Wesson’s Induced Matter Theory with a Weylian Bulk

Abstract

The foundations of Wesson's induced matter theory are analyzed. It is shown that the empty-without matter-5-dimensional bulk must be regarded as a Weylian space rather than as a Riemannian one. Revising the geometry of the bulk, we have assumed that a Weylian connection vector and a gauge function exist in addition to the metric tensor. The framework of a Weyl-Dirac version of Wesson's theory is elaborated and discussed. In the 4-dimensional hypersurface (brane), one obtains equations describing both fields, the gravitational and the electromagnetic. The result is a geometrically based unified theory of gravitation and electromagnetism with mass and current induced by the bulk. In special cases on obtains on the brane the equations of Einstein-Maxwell, or these of the original induced matter theory.

Creation of Neutral Fundamental Particles in the Weyl–Dirac Version of Wesson’s IMT

“…The situation resembles the 4D Weyl geometry (9 -11) , where the light cone is the principal phenomenon describing the space- Following the ideas of Weyl (9,10,11) and Dirac (12) , developed by Nathan Rosen (13) and the present writer (14) , the Weyl-Dirac version of Wesson's IMT was proposed recently (7) and in it the measurability of length was proved (8) .…”

“…Partial differentiation is denoted by a comma (,), Riemannian covariant 4D differentiation by a semicolon (;), and Riemannian 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 2 As shown in a previous paper (7) of the present writer, there are serious reasons for a revision of Wesson's IMT. It was found that the induced geometry on 4D branes is non-integrable.…”

“…Wesson's Induced Matter Theory (IMT) (1,2,3,4,5,6) provides an elegant answer based on the creation of matter by geometry of the 5-dimensional bulk. In the Weyl-Dirac modification (7,8) 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, we are looking for a mechanism of creating fundamental particles.…”

“…Actually, (8) Starting from the 5D equations for the metric AB g and making use of the GaussCodazzi equations the following 4-D equations of gravitation were derived (7) :…”

“…We make use of a recently proposed Weyl-Dirac (W-D) modification of Wesson's IMT (7,8) , where the 5-dimensional (5D) bulk, in addition to the metric tensor AB g , possesses a Dirac gauge function Ω and a Weyl vector C w . On the brane, the five dimensional C w has its four dimensional (4D) counterpart µ w , the latter being regarded as the potential four vector of the Maxwell field.…”