Truncated Moment Formalism for Radiation Hydrodynamics in Numerical Relativity

A truncated moment formalism for general relativistic radiation hydrodynamics, based on the Thorne’s moment formalism, is derived. The fluid rest frame is chosen to be the fiducial frame for defining the radiation moments. Then, zeroth-, first-, and second-rank radiation moments are defined from the distribution function with a physically reasonable assumption for it in the optically thin and thick limits. The source terms are written, focusing specifically on the neutrino transfer and neglecting higher harmonic angular dependence of the reaction angle. Finally, basic equations for a truncated moment formalism for general relativistic radiation hydrodynamics in a closed covariant form are derived assuming a closure relation among the radiation stress tensor, energy density, and energy flux, and a variable Eddington factor, which works well.

💡 Research Summary

The paper presents a systematic derivation of a truncated moment formalism for general‑relativistic radiation hydrodynamics, building on Thorne’s covariant moment approach. Recognizing that solving the full Boltzmann equation (with its six‑dimensional phase space plus time) is computationally prohibitive for multi‑dimensional astrophysical simulations, the authors aim to reduce the radiation field to a small set of moments while preserving causality and the essential physics of neutrino transport.

First, the authors review Thorne’s formalism, defining unprojected moments (M^{\alpha_1\ldots\alpha_k}(\nu)) of the distribution function (f(p^\alpha,x^\beta)) where the frequency (\nu) is measured in the fluid rest frame. The covariant evolution equation for each rank‑(k) moment is given by Eq. (2.9), which contains spatial derivatives, frequency derivatives, and source terms. Importantly, the choice of the fluid rest frame as the fiducial observer simplifies the source‑term evaluation because the interaction rates (absorption, emission, scattering) are naturally expressed in that frame.

In the truncated formalism (Section 3) the authors introduce the zeroth‑, first‑, second‑, and third‑rank moments:

• (J(\nu)) (energy density),
• (H^\alpha(\nu)) (energy flux),
• (L^{\alpha\beta}(\nu)) (radiation pressure tensor),
• (N^{\alpha\beta\gamma}(\nu)) (third‑order moment).
  These are defined by angular integrals over the unit vector (\ell^\alpha) in the fluid rest frame. The authors then assume a physically motivated form for the distribution function in two limiting regimes.

In the optically thick limit the distribution is expanded as (f = f_0 + f_1^\alpha \ell_\alpha + f_2^{\alpha\beta}\ell_\alpha\ell_\beta) with (|f_0|\gg|f_1|,|f_2|). This yields simple relations: (J = 4\pi\nu^3 f_0), (H^\alpha = (4\pi/3)\nu^3 f_1^\alpha), and (L^{\alpha\beta}= (1/3)J h^{\alpha\beta} +) a traceless part proportional to (f_2^{\alpha\beta}). The third‑order moment is then automatically expressed in terms of the first‑order moment (Eq. 3.12).

In the optically thin limit, where radiation streams freely, the authors adopt a delta‑function angular distribution (f = 4\pi f_f(\nu),\delta(\Omega-\Omega_f)). Consequently, all moments factorize as products of the flow direction unit vector (\ell_f^\alpha): (J = 4\pi\nu^3 f_f), (H^\alpha = J \ell_f^\alpha), (L^{\alpha\beta}=J \ell_f^\alpha\ell_f^\beta), etc. This representation captures the ballistic nature of neutrinos in low‑density regions.

The evolution equations for (J) and (H^\alpha) are obtained by projecting the second‑rank moment equation (2.9) and substituting the decompositions (3.5)–(3.6). The resulting equations (3.22)–(3.27) involve the unknown second‑rank tensor (L^{\alpha\beta}); to close the system the authors introduce a variable Eddington factor (\chi) that interpolates between the diffusion limit ((\chi=1/3)) and the free‑streaming limit ((\chi=1)). The closure relation is written as
\