Coframe teleparallel models of gravity. Exact solutions

TL;DR

This paper explores the coframe teleparallel theory of gravity with a general quadratic Lagrangian, deriving exact solutions and analyzing conditions under which it reproduces Einsteinian gravity, including the Schwarzschild solution.

Contribution

It provides a comprehensive analysis of the most general quadratic coframe teleparallel gravity models, identifying conditions for equivalence with general relativity and deriving explicit static spherically symmetric solutions.

Findings

Scalar curvature depends on free parameters, vanishing for a subclass with =0.

The Schwarzschild solution is the unique asymptotic-flat solution with Newtonian limit.

Yang-Mills-type term should be rejected for physical viability.

Abstract

The superstring and superbrane theories which include gravity as a necessary and fundamental part renew an interest to 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 Weitzenb\"{o}ck's quadratic invariants with free dimensionless parameters $\r_1,\r_2,\r_3$. 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. We prove that the sign of the scalar curvature of a metric generated by a static spherical-symmetric solution depends only on a relation between the free parameters. The scalar curvature vanishes only for a subclass of models with $\r_1=0$. 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 $\r_1=0$. Thus the Yang-Mills-type term of the general quadratic coframe Lagrangian should be rejected.