Mathematical Physics and General Relativity Seminar

It has been known for at least 25 years now,due to work of I. Frenkel, J. Lepowsky, A. Meurman, B. Fischer, R. Griess, and others, that there exists a (holomorphic sector) conformal field theory (CFT) with central charge c=24 whose partition function is an elliptic modular function J(z)(well known in number theory), and whose states form a representation of a particular finite group G,called the "monster" (since G has an enormous number of elements,approximately 10 to the 54th power). Remarkably, one can relate factors of degrees of irreducible modules of G to factors of the famous Kac determinant of conformal weight h=10.Also J(z) can be used to explain the weird fact of nature that exp(pi.square root(163)) is "very nearly equal to a whole number". We discuss this circle of ideas, and how the asymptotic growth of J(z) can be used to compute Bekenstein-Hawking black hole entropy in AdS_3 gravity. We point out that entropy correction terms correspond, in fact, to sub-leading asymptotic terms.

1. A moonshine module for the Monster, by I. Frenkel, J. Lepowsky, A. Meurman, Vertex Operators in Mathematics and Physics, MSRI Pub. (1984).
2. Conformal Field Theory by P. Di Francesco, P. Mathieu, D. Senchal, Springer Pub. (1996).
3. 2+1-dimensional gravity revisited,, by E. Witten, arxiv:0706.3359.
4. Note on quantum correction to BTZ instanton entropy, by F. Williams, Proceedings of Science,electronic journal:POS(IC2006)006.
5. Group Theory:The Language of Symmetry in Science and Technology, F. Williams, et. al., National Research Council Panel on Group Theory, National Academy Press(1996).

Komar integral relations in Einstein gravity are useful in deriving certain basic properties of stationary black hole solutions. We will give the basic definition of a Komar integral relation and then ask whether they exist in more general higher derivative gravity theories. We will see that the only purely quadratic gravity theory with a Komar integral relation is the one based on the Gauss-Bonnet Lagrangian. It turns out more generally that Lovelock gravity theories, of which Einstein and Gauss-Bonnet gravity are two examples, can also be shown to have Komar integral relations.

In this talk I will review the Friedmann law of brane models - models where matter is confined to a hypersurface embedded in a space with larger dimension. In particular, I will review the cosmological behavior of compact and non compact codimension-one brane models. I will also discuss, if time allows, the (rather peculiar) properties of codimension-2 models.

1. Nonconventional cosmology from a brane universe by P. Binetruy, C. Deffayet, D. Langlois. LPT-ORSAY-99-25 May 1999. 22pp. Published in Nucl.Phys.B565:269-287, 2000. hep-th/9905012.
2. Single brane cosmological solutions with a stable compact extra dimension by P. Kanti, I. Kogan, K. Olive, M. Pospelov. UMN-TH-1829-99, TPI-MINN-99-54, OUTP-99-62P, Dec 1999. 25pp. hep-ph/9912266. Published in Phys. Rev.D61:106004, 2000.

Although supersymmetry (SUSY), for most of the cases, is introduced on the level of field theory, SUSY in quantum mechanics is simpler, more elegant and yet encompasses a lot of concepts one encounters in field theory. In this chalk talk, I will introduce the supersymmetric Hamiltonian formalism and its use for solving Schrodinger's equation with complicated potentials. After that I will discuss the concept of SUSY breaking. Witten's index and the general factorization method will also be dicussed if time allows.

1. Supersymmetry and Quantum Mechanics, by Fred Cooper, Avinash Khare, Uday Sukhatme.
2. http://arxiv.org/abs/hep-th/9405029

Recently there have been interesting reformulations of Einstein's field equations for scalar field cosmologies, both for isotropic and anisotropic models, in terms of generalized types of Ermakov-Milne-Pinney equations. Inspired by this work, we have discovered an alternative Schrodinger formulation of Einstein's equations in a Friedmann-Lemaitre-Robertson-Walker universe. This provides for an alternate method of obtaining exact solutions of the field equations. After presenting this initial work, I will briefly demonstrate analogous Schrodinger models that have subsequently been found for both Bianchi I and Bianchi V cosmologies.

