Topic
Scalar curvature
About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.
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TL;DR: In this paper, it was shown that for all e small enough there exists a critical point of the Allen-Cahn energy whose nodal set converges to Σ as e tends to 0.
Abstract: Given a nondegenerate minimal hypersurface Σ in a Riemannian manifold, we prove that, for all e small enough there exists ue, a critical point of the Allen-Cahn energy Ee(u) = e2 ∫ |∇u|2 + ∫(1 − u2)2, whose nodal set converges to Σ as e tends to 0. Moreover, if Σ is a volume nondegenerate constant mean curvature hypersurface, then the same conclusion holds with the function ue being a critical point of Ee under some volume constraint.
152 citations
152 citations
TL;DR: In this paper, a non-minimal R-matter coupling is considered for wormhole geometries in curvature-matter coupled modified gravity, and an exact solution is derived by considering an explicit nonminimal coupling between an arbitrary function of the scalar curvature, R, and the Lagrangian density of matter.
Abstract: Wormhole geometries in curvature-matter coupled modified gravity are explored, by considering an explicit nonminimal coupling between an arbitrary function of the scalar curvature, R, and the Lagrangian density of matter. It is the effective stress-energy tensor containing the coupling between matter and the higher order curvature derivatives that is responsible for the null energy condition violation, and consequently for supporting the respective wormhole geometries. The general restrictions imposed by the null energy condition violation are presented in the presence of a nonminimal R-matter coupling. Furthermore, obtaining exact solutions to the gravitational field equations is extremely difficult due to the nonlinearity of the equations, although the problem is mathematically well defined. Thus, we outline several approaches for finding wormhole solutions, and deduce an exact solution by considering a linear R nonmiminal curvature-matter coupling and by considering an explicit monotonically decreasing function for the energy density. Although it is difficult to find exact solutions of matter threading the wormhole satisfying the energy conditions at the throat, an exact solution is found where the nonminimal coupling does indeed minimize the violation of the null energy condition of normal matter at the throat.
150 citations
TL;DR: In this paper, the authors identify an apparently previously unappreciated regularity in the systematics of galactic rotation curves; namely, at the last detected points in galaxies of widely varying luminosity, the centripetal acceleration is found to have the completely universal form (v2/c2R) last = γ 0/2 + γ* N*/2 + β * N*/R2, where γ0 and γ * are new universal constants, β * is the Schwarzschild radius of the Sun, and N* is the total amount of
Abstract: In this paper we identify an apparently previously unappreciated regularity in the systematics of galactic rotation curves; namely, we find that at the last detected points in galaxies of widely varying luminosity, the centripetal acceleration is found to have the completely universal form (v2/c2R)last = γ0/2 + γ*N*/2 + β*N*/R2, where γ0 and γ* are new universal constants, β* is the Schwarzschild radius of the Sun, and N* is the total amount of visible matter in each galaxy. This regularity points to a possible role for the linear potentials associated with conformal gravity, with the galaxy-independent γ0 term being found not to be generated from within individual galaxies at all but rather to be of cosmological origin, being due to the global Hubble flow of a necessarily spatially open universe of 3-space scalar curvature k = -(γ0/2)2 = -2.3 × 10-60 cm-2.
150 citations
TL;DR: In this article, it was shown that for a wide class of analytic Lagrangians which depend only on the scalar curvature of a metric and a connection, the application of the so-called ''Palatini formalism'' leads to universal equations.
Abstract: It is shown that for a wide class of analytic Lagrangians which depend only on the scalar curvature of a metric and a connection, the application of the so--called ``Palatini formalism'', i.e., treating the metric and the connection as independent variables, leads to ``universal'' equations. If the dimension $n$ of space--time is greater than two these universal equations are Einstein equations for a generic Lagrangian and are suitably replaced by other universal equations at bifurcation points. We show that bifurcations take place in particular for conformally invariant Lagrangians $L=R^{n/2} \sqrt g$ and prove that their solutions are conformally equivalent to solutions of Einstein equations. For 2--dimensional space--time we find instead that the universal equation is always the equation of constant scalar curvature; the connection in this case is a Weyl connection, containing the Levi--Civita connection of the metric and an additional vectorfield ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their bifurcations.
150 citations