Tolman–Oppenheimer–Volkoff Equation

Every star is a standoff between gravity pulling inward and a pressure gradient pushing outward. For the Sun that standoff is one line of Newtonian physics — the equation of hydrostatic equilibrium, $\mathrm{d}P/\mathrm{d}r=-Gm\rho/r^2$ . For a neutron star it is not, and the reason is a single dimensionless number: the compactness $2GM/(Rc^2)$ , which is $\sim10^{-6}$ at the Sun's surface but climbs to $\sim0.5$ at a neutron star's. When that number is no longer negligible, gravity stops being a force in flat space and becomes the curvature of spacetime, and three things change at once: pressure, carrying energy, now gravitates; the gravitational field itself carries energy, so gravity feeds on gravity; and proper radial distance is stretched relative to the coordinate $r$ . The Tolman–Oppenheimer–Volkoff (TOV) equation is what hydrostatic equilibrium becomes once all three effects are included — the exact general-relativistic law for a static, spherically symmetric ball of perfect fluid, read straight off Einstein's field equations. Its most consequential feature has no Newtonian analogue. Because the relativistic corrections diverge as a configuration approaches its Schwarzschild radius, adding mass eventually demands infinite central pressure, and causality forbids that. So relativistic stars possess a maximum mass — a hard ceiling above which no static equilibrium exists and collapse to a black hole is inevitable. Discovered by Oppenheimer and Volkoff in 1939, this ceiling is the neutron-star cousin of the Chandrasekhar limit and the pivot on which compact-object astrophysics turns.

Key takeaways

The TOV equation is the general-relativistic replacement for Newtonian hydrostatic equilibrium in a static, spherically symmetric perfect-fluid star, derived directly from Einstein's field equations.
Relative to the Newtonian gradient $-Gm\rho/r^2$ it multiplies by three factors — $(1+P/\rho c^2)$ , $(1+4\pi r^3P/mc^2)$ , and $(1-2Gm/rc^2)^{-1}$ — each $\ge 1$ , so relativistic gravity is always stronger.
Closed by $\mathrm{d}m/\mathrm{d}r=4\pi r^2\rho$ and an equation of state $P(\rho)$ , it integrates outward to the surface where $P=0$ , giving one star per central density.
Because the corrections blow up near $2Gm/rc^2\to1$ while causality caps the sound speed at $c$ , a maximum mass must exist — $\sim0.7\,M_\odot$ for free neutrons, $\sim2.2\,M_\odot$ for realistic stiff matter.
Observed $\sim2\,M_\odot$ pulsars, NICER radii, and GW170817 tidal deformability all read the equation of state through the TOV curve.
Key idea: in general relativity pressure, internal energy, and the gravitational field itself all gravitate; the TOV equation packages this as three factors each $\ge1$ multiplying the Newtonian gradient, which makes equilibrium harder to hold and forces a maximum mass.

The core insight

In general relativity pressure, energy density, and the gravitational field itself all gravitate; the TOV equation encodes this by multiplying the Newtonian hydrostatic gradient by three factors each $\ge 1$ , making equilibrium progressively harder to maintain and forcing the existence of a maximum mass.

Where this lives today

The TOV equation is the quiet engine under every neutron-star measurement. NASA's NICER X-ray telescope models the rotating hot spots on pulsars such as PSR J0740+6620 to pin both mass and radius, and each $(M,R)$ point is a dot a TOV curve must thread — the 2021 result, $R\approx12.4\ \mathrm{km}$ at $2.08\,M_\odot$ , favours moderately stiff matter. In 2017 the gravitational-wave event GW170817 added an independent constraint: the tidal deformability of the inspiralling stars, computed by perturbing the TOV background, squeezed a $1.4\,M_\odot$ radius to roughly $11$ – $13\ \mathrm{km}$ . Together these "multi-messenger" inputs invert the TOV map to reconstruct the pressure of matter at two to five times nuclear density — a regime no laboratory can reach. At the frontier sits the mass gap: the $2.6\,M_\odot$ secondary of GW190814 lies exactly where the heaviest neutron star and lightest black hole blur together, and which it is comes down to $M_{\rm TOV}$ . Six decades on, Oppenheimer and Volkoff's hand integration has become a precision tool for reading the densest matter in the universe.

