Joint institute for nuclear research Bogoliubov Laboratory of Theoretical Physics
Manuscript
Cirilo-Lombardo Diego Julio
Generalizations of the Born-Infeld model and exact solutions
PhD thesis
01.04.02 - theoretical physics
Dubna 2008
ABSTRACT Group theory and geometry play an increasing important role in modern theoretical and mathematical physics. Using the theory of the Lie groups as unifying vehicle, the Born-Infcld (BI) theory is reformulated. New non-abclian supersymmetric models, concepts and results of the BI theory in several fields of physics are expressed in an extremely economical way. Regular exact solutions of the BI theory in curved space-time within this mathematical framework arc presented. Non-abclian supcr-gencralization of the Bt action independent of the gauge group arc proposed in a language where the endomorphism algebras can replace the concept of principal fiber bundle and the gauge theories arc included in the Rieraannian structure. Finally, starting from the superparticle and the string models, the quantum problem of the theories with reparameterization invariant actions is analyzed, the mathematical tools involved are presented and the hints for the further quantum tretmcnt of more complicated theories (e.g. d-branes) arc given. The corresponding physical states and new rclativistic wave equation from these toy models with they relation with the Harmonic Oscillator are shown on several examples.
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Contents
I Problems in Gravitation and Born-Infeld theory viii 1 New spherically symmetric monopole and regular solutions
in Einstein-Born-Infeld theories ix
1.1 Introduction and results: ix
1.2 The Born-Infeld theory: X
1.3 The regularity condition xi
1.4 Statement of the problem: xiii
1.5 Equations for the electromagnetic fields of Born-Infeld in the
tetrad XV
1.6 Reduction and solutions of the system of Einstein-Bom-
Infeld equations xvi
1.6.1 Analysis of the function T (r) from the physical point
of view XV1J
1.6.2 Interesting cases for particular values of n and m . . xix
1.7 Analysis of the metric xix
1.8 Conclusions 1.8.1 Appendix: connections and curvature forms from the xxiii
geometrical Cartanfc formulation xxiii
1.8.2 References XXV
2 Rotating charged Black Holes in Einstein-Born-Infeld theories and their ADM mass. xxvii
iv Contents
2.1 Introduction.................................................xxvii
2.2 Statement of the problem.....................................xxviii
2.3 Analysis of the metric in the Born-infold rotating case . . . xxxiii
2.4 Conclusions..................................................xxxv
2.5 References...................................................xxxv і
3 The Newman-Jams Algorithm, Rotating Solutions and Einstein-Born-Infeld Black Holes xxxvii
3.1 Introduction....................................................xxxvii
3.2 The Born-Infeld theory.........................................xxxviii
3.3 The I4JA and the rotating charged non linear solution ... xli
3.4 Analysis of the energy-morncntum tensor:...................xliv
3.5 Conclusions:.................................................xlvii
3.6 Appendix:....................................................xlvii
3.7 References:..................................................xlix
II The mathematical structure of the Born-Infeld field equations li
3.8 Introduction................................................. lii
3.9 The quatcrnionic structure .................................. liv
3.10 Hamiltonian point of view....................................lvii
3.11 Maxwell equations in the Ricmannian space-time and the Born-Infeld theory.......................................lviii
3.12 Concluding remarks........................................... lix
3.13 References.................................................... Ix
III Non-Abelian Born-Infeld action, geometry and supersymmetry. lxi
3.14 Introduction................................................. lxii
3.15 Geometrical identity and natural non-abelian generalization
of the Born-Infeld action.............................. lxiv
3.16 Requirements for the non-abelian generalization and explicit computation of the determinant............................ Ixv
3.17 Encrgy-momentum tensor......................................lxvi
3.18 Comparison with other prescriptions.........................lxviii
3.19 Topology of the gauge fields and space-time: The reduced lagrangian................................................Ixix
3.20 Equations of motion for the non-al>elian Born-Infeld theory
in curved space-time........................................ lxx
3.21 Wonnhole-instanton solution in NABI theory..................Ixxiii
3.22 Supcrsvmmetric extension....................................lxxvi
3.23 Concluding remarks..........................................lxxx
3.24 Appendix....................................................Ixxxii
Contents V
3.25 References ..........................................................lxxxiii
IV Born-Infeld theory, QFT and Quantization lxxxv
4 Non-compact, groups, Coherent States, Relativistic Wave txpiations and the Harmonic Oscillator Ixxxvii
4.1 Introduction and summary ......................................Ixxxvii
4 2 The superparticle model..........................................Ixxxix
4.3 Hamiltonian treatment in Lanczos formulation...................xcii
4.4 Quantization...................................................xciv
4.5 Mass spectrum and square root of a bispinor....................xcv
4.6 Square root Hamiltonian and the Theory of Semigroups . . c
4.7 Relation with the relativistic Schrödinger equation: compatibility conditions and probability currents.......................... civ
4.8 Relativistic wave equation.....................................cvii
4.9 Concluding remarks.............................................. cxi
4.1Ü References.....................................................cxii
5 Quantum field propagator for extended-objects in the mi-crocanonical ensemble and the S-matrix formulation cxv
5.1 Introduction...................................................cxv
5.2 Microeanonical formulation.....................................cxvi
5.3 Axiomatic S-matrix formulation in QFT: microeanonical description .........................................................cxviii
5.4 The Nambu-Goto action and the microeanonical propagator cxx
5.5 References.....................................................cxxiii
V Concluding remarks and outlook
cxxv
Contents
Preface
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About references and notation. Because the huge amount of new results, models and different analisys of the Born-Infeld theory in several fields of physics in this dissertation, we remain each chapter self contained in the sense that the respective references, notation and appendices are included in it.
Part I
Problems in Gravitation and Born-Infeld theory
New spherically symmetric monopole and regular solutions in Einstein-Born-Infeld theories
1.1 Introduction and results:
The four dimensional solutions with spherical symmetry of the Einstein equations coupled to Horn-Infeld fields have been well studied in the literature1 In particular, the electromagnetic field of the Born Infeld monopole, in contrast to Maxwell counterpart, contributes to the ADM mass of the system (it is, the four momeutum of asymptotic flat manifolds). B. Hofftnann was the first who studied such static solutions in the context of the general relativity with the idea of to obtain a consistent particle-like model2 . Unfortunately, these static Einstein-Born-Infcld (EBI) models generate conical singularities at the origin2-3 that cannot be removed as in global monopoles or other non-local ized defects of the spacetimes"6. With the existence of this type of singularities in the spaco-timc of the monopole we can not identificatc the gravitational with the electromagnetic mass.
In this work a new static spherically symmetric solution with Born-lnfeld charge is obtained. The new' metric, when the intrinsic mass of the system is zero, is reyular everywhere in the sense that was given by B. Hoffinann and L. Infold3 in 1937 and the EBI theory leads to identification of the gravitational with the electromagnetic mass. This means that the metric, the electromagnetic field and their derivatives have not singularities and discontinuities in all the manifold. The fundamental feature of this solution is the lack of conical singularities at the origin. A distant observer will associate with this solution an electromagnetic mass that is a twice of the mass of the electromagnetic geon founded by M. Dernianski1 in 198(i .
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