Quantized Hamiltonian operators for constant magnetic fields on the hyperbolic plane can be expressed in terms of well-known Maass operators which occur in the theory of automorphic forms. A trace formula for these operators projected to a Riemann surface of genus at least 2 can therefore be given-in the form of a generalized Selberg trace formula. We discuss this interesting connection between physics and number theory; many other such connections exist. In particular we consider the influence of a constant magnetic field on the structure of certain zeta functions.

1. Topics in Quantum Mechanics, by F. Williams, Progress in Mathematical Physics Vol 27. Birkhauser (2003).
2. Trace formula for Riemann surfaces with magnetic field, by L.A.Comtet, B.Georgeot, and S.Ouvry, Physical Review Letters 71,(1993),3786-3789.
3. Brownian motion on the hyperbolic plane and Selberg trace formula,, by N.Ikeda and H.Matsumoto, Journal of Functional Analysis 163,(1999),63-110.
4. Group representations,Landau spectra,and magnetic zeta functions, by F. Williams, from Mathematical Methods in Physics,Proceedings of the 1999 Londrina Winter School (2000), 209-226.

In general relativity, the Hamiltonian formalism using the Einstein-Hilbert action is known to give the correct equations of motion, namely the Einstein's equations. However, in this formalism one usually ignores all the surface terms. This produces a Hamiltonian in a pure constraint form which vanishes on solutions. Regge and Teitelboim showed that these surface terms are necessary not only to make the variation of the Hamiltonian well defined but also to correctly produce the Einstein's equations. On solutions, these surface terms give the total energy of the spacetime. Later, Hawking and Horowitz showed that if one includes a surface term in the Einstein-Hilbert action and keeps track of them while carrying out the variation, then the Hamitonian naturally contains the necessary surface terms and one need not add them by hand. On solutions, the value of the surface term gives a definition of the total energy of the spacetime. I intend to discuss the calculation of Hawking and Horowitz and also compare with other definitions of energy of a spacetime.

1. The Gravitational Hamiltonian, Action, Entropy and Surface terms, by Hawking and Horowitz. See arXiv.
2. Role of Surface Integrals In The Hamiltonian Formulation of General Relativity, by Regge and Teitelboim. Annals Phys. 88, 286 (1974)
3. The Dynamics of General Relativity, by Arnowitt, Deser and Misner. See arXiv.

A continuation of the 09 April seminar (see below).

As in Kaluza-Klein models, the Randall Sundrum model is a model of gravity where 4 dimensional gravity emerges at sufficiently large distances, even if spacetime has more than 4 dimensions. However, differentlyl from Kaluza-Klein models, the Randall Sundrum model does not need a compact internal manifold. In this talk I will review the original paper by Randall and Sundrum, showing how 4d Newton's law emerges despite the presence of a noncompact extra dimension. If time allows, I will discuss cosmology in such a scenario.

1. Reference available on the arXiv.

We will give a pedagogical introduction to Hawking's 1975 calculation of emission of quantum mechanical particles from a classical black hole. This semi-classical calculation combines classical general relativity and quantum field theory. This result, in combination with the classical first and second laws of black hole mechanics, show that black holes actually obey a thermodynamics.

1. An Introduction to Black Hole Evaporation, J. Traschen, in Mathematical Methods of Physics, proceedings of the 1999 Londrina Winter School, editors A. Bytsenko and F. Williams, World Scientific (2000). Also available on the arXiv.
2. Hawking (1975)

Classical Kaluza-Klein theory couples Einstein gravity and electromagnetism in 5 dimensions.Using the theory of connections on a principal G-bundle P over M ,one can extend this theory to higher dimensions, which is necessary for the coupling of Yang-Mills fields.Recently I was able to set up a correspondence between Einstein vacuum field equations on P and certain non-vacuum field equations on the base manifold M ,and to relate the corresponding cosmological constants by a simple formula.This result is discussed,and we indicate also how to obtain Yang-Mills solutions for a particular connection on P.

