Area metric gravity and accelerating cosmology

Abstract

Area metric manifolds emerge as effective classical backgrounds in quantum string theory and quantum gauge theory, and present a true generalization of metric geometry. Here, we consider area metric manifolds in their own right, and develop in detail the foundations of area metric differential geometry. Based on the construction of an area metric curvature scalar, which reduces in the metric-induced case to the Ricci scalar, we re-interpret the Einstein-Hilbert action as dynamics for an area metric spacetime. In contrast to modifications of general relativity based on metric geometry, no continuous deformation scale needs to be introduced; the extension to area geometry is purely structural and thus rigid. We present an intriguing prediction of area metric gravity: without dark energy or fine-tuning, the late universe exhibits a small acceleration. INVITATIONA new theoretical concept which, once formulated, naturally emerges in many related contexts, deserves further study. Even more so, if it makes us view well-established theories in a novel way, and meaningfully points beyond standard theory.Area metrics, we argue in this paper, are such an emerging notion in fundamental physics.An area metric may be defined as a fourth rank tensor field which allows to assign a measure to two-dimensional tangent areas, in close analogy to the way a metric assigns a measure to tangent vectors. In more than three dimensions, area metric geometry is a true generalization of metric geometry; although every metric induces an area metric, not every area metric comes from an underlying metric. The mathematical constructions, and physical conclusions, of the present paper are then based on a single principle:Spacetime is an area metric manifold.We will be concerned with justifying this rather bold idea by a detailed construction of the geometry of area metric manifolds, followed by providing an appropriate theory of gravity, which finally culminates in an application of our ideas to cosmology. In the highly symmetric cosmological area metric spacetimes, we can compare our results easily to those of Einstein gravity. We obtain the interesting result that the simplest type of area metric cosmology, namely a universe filled with non-interacting string matter, may be solved exactly and is able to explain the observed [1,2] very small late-time acceleration of our Universe, see the figure on page 40, without introducing any notion of dark energy, nor by invoking fine-tuning arguments.It may come as a surprise, but standard physical theory itself predicts the departure from metric to true area metric manifolds. More precisely, the quantization of classical theories based on metric geometry generates, in a number of interesting cases, area metric geometries: back-reacting photons in quantum electrodynamics effectively propagate in an area metric background [3]; the massless states of quantum string theory give rise to the Neveu-Schwarz two-form potential and dilaton besides the graviton, producing a generalized geometry which may be neatly...

Multimetric extension of the parametrized post-Newtonian formalism: Experimental consistency of repulsive gravity

“…Another close look at the newly obtained equations for c ± 3 and the expansions given in appendix A reveals that all terms containing σ ± − drop out due to the gauge invariance conditions (12). The equations for c ± 3 then turn out to be linearly dependent on the equations we have obtained from the scalar and tensor components of the equations of motion, and thus they are solved identically.…”

“…higher-dimensional models such as [8,9]; and structural extensions such as non-symmetric gravity theory [10] and area metric gravity [11,12]. Of course all alternative theories of gravity have to be tested against the data available from solar system experiments.…”

Gravitational dynamics for all tensorial spacetimes carrying predictive, interpretable, and quantizable matter

“…To aid the reader's intuition, we illustrate each abstract construction in this section immediately for the familiar example of a standard metric geometry, before moving on to the next construction. Occasionally we will also contrast this to area metric geometry [10,11] as a comparatively well-studied example for a non-metric tensorial geometry.…”

Propagation of light in area metric backgrounds

“…where the Fresnel tensor G only depends on the cyclic part C abcd = G abcd − G [abcd] of the inverse area metric [18]. Generically, equation (3) 2 = 0.…”

“…A very natural and surprisingly fruitful way to view all of these generalized geometries (or at least salient aspects in some cases) is in terms of area metric manifolds [18][19][20], where an area metric is a smooth covariant tensor field G of fourth rank that assigns a measure to tangent areas in a way similar to how a metric assigns a measure to tangent vectors. The precise definition is given in section 2.1.…”