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Register a new userA New Non-Inheriting Homogeneous Solution of the Einstein-Maxwell Equations with Cosmological Term
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We find a new homogeneous solution to the Einstein-Maxwell equations with a cosmological term. The spacetime manifold is $R times S^3$. The spacetime metric admits a simply transitive isometry group $G = R times SU(2)$ of isometries and is of Petrov type I. The spacetime is geodesically complete and globally hyperbolic. The electromagnetic field is non-null and non-inheriting: it is only invariant with respect to the $SU(2)$ subgroup and is time-dependent in a stationary reference frame.
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We present a general solution of the coupled Einstein-Maxwell field equations (without the source charges and currents) in three spacetime dimensions. We also admit any value of the cosmological constant. The whole family of such $Lambda$-electrovacuum local solutions splits into two distinct subclasses, namely the non-expanding Kundt class and the expanding Robinson-Trautman class. While the Kundt class only admits electromagnetic fields which are aligned along the geometrically privileged null congruence, the Robinson-Trautman class admits both aligned and also more complex non-aligned Maxwell fields. We derive all the metric and Maxwell field components, together with explicit constraints imposed by the field equations. We also identify the most important special spacetimes of this type, namely the coupled gravitational-electromagnetic waves and charged black holes.
Following the technique of Muller-zum-Hagen, refs [1,2], we show that strictly static and strictly stationary solutions of the Einstein-Maxwell equations are analytic in harmonic coordinates. This holds whether or not the Maxwell field inherits the symmetry.
We present a new bouncing cosmological solution of the non-local theory known as infinite derivative gravity, which goes beyond the recursive ansatz, ${Box R = r_1 R +r_2}$. The non-local field equations are evaluated using the spectral decomposition with respect to the eigenfunctions of the wave operator. The energy-momentum tensor computed for this geometry turns out to be much more sensitive to the choice of the non-local form-factor, since it depends on the value of the function on a continuous infinite interval. We show that this stronger dependence on the form-factor allows us to source the geometry by the perfect fluid with the non-negative energy density satisfying the strong energy condition. We show that this bouncing behaviour is not possible in the local theories of gravity such as in general relativity or $R+R^2$ gravity sourced by a fluid which meets the non-negative energy and strong energy conditions.
We study the most general solution for affine connections that are compatible with the variational principle in the Palatini formalism for the Einstein-Hilbert action (with possible minimally coupled matter terms). We find that there is a family of solutions generalising the Levi-Civita connection, characterised by an arbitrary, non-dynamical vector field ${cal A}_mu$. We discuss the mathematical properties and the physical implications of this family and argue that, although there is a clear mathematical difference between these new Palatini connections and the Levi-Civita one, both unparametrised geodesics and the Einstein equation are shared by all of them. Moreover, the Palatini connections are characterised precisely by these two properties, as well as by other properties of its parallel transport. Based on this, we conclude that physical effects associated to the choice of one or the other will not be distinguishable, at least not at the level of solutions or test particle dynamics. We propose a geometrical interpretation for the existence and unobservability of the new solutions.
The Szekeres system with cosmological constant term describes the evolution of the kinematic quantities for Einstein field equations in $mathbb{R}^4$. In this study, we investigate the behavior of trajectories in the presence of cosmological constant. It has been shown that the Szekeres system is a Hamiltonian dynamical system. It admits at least two conservation laws, $h$ and $I_{0}$ which indicate the integrability of the Hamiltonian system. We solve the Hamilton-Jacobi equation, and we reduce the Szekeres system from $mathbb{R}^4$ to an equivalent system defined in $mathbb{R}^2$. Global dynamics are studied where we find that there exists an attractor in the finite regime only for positive valued cosmological constant and $I_0<2.08$. Otherwise, trajectories reach infinity. For $I_ {0}>0$ the origin of trajectories in $mathbb{R}^2$ is also at infinity. Finally, we investigate the evolution of physical properties by using dimensionless variables different from that of Hubble-normalization conducing to a dynamical system in $mathbb{R}^5$. We see that the attractor at the finite regime in $mathbb{R}^5$ is related with the de Sitter universe for a positive cosmological constant.
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