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In this work the concept of generally covariant duality is treated with the introduction of generalized Levi-Civita tensor within the framework of vierbein formalism. The equations for the attached Lorentz scalar fields are derived from vierbein postulate. It is shown that the masses of the associated particles including that with negative square mass are completely determined by Einstein’s cosmological constant.

Recently there have been many attempts to approach the problem of unification of fundamental interactions on the base of Extended General Relativity [

On the other side, superstring theory [

The aim of this work is to consider the concept of generally covariant duality in General Relativity with the introduction of generalized Levi-Civita tensor and to study the specific properties of the attached fields within the framework of vierbein formalism.

It is shown that the masses of the associated particles, in particular tachyon-like particle, are completely determined by Einstein’s cosmological constant.

In special Relativity the Duality concept is treated by means of the 4-rank Levi-Civita tensor

for electromagnetic field strength tensor

with the identification

for electric

General Relativity requires the generalized version of

and in correspondence the relation (1) is modified to become

where D denotes covariant derivative,

Let the tensor

where

Together with

with the convention

Like for Riemann metric

(

where B(x) and C(x) are some one-component fields and

they transform according to the rule:

J being Jacobian transformation determinant.

The Formula (8) tells that the fields B(x) and C(x) are scalar with respect to Lorentz transformation only, but

instead

space inverse transformation,

From Equations (4)-(7) it follows that the fields B(x) and C(x) have the following vierbein structure:

where

Note also that:

In this sense B(x) and C(x) might be referred to as dual partners.

We now derive the equations for B(x) and C(x), starting from vierbein postulate

From the vierbein structure (4) and (6) this gives:

By inserting (7) into (13) we have:

And hence:

From the expression of

Up to first order in gravitational constant

where

Equations (15) with the expressions (17) inserted gives:

On the other hand, by performing similar calculations for the Ricci tensor we obtain:

Hence, Equations (18) can be rewritten as:

By inserting here the expression of R,

derived from Einstein’s equation with cosmological constant

(

where

Equation (23) tells that the fields B(x) and C(x) have square mass equaling

This corresponds to the following Lagrangian terms describing the fields B(x) and C(x) interacting with the gravitational field:

This also means that one of them is tachyon-like particle unless

In this work we consider the concept of Generally Covariant Duality. The focus point is the generalization of flat Levi-Civita tensor for the case of curved Riemann space-time. This leads to some kind of pseudoscalar fields of cosmological nature with the masses closely related to Einstein’s cosmological constant. In particular among them there is tachyon-like particle having negative square mass. Taking into account that the cosmological constant has a close relation to dark energy, one might think about the possibility for tachyon to be among the candidates for dark matter.