1. Motivation
Recently there have been many attempts to approach the problem of unification of fundamental interactions on the base of Extended General Relativity [1] . The starting insight is that all the fundamental interactions are originated from the nature of space-time itself as the gravitational interaction is.
On the other side, superstring theory [2] [3] is also considered as a promising approach to the construction of the unification theory. Among the most notable theoretical consequences of string theory is the existence of tachyon-scalar particle having negative square mass. In this connection we would mention that according to the results obtained in Refs. [4] - [6] the existence of tachyons could be originated from the compactification of time- like extradimensions.
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.
2. Generally Covariant Duality
In special Relativity the Duality concept is treated by means of the 4-rank Levi-Civita tensor
. The well known example is the relation
(1)
for electromagnetic field strength tensor
with the identification
for electric 
and magnetic 
fields, Equation (1) represents Maxwell’s equations.
General Relativity requires the generalized version of
, which is to be some fully antisymmetric 4-rank tensor denoted by
. In this connection the covariant wedge product of two vectors 
and 
is to be defined as:
(2)
and in correspondence the relation (1) is modified to become
(3)
where D denotes covariant derivative,
being affine connection.
Let the tensor 
under consideration have 
as its vierbein component, namely:
![]()
(4)
where ![]()
stands for vierbein, (![]()
being vierbein indices) statisfying the relations with metric tensors:
![]()
(5)
![]()
and ![]()
being Riemann and Minkowski metric.
Together with ![]()
let us also consider its contravariant partner
![]()
(6)
with the convention
![]()
Like for Riemann metric ![]()
which is expressed as
![]()
(![]()
―gravitational constant,![]()
―gravitational field), here we can put:
![]()
(7)
where B(x) and C(x) are some one-component fields and ![]()
in the limit of flat space-time. Under general transformation.
![]()
they transform according to the rule:
![]()
(8)
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 ![]()
and ![]()
are scalar with respect to general transformation,![]()
. With respect to
space inverse transformation, ![]()
, they both behave like pseudoscalar,
![]()
(9)
From Equations (4)-(7) it follows that the fields B(x) and C(x) have the following vierbein structure:
![]()
(10)
where ![]()
denotes the matrix having ![]()
as element in row a and column![]()
,![]()
―matrix having ![]()
as element in row ![]()
and column![]()
,
![]()
.
Note also that:
![]()
. (11)
In this sense B(x) and C(x) might be referred to as dual partners.
3. Dual Equations
We now derive the equations for B(x) and C(x), starting from vierbein postulate
![]()
(12)
From the vierbein structure (4) and (6) this gives:
![]()
(13)
By inserting (7) into (13) we have:
![]()
(14)
And hence:
![]()
(15)
From the expression of ![]()
we have:
![]()
(16)
Up to first order in gravitational constant ![]()
the calculations give:
![]()
(17)
where![]()
.
Equations (15) with the expressions (17) inserted gives:
![]()
(18)
On the other hand, by performing similar calculations for the Ricci tensor we obtain:
![]()
(19)
Hence, Equations (18) can be rewritten as:
![]()
(20)
By inserting here the expression of R,
![]()
(21)
derived from Einstein’s equation with cosmological constant![]()
,
![]()
(22)
(![]()
denotes energy-momentum tensor of matter field) we have:
![]()
(23)
where
![]()
(24)
Equation (23) tells that the fields B(x) and C(x) have square mass equaling
![]()
(25)
This corresponds to the following Lagrangian terms describing the fields B(x) and C(x) interacting with the gravitational field:
![]()
(26)
This also means that one of them is tachyon-like particle unless![]()
, when they both are massless.
4. Conclusion
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.