LyST: a Scalar-Tensor Theory of Gravity on Lyra Manifold

Abstract

We present a scalar-tensor theory of gravity on a torsion-free and metric compatible Lyra manifold. This is obtained by generalizing the concept of physical reference frame by considering a scale function defined over the manifold. The choice of a specific frame induces a local base, naturally non-holonomic, whose structure constants give rise to extra terms in the expression of the connection coefficients and in the expression for the covariant derivative. In the Lyra manifold, transformations between reference frames involving both coordinates and scale change the transformation law of tensor fields, when compared to those of the Riemann manifold. From a direct generalization of the Einstein-Hilbert minimal action coupled with a matter term, it was possible to build a Lyra invariant action, which gives rise to the associated Lyra Scalar-Tensor theory of gravity (LyST), with field equations for $g_{\mu\nu}$ and $\phi$. These equations have a well-defined Newtonian limit, from which it can be seen that both metric and scale play a role in the description gravitational interaction. We present a spherically symmetric solution for the LyST gravity field equations. It dependent on two parameters $m$ and $r_{L}$, whose physical meaning is carefully investigated. We highlight the properties of LyST spherically symmetric line element and compare it to Schwarzchild solution.