The pressure that weighs down what it holds up

The mass no cold matter can hold up found the limit on a white dwarf. Electrons squeezed into a small volume carry a pressure that has nothing to do with temperature — the pressure that is not a temperature — and it holds the star up until the electrons become relativistic. Then the pressure stiffens no further with density, the mass stops depending on the radius, and there is a largest mass, 1.46 solar masses, above which no cold white dwarf can exist. The argument used Newton’s gravity, and for a white dwarf that is a good approximation: its gravitational potential at the surface is a few parts in ten thousand of $c^2$.

A star above that limit collapses until its electrons are pressed into protons and it becomes a ball of neutrons, the matter whose composition a neutron star cannot choose freely. Neutrons are fermions too and resist compression the same way, with a pressure that stiffens in the same fashion when they become relativistic. Newton’s gravity then gives a limit for a neutron star by the same argument, and it is large: nearly six solar masses. That figure is wrong by a factor of eight, and the reason is not in the neutrons. It is that at the density of a neutron star gravity is no longer Newtonian, and in general relativity the pressure holding up a star is itself heavy.

A star in which pressure has weight #

In Newton’s gravity, a layer of a star is in equilibrium when the difference in pressure across it balances the weight of the layer:

$$\frac{dP}{dr} = -\frac{G m(r)\rho}{r^2},$$

with $m(r)$ the mass inside and $\rho$ the density. Pressure appears once, on the left, as the thing doing the holding.

In general relativity the corresponding equation, derived by Richard Tolman and by Robert Oppenheimer and George Volkoff in 1939, has pressure on both sides:

$$\frac{dP}{dr} = -\frac{G\left(\rho + P/c^2\right)\left(m + 4\pi r^3 P/c^2\right)}{r^2\left(1 - 2Gm/rc^2\right)} .$$

Three changes have been made to Newton’s form, and each says that pressure behaves like mass. The factor $\rho + P/c^2$ says that the pressure contributes to the inertia of the layer, so a more highly pressed layer is harder to support. The factor $m + 4\pi r^3P/c^2$ says that the pressure inside a sphere contributes to the gravity pulling the layer down, in addition to the mass. And $1/(1 - 2Gm/rc^2)$ is the stretch of radial distances that the radius a mass adds that no circumference shows described, which makes each step of radius longer than it looks and the pressure difference across it correspondingly larger.

Pressure has inertia and weight because it is a form of energy density, and in relativity energy of every kind both resists acceleration and gravitates — the box of light that weighs something weighed a box heavier for the light it held. What makes the equation dangerous is the sign of the loop: pressure is what holds the star up, and pressure is part of what it has to hold up.

The heaviest star free neutrons can hold #

Take the simplest possible neutron star: cold, and made of free neutrons that interact only through the exclusion principle. Their pressure at each density is fixed by quantum mechanics alone, and the equation of equilibrium can be integrated outward from any chosen central density to the surface where the pressure falls to zero. Each central density gives one star.

The figure draws the family both ways. Light stars are large and nearly Newtonian, and the two curves lie together below about 0.3 solar masses. As the central density rises the stars become smaller and heavier, and the two curves separate. Newton’s keeps climbing: at a radius of nine kilometres it passes two solar masses and it goes on rising towards the limit set by relativistic neutrons, 5.8 solar masses. Relativity’s curve reaches a highest point, 0.71 solar masses at a radius of 9.5 kilometres, and turns back. Squeezed further, the stars become smaller and lighter, and the curve spirals inward to a point.

The maximum is the Oppenheimer–Volkoff limit, and it is lower than the white-dwarf limit. In Newton’s gravity a star of neutrons should hold four times the mass of a white dwarf, because each neutron both supplies pressure and carries its own mass, where in a white dwarf each pressing electron has to hold up two nucleons. In general relativity it holds half as much as a white dwarf, because it is compact enough for the weight of its own pressure to matter. Oppenheimer and Volkoff published the number in 1939 and concluded that a star heavier than that, having exhausted its fuel, would not stop collapsing — the argument that led the same year to Oppenheimer and Snyder’s paper on what is now called a black hole.

Why the neutrons’ push stops growing fast enough #

The Newtonian limit and the relativistic one have different causes, and it is worth separating them. A cold Fermi gas pushes back because squeezing it forces its particles into states of higher momentum. While the particles are slow, their kinetic energy goes as momentum squared and the pressure grows as the five-thirds power of the density. Once the typical momentum approaches $mc$ — for neutrons, at a density of about $6\times10^{18}$ kilograms per cubic metre — the energy goes as momentum to the first power and the pressure grows only as the four-thirds power. The share that is not half a kT found the same softening in a hot gas whose particles are relativistic: an energy linear in momentum stores more per degree of freedom and pushes back less per unit of compression, dropping the ratio of specific heats to four-thirds.

