Massive vector field as a source of space-time torsion

Conclusions

As we have shown in the foregoing section, an asymmetric connection with nonvanishing torsion must be used to describe the gravitational interaction of a massive vector field. The following question arises: could one not eliminate the non-Reimannian part of the connection by modifying the Lagrangian? In my opinion, this cannot be done by a sufficiently natural modification of the orginal Lagrangian (5) and (9); for if one assumes, for example, that there exists a “direct” interaction of V_k with the gravitational field, i.e., the Lagrangian of the system of these fields has the form

$$K + L_0 + L_v ,$$

where$L_v = {\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}\left( {fV^i V^l \eta _{ik} \wedge \Omega _l^k + gV_i V^i \eta _k^l \wedge \Omega _l^k } \right)$, the resulting equation for the connection will differ from (7a) by the presence on the left-hand side of an additive term of the form

$$D\left( {fV^k V_i \eta _{kj} + gV_k V^k \eta _{ij} } \right).$$

It is readily seen that the presence of such a term cannot cancel the non-Riemannian part of the connection for any choice of f or g.

Finally, we mention that the equation for the field V_k obtained by varying the Lagrangian (9) with respect to V_k can be written with allowance for (13) in the form

where R_i is the term nonlinear in V_i and is a linear combination of terms of the type$V^n \bar D^k V_i \bar D_k V_n$ and$V^n V^k \bar D^n \bar D_k V_i$. We shall not give here the very cumbersome explicit expression for this term (which contains more than twenty terms).

Thus, the torsion of spacetime that results from the gravitational interaction of a massive vector field can be attributed to a nonlinear selfinteraction of the field V_k. The nonlinear term has a much more complicated structure that the corresponding nonlinear term for a spinor field [5, 6, 8].