The physical picture

Isolate a thin fluid shell at areal radius $r$ . In Newtonian gravity its fate is a tug of war between the enclosed mass $m(r)$ pulling it down and the pressure difference holding it up. General relativity keeps the picture but rewrites every term: the pressure inside the shell has energy and adds to the source; the enclosed mass is felt through curved space, which stretches the radial ruler; and the gravitational field, being energetic, gravitates in turn. Each of three multiplicative factors is greater than one, so at fixed $m$ , $\rho$ , and $r$ the relativistic shell always weighs more than the Newtonian one. The first figure shows which factor does what.

Figure

A fluid shell at radius r feels inward gravity balanced by the outward pressure gradient dP/dr. In general relativity three factors each ≥ 1 multiply the Newtonian weight — for a 1.4 M⊙ neutron star the metric factor (1 − 2Gm/rc²)⁻¹ alone reaches ≈ 1.5 near the surface

Now follow the consequence. As matter is piled onto a neutron star it grows more compact, $2Gm/rc^2$ creeps toward unity, and the metric factor $F_3$ runs away toward infinity. To keep the shell in balance the central pressure must diverge, yet no matter can supply unbounded pressure because its sound speed cannot exceed $c$ . What gives is equilibrium itself: beyond a critical mass there is no static solution. The ceiling is not a property of any material but a structural feature of relativistic gravity. The second figure plots TOV integrated over a family of central densities — a mass–radius curve whose turnover is that ceiling.

Figure

Mass–radius curves from the TOV equation: each equation of state traces a curve whose turning point is the maximum mass M_max, beyond which stars are unstable. The upper-left is forbidden by the Schwarzschild limit 2GM = Rc² and by causality; observed ≈ 2 M⊙ pulsars exclude soft equations of state that peak below them

The mass function hides one subtlety. Its form $\mathrm{d}m/\mathrm{d}r=4\pi r^2\rho$ looks Newtonian, but $4\pi r^2\,\mathrm{d}r$ is the coordinate volume, smaller than the proper volume $4\pi r^2\,\mathrm{d}r/\sqrt{1-2Gm/rc^2}$ . Integrating $\rho$ over coordinate volume gives the gravitational mass $M$ a distant orbit measures; over proper volume it gives the larger baryonic mass. Their difference is the gravitational binding energy over $c^2$ , which is why a neutron star weighs about $10$ – $15\%$ less than the neutrons that built it.

The core equations

The TOV equation is the relativistic hydrostatic-equilibrium condition. In factored form it displays the Newtonian gradient explicitly, followed by the three corrections that make it relativistic. Everything downstream — maximum mass, mass–radius relation, redshift — follows from integrating this one line with a mass equation and an equation of state.

\frac{\mathrm{d}P}{\mathrm{d}r}=-\frac{G\,[\rho + P/c^2]\,[m + 4\pi r^3 P/c^2]}{r^2\,[1 - 2Gm/(rc^2)]} \qquad (1)

where: $P$ — local isotropic pressure; $\rho$ — total mass–energy density $\varepsilon/c^2$ (rest mass plus all internal energy), not the baryon rest-mass density; $m(r)$ — gravitational mass inside coordinate radius $r$ ; $r$ — Schwarzschild (areal) coordinate, so a sphere at $r$ has proper area $4\pi r^2$ ; $G,c$ — Newton's constant and the speed of light. Reading it: $[\rho+P/c^2]$ replaces $\rho$ (enthalpy gravitates), $[m+4\pi r^3P/c^2]$ replaces $m$ (pressure adds to the source), and $[1-2Gm/rc^2]^{-1}$ has no Newtonian counterpart (curved space). Each pushes the gradient more negative. The system closes only with a mass equation, which in GR looks identical to the Newtonian one but carries the coordinate-volume subtlety noted above.

