General Relativistic Shock Wave Solutions with Black Hole Formation: The Singular Isothermal Sphere Case

Chien-Ting J. Chen, Michael J. Cai, Fabio Pacucci

First listed 2026-06-30 | Last updated 2026-06-28

Abstract

The rapid emergence at $z\gtrsim 6$ of ubiquitous populations of supermassive black holes (SMBHs) revealed by JWST and of quasars with estimated masses $M_\bullet > 10^{10} M_\odot$ demands efficient pathways for early growth. The smooth collapse of a singular isothermal sphere (SIS) has been solved analytically in full general relativity, but the shock waves that inevitably accompany such collapse have not. Here, we derive general-relativistic self-similar shock-wave solutions for the collapse of an SIS to a black hole, extending the framework of Cai \& Shu (2005) to discontinuous flows. We obtain the general relativistic jump conditions for an isothermal fluid and show that they connect interior collapse solutions to exterior envelopes that may be static, expanding, or collapsing, yielding a rich family of shocks propagating at up to $\sim$40\% the speed of light; the available exterior types narrow with increasing sound speed. A coordinate-matching technique that uses the zero-velocity surface uniquely bridges the Schwarzschild and comoving self-similar descriptions, completing the characterization of the growing black hole. The central accretion rate is set by the interior collapse alone and is suppressed by a factor of $\sim$5--7 relative to the smooth expansion-wave solution, while the energy released at the shock reaches $\sim$10\% of the enclosed rest mass -- nearly twice the 5.7\% radiative efficiency of Schwarzschild accretion. These results provide an analytical energy budget for direct-collapse black hole formation, with implications for SMBH seed assembly, the dense cocoons around nascent high-redshift black holes, the recently discovered JWST's Little Red Dots, and relativistic transients such as gamma-ray bursts.

Short digest

This paper extends the general-relativistic singular isothermal sphere collapse problem to discontinuous flows, deriving self-similar shock solutions that connect a black-hole-forming interior to static, expanding, or collapsing outer envelopes. The main physical result is that the interior collapse fixes the black hole feeding rate, while shocks can move at up to about 0.4c and reduce the central accretion rate by roughly a factor of 5-7 relative to the smooth expansion-wave solution. Using a zero-velocity-surface coordinate match between Schwarzschild and comoving descriptions, the authors complete the global solution and show that the shock can release about 10% of the enclosed rest-mass energy, exceeding the canonical 5.7% Schwarzschild accretion efficiency. That gives an analytic energy budget for direct-collapse seed formation and a concrete framework for thinking about dense cocoons, early SMBH assembly, Little Red Dots, and other relativistic transients tied to nascent black holes.

Key figures to inspect

• Figure 1 is the cleanest entry point for the new solution class because it shows how the collapse solutions with critical points connect a supersonically collapsing interior to a hydrostatic outer singular isothermal sphere through a shock, while also comparing directly to the older smooth expansion-wave solution. It makes the paper's core claim visually explicit: black-hole-forming SIS collapse in general relativity admits shocked, not just smooth, self-similar continuations.
• Figure 2 is important because it broadens the result beyond a static envelope and demonstrates the full family of allowed shocked exteriors at fixed sound speed, including envelope collapse, breeze, and envelope expansion solutions. This is where the paper's statement about a rich shock solution space is most concrete, and the inset highlights how the shock links the shared collapsing interior to distinct outer flows.
• Figure 3 best captures the coordinate-matching and interior-universality result by showing the comoving solutions anchored at the zero-velocity bridge, where the Schwarzschild and comoving descriptions coincide. It also makes clear that different hydrodynamic exteriors share the same comoving interior and differ mainly by the shock-shell location, which is central to the paper's global characterization of the growing black hole.
• Figure 5 is the key quantitative summary for black-hole growth because it shows that the central accretion rate lies on a single curve set only by the interior parameter, independent of which shocked exterior is attached. This is the figure that most directly supports the headline conclusion that shocks suppress the black hole mass supply by about a factor of 7 relative to the smooth expansion-wave collapse.
• Figure 6 is the strongest synthesis figure for the paper's astrophysical payoff because it ties sound speed to shock velocity, Mach number, and energy extraction efficiency in one place. It is the figure that most clearly supports the claims that shocks can reach relativistic speeds and that their energy release can approach about 10% of the enclosed rest-mass energy, exceeding the benchmark Schwarzschild radiative efficiency.