Lagrange multipliers and gravitational theory

Abstract

The Lagrange mUltiplier version of the Palatini variational principle is extended to nonlinear Lagrangians. where it is shown in the case of the quadratic Lagrangians, as expected, that this version of the Palatini approach is equivalent to the Hilbert variational method. The (nonvanishing) Lagrange multipliers for the quadratic Lagrangians are then explicitly obtained in covariant form. It is then pointed out how the Lagrange multiplier approach in the language of the (3+ I)-formalism developed by Arnowitt, Oeser, and Misner permits the recasting of the equations of motion for quadratic and general higher-order Lagrangians into the ADM canonical formalism. In general without the Lagrange multiplier approach, the higher order ADM problem could not be solved. This is done explicitly for the simplest quadratic Lagrangian (g 1/2R 2) as an example.

Stability in quadratic torsion theories

“…[23,24]. Therefore, the two methods coincide without imposing the Lagrange multiplier when after solving the field equations the related quantity turns out to be zero.…”

“…[34]. Back to the Lagrangian density (24), when the metric corresponds to the Minkowski spacetime the expression reduces to…”

“…As we want to get a set of stability condition on the parameters of the theory when g μν = η μν , we take the expression of the curvature tensors (D.27), (D.28) and (D.29) to compute the scalars in the Lagrangian density (24). These are…”

“…In the spirit of investigating only slight modifications of GR, we assume that they are the only non-vanishing torsion components for a minimal modification over the FLRW background. Under these considerations, we will now impose the absence of ghost and tachyon instabilities for the theory given by the Lagrangian density (24). The quadratic Riemann and torsion terms that appear in this Lagrangian density are computed in Appendix D.…”

“…The field equations of the Lagrangian density (24) have to be obtained, as usual, from a variational principle where the action is extremised with respect to the dynamical variables. However, different sets of dynamical variables can be chosen and different field equations will be obtained accordingly.…”