1. Einstein Manifolds, A. Besse, Springer-Verlag Pub. (1987)
2. Gauge theory and variational principles, D. Bleecker, Addison-Wesley Pub. (1981)
3. Higher-dimensional unifications of gravitation and gauge theories, Y. Cho, J. Math. Phys. 16 (1975) 2029
4. Kaluza and Klein's five-dimensional relativity, J. Miller, in Quantum Theory and Gravitation, Ed. A. Marlow, Academic Press (1980)
5. Remarks of Einstein field equations and non-abelian gauge fields in higher dimensions, F. Williams, paper presented on the occassion of the 100th year celebratio of Einstein Relativity (1905-2005), Amravati University, India.

I will start with a brief introduction to Lovelock gravity and the Gauss Bonnet term arising from this "generalized" Einstein's Equations. Then I will discuss about the spherical thin shells in Gravity and the solution which is a "thin shell of nothingness". This will show how we don't the need stress energy as a source for the wormhole solutions to exist.The talk will be based on the recent paper by Elias Gravanis and Steven Willisons, titled, "Mass without mass from thins shells in Gauss-Bonnet gravity".

1. Mass without mass from thin shells in Gauss-Bonnet gravity, E. Gravanis and S. Willisons. Complete article

Wave Maps are harmonic maps from a Minkowski space into a Riemannian manifold. We will give a variational definition of the wave maps, and derive the wave maps system in covariant form, as well as in local coordinates on the target manifold. It will also be shown how the Wave Maps system arises in the reduction of the Eistein Vacuum Equations under special symmetry conditions, as well as the connection with the Yang-Mills system. Global existence, regularity and stability questions for the wave maps system, in both geometric and analytic contexts, will then be discussed. Some recent large data blow up results will also be mentioned.

1. J. Shatah, M. Struwe, Geometric wave equations. Courant Lecture Notes in Mathematics, 2. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1998
2. T. Tao Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006
3. S. Klainerman, On the regularity of classical field theories in Minkowski space-time R^{3+1}, Prog. in Nonlin. Diff. Eq. and their Applic., 29, (1997), Birkhauser, 113-150
4. T. Cazenave, J. Shatah, A.S. Tahvildar-Zadeh, Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields, Ann. Inst. H. Poincar Phys. Thor.68 (1998), 315-349

In the theme of the last two lectures, we discuss solutions of two-dimensional string theory. In particular, we shall explore a Schwarzschild-type gauge in which the field equations may be solved explicitly, producing a classical solution (as opposed to a distributional one). We shall also explore the relationship between such solutions and soliton metric solutions of the string equations. My recent work [3] constructs explicitly a transformation between a particular soliton gauge (sine-Gordon-type coordinates) and black hole reformulation of the Schwarzschild gauge. Time permitting, we will indicate the key ideas in constructing this transformation.

1. Classical soutions of 2-dimensional string theory, by G. Mandal, A. Sengupta, and S. Wadia. Mod. Phys. Lett A. Vol 6 #18 (1991) pp.1685-1692
2. Relating black holes in two and three dimensions, by A. Achucarro and M. Ortiz. Phys.Rev. D48 (1993) 3600-3605.
3. F. Williams and S. Beheshti, (2005-2006 communications)

We present distributional solutions of the Einstein equations that are cosmic strings. The mathematics is similar to that of Mohamed's lecture last week on gravitational shock waves.

1. Solitons in field theory and nonlinear analysis, by Yisong Yang, Springer Pub.
2. A. Vilenken, Phys. Report 121 (1985), p.263
3. T. Kibble, J. Phys. A9 (1076), p.1387
4. A. Comtet, G. Gibbons, Nucl. Phys. B 299 (1988) p.719
5. D. Forster, Nucl. Phys. B 81 (1974), p.84

Gravitationa shock waves are exact solutions of Einstein equations that are sourced by massless particles moving along null surfaces. In this talk, I will review the construction of these solutions in curved spacetimes using two methods: by boosting exact solutions of Einstein equations (Aichelburg and Sexl Method) and by using Dray and 't Hooft cut and paste method. In addition, I will talk briefly about the importance of such solutions in physics.