\frac{\mathrm{d}m}{\mathrm{d}r}=4\pi r^2\rho,\qquad m(0)=0 \qquad (2)

where: $m(r)$ — gravitational mass enclosed, integrating the total mass–energy density over coordinate volume; $\rho=\varepsilon/c^2$ . Reading it: given $\rho_c$ and an equation of state $P(\rho)$ , equations (1) and (2) integrate outward as a coupled pair until the pressure reaches zero — that radius is the surface. The two structure equations descend from a metric. For a static spherical spacetime the line element has just two unknown functions, one of them the mass function itself.

\mathrm{d}s^2 = -e^{2\Phi(r)}c^2\mathrm{d}t^2 + \left(1-\frac{2Gm(r)}{rc^2}\right)^{-1}\mathrm{d}r^2 + r^2\mathrm{d}\Omega^2 \qquad (3)

where: $\Phi(r)$ — the metric (time) potential, tending to the Newtonian potential over $c^2$ in the weak field; $m(r)$ — mass function fixing $g_{rr}$ ; $\mathrm{d}\Omega^2=\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2$ . Reading it: matching this interior at $r=R$ onto the vacuum Schwarzschild exterior forces $e^{2\Phi(R)}=1-2GM/Rc^2$ , tying the potential to the observable mass. Stress–energy conservation turns the potential gradient into the pressure gradient — the relativistic Euler equation, from which (1) follows.

\frac{\mathrm{d}P}{\mathrm{d}r}=-\left(\rho c^2+P\right)\frac{\mathrm{d}\Phi}{\mathrm{d}r},\qquad \frac{\mathrm{d}\Phi}{\mathrm{d}r}=\frac{G\,[m+4\pi r^3P/c^2]}{r^2c^2\,[1-2Gm/rc^2]} \qquad (4)

where: the combination $\rho c^2+P$ is the enthalpy density, which replaces the mass density of Newtonian hydrostatics. Reading it: the fluid is pushed not by its rest mass alone but by its inertia including pressure, which is precisely why $P$ appears inside the first bracket of (1). Once solved, the star advertises its compactness through the redshift of light leaving its surface — a directly observable diagnostic.

z_{\rm surf} = \left(1-\frac{2GM}{Rc^2}\right)^{-1/2}-1 \qquad (5)

where: $z_{\rm surf}$ — fractional wavelength shift of a photon climbing out of the surface potential; $M=m(R)$ , $R$ — gravitational mass and areal radius. Reading it: a line at rest wavelength $\lambda_0$ arrives at $\lambda_0(1+z)$ , so measuring $z$ measures $M/R$ directly. Two inequalities bound every TOV solution: a causality limit on stiffness, and Buchdahl's theorem capping compactness.

c_s^2=\frac{\mathrm{d}P}{\mathrm{d}\rho}\le c^2,\qquad \frac{2GM}{Rc^2}\le \frac{8}{9} \qquad (6)

where: $c_s$ — adiabatic sound speed, whose bound $c$ caps the stiffness of any physical equation of state; the $8/9$ ceiling holds for any star whose density does not increase outward. Reading it: together these turn "the corrections diverge" into a finite, unavoidable maximum mass.

Derivation ADeriving the TOV equation from the Einstein field equations

The goal is to obtain equation (1) assuming nothing beyond a static, spherically symmetric spacetime filled with a perfect fluid. We feed the metric ansatz (3) into Einstein's equations $G_{\mu\nu}=(8\pi G/c^4)T_{\mu\nu}$ with the perfect-fluid tensor $T^{\mu\nu}=(\rho c^2+P)u^\mu u^\nu/c^2+Pg^{\mu\nu}$ , read off two field-equation components, add local energy–momentum conservation, and combine — the route Tolman, Oppenheimer, and Volkoff took in 1939.

