Multicenter solutions in Eddington-inspired Born–Infeld gravity

We find multicenter (Majumdar–Papapetrou type) solutions of Eddington-inspired Born–Infeld gravity coupled to electromagnetic fields governed by a Born–Infeld-like Lagrangian. We construct the general solution for an arbitrary number of centers in equilibrium and then discuss the properties of their one-particle configurations, including the existence of bounces and the regularity (geodesic completeness) of these spacetimes. Our method can be used to construct multicenter solutions in other theories of gravity.

Geodesic completeness of the left-invariant metrics on ℝ H n

UDC 514 We give the full classification of left-invariant metrics of an arbitrary signature on the Lie group corresponding to the real hyperbolic space. We show that all metrics have constant sectional curvature and that they are geodesically complete only in the Riemannian case.

Designing metrics; the delta metric for curves

In the first part, we revisit some key notions. Let M be a Riemannian manifold. Let G be a group acting on M. We discuss the relationship between the quotient M∕G, “horizontality” and “normalization”. We discuss the distinction between path-wise invariance and point-wise invariance and how the former positively impacts the design of metrics, in particular for the mathematical and numerical treatment of geodesics. We then discuss a strategy to design metrics with desired properties. In the second part, we prepare methods to normalize some standard group actions on the curve; we design a simple differential operator, called the delta operator, and compare it to the usual differential operators used in defining Riemannian metrics for curves. In the third part we design two examples of Riemannian metrics in the space of planar curves. These metrics are based on the “delta” operator; they are “modular”, they are composed of different terms, each associated to a group action. These are “strong” metrics, that is, smooth metrics on the space of curves, that is defined as a differentiable manifolds, modeled on the standard Sobolev space H2. These metrics enjoy many important properties, including: metric completeness, geodesic completeness, existence of minimal length geodesics. These metrics properly project on the space of curves up to parameterization; the quotient space again enjoys the above properties.