Four-thirds is the exponent at which a Newtonian star’s mass stops depending on its radius, which is the whole of the white-dwarf limit. A Newtonian neutron star approaches that exponent only asymptotically, as its centre becomes more and more relativistic, and its mass creeps up towards 5.8 solar masses without ever turning. The relativistic limit arrives much earlier. At the Oppenheimer–Volkoff maximum the neutrons at the centre have a Fermi momentum of about $0.8\,m_nc$: they are only mildly relativistic, their pressure is still growing faster than the four-thirds power, and in Newton’s gravity the star would be nowhere near a limit. The collapse is caused by the weight of the pressure, not by any softening of the gas.

That also explains why the maximum sits at a compactness where general relativity is large. At 0.71 solar masses in 9.5 kilometres, $2GM/Rc^2$ is 0.22 — the surface clock of such a star runs 12 per cent slow — and the corrections in the equation of equilibrium are all of that order. The mass no cold matter can hold up could ignore them because a white dwarf’s compactness is a few ten-thousandths. A neutron star cannot, and the limit it runs into is set by gravity rather than by the matter.

Squeezing harder, and getting less #

The turn in the curve is easiest to see against central density rather than radius.

In Newton’s gravity every increase in central density buys more mass. The curve flattens as the neutrons become relativistic and their pressure grows more slowly with density, which is exactly the mechanism of the white-dwarf limit, but it never turns down. In general relativity the curve peaks at a central density of $3.6\times10^{18}$ kilograms per cubic metre — some fifteen times the density of an atomic nucleus — and then falls: past the peak, a star compressed further is a lighter star.

The peak is also where stability is lost. A star on the rising branch, squeezed slightly, finds its pressure growing faster than the gravity it has to resist, and springs back. A star on the falling branch, squeezed slightly, finds the extra weight of the extra pressure outrunning the extra support, and collapses further. The same turning-point rule governs white dwarfs and every other cold self-gravitating family: stability changes where the mass is stationary against the central density. The disturbance that grows instead of travelling found the Jeans instability in a uniform gas by asking whether a compression is restored or amplified; this is the same question, asked of a whole star.

Three factors, all in the same direction #

It is worth looking inside the heaviest star to see how much each of the three relativistic changes contributes.

The figure plots the three factors through the heaviest star of free neutrons. Near the centre the pressure is about a tenth of the energy density, and its inertia adds nine per cent to the load. The pull of the pressure adds more, twenty-seven per cent at the centre, because a sphere’s gravity counts its pressure three times — once for each direction in which it pushes — so that near the centre the factor is $1 + 3P/\varepsilon$. Towards the surface the pressure falls and both of those fade, while the stretch of space, which depends on how much mass is enclosed, grows to 37 per cent. The product stays between about 1.4 and 1.5 all the way out.

A pressure gradient half as steep again as Newton’s would be tolerable on its own. The trouble is the loop. The extra gradient needs a higher central pressure, which adds to all three factors, which demands a steeper gradient still. For a light star the loop converges quickly; at the maximum mass it converges no longer, and adding matter makes it diverge. The limit is not reached because the neutrons fail to push. It is reached because every increment of push is also an increment of load.

A ceiling no matter can break #

How far could stiffer matter move the limit? The cleanest answer comes from the stiffest body imaginable: one of uniform density, incompressible, with a sound speed that is infinite.

Karl Schwarzschild solved this case exactly in 1916. The pressure needed at the centre of a uniform star, as a fraction of its energy density, is $(1 - \sqrt{1 - s})/(3\sqrt{1 - s} - 1)$, where $s$ is the compactness $2GM/Rc^2$. At small compactness this is $s/4$, Newton’s answer. As the compactness grows it pulls away, and at $s = 8/9$ the denominator reaches zero and the pressure becomes infinite. Nothing, however stiff, can hold up a static body of uniform density smaller than nine-eighths of its Schwarzschild radius. In 1959 Hans Buchdahl showed that the same bound holds for any body whose density does not increase outwards: the most compact static star possible has a radius of $9GM/4c^2$ — inside the photon sphere at $3GM/c^2$, so that light could circle outside such a star’s surface, as the circle light cannot leave found it doing around a black hole.

The bound is purely geometric. It does not depend on what the star is made of, only on the fact that pressure gravitates. It also means there is a gap between the most compact star and a black hole: a body can sit at $R = 9GM/4c^2$ or it can be inside $2GM/c^2$, but nothing static can be in between.

Reading the inside of a neutron star from its mass #

The free-neutron limit of 0.71 solar masses would be only a historical curiosity if neutron stars obeyed it. They do not.