Step 1. The time–time component of the field equations reduces to $\frac{1}{r^2}\frac{\mathrm{d}}{\mathrm{d}r}\!\left[r\left(1-e^{-2\lambda}\right)\right]=\frac{8\pi G}{c^4}\varepsilon$ , where $e^{-2\lambda}\equiv1-2Gm/rc^2$ defines the mass function. With $\varepsilon=\rho c^2$ this integrates immediately to $\mathrm{d}m/\mathrm{d}r=4\pi r^2\rho$ , equation (2). Why: only the energy density sources $g_{rr}$ , so the $tt$ -equation is what promotes the coordinate label $m(r)$ into a genuine gravitational mass.

Step 2. The radial–radial component gives the potential gradient $\dfrac{\mathrm{d}\Phi}{\mathrm{d}r}=\dfrac{G\,[m+4\pi r^3P/c^2]}{r^2c^2\,[1-2Gm/rc^2]}$ , equation (4, right). Why: pressure enters the source through $4\pi r^3P/c^2$ — the field equations, not an assumption, make pressure gravitate.

Step 3. Local conservation $\nabla_\mu T^{\mu r}=0$ for a static fluid yields the relativistic Euler equation $\dfrac{\mathrm{d}P}{\mathrm{d}r}=-(\rho c^2+P)\dfrac{\mathrm{d}\Phi}{\mathrm{d}r}$ , equation (4, left). Why: this is the equilibrium condition, and where the enthalpy density $\rho c^2+P$ — not the mass density — appears as the inertia being pushed.

Step 4. Substitute the Step 2 gradient into the Step 3 relation and divide the enthalpy by $c^2$ , writing $(\rho c^2+P)/c^2=\rho+P/c^2$ . The result is exactly equation (1). Why: "pressure gravitates" (Step 2) and "enthalpy is pushed" (Step 3) stack the first two factors, while the $g_{rr}$ metric supplies the third. *As a check: in the weak-field limit $P\ll\rho c^2$ and $Gm\ll rc^2$ , equation (1) reduces exactly to the Newtonian $-Gm\rho/r^2$ , confirming the derivation is consistent.*

Go deeper · Derivation B · 1 minThe Newtonian limit and why a maximum mass is inevitable

Derivation BThe Newtonian limit and why a maximum mass is inevitable

The goal is to show two things at once: that TOV collapses smoothly onto Newtonian hydrostatics when the star is dilute and weakly bound, and that when it is not, the corrections drive the required central pressure to infinity at finite mass. We rewrite (1) as the Newtonian gradient times three explicit factors, take each to its weak-field value, then track the opposite limit as compactness rises toward the Schwarzschild condition.

Step 1. Factor equation (1) as $\dfrac{\mathrm{d}P}{\mathrm{d}r}=-\dfrac{Gm\rho}{r^2}\big(1+\tfrac{P}{\rho c^2}\big)\big(1+\tfrac{4\pi r^3P}{mc^2}\big)\big(1-\tfrac{2Gm}{rc^2}\big)^{-1}$ , so the corrections are $(1+\epsilon_1)(1+\epsilon_2)(1+\epsilon_3)$ with every $\epsilon_i\ge0$ . Why: written as manifestly non-negative additions, they prove at a glance that the relativistic gradient never falls below the Newtonian one — gravity is strictly stronger.

Step 2. In the weak field all three shrink: $\epsilon_1\sim P/\rho c^2\to0$ , $\epsilon_2\sim4\pi r^3P/mc^2\to0$ , $\epsilon_3\sim2Gm/rc^2\to0$ , leaving $\mathrm{d}P/\mathrm{d}r=-Gm\rho/r^2$ . For the Sun the largest is $GM/Rc^2\approx2\times10^{-6}$ . Why: this controlled recovery licenses Newtonian physics wherever the compactness is small, and pinpoints the single number that measures the error.

Step 3. Now push the opposite way. As a star is made more compact, $2Gm/rc^2\to1$ and $\epsilon_3=(1-2Gm/rc^2)^{-1}-1\to\infty$ , while the growing pressure inflates $\epsilon_1$ and $\epsilon_2$ . Holding up a more compact configuration then demands an ever-steeper, ultimately diverging pressure gradient, so $P_c\to\infty$ at finite mass. Why: the runaway is geometric, not material — it comes from the metric factor, so no choice of matter can escape it.

