Related papers: Singular Solutions for the Conformal Dirac-Einstei…
Beginning with the self-dual two-forms approach to the Einstein equations, we show how, by choosing basis spinors which are proportional to solutions of the Dirac equation, we may rewrite the vacuum Einstein equations in terms of a set of…
To study the problem of the assigned Gauss curvature with conical singularities on Riemanian manifolds, we consider the Liouville equation with a single Dirac measure on the two-dimensional sphere. By a stereographic projection, we reduce…
In this work we initiate the mathematical study of naked singularities for the Einstein vacuum equations in $3+1$ dimensions by constructing solutions which correspond to the exterior region of a naked singularity. A key element is our…
A large class of solutions of the Einstein-conformal scalar equations in D=2+1 and D=3+1 is identified. They describe the collisions of asymptotic conformal scalar waves and are generated from Einstein-minimally coupled scalar spacetimes…
There are a number of publications on relativistic objects dealing either with black holes or naked singularities in the center. Here we show that there exist static spherically symmetric solutions of Einstein equations with a strongly…
Einstein's field equations in general relativity admit a variety of solutions with spacetime singularities. Numerical relativity has recently revealed the properties of somewhat generic spacetime singularities. It has been found that in a…
New formulas on the inverse problem for the continuous skew-self-adjoint Dirac type system are obtained. For the discrete skew-self-adjoint Dirac type system the solution of a general type inverse spectral problem is also derived in terms…
In this paper, we are exploring some of the properties of the self-similar solutions of the first kind. In particular, we shall discuss the kinematic properties and also check the singularities of these solutions. We discuss these…
The two-dimensional case occupies a special position in the theory of critical phenomena due to the exact results provided by lattice solutions and, directly in the continuum, by the infinite-dimensional character of the conformal algebra.…
This work deals with the conformal transformations in six-dimensional spinorial formalism. Several conformally invariant equations are obtained and their geometrical interpretation are worked out. Finally, the integrability conditions for…
A new proof of the conformal covariance of the powers of the flat Dirac operator is obtained. The proof uses their relation with the Knapp-Stein intertwining operators for the spinorial principal series. We also treat the compact picture,…
Observing the list of compatible second order equations of Absolute Parallelism (AP) found by Einstein and Mayer (they used D=4), we choose the one-parameter class of equations which take on a 3-linear form (when contra-frame density of…
We solve the classical conformal welding problem for a composition of two random homeomorphisms generated by independent Gaussian multiplicative chaos measures with small parameter values. In other words, given two such measures on the…
We obtain an exact solution for the Einstein's equations with cosmological constant coupled to a scalar, static particle in static, "spherically" symmetric background in 2+1 dimensions.
Making use of integral representations, we develop a unified approach to establish blow up profiles, compactness and existence of positive solutions of the conformally invariant equations $P_\sigma(v)= Kv^{\frac{n+2\sigma}{n-2\sigma}}$ on…
In this paper, we elucidate the problem of gravitating Skyrmion governed by field equations of the Einstein-Skyrme system with no potential term in the Bondi coordinate. The spherical symmetry has to be assumed and both the metric functions…
It is shown, for the self-consistent system of scalar, electro-magnetic and gravitational fields in general relativity, that the equations of motion admit a special kind of solutions with spherical or cylindrical symmetry. For these…
A conformal geometry determines a distinguished, potentially singular, variant of the usual Yamabe problem, where the conformal factor can change sign. When a smooth solution does change sign, its zero locus is a smoothly embedded…
In a previous work [I. Rodnianski and Y. Shlapentokh-Rothman, Naked Singularities for the Einstein Vacuum Equations: The Exterior Solution, arXiv:1912.08478] we constructed solutions to the Einstein vacuum equations in 3+1 dimensions which…
We consider the conformal wave equation on the Einstein cylinder with a defocusing cubic non-linearity. Motivated by a method developed by Rostworowski-Maliborski on the existence of time periodic solutions to the spherically symmetric…