure of all possible
self-dual spaces (H-spaces) with complex or Euclidean signature. A natural
generalization of this is the study of HH-spaces, which may possess Minkowski
signature, real, algebraically-degenerate cross-sections. Work on HH-spaces
is proceeding by division into distinct Petrov types, leading particularly to
the study of gravitational waves. Before his death much of this work was
done in active collaboration
with Jerzy Plebanski at the Instituto de Investigaciones y Estudios Avanzados
del Instituto Politecnico Nacional in Mexico City and the University of Warsaw
in Poland.
Some publications in this area are the following:
- "Third-order ODE's for Twisting, Type-N Vacuum Solutions,"
Class. Qu. Grav. 11 (1994) 157-166, with J. F. Plebanski and
M. Przanowski.
- "The Involutive Prolongation of the (Complex) Twisting, Type-N
Vacuum Field Equations," Proc. of the International Conf. of Aspects
of General Relativity and Mathematical Physics, N. Breton,
R. Capovilla and T. Matos (Eds.), published by CINVESTAV,
Mexico City, 1994, with Andrew Price.
- "An iterative approach to twisting and diverging, type-N, vacuum
Einstein equations: a (third-order) resolution of Stephani's
`paradox'," Class. Qu. Grav. 14 (1997) 489-497, with J. F. Plebanski and
Maciej Przanowski.
- "Killing-Vector Reductions for Complex-Valued, Twisting, Type-N
Vacuum Solutions," to be published in the proceedings of the
9th Marcel Grossmann Conference, held in Rome, July, 2000.
Published in Proceedings of the Ninth
Marcel Grossmann Meeting on General Relativity, V.G. Gurzadyan, R.T. Jantzen, and
Remo Ruffini (Eds.) (World Scientific, 2002), p. 839-841.
Also available
online in 2001.
The pdf-version is available here.
- "Equations for Complex-Valued, Twisting, Type N, Vacuum Solutions,
with one or two Killing/homothetic vectors," from a talk at GR16, Durban, South Africa, July, 2001.
A pdf-version is here, which is also available from the
archive.
- "Asymptotic properties of the C-metric," with Pavel Sladek (Charles University, Prague,
Czech Republic). Class. Qu. Grav. 27 205020 (2010). A pdf-version is
available here.
- "Lower-order ODEs to determine new twisting type N Einstein spaces via CR geometry," with Xuefeng Zhang.
Class. Qu. Grav. 29 (2012) 065010. A pdf-copy is
available here.
- "Interpretation of twisting type N vacuum solutions with cosmological constant," with Xuefeng Zhang.
Class. Qu. Grav. 30 (2013) 075021. A pdf-copy is
available here.
- "CR structures and twisting vacuum spacetimes with two Killing vectors and cosmological
constant: type II and more special," with Xuefeng Zhang.
Class. Qu. Grav. 30 (2013) 115006. A pdf-copy is
available here.
Areas of investigation in nonlinear physics include methods of finding
families of particular, exact solutions of many different systems of nonlinear
equations, motivation for the choices being found in problems in hydrodynamic
flow, chemical mixing, nonlinear optics, three-level laser systems, and
gravitational waves. These methods are used to generate families of soliton
or breather solutions or Bäcklund transformations between equations, and to
establish connections with many (other) interesting mathematical structures,
including complex manifold theory, twistor methods,
infinite solvable algebras, Kac-Moody algebras, and larger algebras (of
infinite growth). John K. McIver is a collaborator
in this work.
Some publications in this area are the following:
- "Infinite-dimensional Estabrook-Wahlquist Prolongations for the
Sine-Gordon Equation," J. Math. Phys. 36 (1995), 5707-5734,
with John K. McIver.
- "The Robinson-Trautman Type III Prolongation Structure Contains
K2,"
Commun. Math. Phys. 178 (1996) 375-390. This identifies
carefully an important, still-unsolved problem in both general
relativity and in infinite-dimensional Lie algebras (of infinite
growth). Although published, here is
a pdf-version if you want.
- "Estabrook-Wahlquist Prolongations and Infinite-Dimensional Algebras,"
Symmetry Methods in Physics, VII International Conference,
Dubna, Russia, 1995, edited by A.N. Sissakian and G.S. Pogosyan,
published by the Joint
Institute for Nuclear Research, Dubna, 1996, p. 203-211.
A pdf-version may be found by
clicking here.
- "Twisting gravitational waves and eigenvector fields for SL(2,C)
on an infinite jet," Electron. J. Diff. Eqns., Conf. 04,
(2000) 75-85. This brings together intimately the difficulties in
the two different areas of interest.
This may be accessed on line, since this is an electronic journal, at
the following URL
http://www.ma.hw.ac.uk/EJDE/conf-proc/04/f2/abstr.html , or via
an Acrobat-readable (*.pdf) file right here.
- A sequence of papers concerning "progress" on the general solution
of the field equations for self-dual, vacuum Einstein spaces with one
(rotational) Killing vector,
usually called the SDiff(2) Toda equation:
- "Difficulties with the SDiff(2) Toda Equation," CRM Proceedings and
Lecture Notes (Proceedings of a Conference in Halifax, June, 1999),
vol. 29 (American Mathematical Society, 2001), p. 217-224.
The Acrobat-readable file
is available here.
- "Infinite-Dimensional Symmetry Algebras as a Help Toward Solutions
of the Self-Dual Field Equations with one Killing Vector," with John K.
McIver, in the proceedings of the 9th Marcel Grossmann
Conference, held in Rome, July, 2000. Published in Proceedings of the Ninth
Marcel Grossmann Meeting on General Relativity, V.G. Gurzadyan, R.T. Jantzen, and
Remo Ruffini (Eds.) (World Scientific, 2002), p. 871-879. Also available
online in 2001.
The Acrobat-readable file
is available here.
- "Generalized Symmetries for the sDiff(2) Toda Equation," with John K. McIver, in the Proceedings
of the 2002 International Conference, Topics in Mathematical Physics, General Relativity and Cosmology,
in Honor of Jerzy Plebañski, H. Garca, B. Mielnik, M. Montesinos & M. Przanowski, Eds., World
Scientific Publ. Co., Hackensack, N.J., 2006, p. 177-191. A pdf-version of the content of the
talk can be found at
this link, while
this link provides the published format.
- "Non-Abelian Infinite Algebra of Generalized
Symmetries for the SDiff(2)Toda Equation" does (finally)
provide some real progress with this
equation, describing the complete, non-Abelian algebra of generalized symmetries (or Lie-
Bäcklund transformations) for this equation. The link in the previous sentence is
our original version; this link provides a pdf-file
of the version published in J. Phys. A 37 (2004) 5825-5847.
- "Solutions of the sDiff(2)Toda equation
with SU(2) Symmetry" provides the
general solution of this equation when the algebra of Killing vectors is increased to
be the Lie algebra for SU(2), which is the same as the algebra for SO(3). It was published
as Class. Quantum Grav. 27 (2010) 145001, with abstract and connection to the complete paper
at
this link.
Last updated/modified: 30 May, 2013
finley@unm.edu,
finley@phys.unm.edu