Shock waves during collapse into a black hole: a new look at the growth of supermassive objects ✦
Self-similar solutions with shock waves in general relativity were constructed for the collapse of a singular isothermal sphere into a black hole, extending the Cai & Shu (2005) approach to discontinuous flows. General relativistic jump conditions for an isothermal fluid were derived, linking inner collapsing flows to an outer shell that can be static, expanding, or collapsing; shock wave speeds reach ~0.4c, and the set of possible outer types narrows as the sound speed increases. A coordinate matching method across the zero-velocity surface uniquely connects the Schwarzschild and comoving self-similar descriptions. The central accretion rate, determined solely by the inner collapse, is suppressed by a factor of 5–7 compared to the expansion-wave solution, and the energy released at the shock front reaches ~10% of the enclosed rest mass — nearly twice the radiative efficiency of Schwarzschild accretion (5.7%).
Context
The James Webb Space Telescope has detected actively accreting black holes with masses of billions of solar masses just 500 million years after the Big Bang (at redshifts z~10–11). Such 'little red dots' and powerful active galactic nuclei challenge gradual growth models. One plausible scenario is the direct collapse of low-metallicity gas into a massive seed. However, the dynamics of such collapse within general relativity had not been analytically explored for the case of shock waves. The previous Einstein-Schwarzschild solution for smooth collapse of an isothermal sphere (Cai & Shu 2005) did not include discontinuities, which, as the second law of thermodynamics suggests, should arise due to entropy increase.
Methods
Using self-similar variables in the Schwarzschild metric, the authors derive general relativistic jump conditions for an isothermal fluid, generalizing the classical Landau-Lifshitz conditions. These conditions link the inner collapsing region with an outer envelope that can be static, expanding, or contracting. Calculations are carried out for various values of the square of the sound speed γ (up to ~0.15, corresponding to a sound speed up to ~0.39 of the speed of light). To match Schwarzschild and comoving coordinates, the zero-velocity surface (where spacetime curvature allows a static observer) is used. Solutions are obtained by numerically integrating ordinary differential equations with singular points.
Results
A family of shock-wave solutions is found. For a static outer envelope, the shock speed monotonically increases with γ, reaching ~0.4c at the limiting γ≈0.154; above this threshold, the shock transition becomes subsonic. The Mach number at small γ approaches the Newtonian value of ~1.26. For hydrodynamic envelopes (contraction, expansion, 'breeze'), a continuous family of solutions exists, and with increasing γ, the types of available envelopes decrease. The central accretion rate is determined solely by the inner solution and turns out to be ~5–7 times lower than in the smooth case, due to gas 'fountaining': the shock first throws matter outward, and only when the expanding zero-velocity surface catches up with the gas does it turn around and fall unimpeded into the singularity. The energy extraction efficiency η = E_shock / M_matter c^2, following from mass-energy equivalence, reaches ~10% for collapsing envelopes — nearly twice the limit for an accretion disk around a Schwarzschild black hole (5.7%).
Implications
Such powerful shock waves can serve as a feedback source in the formation of supermassive black hole seeds in the early Universe. Energy release on the order of 10^57–10^58 erg can heat and inflate the dense gas envelope around a newborn black hole, explaining the observed properties of JWST 'little red dots' as objects embedded in ionized cocoons. Moreover, the suppression of accretion itself means that black hole growth can be self-regulated: part of the gravitational energy that would otherwise go into mass growth is carried away by the shock. This could help reconcile the observed correlations between black hole and galaxy masses even at high redshifts.
Future development
Future work should go beyond the isothermal approximation, include rotation and magnetic fields. For a direct connection to the collapsar model of gamma-ray bursts, realistic equations of state need to be considered. Additionally, numerical simulations based on full GR will verify whether the found self-similar solutions are attractors in the nonlinear dynamics. It is also important to assess potential gravitational-wave emission from non-spherical perturbations, which will be a target for future detectors like LISA.
Impact
The results will influence the theory of supermassive black hole formation, interpretation of JWST data on 'little red dots', modeling of gamma-ray bursts, and development of general relativistic hydrodynamics codes, for which analytical solutions serve as benchmark tests.
Next steps
The immediate next steps are implementing numerical simulations of collapse in GR with the same initial conditions and comparing with analytical profiles, as well as accounting for non-thermal processes and radiation transfer.
Key open problems
The article directly addresses a key problem in astrophysics — the rapid growth of supermassive black holes in the early Universe, as well as the nature of the mysterious JWST 'little red dots' and the energetics of relativistic jets in gamma-ray bursts.
🎯 In shock-wave solutions, the gas doesn't just fall into the black hole but performs a 'fountain' maneuver: the shock wave first flings it outward, and only when the expanding zero-velocity surface catches up with the gas does it turn around and rush unimpeded towards the singularity.
Key numbers
- energy extraction efficiency: up to ~10%
- shock wave speed: up to ~0.4c
- accretion rate suppression: by a factor of ~5–7
- redshift of detected black holes: z ~ 10–11
- maximum accretion efficiency for a Schwarzschild black hole: 5.7%