Step 4. Impose causality, $c_s^2=\mathrm{d}P/\mathrm{d}\rho\le c^2$ . A finite sound speed means finite pressure at finite density, so the diverging demand of Step 3 cannot be met beyond some point: a maximum static mass $M_{\rm TOV}$ exists. Why: the maximum mass is the collision of a geometric divergence (Step 3) with a physical ceiling on stiffness (Step 4) — remove either and it disappears, which is why Newtonian polytropes with $\Gamma>4/3$ have none.

Go deeper · Derivation C · 1 minThe incompressible star and the Buchdahl 8/9 bound

Derivation CThe incompressible star and the Buchdahl 8/9 bound

The goal is to exhibit the maximum-compactness ceiling analytically in the one case where TOV solves in closed form — a star of uniform density $\rho=\text{const}$ , the Schwarzschild interior solution of 1916. Constant density is the stiffest conceivable equation of state, so if even this configuration caps out at a finite compactness, every physical star must obey a bound at least as tight. We integrate TOV, read off the central pressure, and find where it diverges.

Step 1. With $\rho$ constant, equation (2) gives $m(r)=\tfrac{4}{3}\pi r^3\rho$ , so $2Gm/rc^2=(2GM/Rc^2)(r/R)^2$ grows quadratically from the centre. Why: uniform density makes the enclosed mass a pure cubic, which is what lets the otherwise nonlinear TOV integral be done exactly.

Step 2. Substituting into (1) and integrating inward from the surface ( $P=0$ at $r=R$ ) gives $P(r)=\rho c^2\,\dfrac{\sqrt{1-2GMr^2/R^3c^2}-\sqrt{1-2GM/Rc^2}}{3\sqrt{1-2GM/Rc^2}-\sqrt{1-2GMr^2/R^3c^2}}$ . Why: this closed form steepens toward the centre as the metric factor bites — unlike the Newtonian parabola.

Step 3. Set $r=0$ : $P_c=\rho c^2\,\dfrac{1-\sqrt{1-2GM/Rc^2}}{3\sqrt{1-2GM/Rc^2}-1}$ . This diverges when the denominator vanishes, i.e. when $\sqrt{1-2GM/Rc^2}=\tfrac13$ , which is $\dfrac{2GM}{Rc^2}=\dfrac{8}{9}$ . Why: the divergence of $P_c$ at a purely geometric compactness is the sharp form of the runaway argued qualitatively before.

Step 4. Therefore no static uniform star can exceed compactness $8/9$ ; Buchdahl's theorem extends this to any star with outward-decreasing density, so $2GM/Rc^2\le8/9$ universally, equivalently $R\ge\tfrac{9}{8}R_s$ and $z\le2$ . Why: uniform density is the stiffest allowed, so its ceiling is the loosest possible — real matter is bounded even more strongly, which is why observed neutron stars sit near compactness $0.3$ – $0.5$ . *As a check: the central-pressure formula of Step 2 reproduces the Schwarzschild interior solution, and the divergence $P_c\to\infty$ at $2GM/Rc^2=8/9$ is precisely Buchdahl's bound.*

Limiting cases

The TOV equation spans ordinary stars to the brink of collapse; the table reads off what it becomes in each regime.

Case	Result	Meaning
Newtonian limit	$P\ll\rho c^2,\ Gm/rc^2\ll1\Rightarrow\mathrm{d}P/\mathrm{d}r=-Gm\rho/r^2$	All three factors $\to1$ ; recovers hydrostatic equilibrium for main-sequence stars and white dwarfs, where $GM/Rc^2\lesssim10^{-4}$ .
Compactness (Buchdahl) bound	$2GM/Rc^2\le8/9$	Any density-decreasing star has finite central pressure only below this compactness; at $8/9$ even an incompressible star's $P_c$ diverges.
Incompressible star	$P_c=\rho c^2\dfrac{1-\sqrt{1-2GM/Rc^2}}{3\sqrt{1-2GM/Rc^2}-1}$	Exact Schwarzschild-interior solution; the analytic window onto the compactness limit, with $P_c\to\infty$ as $2GM/Rc^2\to8/9$ .
Maximum-mass configuration	$\mathrm{d}M/\mathrm{d}\rho_c=0$	The turning point of the mass–central-density curve; marks the onset of radial instability and the boundary between neutron stars and black holes.