The first binary pulsar gave a neutron star mass of 1.44 solar masses in the 1970s, already twice the free-neutron limit. Radio timing of the millisecond pulsar J0740+6620, whose white-dwarf companion delays its pulses as they pass close by, gave 2.08 solar masses in 2021, and a few less precise measurements go somewhat higher. Something must make neutron-star matter far stiffer than a gas of free neutrons, and that something is the strong force: at densities above that of a nucleus, nucleons repel one another strongly at short range, and the repulsion adds a pressure that free particles do not have.

The measured masses are therefore a statement about nuclear physics at densities no laboratory can reach. A proposed equation of state for dense matter predicts a maximum mass by exactly the integration drawn in the first figure, and any equation whose maximum falls below the heaviest measured star is ruled out. Many were, when the two-solar-mass pulsars were found — in particular, several that allowed the neutrons to turn into exotic particles such as hyperons, which soften the pressure.

At the other end, Clifford Rhoades and Remo Ruffini asked in 1974 how heavy a neutron star could be if the matter inside were as stiff as causality allows: known nuclear physics up to nuclear density, and above it a sound speed equal to that of light. The answer is about 3.2 solar masses. Stiffer than that is not possible, because the sound speed would exceed $c$. Newton’s figure of 5.8 for free neutrons is out of reach not because the neutrons are too soft but because relativity forbids any matter from being stiff enough.

What a star above the ceiling becomes #

A cold star above whatever the true maximum is has no static configuration at all, and the equation of equilibrium says so by having no solution. The core of a massive star that collapses at the end of its life therefore either stops at a neutron star below the maximum, rebounding and driving off the outer layers as a supernova, or continues inward. Oppenheimer and Snyder followed the second case in 1939 for a cloud with no pressure at all, and found it closing within its own Schwarzschild radius in a finite time on a falling clock and an infinite time as seen from outside — the surface that only lets things in is the end point.

The Buchdahl bound makes the transition abrupt. There is no family of static stars that shrinks smoothly from $9GM/4c^2$ down to a horizon at $2GM/c^2$; between the two there is nothing that can stand still. A collapse that passes the bound cannot be stopped by any pressure, because a pressure large enough to stop it would be a mass large enough to require more pressure.

A neutron star just below its maximum is correspondingly delicate. Accretion from a companion adds mass slowly, and a star that reaches the peak of the curve drawn above will collapse; such events have been proposed as the origin of some of the black holes found with masses just above the heaviest neutron stars. The crust on top — the few hundred metres of neutron-rich nuclei whose strength sets the mountain a spinning star is allowed — plays no part in the balance, which is decided entirely in the core.

Where the argument stands and does not #

The figures use the simplest equation of state there is and one that is known to be wrong for real neutron stars, and they do so on purpose: the free-neutron star shows what general relativity does to a star independently of the unknown nuclear physics. The real equation of state above nuclear density is uncertain by a factor of a few in pressure, and the radius of a real 1.4-solar-mass neutron star, measured by X-ray timing of hot spots on its surface and by the deformation of neutron stars in the merger GW170817, is around twelve kilometres, not the nine and a half the free-neutron model gives at its maximum.

The figures also assume the star is cold, static and not rotating. Rotation holds a star up partly by centrifugal support and raises the maximum mass by up to about twenty per cent for a star spinning near its breakup rate. A hot newborn neutron star, or the remnant of a merger, can be temporarily supported above the cold maximum and then collapse as it cools or spins down. And the uniform-density star is an idealisation useful only because it can be solved exactly: real matter compresses, and the Buchdahl bound is a bound, not a description.

Still open: what the heaviest neutron star is #

The maximum mass of a cold, non-rotating neutron star is not known. It lies between the heaviest measured, about 2.1 solar masses, and the causal ceiling of about 3.2, and several lines of evidence — the absence of a long-lived remnant after GW170817, the masses of objects found in the gap between neutron stars and black holes — point to somewhere between 2.2 and 2.5. Measurements of neutron-star radii to a few per cent, now being made by timing the rotation of X-ray hot spots, would pin down the equation of state and with it the maximum. Whether the core of the heaviest neutron stars is made of nucleons, of hyperons, or of free quarks is an open question in nuclear physics that will be answered, if at all, by weighing and measuring stars.

The habit worth carrying away is to ask whether the thing that supports a structure is also part of its load. In Newton’s gravity pressure only holds a star up; in general relativity it also weighs it down, three times over, so beyond a certain compactness pressing harder makes things worse, and no material of any stiffness can hold a static body inside nine-eighths of its Schwarzschild radius. A neutron star twice as heavy as free neutrons allow is therefore a measurement of what fills it.