A Geometrical Interpretation of Parity Violation in Gravity with Torsion

In a space-time with torsion, the action for the gravitational field can be extended with a parity-violating piece. We show how to obtain such a piece from geometry itself, by suitably modifying the affine connection so as to include a pseudo-tensorial part. A consistent method is thus suggested for incorporating parity-violation in the Lagrangians of all matter fields with spin in a space-time background with torsion. PACS Nos. : 11.30.Er, 04.20Cv, 11.10.Ef Electronic address: [email protected] Electronic address: [email protected] Theories of gravitation in a space-time with torsion have been under investigation for a long time. The presence of torsion modifies the spacetime geometry through nonsymmetric connections[1]. Looking from another angle, the commutator of the covariant derivatives acting on a scalar does not vanish in a torsioned space. Torsion has been incorporated in theories of gravitation, which range from ones as old as the EinsteinCartan model to delvelopments as recent as superstring theories. It has been argued that torsion is an inescapable consequence if the matter fields giving rise to space-time curvature are possessed with spin[2]. Various consequences of torsion have already been explored in the literature, including those involving complex connections[3], and aiming, for example, to present the electromagnetic potential as the trace of the torsion tensor[4]. One of the most important observations in this context is that the presence of torsion destroys the cyclic property of the Riemann-Christoffel tensor. As a result, the standard Einstein-Hilbert action admits of an additional term which is parity-violating[5]. Thus a generalised theory of gravity can be conceived of as having the possibility of parity violation inbuilt in it. An immediate extension of this idea leads to the expectation that the Lagrangians of various types of matter fields coupled to a torsioned space-time background[6, 7] will also contain parity-violating terms. Although such terms are a priori weaker than those corresponding to weak interacton (which is an already established source of parity violation), they can be significant in astrophysical and cosmological contexts[8], particularly in relation to coherent effects. And, above all, the sheer universality of gravity compels us to attach sufficient importance to such new physics possibilities. The question that perhaps may be raised is: is there a consistent way of introducing parity violation in a spacetime with torsion, which will be applicable to both pure gravity and the various types of matter fields with spin? In this note, we try to answer this question by seeking the origin of parity (or any discrete symmetry) violation in the geometry of space-time. We point out that just as the original Einstein-Cartan theory follows by adding an antisymmetric tensor to the hitherto symmetric affine connnections, the incorporation of a pseudo-tensorial connection introduces parity violation in the theory. This enables one to start from the original Einstein-Hilbert action, with the scalar curvature R suitably modified by the new form of the covariant derivative. We show that, with the most general choice of the pseudo-tensorial connection which is linear in the torsion field, the induced parity-violating term has close resemblance to what arises in earlier works from a separate piece in the Lagrangian. This generalised connection also enables us to obtain the Lagrangians for spin-1 and spin-1/2 fields, with built-in parity-violating interactions with the torsion field. Thus one can trace all parity-violating effects in torsioned space to a common origin. The Einstein-Hilbert action for pure gravity is given by S = ∫ √ −gRdx (1) where R is the scalar curvature, defined as R = Rαμβνg g . Rαμβν is the RiemannChristoffel tensor: R μνλ = Γ κ νλ,μ − Γκμλ,ν + ΓκμσΓσνλ − ΓκνσΓσμλ (2) In the absence of torsion, the Γ’s are the usual Christoffel symbols, symmetric in the two lower indices. If there is torsion, Γ needs to be replaced by Γ̃, where Γ̃μνλ = Γ μ νλ −H μ νλ (3) H being antisymmetric in the two lower indices. Further, the requirement that the metric be covariantly conserved (the so-called ‘metricity condition’) restricts H to a form where it is antisymmetric in all three indices, a form in which it is commonly known as the ‘contortion tensor’. It is related to the torsion field S through H μν = −S μν + Sνλμ − S μν . The inclusion of H destroys the cyclicity of Rαβμν in any three of its four indices. Consequently, a term of the form ǫRαβμν (which, in a torsion-free scenario, would have been forced to vanish by the cyclicity condition) can be added to the scalar curvature [5] , in perfect consistence with general covariance. The latter is manifestly parity-violating. Thus one is led to the conclusion that unless the conservation of parity is artificially imposed on the theory, there is no reason for it to hold generally in a scenario with torsioned space-time. However, the above way of introducing the parity-violating terms under torsion is of a somewhat ad hoc nature. Nor does it provide one with a guideline as to how to extend this to the gravitational interactions of particles of different spins, once parity ceases to be a symmetry in the pure gravity sector. As we have already mentioned, our proposal is to seek the very essence of parity violation in geometry itself, by introducing it in the covariant derivative in a curved space-time. That is to say, the most general connection can be made to include, in addition to the symmetric and antisymmetric parts, a pseudo-tensorial part as well. The commutators of covariant derivatives acting on a scalar in such a case is still non-vanishing, and will depend on both the torsion field and its pseudo-tensorial extension. Remembering that the latter should vanish in the limit of zero torsion, a general form in the minimal extension scheme is Γ̃κνλ = Γ κ νλ −H νλ − q(ǫ γδ νλH κ γδ − ǫ βλHβ να + ǫ βνH β λα) (4) Here q is a parameter determining the extent of parity violation, depending, presumably, on the matter distribution. It is extremely important to note at this juncture that the covariant derivative of the metric (Dμgνλ = 0, with Dμ defined in terms of Γ̃) automatically vanishes so long as H is antisymmetric in all three indices. This means that the same condition for metricity as the one in ordinary Einstein-Cartan theory also suffices when a pseudo-tensorial part is included in the connection. However, the relative signs of the three additional terms get fixed by metricity. The same requirement also leads us to the conclusion that all the three aforementioned terms have to be coupled through the same charge q. Next, we calculate the curvature scalar R̃ using the above connection and the metricity condition. After some algebra, one obtains