Commutative deformations of general relativity: nonlocality, causality, and dark matter

Abstract

Hopf algebra methods are applied to study Drinfeld twists of (3 + 1)-diffeomorphisms and deformed general relativity on commutative manifolds. A classical nonlocality length scale is produced above which microcausality emerges. Matter fields are utilized to generate self-consistent Abelian Drinfeld twists in a background independent manner and their continuous and discrete symmetries are examined. There is negligible experimental effect on the standard model of particles. While baryonic twist producing matter would begin to behave acausally for rest masses above ∼1-10 TeV, other possibilities are viable dark matter candidates or a right-handed neutrino. First order deformed Maxwell equations are derived and yield immeasurably small cosmological dispersion and produce a propagation horizon only for photons at or above Planck energies. This model incorporates dark matter without any appeal to extra dimensions, supersymmetry, strings, grand unified theories, mirror worlds, or modifications of Newtonian dynamics.

Commutatively deformed general relativity: foundations, cosmology, and experimental tests

“…A few years ago Hopf algebra methods for commutatively deformed classical curved manifolds were studied and applied to investigate spacetime physics and gravitation [1,2]. This approach differs from quantum non-commutative models, where coordinates x μ are promoted to quantum operatorsx μ obeying [x μ ,x ν ] = iθ μν for some background a e-mail: Paul.deVegvar@post.harvard.edu (corresponding author) field θ μν [3][4][5][6].…”

“…Instead, commutatively deformed classical manifolds possess a commutative -product, ( f g)(x) = (g f )(x) [9][10][11]. In the early work [1], f g was defined via a Drin'feld differential twist (DDT):…”

“…For the deformed action to retain gauge invariance(s), X itself must be gauge invariant. It was shown in earlier work [1] that if X is derived from a matter current, 0123456789(). : V,-vol then that timelike current is either a U (1) current or gaugeless, and the twist producing substance is called Groenewold-Moyal (GM) matter.…”

“…Because contains X and X contains , X is defined recursively or self-consistently (sc) as a solution of X = X sc [φ, * (X )], where X sc is an expression such as (3) or (4), and will be discussed in more detail below. Investigating these GM gauge current forms for the generators in [1] led to the conclusion that GM matter could not be standard model (SM) matter, rather it had to be either a sterile neutrino or a viable dark matter (DM) candidate. A -Einstein field equation for the -metric tensor was derived that included an uncalculated O(λ 1 ) correction term μν to the -Einstein tensor coming from variations of the fields contained within (X ).…”

“…These early steps raise questions about the foundations of DGR along with its cosmological and phenomenological implications. For instance, when using the the DDT form (1) for the twist the -Einstein equation, and indeed any -Euler-Lagrange or -wave equation, all become infinite order partial differential equations. The theory then appears to lose predictive power, since knowledge of derivatives of all orders at some boundary condition becomes necessary to determine the dynamics.…”