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Creation of fundamental particles in Wesson's IMT

Fundamental particles, regarded as the constituents of quarks and leptons, are described classicaly in the framework of the Weyl-Dirac version of Wesson's Induced Matter Theory. There are neutral particles and particles having charge Q=+/-(1/3e). The particles appear on the 4D brane, our universe, and are filled with a substance induced by the 5D bulk. This substace is taken to have mass density, charge density, pressure and is characterized by the prematter eq. of state P+\rho=0. The interior is separated from the surrounding vacuum by a boundary surface where the 4D metric tensor satisfies an a'la Schwarzschild condition. Outside of the boundary holds the Schwarzschild, or the Reissner-Nordstroem Metric, while the particles are characterized by mass, radius, charge.


INTRODUCTION
Matter and field are basic concepts of classical field theories. They play a fundamental role in the general relativity theory [1,2], where the Einstein tensor ν µ G is expressed in terms of the geometry of space-time, and the matter is represented by its momentumenergy density tensor ν µ T . These two intrinsic concepts are connected by the Einstein field equation Theory (IMT) [3][4][5][6][7][8] provides an elegant answer based on the creation of matter by geometry of the 5-dimensional (5D) bulk. In the Weyl-Dirac modification [9, 10] of Wesson's IMT the bulk induces on the 4D brane both, gravitation and electromagnetism, as well gravitational matter and electric current. Now, as a considerable amount of conventional matter appears in the form of particles, it would be interesting to look for a mechanism of creating fundamental particles in the framework of the Weyl-Dirac modification of Wesson's IMT (cf. [11].
In the present note we investigate the possibility of creation 4D neutral and electrically charged particles, induced by the 5D bulk in the framework of the Weyl-Dirac modification of Wesson's theory.
These particles are to be regarded as constituents of elementary particles (like quarks and leptons) and are characterized by their charge being e 3 1 ; 0 ± , with e -the electron charge, as well by mass. Thus every quark or lepton is made up of three of these particles.
These fundamental classical particles having charge and mass are taken to be spinless and to have spherical symmetry. It is expected that, when they are quantized, they will acquire a spin, as in the case of a point particle described by the Dirac equations.
Presumably the particles have other properties such as color hypercolor etc. However, these will be left to be dealt with in the future.
In the present work following conventions are valid: Uppercase Latin indices run from 0 to 4; lowercase Greek indices run from 0 to 3. Partial differentiation is denoted by a comma (,), Riemannian covariant 4D differentiation by a semicolon (;), and Riemannian covariant 5D differentiation by a colon (:). Further, the 5D metric tensor is denoted by AB g , its 4D counterpart by µν h ; sometimes 5D quantities will be marked by a