Common trapthe misconception

It is tempting to read $\rho$ in the TOV equation as the ordinary (baryon rest-mass) density and $m(r)$ as "the mass of the stuff inside $r$ ," as in Newtonian physics. Both are wrong. The density $\rho=\varepsilon/c^2$ is the total mass–energy density, including kinetic and interaction energy, which at nuclear densities is a sizeable fraction of the rest mass. And $m(r)=\int4\pi r^2\rho\,\mathrm{d}r$ integrates over coordinate volume, so it is the gravitational mass — it already has the negative binding energy folded in. It is smaller than the baryonic mass $\int4\pi r^2\rho\,(1-2Gm/rc^2)^{-1/2}\mathrm{d}r$ , and the deficit is why a star's gravitational mass, not its baryon number, is what gravity outside measures.

Worked examples

1 · Compactness and surface redshift of a two-solar-mass pulsar (fully worked)

NICER and radio timing place PSR J0740+6620 at $M=2.08\,M_\odot$ , $R\approx12.4\ \mathrm{km}$ . Find its compactness, surface redshift, and check against Buchdahl. Take $G=6.674\times10^{-11}$ , $c=2.998\times10^8$ , $M_\odot=1.989\times10^{30}$ kg.

Step 1. Mass in kilograms: $M=2.08\times1.989\times10^{30}=4.137\times10^{30}\ \mathrm{kg}$ , so $GM=2.761\times10^{20}$ and the Schwarzschild radius is $R_s=2GM/c^2=\mathbf{6.14\ km}$ .

Step 2. Compactness: $\dfrac{2GM}{Rc^2}=\dfrac{R_s}{R}=\dfrac{6.14}{12.4}=\mathbf{0.496}$ . The radius is about twice the Schwarzschild radius — one of the most compact objects known short of a black hole.

Step 3. Surface redshift from equation (5): $z=(1-0.496)^{-1/2}-1=1.409-1=\mathbf{0.41}$ . A $6.4\ \mathrm{keV}$ iron line at the surface would be received near $4.5\ \mathrm{keV}$ .

Step 4. Buchdahl check: $0.496/0.889=0.56$ , so the star sits at $56\%$ of the maximum allowed compactness — safely stable, with $R_{\min}=\tfrac98R_s=6.9\ \mathrm{km}$ , well inside the measured $12.4\ \mathrm{km}$ . Result: compactness ≈ 0.50, z_surf ≈ 0.41, comfortably below the Buchdahl bound.

2 · Why the maximum mass is about one solar mass (order-of-magnitude)

Show from fundamental constants alone that the Oppenheimer–Volkoff maximum mass must land near a solar mass, without integrating anything — the maximum mass of a degenerate gas is set by the same balance that fixes the Chandrasekhar mass.

Step 1. For a relativistic degenerate gas, equilibrium against gravity is scale-free, and the mass at which it fails is the Chandrasekhar-type combination $M_{\max}\sim\left(\dfrac{\hbar c}{G}\right)^{3/2}\dfrac{1}{m_n^2}=\dfrac{M_{\rm Pl}^3}{m_n^2}$ , where $M_{\rm Pl}=(\hbar c/G)^{1/2}$ is the Planck mass and $m_n$ replaces the nucleon mass of the white-dwarf problem.

Step 2. Numerically $M_{\rm Pl}=2.18\times10^{-8}\ \mathrm{kg}$ and $m_n=1.675\times10^{-27}\ \mathrm{kg}$ , so $M_{\max}\sim(2.18\times10^{-8})^3/(1.675\times10^{-27})^2=3.7\times10^{30}\ \mathrm{kg}$ .

