Scalar Green function of the Kerr spacetime

Abstract

In this paper we study the scalar Green function in the Kerr spacetime using Wentzel-Kramers-Brillouin (WKB) methods. The Green function can be expressed by Fourier-transforming to its frequency-domain counterpart, and with the help of complex analysis it can be divided into parts: 1) the "direct part," which propagates on the light cone and dominates at very early times; 2) the "quasinormal-mode part," which represents the waves traveling around the photon sphere and is important at early and intermediate times; and 3) the "tail part," which is due to scattering by the Coulomb-type potential and becomes more important at later times. We focus on the "quasinormal-mode part" of the Green function and derive an approximate analytical formula for it using WKB techniques. This approximate Green function diverges at points that are connected by null geodesics, and it recovers the fourfold singular structure of Green functions that are seen in Schwarzschild and other spacetimes. It also carries unique signatures of the Kerr spacetime such as frame dragging. Along the way, we also derive approximate quasinormal-mode wave functions and expressions for the black hole excitation factors in the Kerr spacetime. We expect this work to benefit the understanding of both wave propagation and the problem of self-force in the Kerr spacetime.

Large-scale stability and astronomical constraints for coupled dark-energy models

“…This view has already been confirmed by many investigations [44,45,46,47,48,49,50,51,52,53]. One conclusion from these studies was that joint measurements of the geometry and dynamical observations found that the interaction rate, Q, was zero at about 1σ [30,31,32,33]. Unfortunately, this conclusion has been drawn using the intervals w x ≤ −1 and w x ≥ −1 separately.…”

“…Following this, interaction term Q = 3Hξρ x needs to be tested against the observations with two intervals for dark-energy equation of state w x ≤ −1 and w x ≥ −1 [29,30,31,32,33]. We note that w x = −1 is the limiting case, see [29] for details.…”

“…Until now, there have been many interacting dark energy models based on different proposals for the form of energy exchange term Q. A series of investigations have been performed using observational data with interesting results [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. Aside from the specific issue of dark matter-dark energy interactions, we can also view the interaction, Q, as an energy exchange between any two barotropic fluids, see [36].…”

“…However, the analysis of inhomogeneous cosmological perturbations is essential to provide a fuller picture of these models, to determine if they are stable or unstable components of the large scale structure of the universe. For example, a simple energy exchange term Q ∝ ρ c leads to an instability in the dark matter perturbations at early times since the curvature perturbation blows up on super-Hubble scales [27].In order to derive a stable perturbation evolution, another simple interaction term, Q ∝ ρ x needs to be tested by the observations with two intervals of possible dark energy equations of state: w x ≤ −1 and w x ≥ −1 [29,30,31,32,33]. Therefore, the principal motivation of this paper is to find a form of energy transfer, Q, which could alleviate the perturbative instability.…”

Cosmological determination to the values of the pre-factors in the logarithmic corrected entropy-area relation

“…The second correction term has also appeared in the entropic cosmology introduced in [42]. While positive and negative values have been suggested for α and β by some authors [43]- [47], it has also been argued that the best guess to the logarithmic term might simply be zero [48]. A stable flat entropy-corrected FRW universe with a decelerationacceleration transition has been constructed in [49] for zero values of both pre-factors.…”