General Relativistic Hydrodynamics in Discrete Spacetime: Perfect Fluid Accretion onto Static and Spinning Black Holes

Abstract: We study the problem of a spherically-symmetric distribution of a perfect relativistic fluid accreting onto a (potentially spinning) black hole within a fully discrete spacetime setting. This problem has previously been studied extensively in the context of continuum spacetimes, beginning with the purely analytic work of Bondi in the spherically-symmetric Newtonian case, Michel in the spherically-symmetric general relativistic case, and Petrich, Shapiro and Teukolsky in the axially-symmetric general relativistic case relevant for spinning black holes. However, the purpose of the present work is to determine the effect of discretization of the underlying spacetime upon the mass/energy and momentum accretion rates, the overall morphology and characteristics of the accretion flow, and the drag force exerted on the black hole in the case of non-zero spin. In order to achieve this, we first develop a novel formulation of the equations of general relativistic hydrodynamics that is more directly amenable to rigorous analysis within a discrete spacetime setting, and we then proceed to implement this formulation into the Gravitas computational general relativity framework. Through a combination of mathematical analysis and explicit numerical simulation in Gravitas, we discover that the mass/energy and momentum accretion rates both decrease monotonically as functions of the underlying spacetime discretization scale, with this effect becoming more pronounced for higher values of the black hole spin parameter, higher fluid temperatures, and stiffer equation of state parameters. We also find that the exerted drag force is highly sensitive to the value of the underlying discretization scale in the case of spinning black hole spacetimes, with certain instabilities becoming significantly more pronounced at certain critical values of the discretization parameter.

• The paper introduces a novel formulation of general relativistic hydrodynamics in discrete spacetime to simulate perfect fluid accretion onto static and spinning black holes.
• It employs advanced numerical methods, including a conservative-to-primitive reconstruction algorithm, validated against standard relativistic hydrodynamics problems.
• Simulation results reveal that increased spacetime discretization decreases mass/energy and momentum accretion rates, particularly for high black hole spins.

Overview

A recent study investigates the hydrodynamics of perfect fluid accretion onto black holes within the framework of a discretized spacetime. The research, conducted at Princeton University, considers both static and spinning black holes and aims to determine how spacetime discretization influences mass/energy and momentum accretion rates, accretion flow characteristics, and drag force on the black hole.

Formulation and Computational Implementation

The study presents a novel formulation of the equations of general relativistic hydrodynamics for discrete spacetime analysis. This development is necessary for simulations where spacetime is not fixed in advance but evolves together with fluid variables. Implemented using the Gravitas computational general relativity framework, the paper details the hyperbolic evolution equations amended suitably for numerical relativity simulations. An innovative conservative-to-primitive variable reconstruction algorithm is employed to transition from evolved to computationally necessary variables. The algorithm's efficacy is validated against a standard relativistic hydrodynamics problem.

Simulation Outcomes

Accretion onto black holes is simulated using Schwarzschild (non-spinning) and Kerr (spinning) metrics. Different black hole spin parameters are considered with adaptations for various coordinate systems, highlighting the flexibility of the formulated hydrodynamics equations. Results indicate that mass/energy and momentum accretion rates decrease monotonically with increasing spacetime discretization, an effect amplified by higher black hole spins and stiffer equation of state parameters. The simulations also hint at a potential instability in the spacetime structure at certain discretization scales, particularly at higher Mach numbers.

Implications and Future Research

The findings theorize that the astrophysical observation of black hole accretion could reveal properties of spacetime discreteness posited by various quantum gravity models. With the observed decrease in accretion rates contingent on discretization scales, there's a path forward for exploring the astrophysical detectability of such spacetime effects. However, the research remains limited by several assumptions, particularly the use of ideal gas equations of state valid primarily in non-relativistic or extreme relativistic regimes. More physically sound equations are needed for generalizing findings.

Subsequent research endeavors should aim to expand on the idealized accretion scenarios currently studied to include more astrophysically relevant conditions and equations of state. Numerical methods must also evolve to simulate more complex scenarios like binary neutron star mergers within discrete spacetime. The potential effects of quantum gravitational theories on accretion dynamics and the global spacetime topology distinctive to discrete spacetime frameworks are propitious areas for further investigation.