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

Point particle singularityover 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.

Formulation of a problem

• Construction of an extensible model of an electron within the framework of unified Brane Action principle.

Formulation of a problem

• Construction of an extensible model of an electron within the framework of unified Brane Action principle.

• Remove source singularity.

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).

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).

Dirac’s “Extensible model of the electron”

- No gravity effects.

3 4 MNMNS d x g d y GF F

- Start with the following Action Principle

- Accompanied by the constraints

( ) 0 ( ) 0MA y f y

Dirac’s Variation Principle

InOut

Dirac’s Variation Principle

The variation of the action with respect to embedding vector of the surface is not linear and therefore is not correct.

InOut

The surface of the electron must not deformed during the variation.

( )f y

• 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

Gravitational extension of Dirac’s model for the electron

The action ( ) ( ) ( ) ( )in in out outEH HG brane HG EHI I I I I I

Gravitational extension of Dirac’s model for the electron

The action

( ) ( ) ( ) ( ) 4

( )

( ) ( ) 3

1

16

1

8

i i i iEH m

Ni

i iHG

N

I R L G d yG

I K gd xG

( ) ( ) ( ) ( )in in out outEH HG brane HG EHI I I I I I

3 ( )

( ) ( ) ( ) ( ) ( ) ( ) 3( ) , , ( ) , ( ) ( ) ( )

,

1

16

[ ( 1)]

bbrane m

bbrane

i A i B i A i i i A Bi AB i A i AB i i

i in out brane

I d x gLG

g G y y y n G n n gd x

Dirac’s Variation Principle with Gravity

0 0TOT TOTab

ab ab

I IG

G G

( ) ( ) ( )

,

1 10

2 16b i i

i in outN

S T K K gG

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)

g

ABG

0 0abab ab

L LG

G G

Relaxation on variation of on the brane

leads to generalized Israel Junction Conditions

ABG

Dirac’s Variation Principle with Gravity

0 0abab ab

L LG

G G

*( )

,i

i in out

S

Relaxation on variation of on the brane

leads to generalized Israel Junction Conditions

ABG

Dirac’s Variation Principle with Gravity

( )( ) ( ) ;0, 0ii iK

0 0abab ab

L LG

G G

*( )

,i

i in out

S

Relaxation on variation of on the brane

leads to generalized Israel Junction Conditions

ABG

( ) ( )

1

16 i iN

KG

Dirac’s Variation Principle with Gravity

Gravitational extension of Dirac’s model for the electron

2Z

( )1,

4AB brane

m AB mL F F L

22 2 2 2

(2)

2

2

( )( )

2( ) 1 N N

dRds f R dR R d

f R

G M G ef R

R R

symmetry In=Out

Reissner-Nordstrom metric in the bulk

Gravitational extension of Dirac’s model for the electron

, ,A B

ABg y y G

The embedding of the bubble

R a

T T

The induced metric

The extrinsic curvature1

2 nK L g

Lagrange multipliers( )

( ) ( ) ;0, 0ii iK

symmetry wormhole2Z

Out

Brane

In

Effective potential

2 ; 0effa V a M

Effective potential

222 2

2 3

2 2( ) 1

2 2N N

eff NN

G M G QV a G a

a a G a

2 ; 0effa V a M

a

Mass Ueff

Global Minimum

Beyond surface tension on the bubble

( ) 3

4brane

mS d x g f f J A

, ,, , MMf a a J a A y A

U(1) in the bulk couples to U(1)g on the brane

Beyond surface tension on the bubble

( ) 3

4brane

mS d x g f f J A

, ,, , MMf a a J a A y A

1 cos ; where , 2M

e nA a g eg n

R

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

External small gravitational perturbation

Position of the brane is described by one degree of freedom (radion field) A Ay n

Fluctuation equation

*

* 0

S

K

Fluctuation equation

*

* * 0

S

K K

*

* 0

S

K

Fluctuation equation

*S K K

*

* * 0

S

K K

*

* 0

S

K

Perform contraction to obtain a single equation

Fluctuation equation

*S K K

*

* * 0

S

K K

*

* 0

S

K

h

Perform contraction to obtain a single equation

Split into perturbation of gravitational field in the bulk and into brane bending

Fluctuation equationPerturbation for position of the brane,

* *

h

h h

J

S K K S K K

Fluctuation equation

, ,( ) ( ) hA a B a C a J

Perturbation for position of the brane,

* *

h

h h

J

S K K S K K

Fluctuation equation

, ,( ) ( ) hA a B a C a J

Perturbation for position of the brane,

* *

h

h h

J

S K K S K K

Perturbation for metric in the bulk:

2 0

2AB

AB ABbrane

h

h K

Stability

We can choose such that we obtaintime independent fluctuation equation

, ,( ) ( ) hA a B a C a J

,( ) ( ) hB a C a J

Mass correction (splitting)

ADM mass formula

2 2, , (2)

1

16j i

i i jN

M h h n r dG

Mass correction (splitting)

ADM mass formula

2 2, , (2)

1

16j i

i i jN

M h h n r dG

M eg

Mass correction (splitting)

ADM mass formula

2 2, , (2)

1

16j i

i i jN

M h h n r dG

M eg

1 1( ) ( )2 2l eg l

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.

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.

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