Step 3. In solar units that is $3.7\times10^{30}/1.989\times10^{30}=\mathbf{1.85\,M_\odot}$ . The full free-neutron integration carries an $O(1)$ coefficient pulling this to Oppenheimer and Volkoff's $0.71\,M_\odot$ ; nuclear repulsion stiffens real matter back to $\sim2.2\,M_\odot$ . Result: dimensional analysis alone fixes the scale at ≈ 2 M⊙ — the maximum mass is written in ℏ, c, G, and the neutron mass.

Physics of the portrait

The TOV equation is the meeting point of three careers and one revolutionary idea: that gravity, taken seriously, can defeat any pressure and end a star's life in collapse. The portrait belongs to the man who first computed that collapse.

Origins · 1 minJ. Robert Oppenheimer and how this was discovered

J. Robert Oppenheimer

1904–1967 · general relativity, nuclear physics, and gravitational collapse

Before the Manhattan Project made him famous, J. Robert Oppenheimer wrote a short run of papers in 1938–1939 that founded the physics of compact objects. Building on Richard C. Tolman's general static-fluid-sphere solutions of Einstein's equations — Tolman, a Caltech physical chemist turned relativist, had catalogued the interior metrics — Oppenheimer and his student George Volkoff set up the relativistic equilibrium of a ball of degenerate free neutrons and integrated it numerically by hand. They found what no Newtonian calculation could show: above roughly $0.7\,M_\odot$ there is no equilibrium at all. The following year, with Hartland Snyder, Oppenheimer described "continued gravitational contraction" — a star collapsing through its own Schwarzschild radius, the first modern picture of a black hole. The community, Einstein and Wheeler included, resisted these conclusions for decades; Oppenheimer himself turned to weapons work and never returned. The $0.7\,M_\odot$ figure was too low — free neutrons are far too soft, and nuclear forces stiffen real matter to $\sim2\,M_\odot$ — but the structure of the argument was exactly right. The existence of a maximum mass, and collapse beyond it, is Oppenheimer's most enduring contribution to physics, vindicated when pulsars were found in 1967, the year he died.

History

1916

— Schwarzschild solves the field equations for a static sphere, giving both exterior and uniform-density interior metrics

1934

— Baade and Zwicky propose that supernovae leave behind neutron stars

1939

— Tolman publishes general static-fluid solutions; Oppenheimer and Volkoff derive the neutron-star maximum mass near 0.7 M⊙

1959

— Buchdahl proves the universal 8/9 compactness bound

2010

— Demorest and colleagues measure a 1.97 M⊙ pulsar, excluding the softest equations of state

Going further

Texts. Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §16 (neutron stars and general relativity) · Shapiro and Teukolsky, Black Holes, White Dwarfs, and Neutron Stars, Ch. 5 (the definitive TOV treatment) · Weinberg, Gravitation and Cosmology, §11.1 and §11.6 · Glendenning, Compact Stars, 2nd ed. · Oppenheimer and Volkoff, Phys. Rev. 55, 374 (1939); Tolman, Phys. Rev. 55, 364 (1939). Key results to keep in mind: TOV reduces to $-Gm\rho/r^2$ in the weak-field limit; the incompressible central-pressure formula diverges at $2GM/Rc^2=8/9$ ; the worked numbers are compactness $0.50$ and $z\approx0.41$ for $2.08\,M_\odot$ at $12.4\ \mathrm{km}$ , and the dimensional maximum-mass scale $M_{\rm Pl}^3/m_n^2\approx1.85\,M_\odot$ ; the quoted masses are $\sim0.7\,M_\odot$ for a free-neutron gas and $\sim2\,M_\odot$ for observed pulsars.

Prerequisites. Neutron Stars · Equation of State of Dense Matter · Chandrasekhar Mass Limit · Electron Degeneracy Pressure. Leads to. Neutron Star Structure and Interior · Pulsars · Magnetars · Millisecond and Recycled Pulsars.

