COFRAME TELEPARALLEL MODELS OF GRAVITY: EXACT SOLUTIONS

Abstract

The superstring and superbrane theories include gravity as a necessary and fundamental part of a (future) unified field theory. Thus it is important to consider the alternative representations of general relativity as well as the alternative models of gravity.We study the coframe teleparallel theory of gravity with a most general quadratic Lagrangian. The coframe field on a differentiable manifold is a basic dynamical variable. A metric tensor as well as a metric compatible connection is generated by a coframe in a unique manner.The Lagrangian is a general linear combination of Weitzenböck's quadratic invariants with free dimensionless parameters ρ1, ρ2, ρ3. Every independent term of the Lagrangian is a global SO(1,3)-invariant 4-form. For a special choice of parameters which confirms with the local SO(1,3) invariance this theory gives an alternative description of Einsteinian gravity — teleparallel equivalent of GR.The field equations of the theory is studied by a "diagonal" coframe ansatz which is a subclass of a most general spherical-nsymmetric Einstein–Mayer ansatz. The restricted Lagrangian depends only on two free parameters ρ1, ρ3.We obtain a formula for scalar curvature of a pseudo-Riemannian manifold with a metric constructed from the static "diagonal" solution of the field equation. It is proved that the sign of the scalar curvature depends only on a relation between the parameters ρ1and ρ3. Thus by a specific choice of free parameters a manifold of positive or negative curvature can be obtained. The scalar curvature vanishes only for a subclass of models with ρ1=0. This subclass includes the teleparallel equivalent of GR.We obtain the explicit form of all spherically symmetric static solutions of the "diagonal" type to the field equations for an arbitrary choice of free parameters. We prove that the unique asymptotic-flat solution with Newtonian limit is the Schwarzschild solution that holds for a subclass of teleparallel models with ρ1=0. Thus the Yang–Mills-type term of the general quadratic coframe Lagrangian should be rejected.