THE EMBEDDING FORMALISM. THE FIELD EQUATIONS
Following the ideas of Weyl [12,13] and Dirac [14], developed by Nathan Rosen [15] and the present writer [16,17], the Weyl-Dirac version of Wesson's IMT was proposed recently [9,10]. As shown in a previous paper [9] of the present writer, there are serious reasons for a revision of Wesson's IMT. It was found that the induced geometry on a 4D brane is non-integrable.
This non-integrability follows from the structure of the bulk. In Wesson's original 5D IMT, one regards the bulk as pure geometry without any additional fields. The geometry is described by the metric tensor AB g . Thus, the principal phenomenon, which carries information, is a metric perturbation propagating in the form of a gravitational wave. In order to avoid misinterpretations one must assume that all gravitational waves have the same speed. Therefore, in the 5D bulk, the isotropic interval between two space-time events (Cf. [12,13] Below follows a concise description of the general embedding formalism. The notations as well as the geometric construction given below accord to these given in works of Paul Wesson and Sanjeev S. Seahra [3][4][5][6][7], as well in works of the present writer [9,11]. As mentioned above, one considers a 5-dimensional manifold { M } (the "bulk") with the metric AB g . The latter has the signature ( ) The manifold is mapped by coordinates { A x } and described by the line-element A given 5D quantity (vector, tensor) in the bulk has a 4D counterpart located on the brane. These counterparts may be formed by means of the following system of basis vectors, which are orthogonal to A n The brane l Σ is stretched on four ( ) Let us consider a 5D vector in the bulk {M}. Its 4D counterpart on the brane l Σ is given by On the other hand the 5D vector may be written as Further, the 5D metric tensor, . The vector A w induces on the brane its counterpart µ w (cf. (6)), which is regarded as the potential vector of the 4D Maxwell field , the latter being also the 4D counterpart of AB W . On the 4D brane one has also the metric µν h (cf. (8)) and the Dirac gauge function Ω .
Starting from the 5D equations for the metric AB g and making use of the Gauss-Codazzi equations the 4-D equations of gravitation was derived recently [9,10] ( ) Further from the equation of the source-free 5D Weylian field in the bulk ( ) 0 the 4D equation for the Maxwell field µν W on the brane was derived in [9] ( ) In the principal equations (10, 11) appear the following quantities: a) The conventional energy-momentum density tensor of the 4D electromagnetic field It is believed that the entity is restricted by a spherical boundary surface of radius b r r = ; the interior ( b r r ≤ ) is filled with a substance induced by the bulk and described by matter density ρ , charge density e ρ and pressure P . These three characteristic functions have The bulk is be mapped by . and the 5D line element is given by Let us assume that for our metric functions the dependence on r and on l, may be separated, so that Hereafter, we denote a partial derivative with respect to r by a prime and that with respect to the fifth coordinate l by a dot. Without any restriction we can impose the condition ( ) ( ) ( ) 0 for the values on the brane 0 l l = -our 4D spacetime.
The basic vectors, the metrics as well the Christoffel symbols of (13-15) are given by Besides the metric tensor AB g , the bulk possesses the Weyl vector A w , which we take having the following non-zero components and as ( ) ( ) ( ) 0 we have for the 4D Maxwell field on the brane (cf. (6)) There is also the Dirac gauge function Ω and its partial derivative It must be emphasized that the 5D bulk is empty -it possesses no matter or other field sources. The functions Ω ,  11)).
It is convenient to write the gravitational equation (10) in its co-contravariant form.
Further, we take into account that by (19) and The quantities appearing in (20) and listed in (12a-12e) may be accounted making use of (16)(17)(18), as well of (A-3) -(A-5). The result is listed in (A-6) Turning to an auxiliary gauge function ( ) ( ) r r Ω = ln ω , and making use of (A-6a-A-6e), we obtain from (20) the explicitly written gravitational equations on the brane Integrating (24) one obtains In the above procedure are 4 equations (21 -23, 25) for six functions, The quantity ) ( r q is regarded as the effective charge inside a sphere of radius r and according to (21) and (25)) it is given by (The constant term in (25), which leads to a singular point charge, was discarded.) The term in (21a-23a) , which includes the remaining terms in the RHS of (21), is the matter density inside the spherically symmetric entity, ( ) r P n is the radial pressure and ( ) r P τ stands for the tangential pressure.
We are looking for a non-rotating entity filled with perfect fluid, therefore we impose The second condition will be imposed in order to get the prematter equation of state 2 The prematter condition (28) leads to There is, however, a restriction. For the metric as given in (13) 2 Following previous papers [18,19] we will refer to matter in such a state as "prematter" and regard it as a primary substance.
Equations (21)(22)(23)25) with conditions (29, 30) describe a charged spherically symmetric, static prematter entity. Below we will make use of these equations in order to get models of neutral and charged particles.

