Newtonian hydrodynamic equations with relativistic pressure and velocity

Abstract

We present a new approximation to include fully general relativistic pressure and velocity in Newtonian hydrodynamics. The energy conservation, momentum conservation and two Poisson's equations are consistently derived from Einstein's gravity in the zero-shear gauge assuming weak gravity and action-at-a-distance limit. The equations show proper special relativity limit in the absence of gravity. Our approximation is complementary to the post-Newtonian approximation and the equations are valid in fully nonlinear situations.PACS numbers: 04.25.Nx, 95.30.Lz, 95.30.Sf 1. Introduction: Considering the enormous practical and conceptual difficulties in handling general relativistic astrophysical situations using numerical simulations of full Einstein's gravity [1], it is always welcome to have an approximation method. The post-Newtonian (PN) approximation is one such method [2-5] where we restore the good and old absolute space and absolute time, and regard Einstein's gravity effects as corrections to the Newtonian equations. In this way we can handle weak but relativistic effects of gravity in Newtonian style, and the resulting equations are fully nonlinear.Here, we provide a complementary approximation which can handle the fully relativistic pressure and velocity in the weak gravity and action-at-a-distance limit: for our assumptions see Eq. (7). We present the energy conservation, momentum conservation and Poisson's equation which allow us to handle such astrophysical situations in Newtonian manner. In this approximation also the equations are valid to fully nonlinear orders. Our derivation is based on the zero-shear gauge which will be explained later. We ignore the transverse-tracefree tensor-type perturbation in the spatial metric, and ignore the anisotropic stress.2. Result: A closed form of new hydrodynamic equations we are proposing is

A Short Review on Clustering Dark Energy

“…provides the correct growing modes for linear perturbations of fluids with relativistic pressure. The PNC was extensively studied and applied in the context of cosmological perturbations Abramo et al [46], Reis et al [68], Reis [113], chan Hwang and Noh [114], Fabris et al [115], Velten et al [116], Hwang et al [117]. In particular, it was shown that PNC growing modes for DE perturbations agrees with the relativistic analysis Abramo et al [46].…”

A short review on clustering dark energy

“…The PNC was extensively studied and applied in the context of cosmological perturbations Abramo et al [46], Reis et al [68], Reis [113], chan Hwang and Noh [114], Fabris et al [115], Velten et al [116], Hwang et al [117]. In particular, it was shown that PNC growing modes for DE perturbations agrees with the relativistic analysis Abramo et al [46].…”

Cosmological hydrodynamics with relativistic pressure and velocity

“…where we changed signs of Φ and Ψ compared with [8,9]. The index of χ i is raised and lowered by δ ij as the metric.…”

“…In [8,9] we also have considered the zero-shear gauge (often termed as longitudinal or conformal Newtonian gauge) which sets χ ≡ 0 as the temporal gauge condition. In the presence of relativistic pressure this gauge condition leads to a trouble by failing to match with spherically symmetric solution of Tolman-Oppenheimer-Volkoff [14,15], missing a 3p-term in the Poisson's equation, compare Eq.…”

“…We have successfully applied the formulation to Newtonian limit [5], first-order post-Newtonian limit [6], and Newtonian theory with relativistic pressure [7]. In [8,9] we applied the formulation to special relativistic hydrodynamics combined with weak gravity in Minkowski background. The resulting equations in these formulations are still fully nonlinear and exact.…”