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.

Combinatorial curvature for planar graphs

Abstract: Regarding an infinite planar graph G as a discrete analogue of a noncompact simply connected Riemannian surface, we introduce the combinatorial curvature of G corresponding to the sectional curvature of a manifold. We show this curvature has the property that its negative values are bounded above by a universal negative constant. We also prove that G is hyperbolic if its curvature is negative. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 220–229, 2001

Evolution of axially symmetric anisotropic sources in f(R, T) gravity

Abstract: We discuss the dynamical analysis in f(R, T) gravity (where R is the Ricci scalar and T is the trace of the energy momentum tensor) for gravitating sources carrying axial symmetry. The self-gravitating system is taken to be anisotropic and the line element describes an axially symmetric geometry avoiding rotation about the symmetry axis and meridional motions (zero vorticity case). The modified field equations for axial symmetry in f(R, T) theory are formulated, together with the dynamical equations. Linearly perturbed dynamical equations lead to the evolution equation carrying the adiabatic index $$\Gamma $$ , which defines the impact of a non-minimal matter to geometry coupling on the range of instability for Newtonian and post-Newtonian approximations.

General class of brane-world black holes

Abstract: We use the general solution to the trace of the 4-dimensional Einstein equations for static, spherically symmetric configurations as a basis for finding a general class of black hole (BH) metrics, containing one arbitrary function ${g}_{\mathrm{tt}}=A(r)$ which vanishes at some ${r=r}_{h}g0,$ the horizon radius. Under certain reasonable restrictions, BH metrics are found with or without matter and, depending on the boundary conditions, can be asymptotically flat or have any other prescribed asymptotic. It is shown that our procedure generically leads to families of globally regular BHs with a Kerr-like global structure as well as symmetric wormholes. Horizons in space-times with zero scalar curvature are shown to be either simple or double. The same is generically true for horizons inside a matter distribution, but in special cases there can be horizons of any order. A few simple examples are discussed. A natural application of the above results is the brane world concept, in which the trace of the 4D gravity equations is the only unambiguous equation for the 4D metric, and its solutions can be continued into the 5D bulk according to the embedding theorems.

Reconstructing the Distortion Function for Nonlocal Cosmology

Abstract: We consider the cosmology of modified gravity models in which Newton's constant is distorted by a function of the inverse d'Alembertian acting on the Ricci scalar. We derive a technique for choosing the distortion function so as to fit an arbitrary expansion history. This technique is applied numerically to the case of LambdaCDM cosmology, and the result agrees well with a simple hyperbolic tangent.

A momentum construction for circle-invariant Kähler metrics

Abstract: Examples of Kahler metrics of constant scalar curvature are relatively scarce. Over the past two decades, several workers in geometry and physics have used symmetry reduction to construct complete Kahler metrics of constant scalar curvature by ODE methods. One fruitful idea-the Calabi ansatz-is to begin with an Hermitian line bundle p: (L, h) → (M, g M ) over a Kahler manifold, and to search for Kahler forms ω = p*ωM + dd c f(t) in some disk subbundle, where t is the logarithm of the norm function and f is a function of one variable. Our main technical result (Theorem A) is the calculation of the scalar curvature for an arbitrary Kahler metric g arising from the Calabi ansatz. This suggests geometric hypotheses (which we call a-constancy) to impose upon the base metric g M and Hermitian structure h in order that the scalar curvature of g be specified by solving an ODE. We show that σ-constancy is necessary and sufficient for the Calabi ansatz to work in the following sense. Under the assumption of σ-constancy, the disk bundle admits a one-parameter family of complete Kahler metrics of constant scalar curvature that restrict to g M on the zero section (Theorems B and D); an analogous result holds for the punctured disk bundle (Theorem C). A simple criterion determines when such a metric is Einstein. Conversely, in the absence of σ-constancy the Calabi ansatz yields at most one metric of constant scalar curvature, in either the disk bundle or the punctured disk bundle (Theorem E). Many of the metrics constructed here seem to be new, including a complete, negative Einstein-Kahler metric on the disk subbundle of a stable vector bundle over a Riemann surface of genus at least two, and a complete, scalar-flat Kahler metric on C 2 .