1. General relativity and gravitation, by Aichlburg and Sexel, Vol 2, pp 303-312.
2. Dray and 't Hooft, Nuclear Physics B 253, pp 173-188
3. Sfetsos arXiv paper hep-th-9408169
   

Constructions of Lawson, Kapouleas, and others give examples of surfaces of any genus harmonically mapped into the three sphere. It is not clear from the methods used, however, whether these surfaces occur within any smoothly varying family of harmonic surfaces when the genus is greater than one. I would like to suggest a new method for at least showing existence of (and perhaps constucting) such surfaces which follows Hitchin's proof that the purely algebraic data of a stable Higgs bundle over a Riemann surface give rise to a solution of the dimesionally reduced self-dual SU(2) Yang-Mills equations over the surface. These equations bear a formal resemblence to the equations for a harmonic map of the surface into SU(2) = S^3, and it is hoped that by considering also the data of semistable Higgs bundles, some of the structure of Hitchin's results can be carried over to the harmonic case.

1. Harmonic maps from a 2-torus to the 3-sphere, by N.J. Hitchin. Journal of Diff. Geom. (1990)
2. The self-duality equations on a Riemann surface, by N.J. Hitchin. Proc. London Math. Soc. (1987)

In this talk I will present a non-linear Schrodinger type formulation of Einstein's equations for a Friedmann-Lemaitre-Robertson-Walker universe with scalar field and perfect fluid matter source. This provides for an alternate method of obtaining exact solutions of the Einstein field equations for a homogeneous, isotropic universe. Some examples of exact solutions obtained in this way will be demonstrated, and I will mention analogous work that is currently being done with a Bianchi metric in an anisotropic universe.

1. The Ermakov-Pinney equation in scalar field cosmologies, by R. Hawkins, J. Lidsey. Physical Review D, 66 (2002), 023523-023531.
2. On 2+1-dimensional Friedmann-Robertson-Walker universes: an Ermakov-Pinney equation approach, by P. Kevrekidis, F. Williams. Classical and Quantum Gravity, 20 (2003) L177-L184.

In recent years, the soliton metric given by ds^2=cos(w)^2 dx^2 - sin(w)^2 dt^2 has been of particular interest to physicists studying the Jackiw-Teitelboim model for gravity [2]. In this talk, I will provide a transformation from such a metric to a two-dimensional slice of the Banados-Teitelboim-Zanelli (BTZ) black hole metric. In fact, this transformation will allow us to reduce a difficult, highly nonlinear eigenvalue problem to a slightly simpler one which is solved by using hypergeometric functions.

1. Further Thoughts on first generation solitons and J-T gravity, by F. Williams, in Trends in Soliton Research, L. Chen, Editor. Nova Science Pub (2006).
2. Geometrodynamics of sine-Gordon solitons, by J. Gegenberg, G. Kunstatter. Phys. Rev. D, Vol 58 124010.
3. 2D extremal black holes as solitons, by M. Cadoni. See hep-th/9803257.
4. Special Functions of Mathematical Physics, by A. Nikiforov, V. Uvarov. Birkhauser, 1988.

In this talk we shall present some geometrical properties of Dirac operators for manifolds which admit spinorial structure. We shall also deal with qualitative aspects of the Atiyah-Singer Index Theorem (for loop spaces) on which the validity of the Atiyah-Witten conjecture is based. This conjecture involves a supersymmetric derivation of the Duistermaat-Heckman Theorem for infinite dimensional spaces.

1. L. Alvares-Gaumi (1983), Commun. Math. Phys., 90, p.161.
2. M. F. Atiyah (1985), Asterisque, 131, p.43.
3. A. Heitamaki, A. Yu. Morozov, A. J. Niemi, K. Palo (1991), Phys. Lett. B, 263, p.417.
4. M. Nakahara (2003), Geometry, Topology and Physics, IOP, second edition.

We consider some exact field equation solutions in 2-dimensional J-T gravity with a cosmological constant.Particular attention will be focused on dilaton structure-in the case of minimal coupling to matter.The interesting case of conformal coupling will be deferred to a later talk.

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