arXiv subjects

Adrian Sotomayor

Publications and source records attributed to Adrian Sotomayor.

4 recordsLinked to original sources

arXiv2018

A new integrable equation valued on a Cayley-Dickson algebra

We introduce a new integrable equation valued on a Cayley-Dickson (C-D) algebra. In the particular case in which the algebra reduces to the complex one the new interacting term in the equation cancells and the equation becomes the known Korteweg-de Vries equation. For each C-D algebra the equation has an infinite sequence of local conserved quantities. We obtain a Bäcklund transformation in the sense of Walhquist-Estabrook for the equation for any Cayley-Dickson algebra, and relate it to a generalized Gardner equation. From it, the infinite sequence of conserved quantities follows directly. We give the explicit expression for the first few of them. From the Bäcklund transformation we get the Lax pair and the one-soliton and two-soliton solutions generalizing the known solutions for the quaternion valued KdV equation. From the Gardner equation we obtain the generalized modified KdV equation which also has an infinite sequence of conserved quantities. The new integrable equation is preserved under a subgroup of the automorphisms of the C-D algebra. In the particular case of the algebra of octonions, the equation is invariant under $SU(3)$.

math-ph↗

arXiv2015

Wormholes in Horava gravity with cosmological constant

By combining analytical and numerical methods we find that the solutions of the complete Horava theory with negative cosmological constant that satisfy the conditions of staticity, spherical symmetry and vanishing of the shift function are two kinds of geometry: (i) a wormhole-like solution with two sides joined by a throat and (ii) a single side with a naked singularity at the origin. We study the second-order effective action. We consider the case when the coupling constant of the (partial ln N)^2 term, which is the unique deviation from general relativity in the effective action, is small. At one side the wormhole acquires a kind of deformed AdS asymptotia and at the other side there is an asymptotic essential singularity. The deformation of AdS essentially means that the lapse function N diverges asymptotically a bit faster than AdS. This can also be interpreted as an anisotropic Lifshitz scaling that the solutions acquire asymptotically.

gr-qc↗

arXiv2014

Wormholes and naked singularities in the complete Horava theory

We find the static spherically symmetric solutions (with vanishing shift function) of the complete nonprojectable Horava theory explicitly, writing the space-time metrics as explicit tensors in local coordinate systems. This completes previous works of other authors that have studied the same configurations. The solutions depend on the coupling constant alpha of the (partial_i ln N)^2 term. The lambda = 1/3 case of the theory does not possess any extra mode, hence the range of alpha is in principle not limited by the linear stability of any extra mode. We study the full range of alpha, both in the positive and negative sectors. We find the same wormhole solutions and naked singularities that were found for the Einstein-aether theory in a sector of the space of alpha. There also arise wormholes in other sector of alpha. Our coordinate systems are valid at the throats of the wormholes. We also find the perturbative solutions for small alpha. We give this version of the solutions directly on the original radial coordinate r, which is particularly suitable for representing the exterior region of solutions with localized sources.

gr-qc↗

arXiv2012

Non-perturbative analysis of the constraints and the positivity of the energy of the complete Horava theory

We perform a non-perturbative analysis to the Hamiltonian constraint of the lowest-order effective action of the complete Horava theory, which includes a (\partial_i \ln N)^2 term in the Lagrangian. We cast this constraint as a partial differential equation for N and show that the solution exists and is unique under a condition of positivity for the metric and its conjugate momentum. We interpret this condition as the analog of the positivity of the spatial scalar curvature in general relativity. From the analysis we extract several general properties of the solution for N: an upper bound on its absolute value and its asymptotic behavior. In particular, we find that the asymptotic behavior is different to that of general relativity, which has consequences on the evolution of the initial data and the calculus of variations. Similarly, we proof the existence and uniqueness of the solution of the equation for the Lagrange multiplier of the theory. We also find a relationship between the expression of the energy and the solution of the Hamiltonian constraint. Using it we prove the positivity of the energy of the effective action under consideration. Minkowski spacetime is obtained from Horava theory at minimal energy.

hep-th↗