Ask the tutor: Ask me to integrate the TOV equation for a polytropic or realistic equation of state and watch the mass–radius curve turn over at the maximum mass.

What exactly is the difference between the TOV equation and Newtonian hydrostatic equilibrium?

Both say the pressure gradient at each radius holds up the weight above, but they compute that weight differently. Newton uses $\mathrm{d}P/\mathrm{d}r=-Gm\rho/r^2$ : mass density pulled by enclosed mass through flat space. TOV multiplies this by three factors, each $\ge1$ : $(1+P/\rho c^2)$ because pressure gravitates, $(1+4\pi r^3P/mc^2)$ because pressure adds to the source, and $(1-2Gm/rc^2)^{-1}$ because space is curved. For the Sun these are parts per million and Newton is exact. For a neutron star they are order unity, and ignoring them gets the structure — and the very existence of a maximum mass — completely wrong.

Why is there a maximum mass, in one sentence?

Because making a star more compact drives the correction factor $(1-2Gm/rc^2)^{-1}$ toward infinity, demanding infinite central pressure to stay in balance, and no real matter can supply that since its sound speed — hence its stiffness — is capped at $c$ . The maximum mass is where these two facts collide. It is a structural feature of relativistic gravity, not a property of neutrons, which is why quark and boson stars have their own TOV maximum masses. Newtonian gravity has no such curvature factor, so a stiff enough equation of state supports any mass — there is no Newtonian analogue of the ceiling.

Do I need to know the equation of state to write down the TOV equation?

No — and this split is the whole reason the equation is useful. The TOV equation and the mass equation are pure gravity: they follow from Einstein's equations and hold for any perfect fluid. All the uncertain nuclear and particle physics is quarantined into one extra relation, the equation of state $P(\rho)$ , which you supply separately. Sweeping the central density traces a mass–radius curve for that equation of state, and comparing different curves with measured masses and radii is precisely how observations constrain the microphysics. Gravity is known exactly; the matter is what we are trying to learn.

Is the gravitational mass the same as the mass of all the neutrons added up?

No, and the difference is physically important. The gravitational mass $M=\int4\pi r^2\rho\,\mathrm{d}r$ integrates the total energy density over coordinate volume and is what an orbiting companion reports. The baryonic mass integrates the same matter over the larger proper volume, weighted by $(1-2Gm/rc^2)^{-1/2}$ , counting the rest energy of every baryon as if dispersed to infinity. The gravitational mass is smaller, by the binding energy over $c^2$ — typically $10$ – $15\%$ . That deficit is released as neutrinos and gravitational waves when the star forms or merges, and it is baryon number, not gravitational mass, that is conserved when tracking a star through collapse.

Why did Oppenheimer and Volkoff get only 0.7 solar masses when real neutron stars reach two?

Because they used the simplest possible matter — an ideal gas of free, non-interacting neutrons — which is far too soft. A soft equation of state provides little pressure per unit density, so gravity wins early and the maximum mass is low. Real neutron-star matter is stiffened by the strong nuclear force, which is powerfully repulsive at short range once neutrons are packed to nuclear density. That extra pressure holds up much more mass, pushing the maximum to roughly $2.2$ – $2.3\,M_\odot$ . Their number was wrong, but their method and their central discovery — that a maximum mass exists and collapse follows — were exactly right.

What does the surface redshift actually tell an observer?

The surface redshift $z=(1-2GM/Rc^2)^{-1/2}-1$ is a direct readout of the compactness $M/R$ . A spectral line emitted at the surface climbs out of the gravitational well and arrives stretched by $1+z$ , so measuring the shift of an identified line fixes $M/R$ in one stroke — independent of any equation of state. For a canonical neutron star ( $1.4\,M_\odot$ , $12\ \mathrm{km}$ ) this is about $0.24$ ; for the most compact stars it approaches $0.4$ . Combined with an independent mass it yields the radius, and against the Buchdahl ceiling $z\le2$ it bounds how compact any star can be.