Tolman–Oppenheimer–Volkoff limit
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| Tolman–Oppenheimer–Volkoff limit | |
|---|---|
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| Name | Tolman–Oppenheimer–Volkoff limit |
| Discovered | 1939 |
| Discoverer | Richard Tolman; J. Robert Oppenheimer; George Volkoff |
| Field | Astrophysics; Relativity |
Tolman–Oppenheimer–Volkoff limit The Tolman–Oppenheimer–Volkoff limit is the theoretical maximum mass for a cold, nonrotating, spherical compact star supported by degeneracy pressure against gravitational collapse into a black hole; it arises from combining Albert Einstein's General relativity with quantum degeneracy and nuclear interactions. The limit was formulated in 1939 by Richard C. Tolman, J. Robert Oppenheimer, and George Volkoff and remains central to studies of neutron star structure, supernova remnants, and compact-object population synthesis.
Introduction
The Tolman–Oppenheimer–Volkoff concept unites results from Einstein field equations, the Fermi–Dirac statistics underpinning Wolfgang Pauli's exclusion principle, and nuclear physics models such as those developed by Enrico Fermi, Hans Bethe, and Lev Landau. It complements limits in other domains, including the Chandrasekhar limit obtained by Subrahmanyan Chandrasekhar for white dwarfs and theoretical bounds considered by Kip Thorne and John Archibald Wheeler for compact objects. The limit informs interpretations of observations from facilities like the Hubble Space Telescope, Arecibo Observatory, and gravitational-wave detectors such as LIGO and VIRGO.
Historical development
The original papers by Tolman, Oppenheimer, and Volkoff applied methods later used by Oppenheimer and Snyder and discussed by contemporaries including Arthur Eddington and P. A. M. Dirac; subsequent developments involved researchers like Stanley Chandrasekhar, Subrahmanyan Chandrasekhar's contemporaries, and postwar efforts by George Gamow and Hermann Bondi. Mid-20th century work by John Wheeler and Wheeler's collaborators connected the limit to concepts explored at institutions such as Princeton University, University of Chicago, and Cambridge University. Late-20th and early-21st century refinements incorporated input from research groups at Max Planck Society, Caltech, and Massachusetts Institute of Technology.
Theoretical derivation
Derivation begins with the Tolman–Oppenheimer–Volkoff equations (derived from the Einstein field equations) applied to a static, spherically symmetric metric studied by Karl Schwarzschild and later by Willem de Sitter; the differential equations couple pressure, mass-energy density, and metric functions as in treatments by Oppenheimer and Volkoff. Inputs include microphysical equations of state developed by nuclear theorists such as Sergei Matveenko, Harry Bethe, and James Lattimer along with many-body approaches advanced at institutions like Oak Ridge National Laboratory, Los Alamos National Laboratory, and Lawrence Livermore National Laboratory. The balance between pressure gradients and spacetime curvature yields a maximum stable mass when central density and relativistic corrections, discussed by Lev Landau and Paul Dirac, drive an instability to collapse toward configurations described by Schwarzschild black hole solutions.
Dependence on equation of state
Numerical value depends sensitively on the high-density equation of state proposed by models from groups including those led by James Lattimer, Madappa Prakash, and Fridolin Weber, and by frameworks such as relativistic mean-field theories developed by John Walecka and quantum Monte Carlo methods used by Steven Reddy. Competing microphysical inputs—nucleonic interactions from work by Hans Bethe, hyperonic degrees of freedom explored by John Millener, meson-exchange descriptions advanced by Maurice Goldhaber, and deconfined quark matter scenarios influenced by Niels Bohr-inspired approaches—lead to divergent predictions cited in studies at CERN, Brookhaven National Laboratory, and RIKEN. Observational programs directed by teams at European Southern Observatory, National Radio Astronomy Observatory, and Keck Observatory constrain candidate equations of state.
Numerical estimates and observational constraints
Current estimates from analyses combining electromagnetic and gravitational-wave data, including results from PSR J0740+6620 timing by groups at Arecibo Observatory and Green Bank Telescope, and the GW170817 binary neutron star merger observed by LIGO and VIRGO, place the limit near values inferred by James Lattimer and Andrew Steiner's compilations. Mass measurements of pulsars such as PSR J1614−2230 and PSR J0348+0432 reported by teams at Max Planck Institute for Radio Astronomy and Swinburne University provide lower bounds, while kilonova modeling from groups at University of Edinburgh, Monash University, and University of Barcelona inform upper bounds. Constraints are also refined by X-ray pulse-profile modeling from the Neutron Star Interior Composition Explorer mission managed by NASA and collaborating institutions like Goddard Space Flight Center.
Implications for compact object evolution
The limit determines outcomes of stellar evolution modeled by codes developed at MESA-affiliated groups and used by researchers at University of Arizona, University of California, Santa Cruz, and University of Chicago; it controls thresholds for collapse during core-collapse supernovae studied by teams at Max Planck Institute for Astrophysics, Princeton Plasma Physics Laboratory, and University of Tokyo. It affects remnant populations in galaxies examined by astronomers at Harvard–Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, and Royal Observatory Edinburgh, and it influences scenarios for short gamma-ray bursts interpreted by groups at Caltech, MIT, and NASA Goddard. The limit also bears on proposed pathways to form intermediate-mass black holes investigated by researchers at European Space Agency and Space Telescope Science Institute.
Alternative formulations and extensions
Extensions consider rotation, magnetic fields, and thermal effects implemented in models by Stergioulas and incorporated in codes from RNS collaboration and LORENE teams; these modifications relate to results by Shapiro and Teukolsky and torque-transfer studies influenced by Evanthia Hatziminaoglou's collaborators. Alternative approaches explore maximum mass in the presence of deconfined quark phases investigated by Edward Witten and Kurkela-affiliated groups, or in modified-gravity theories studied by teams at Perimeter Institute, Institute for Advanced Study, and CERN. Observational tests of extended formulations are pursued by collaborations including LIGO Scientific Collaboration, NICER team, and surveys from European Southern Observatory instruments.