Radiation, Neutrinos, Elasticity, and Transport

Compact objects contain carriers of stress and energy that do not fit a single perfect-fluid closure. Photons and neutrinos occupy angular and spectral phase space; solids remember strain; viscous and conductive fluxes relax toward gradient-driven values. This chapter develops each sector from covariant conservation and then shows how optical depth, mean free path, reaction time, and relaxation time determine which description is controlled.

Helpful background. Hilbert Stress–Energy from Metric Variation supplies the shared stress and exchange language; The Einstein Equation supplies gravitational coupling; and Relativistic Matter and Compact Objects gives the volume map.

From rays to radiation stress

Covariant Radiative Transfer begins on the future null shell. A local observer measures photon energy $\epsilon=-p_au^a=h\nu=2\pi\nu$ in our $\hbar=1$ units, where $\nu$ is ordinary frequency. The scalar occupation implies the invariant

$$\mathcal I=\frac{I_\nu}{\nu^3}.$$

It controls redshift, boosts, and vacuum propagation without confusing intensity transformation with lens magnification. Emission, absorption, and scattering are local collision terms evaluated in a declared material frame. Constant-coefficient attenuation and thermal detailed balance provide sign and equilibrium checks. The invariant transfer equation is developed in Younsi, Wu, and Fuerst 2012, §§ 2–2.1, Eqs. (5)–(18).

Radiation Moments and Closure integrates angular information into energy $E$, flux $F^a$, and pressure $P^{ab}$. The moment equations are exact, but closure is not. The Levermore M1 closure reproduces isotropic radiation and a single beam and is defined on the physical domain $\lvert F\rvert\leq E$; a numerical scheme must also preserve that domain. Two opposed beams prove that $E$ and $F$ cannot encode every angular distribution. That counterexample distinguishes closure uncertainty from numerical truncation error. See Foucart 2023, §§ 4.2.1–4.2.2, 4.2.5, and 4.2.9.

Neutrinos add flavor and reactions

Neutrino Transport in Dense Matter retains energy groups, species, weak interactions, blocking, and lepton-number exchange. Electron neutrinos and antineutrinos participate differently in charged-current reactions; all flavors scatter and undergo pair processes. Opacity depends on energy and material state, so a single grey decoupling surface is a coarse approximation.

Detailed balance links emissivity and absorption to the same Fermi–Dirac equilibrium and equation-of-state chemical potentials. With coherent flavor conversion excluded, charged-current updates must balance matter and neutrino electron lepton number as well as energy. The page derives a one-zone relaxation solution and proposes diffusion, free-streaming, and source-conservation tests with separate spatial, spectral, and tolerance studies. Foucart compares transport approximations in Foucart 2023, §§ 3.2 and 4.1–4.3.

Solids and dissipative fluids

Relativistic Elasticity introduces material labels $X^A$ carried along worldlines and a pulled-back relaxed metric. Stored energy depending on compression and shear produces stress by metric variation. Removing shear dependence recovers a perfect fluid; retaining it produces transverse waves and direction-dependent stress. Positive moduli are not enough for relativistic admissibility: every elastic characteristic must remain causal in the strained state. See Karlovini and Samuelsson 2003, §§ 2 and 5–6.

Causal Viscosity and Heat Conduction contrasts conventional relativistic Navier–Stokes laws with transient relaxation. Instantaneous shear response produces diffusion; giving shear stress a relaxation time produces a telegraph equation with a finite characteristic speed and an extra decaying mode. Navier–Stokes behavior emerges at long wavelengths after the transient decays. The page also explains why modern BDNK theories are a causal first-order alternative: derivative order alone is not the deciding criterion. See Rocha et al. 2024, preprint §§ III.B, III.D, and IV.C.

Both solids and dissipative fluids have calibrated domains. Elasticity fails after yielding or melting; a near-equilibrium viscous expansion fails when dissipative stresses become comparable with equilibrium stress. Stable numerical integration cannot extend a constitutive model beyond those boundaries.

Choosing a regime and recognizing breakdown

Multiphysics Coupling and Regime Selection assembles the chapter into a scale map. Optical depth and Knudsen number govern transport angular structure; reaction and relaxation ratios govern chemical and constitutive equilibrium; magnetization and field invariants govern MHD versus force-free limits; melting and shear times govern elastic response.

The most important coupling rule is exact cancellation of exchange terms. Radiation energy lost must enter matter, charged-current sources must balance electron lepton number, and switching variables must preserve common stress and currents. Trapped radiation or neutrino pressure included in an equation of state cannot also be evolved as an independent full stress without subtracting the overlapping contribution. Likewise, a hard regime switch can create flux jumps even if both component models pass isolated tests.

The chapter therefore favors asymptotic overlap. A transport method should recover diffusion when mean free paths shrink and streaming when opacity vanishes. Transient dissipation should recover Navier–Stokes only at long times. Elastic stress should recover fluid stress as shear modulus vanishes. Hybrid refinement must vary switching thresholds and overlap widths independently of grid spacing.

A practical decision sequence

First identify the observables and conserved totals: energy–momentum, charge, baryon number, and any lepton or composition currents. Second list the microscopic lengths and times against resolved $L$ and $t_{\rm dyn}$. Third choose variables that retain every scale near unity. Fourth state equilibrium, closure, and frame assumptions. Fifth verify thick, thin, fluid, elastic, ideal, and free-streaming limits as applicable.

Finally, test the coupling rather than only modules. Track source cancellation, boundary fluxes, and contributions from floors or regime changes; include the geometric terms when the background is not flat and stationary. A convergence plateau can be modeling error, and a smooth answer can still double count pressure. Start with the analytically controlled limits on the individual pages, then compare the coupled calculation with a transport reference.

Review the chapter

A hot layer is optically thick to high-energy neutrinos but transparent to low-energy neutrinos, with charged-current reaction time comparable to the dynamical time. Is one grey equilibrium neutrino component sufficient?

Solution

No. Energy-dependent optical depths demand spectral transport or energy groups, and a reaction time comparable to dynamics means chemical equilibrium cannot be imposed everywhere. High-energy groups may diffuse while low-energy groups stream; electron-flavor source terms must evolve matter energy and $Y_e$ consistently. A grey equilibrium component would erase both the spectral decoupling and nonequilibrium composition response.

References

• Foucart, François. “Neutrino Transport in General Relativistic Neutron Star Merger Simulations.” Living Reviews in Computational Astrophysics 9, 1 (2023). DOI.
• Karlovini, Max, and Lars Samuelsson. “Elastic Stars in General Relativity: I. Foundations and Equilibrium Models.” Classical and Quantum Gravity 20, 3613–3648 (2003). DOI. Open preprint.
• Rocha, Gabriel S., David Wagner, Gabriel S. Denicol, Jorge Noronha, and Dirk H. Rischke. “Theories of Relativistic Dissipative Fluid Dynamics.” Entropy 26, 189 (2024). DOI. Open preprint.
• Younsi, Ziri, Kinwah Wu, and Steven V. Fuerst. “General Relativistic Radiative Transfer: Formulation and Emission from Structured Tori Around Black Holes.” Astronomy & Astrophysics 545, A13 (2012). DOI. Open preprint.