1. Gravitational Dirac Bubbles: Stability and Mass Splitting Speaker Shimon Rubin ( work with Aharon Davidson) Ben-Gurion University of the Negev Miami, 16-21 December 2008

2. Point particle singularity over the years… Abraham(1903)-Lorentz(1904)-Poincare(1906): Electron is a sphere whose rest mass is of electromagnetic origin. Einstein(1919,1935): Gravitational fields play an essential part in the structure of the elementary particles of matter. Wheeler(1955): Classical, divergence free, Newtonian concept of the body is realized by gravitational- electromagnetic wave (Geon). Dirac(1962): Electron and Muon are different energy excitations of a conducting surface with surface tension in Electromagnetic field.

3. Why Classical Model? (without quantum mechanics) - We do not know under what extent the methods of GR may be applied. - We do not know how to build a theory which combines GR and Quantum Mechanics.

4. Formulation of a problem Construction of an extensible model of an electron within the framework of unified Brane Action principle.

5. Formulation of a problem Construction of an extensible model of an electron within the framework of unified Brane Action principle. Remove source singularity.

6. Formulation of a problem Construction of an extensible model of an electron within the framework of unified Brane Action principle. Remove source singularity. Possibility to construct solution with arbitrary small mass for arbitrary small radius. (Solve the problem of the classical radius of the electron).

7. Formulation of a problem Construction of an extensible model of an electron within the framework of unified Brane Action principle. Remove source singularity. Stability of the configuration (with respect to radial fluctuations and local shape deformations). Possibility to construct solution with arbitrary small mass for arbitrary small radius. (Solve the problem of the classical radius of the electron).

8. Dirac’s “Extensible model of the electron” - No gravity effects. - Start with the following Action Principle - Accompanied by the constraints

9. Dirac’s Variation Principle

10. The variation of the action with respect to embedding vector of the surface is not linear and therefore is not correct. The surface of the electron must not deformed during the variation.

11. Extend Dirac’s ‘extensible model’ to include gravitational fields. Extend Dirac’s variation principle for gravitational fields. Gravitational extension of Dirac’s model – Gravitational bubble

12. Gravitational extension of Dirac’s model for the electron The action

13. Gravitational extension of Dirac’s model for the electron The action

14. Dirac’s Variation Principle with Gravity Variation with respect to in the bulk leads to Einstein’s field equations (no relaxation) Variation with respect to leads to Israel junction conditions (no relaxation)

15. Relaxation on variation of on the brane leads to generalized Israel Junction Conditions Dirac’s Variation Principle with Gravity

16. Relaxation on variation of on the brane leads to generalized Israel Junction Conditions Dirac’s Variation Principle with Gravity

17. Relaxation on variation of on the brane leads to generalized Israel Junction Conditions Dirac’s Variation Principle with Gravity

18. Gravitational extension of Dirac’s model for the electron symmetry In=Out Reissner-Nordstrom metric in the bulk

19. Gravitational extension of Dirac’s model for the electron The embedding of the bubble The induced metric The extrinsic curvature Lagrange multipliers

20. symmetry wormhole

21. Effective potential

22. Global Minimum

23. Beyond surface tension on the bubble U(1) in the bulk couples to U(1) g on the brane

24. Beyond surface tension on the bubble U(1) in the bulk couples to U(1) g on the brane Radially symmetric electrical field in the bulk magnetic monopole on the two-sphere

25. External small gravitational perturbation Position of the brane is described by one degree of freedom (radion field)

26. Fluctuation equation

28. Perform contraction to obtain a single equation

29. Fluctuation equation Perform contraction to obtain a single equation Split into perturbation of gravitational field in the bulk and into brane bending

30. Fluctuation equation Perturbation for position of the brane,

31. Fluctuation equation Perturbation for position of the brane,

32. Fluctuation equation Perturbation for position of the brane, Perturbation for metric in the bulk:

33. Stability We can choose such that we obtain time independent fluctuation equation

34. Mass correction (splitting) ADM mass formula

35. Mass correction (splitting) ADM mass formula

36. Mass correction (splitting) ADM mass formula

37. Conclusions In the framework of brane gravity we extend Dirac’s "Extensible model of the electron" to include gravitational field. The effective potential for the bubble radius which we derive possesses a global minimum and at the same time permits arbitrary small radius and arbitrary small mass without fine-tuning, thereby avoid the problem of classical radius of the electron. Local stability with respect to perturbation of external fields.

38. References -H. Poincare, C. R. Acad. Sci. 140, 1504, (1905). -A. Einstein, Siz. Preuss. Acad. Scis., (1919); Do Gravitational Fields Play an essential Role in the Structure of Elementary Particle of Matter,îin The Principle of Relativity A Collection of Original Memoirs on the Special and General Theory of Relativity by A. Einstein et al (Dover, New York, 1923), pp. 191-198, English translation. -A. Einstein and N. Rosen. "The Particle Problem in the General theory of Relativity", Phys. Rev. 48, 73 (1935). -P. A. M. Dirac, An extensible model of the electron, Proc. Roy. Soc. of London A268, No.1332 (Jun. 19, 1962), pp. 57-67. -P. A. M. Dirac, C. Moller, A. Lichnerowicz, "Particles of Finite Size in the Gravitational Field [and Discussion]", Proc. Roy.Soc. of London. Series A270, No. 1342 (Nov. 27, 1962), pp. 354- 356. -W. Israel, Nuovo Cimento B44 1 (1966). -A. Davidson and, I. Gurwich. Phys. Rev. D74, 044023 (2006). -A. Davidson and E.I. Guendelman, Phys. Lett. 251B (1990) 250. -N. Deruelle, T. Dolezel, and J. Katz Phys. Rev. D63, 083513 (2001). -J. Garriga and, T. Tanaka, Phys. Rev. Lett. 84 (2000) 2778. -P. Gnadig, Z. Kunst, P. Hasenfratz and J. Kuti, Ann. Phys. (N. Y.) 116, 380, (1978).