We provide a Lie algebra expansion procedure to construct three-dimensional higher-order Schrödinger algebras which relies on a particular subalgebra of the fourdimensional relativistic conformal algebra. In particular, we reproduce the extended Schrödinger algebra and provide a new higher-order Schrödinger algebra. The structure of this new algebra leads to a discussion on the uniqueness of the higher-order non-relativistic algebras. Especially, we show that the recent d-dimensional symmetry algebra of an action principle for Newtonian gravity is not uniquely defined but can accommodate three discrete parameters. For a particular choice of these parameters, the Bargmann algebra becomes a subalgebra of that extended algebra which allows one to introduce a mass current in a Bargmann-invariant sense to the extended theory.April 15, 2020

We searched for a resolution of the flat galactic rotation curve problem from geometry instead of assuming the existence of dark matter. We observed that the scale independence of the rotational velocity in the outer region of galaxies could point out to a possible existence of local scale symmetry and therefore the gravitational phenomena inside such regions should be described by the unique local scale symmetric theory, namely Weyl's theory of gravity. We solved field equations of Weyl gravity and determined the special geometry in the outer region of galaxies. In order to understand the scale dependent description of gravitational phenomena, we compared individual terms of so called Einstein-Weyl theory and concluded that while the outer region of galaxies are described by the Weyl term, the inner region of galaxies are described by the Einstein-Hilbert term.

We calculate the deflection angle of light from a distant source by a galaxy cluster in Weyl's conformal gravity. The general method of calculation is first applied to calculate the deflection angle in Schwarzschild-de Sitter (Kottler) spacetime. The deflection angle calculated in Kottler spacetime includes the contribution of the cosmological constant, which quantitatively agrees with one work and disagrees with many works in the literature. We then calculate the deflection angle in Mannheim-Kazanas spacetime in two conformally related coordinate systems and find that the result includes contributions from both the cosmological constant and the Mannheim-Kazanas parameter. There are conflicting results on the deflection angle for light in Weyl gravity in the literature. We point out a possible reason for the discrepancy between our work and the others.

We discuss a general methodology to provide rigid, off-shell matter multiplets and actions for recently constructed non-relativistic superalgebras. The technique is based on the Lie algebra expansion, which, in the context of supersymmetry, has so far been used to obtain non-relativistic on-shell supergravity models. We first explain how the Lie algebra expansion can be implemented to generate off-shell rigid multiplets on a flat background by developing an auxiliary framework where only the lowest order behavior of the spatial and temporal vielbein is relevant. We then provide explicit examples for the field content, the transformation rules as well as action principles for certain multiplets in three and four dimensions.

We aim to explain some astrophysical phenomena in low density astronomical regions without invoking dark matter. We propose that the effective theory of gravity could have individual terms which become dominant in respective regions, and effectively describe the gravitational phenomena there. In the outer region of galaxies, the effective theory of gravity could be the Weyl gravity, which is the unique local scale symmetric metric theory. Inside the bulge, however, general relativity is the effective description of gravity. Hence, we propose the Einstein–Weyl gravity as a scale-dependent effective description of the gravitational phenomena. At high densities or high curvature regions, the effective description for gravity is provided by Einstein’s general relativity, whereas in the low density regions it is provided by the Weyl gravity. We then solve the field equations of Weyl gravity in the outer region of galaxies and determine the geometry in which stars move with the same rotational velocity independent of their distance from the center of the galaxy. We then utilize this metric solution to calculate the deflection angle of light from a distant source by a galaxy cluster. The deflection angle calculated in Schwarzschild–de Sitter-like space includes contributions from both the cosmological constant [Formula: see text] and the Mannheim–Kazanas parameter [Formula: see text]. There are conflicting results on the deflection angle for light in Weyl gravity in the literature. Our result agrees with some and disagrees with many. We point out possible reasons for this discrepancy.