A$\overline {Poincar\acute{e}}$ gauge theory of gravity

Abstract

We develop a gauge theory of gravity on the basis of the principal fiber bundle over the four-dimensional space-timeM with the covering groupP¯ ₀ of the proper orthochronous Poincaré group. The field components$e^k _\mu$ are constructed with the connection coefficients$A^k _\mu$,$A^{kl} _\mu$ and with a Higgs-type fieldψ. A Lorentz metricg is introduced with$e^k _\mu$, which are then identified with the components of duals of the Vierbein fields. Associated with ψ there is a spinor structure onM. For Lagrangian densityL, which is a function of$A^k _\mu$,$A^{kl} _\mu$,ψ, matter field φ, and oftheir first derivatives, we give the conditions imposed by the requirement of the$\overline {Poincar\acute{e}}$ gauge invariance. The Lagrangian densityL is restricted to be of the formL =L ^tot(ψ, T ^klm,R ^klmn,Δ _k φ, φ), in whichT ^klm,R ^klmn are the field strengths of$e^k _\mu$,$A^{kl} _\mu$, respectively. Identities and conservation laws following from the$\overline {Poincar\acute{e}}$ gauge invariance are given. Particularly noteworthy is the fact that the energy momentum conservation law follows from theinternal translational invariance. The field equation of ψ is automatically satisfied, if those of$A^k _\mu$ and of φ are both satisfied. The possible existence of matter fields with “intrinsic” energy momentum is pointed out. When φ is a field with vanishing “intrinsic” energy momentum, the present theory practically agrees with the conventional “Poincaré gauge” theory of gravity, except for the seemingly trivial terms in the expression of the spin-angular momentum density. A condition leading to a Riemann-Cartan space-time is given. The fieldψ holds a key position in the formulation.