Theory and phenomenology of area-metric gravity

Abstract

Area metrics are generalised geometric structures to describe spacetime. They feature additional non-metric degrees of freedom beyond the metric degrees of freedom of classical gravity at low energies. As such, area-metric gravity is a candidate effective field theory for the continuum limit of loop quantum gravity and spin foams which accounts for the extended gravitational configuration space of these approaches in their semiclassical regime. On these grounds, following a bottom-up approach, we construct area-metric gravity perturbatively guided by the principle of general covariance. The most general local and diffeomorphism-invariant action quadratic in area-metric perturbations and of second order in derivatives contains four free parameters. These are the two masses of the right-handed and left-handed non-metric degrees of freedom, and the two interaction couplings between these and the metric degrees of freedom of the area metric. Linearised area-metric gravity violates parity for generic values of these parameters. The effective metric actions obtained after integrating out the non-metric degrees of freedom are quasi-local linearised Einstein-Weyl actions. For special choices of couplings, the spin-2 propagator is ghostfree and exhibits only the pole associated with the massless graviton. The corresponding two-parameter subclass of linearised area-metric actions is characterised by a shift symmetry in the kinetic term. The physical spectrum of these theories consists of two massless transverse-traceless modes and five additional massive modes. The Hamiltonian dynamics mixes the two massless transverse-traceless modes in the linear polarisation basis as a result of parity violation in the original area-metric Lagrangian. Extending the analysis of area-metric actions, we show that modified Plebanski theories provide a natural framework for non-linear area-metric gravity. In these theories, a subset of the simplicity constraints on the bivector field in the original Plebanski action is replaced by a potential. This mechanism may be viewed as a continuum analogue of the weak imposition of second-class constraints in the spin-foam path integral. The Immirzi parameter γ, defined as the inverse coupling in front of the Holst action, and appearing in the commutator between second-class constraints in the quantum theory, is identified as a parity-violating coupling in area-metric gravity. Different from classical metric gravity, γ enters the dynamical equations of motion in area-metric gravity. Based on these results, we proceed to analyse aspects concerning the phenomenological viability of area-metric gravity as a quantum and classical effective field theory. Considering area-metric gravity as a local quantum effective field theory in a regime below the cutoff scale of a fundamental theory of quantum gravity, we evaluate its renormalisation-group flow towards the infrared regime. The non-metric degrees of freedom generically decouple as a result of their heavy masses at low energies. However, simultaneously growing interaction couplings between these and the metric degrees of freedom may leave an imprint in the low-energy effective action for the metric. In addition, parity violation at high energies is dynamically enhanced at low energies. The flow of the Immirzi parameter exhibits fixed points and zero and infinite γ. Finally, we consider non-linear quasi-local Einstein-Weyl gravity as a classical effective field theory for the metric degrees of freedom in area-metric gravity. After localising the action and restricting to static spherical symmetry, we show that solutions in the weak-field regime are characterised by an effective mass parameter which reduces to the mass of the spin-2 ghost in local Einstein-Weyl gravity when the non-locality in the original action is taken to zero. Additionally, we derive a regular Frobenius solution family at the radial center as the first step towards a future classification of Frobenius solutions around a generic expansion point.