A NEUTRAL PARTICLE IN THE EINSTEIN GAUGE
In this section a spatially restricted entity in the Einstein gauge will be considered.
Consequently we set 0 = Λ and 0 Let us take the coordinates on the 4D brane as well the static, spherically symmetric line-element as given by (13). But in (15) (33) and the 5D metric tensor is On the brane one has for the 4D Weyl vector Taking into account (33-37) one obtains from (21-23) the gravitational EQ-s on the brane The 4D Maxwell EQ. (25) in our case (cf. (33, 35)) is Thus, (41a) is satisfied identically 3 and we are left with From (44 -46) we have for the matter density and for the pressure We recall that the mass density is given by

A NEUTRAL PARTICLE IN AN APPROPRIATE GAUGE
In this section a neutral spherically symmetric entity with an arbitrary gauge function will be considered. As we are interested in spatially restricted entities, we neglect the From (20a) (or alternatively from (21) -(23)) one obtains the gravitational equations.
, is also arbitrary. In order to have a spherically symmetric non-rotating entity one equates the RHS of (55) and (56) obtaining the , on the brane 0 l Σ .
Then we obtain a very simple gauge condition Instead of solving (56a) one can make use of the equilibrium equation This is formally identical with that obtained in the previous model (cf. (51)). One sees that there is a de Sitter universe, and if one introduces The described entity may be regarded as a classical model of a neutral fundamental particle induced by the 5D bulk. It must be emphasized that the present model is obtained by the choice (59) of the gauge function and that the constant mass density inside the particle according to (61) is given by It is believed that more models of neutral particles may be found in addition to the two presented in Sec. 4 and 5.

A CHARGED PARTICLE
To get an entity, which may be regarded as a charged particle, we will adopt the static spherically symmetric 4D line element (13), but for the metric functions given in (15) we will set ( ) ( ) ; Thus, the 5D line element is This results in the very simple gauge condition (cf. (59)) ψ ω ψ ω  ( ) ( ) From (68) and (69) As noted above, the entity is restricted by a sphere of radius b r . Inside there is the prematter substance, outside one has vacuum. Introducing the function ( )  In the present section a plausible model of charged fundamental particles created by the bulk in the Weyl-Dirac modification of Wesson's IMT was obtained. It is believed that more models may be found besides the considered above.

DISCUSSION
Is it possible to describe singularity-free particles from the classical (non-quantum) standpoint? Einstein and collaborators were certain that particles having inner structure can be considered in the framework of general relativity. As long ago in 1935 Albert Einstein and Nathan Rosen in their celebrated work [20] presented an interesting solution to the problem, with a charged particle described as a "bridge" 4 .
Later, in 1991, N, Rosen and the present writer presented general relativistic models [21,22] of fundamental particles consisting of prematter, the latter satisfying the equation In the present paper, models of fundamental neutral and charged particles in the Weyl -Dirac version [9, 10] of Wesson's IMT [3][4][5][6][7][8] are presented. These are induced by the 5D Weyl-Dirac-Wesson bulk in the empty 4D brane, our universe. In this framework models of neutral and electrically charged fundamental particles are carried out. In all considered models, the interior is filled with a substance, being in the state of prematter (cf. Ref. [18,19]).
The reason for taking prematter as a substance suitable for describing the inside of particles is the following. Let us suppose one is looking for extremely small fundamental particles having a noticeable mass. This seems to be possible only with an enormous mass density ρ . One can expect that at such densities the properties of matter will be very different from those, with which we are acquainted. Bearing in mind that we lack any knowledge whatsoever of the constitution of matter and its behavior under such extreme conditions, let us assume that inside the particle the matter tensor is simply related to the metric tensor in the sense that (This approach was used first by E. Gliner [20,21] in the seventies.) From (93) one is led to ; ; and finally to 0 = + P ρ . It must be emphasized that inside the entity one has an enormous tension, making for the particle's stability.
In carrying out the models we started from the static spherically symmetric lineelements (13)(14)(15). The interior of neutral particles, considered in the present work, is In SEC. 6, a model of charged particles was considered as a spherically symmetric entity filled with induced charged prematter in the brane. This entity is restricted by a , the prematter filling the interior is characterized by a mass density and pressure-tension . It may also be that there are two neutral particles with each of these masses; although from the aesthetic point of view it seems desirable to have as few different fundamental particles as possible.

APPENDIX
The metric tensors as given in EQ-s (13), (14) are ϑ λ ν Hereafter a dot will denote partial differentiation with respect to l , while a prime will stand for the partial derivative with respect to r . Taking into account the r, l separation (cf. (15)