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Register a new userHyperboloidal evolution of test fields in three spatial dimensions
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We present the numerical implementation of a clean solution to the outer boundary and radiation extraction problems within the 3+1 formalism for hyperbolic partial differential equations on a given background. Our approach is based on compactification at null infinity in hyperboloidal scri fixing coordinates. We report numerical tests for the particular example of a scalar wave equation on Minkowski and Schwarzschild backgrounds. We address issues related to the implementation of the hyperboloidal approach for the Einstein equations, such as nonlinear source functions, matching, and evaluation of formally singular terms at null infinity.
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We study the fall-off behaviour of test electromagnetic fields in higher dimensions as one approaches infinity along a congruence of expanding null geodesics. The considered backgrounds are Einstein spacetimes including, in particular, (asymptotically) flat and (anti-)de Sitter spacetimes. Various possible boundary conditions result in different characteristic fall-offs, in which the leading component can be of any algebraic type (N, II or G). In particular, the peeling-off of radiative fields F=Nr^{1-n/2}+Gr^{-n/2}+... differs from the standard four-dimensional one (instead it qualitatively resembles the recently determined behaviour of the Weyl tensor in higher dimensions). General p-form fields are also briefly discussed. In even n dimensions, the special case p=n/2 displays unique properties and peels off in the standard way as F=Nr^{1-n/2}+IIr^{-n/2}+.... A few explicit examples are mentioned.
In this paper we present the equations of the evolution of the universe in $D$ spatial dimensions, as a generalization of the work of Lima citep{lima}. We discuss the Friedmann-Robertson-Walker cosmological equations in $D$ spatial dimensions for a simple fluid with equation of state $p=omega_{D}rho$. It is possible to reduce the multidimensional equations to the equation of a point particle system subject to a linear force. This force can be expressed as an oscillator equation, anti-oscillator or a free particle equation, depending on the $k$ parameter of the spatial curvature. An interesting result is the independence on the dimension $D$ in a de Sitter evolution. We also stress the generality of this procedure with a cosmological $Lambda$ term. A more interesting result is that the reduction of the dimensionality leads naturally to an accelerated expansion of the scale factor in the plane case.
Screening mechanisms for a three-form field around a dense source such as the Sun are investigated. Working with the dual vector, we can obtain a thin-shell where field interactions are short range. The field outside the source adopts the configuration of a dipole which is a manifestly distinct behaviour from the one obtained with a scalar field or even a previously proposed vector field model. We identify the region of parameter space where this model satisfies present solar system tests.
Geometrical symmetry in a spacetime can generate test solutions to the Maxwell equation. We demonstrate that the source-free Maxwell equation is satisfied by any generator of spacetime self-similarity---a proper homothetic vector---identified with a vector potential of the Maxwell theory. The test fields obtained in this way share the scale symmetry of the background.
Numerical simulations are performed of a test scalar field in a spacetime undergoing gravitational collapse. The behavior of the scalar field near the singularity is examined and implications for generic singularities are discussed. In particular, our example is the first confirmation of the BKL conjecture for an asymptotically flat spacetime.
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