Merger and Remnant Classification

A binary-neutron-star merger can collapse promptly to a black hole or produce a differentially rotating remnant that survives for milliseconds, seconds, or much longer. “Hypermassive”, “supramassive”, and “stable” compare the remnant with cold equilibrium limits; they are not synonyms for chosen lifetime bins. Classification requires a consistent mass definition, equation of state, support mechanisms, and a resolved horizon diagnostic, together with a stated time window for prompt collapse.

Helpful background. Stress–Energy and Conservation supplies mass and angular-momentum budgets; the Einstein equations supply collapse diagnostics; Relativistic Two-Body Dynamics supplies binary masses and orbit; Relativistic Matter and Compact Objects supplies equilibrium limits; Gravitational Radiation and Waveforms supplies radiated losses; and Computational Relativity supplies merger evolution and convergence tests.

Cold limits and transient support

For one cold equation of state, define $M_{\rm TOV}$ as the maximum gravitational mass of the nonrotating sequence. To classify a remnant by a conserved quantity, separately determine $M_{b,\rm TOV}$, the largest baryon mass on the stable nonrotating branch, and $M_{b,\rm uni}$, the largest baryon mass supportable on the stable uniformly rotating family, bounded by mass shedding and instability. These are EOS-dependent limits, not a universal conversion of gravitational mass. Then, schematically:

• a remnant with $M_{b,\rm rem}\lt M_{b,\rm TOV}$ can in principle settle to a cold nonrotating star;
• a supramassive remnant has $M_{b,\rm TOV}\lt M_{b,\rm rem}\lt M_{b,\rm uni}$ and is supportable by uniform rotation at sufficiently large angular momentum;
• a hypermassive remnant has $M_{b,\rm rem}\gt M_{b,\rm uni}$ and therefore needs differential rotation, thermal pressure, or other transient support;
• prompt collapse forms a black hole on a declared dynamical postmerger timescale, without an extended material-remnant phase.

These comparisons must use baryon mass consistently. Gravitational mass changes through gravitational waves, neutrinos, ejecta, and disk formation; adding two isolated gravitational masses is not the remnant’s conserved mass. Thermal pressure and composition make the early remnant unlike the cold comparison sequences, so the labels identify support requirements rather than exact collapse clocks. Baiotti and Rezzolla review remnant classes, support mechanisms, and numerical results in Baiotti and Rezzolla 2017, §II and Fig. 1.

The conventions are $(-,+,+,+)$, explicit $G$, and $c=\hbar=k_B=1$. Useful dimensionless combinations are compactness $GM/R$ and mass-scaled frequency $GM\Omega$; every threshold fit must state whether its input is ADM, isolated gravitational, or baryon mass.

Prompt-collapse thresholds

For a fixed EOS family and binary class, simulations often summarize the threshold total mass as

$$M_{\rm th}=k_{\rm th}M_{\rm TOV},$$

where the total binary mass is $M_1+M_2$, the sum of the isolated stars’ gravitational masses in the specified initial-data convention. It is not the postmerger core’s baryon mass used in the classification above. The factor $k_{\rm th}$ correlates with stellar compactness or radius. This is an empirical calibration, not a universal identity. Mass ratio, spins, thermal closure, neutrino treatment, initial eccentricity, and the operational prompt time can shift the boundary. Bauswein, Baumgarte, and Janka demonstrate the EOS correlation for equal-mass, irrotational binaries using smoothed-particle hydrodynamics, a conformally flat gravity approximation, and temperature-dependent nuclear EOSs in Bauswein, Baumgarte, and Janka 2013, Eq. (1), Table I, and Fig. 1.

A threshold study brackets the transition with multiple masses for each EOS and refines near the boundary. Quoting one exploding—or one collapsing—binary does not locate $M_{\rm th}$. Applying a fit beyond its calibration range in mass ratio, spin, EOS compactness, or microphysics can silently change the question.

Prompt collapse should be defined through formation of a resolved apparent horizon within a specified interval after the waveform or density merger time. Lapse collapse or central-density growth alone is gauge-dependent. Horizon formation, constraint behavior, and matter outside the horizon should be reported together. The subsequent generic horizon evolution is developed in Black-Hole Dynamics.

Support timescales and delayed collapse

Differential rotation can support a hypermassive core, while thermal pressure supports shocked layers. Gravitational-wave torques, hydrodynamic redistribution, magnetic winding and turbulence, viscosity, neutrino cooling, and mass ejection change angular momentum and thermal profiles on different timescales. A simulation that omits magnetic fields or stops before cooling cannot predict the lifetime controlled by those processes.

Supramassive remnants can survive differential-rotation removal but collapse after enough uniform rotational support is lost. A remnant below the cold TOV baryon-mass limit can in principle settle to a stable star, provided fallback or later phase evolution does not move it across the limit. Disk and ejecta masses must be separated from the central remnant when making these comparisons.

An independent classification uses both instantaneous support and controlled evolution: compare conserved baryon mass and angular momentum with cold nonrotating and uniformly rotating sequences, inspect rotation and entropy profiles, and monitor whether an apparent horizon forms. A lifetime category without these quantities is phenomenological and model-dependent.

Computational validation

Continuum inputs include component baryon masses, mass ratio, spins, EOS table and thermal/composition extension, constraint-solved initial data, magnetic and neutrino assumptions, evolution formulation, atmosphere, grid and boundaries, and definitions of merger, prompt interval, remnant mass, disk, ejecta, and horizon. The software environment must pin code commit, compiler and libraries, precision, EOS and microphysics checksums, reconstruction and Riemann solver, gauge, refinement hierarchy, horizon finder, time-step and solver tolerances, and deterministic settings.

A useful benchmark reproduces a published equal-mass merger at matched dimensionless mass and closes baryon, energy, and angular-momentum budgets. It checks horizon formation against an independent finder or surface diagnostic and classifies support relative to equilibrium sequences built with the same EOS. Output should include collapse time or a lower bound, central-remnant, disk and ejecta masses, angular momentum, maximum density, and constraints.

Convergence studies refine grid and outer boundary, atmosphere, thermal interpolation, neutrino prescription, and horizon resolution. Near a threshold, resolution-induced collapse or survival is a principal uncertainty; stable classification requires bracketing at multiple resolutions, not extrapolation from a single trajectory.

Each failure mode must be exposed: atmosphere mass loading, thermal-EOS inconsistency, unclosed angular momentum, gauge-only collapse indicators, unresolved horizon formation, magnetic or neutrino timescale truncation, and a threshold fit used outside its calibration domain. Report uncertainty rather than forcing a binary label when the resolution bracket straddles collapse.

No merger was run for this page. A reproducible companion would provide one documented rerun command that regenerates machine-readable equilibrium, benchmark, budget, horizon, and refinement tables from a pinned environment and fails when admissibility, conservation, reference, constraint, or convergence tolerances are missed.

Check a remnant label

For one hypothetical EOS, suppose the cold stable-family baryon-mass limits are $M_{b,\rm TOV}=2.40\,M_\odot$ and $M_{b,\rm uni}=2.85\,M_\odot$. A central remnant of baryon mass $M_{b,\rm rem}=3.00\,M_\odot$, excluding the bound disk and ejecta, forms no horizon during the prompt interval and is strongly differentially rotating. Which support class describes it?

Solution

Because $3.00\,M_\odot\gt M_{b,\rm uni}$, neither a cold nonrotating nor a cold uniformly rotating equilibrium can support the stated mass. Its temporary survival therefore places it in the hypermassive class. The label does not predict its exact lifetime: thermal pressure, angular-momentum redistribution, cooling, magnetic fields, and mass loss determine when or whether it crosses a dynamical stability boundary. All three masses here use the same baryon-mass definition. Comparing $M_{b,\rm rem}$ directly with a gravitational-mass $M_{\rm TOV}$ would be invalid.

References

• Baiotti, Luca, and Luciano Rezzolla. “Binary Neutron Star Mergers: A Review of Einstein’s Richest Laboratory.” Reports on Progress in Physics 80 (2017): 096901. arXiv.
• Bauswein, Andreas, Thomas W. Baumgarte, and Hans-Thomas Janka. “Prompt Merger Collapse and the Maximum Mass of Neutron Stars.” Physical Review Letters 111 (2013): 131101. DOI.