Pressure pulls too, and that is why there is a bound – a census of Buchdahl’s limit.
A static star cannot be made arbitrarily small. This paper writes why that bound exists, what it demands and does not demand, and what exactly it bounds. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Tolman (1939), Oppenheimer-Volkoff (1939), Buchdahl (1959) and the interior Schwarzschild solution are all standard.
First, and this is the core. In general relativity, pressure pulls too. Newtonian hydrostatic balance sets the pressure gradient equal to minus the gravitational mass times the density over the square of the radius. Its general-relativistic counterpart, the Tolman-Oppenheimer-Volkoff equation, multiplies this by three factors. The first is one plus the pressure divided by the density times the square of the speed of light, expressing that the pressure itself carries energy. The second is one plus four pi times the cube of the radius times the pressure, divided by the mass times the square of the speed of light, expressing that pressure is a source of gravity. The third is the reciprocal of one minus twice the gravitational constant times the mass over the radius times the square of the speed of light, expressing the curvature of space. So long as the pressure is positive and one is outside the horizon, all three exceed one. In Newtonian gravity pressure appears only on the supporting side, whereas in general relativity it supports and also pulls. The move of raising the pressure when a thing is about to collapse therefore stops working partway: raising the pressure raises the first two factors and strengthens the gravity too, and somewhere the added gravity beats the added support. That is why a bound exists. In Paper 141 the bound arose because two forces carried the same power of the radius; here it arises because the supporting side also joins the pulling side. One word, limit, naming two different mechanisms (Paper 17).
Second, the location of the bound follows in closed form. In the uniform-density interior Schwarzschild solution, the central pressure divided by the density times the square of the speed of light equals one minus the square root of one minus the compactness, all over three times that square root minus one, where the compactness is twice the gravitational constant times the mass over the radius times the square of the speed of light. Solving symbolically for the vanishing of the denominator gives a compactness of exactly eight ninths, so the radius must exceed nine eighths of the Schwarzschild radius: Buchdahl’s bound. Numerically the central pressure ratio is 0.0278 at a compactness of 0.1, 0.2612 at 0.5, 1.6180 at 0.8, 16.6603 at 0.88 and 166.666 at 0.888, diverging as eight ninths is approached. A static star cannot be more compact than that, whatever it is made of. Note the direction: the bound has the radius as the larger quantity. And the nine eighths is not a geometric quantity, following from the condition that the square root of one minus the compactness equal one third, a third that comes from the structure of the interior solution and from neither a sphere nor a solid angle (the discipline of Papers 35, 45).
Third, this bound needs no equation of state. Buchdahl’s theorem demands only that the star be static, spherically symmetric, of isotropic pressure, with density not increasing outward and not negative. What the star is made of is not asked. This is unusual: the Chandrasekhar bound of Paper 141 required a specific equation of state, that of an ultra-relativistic degenerate gas, and here none is needed. But a light demand buys a light conclusion (Paper 126): Buchdahl speaks only of a ratio and says nothing whatever about the mass.
Fourth, what is bounded is a ratio and not a mass. Scaling the mass and the radius by the same factor leaves the compactness unchanged, so Buchdahl’s condition is satisfied in the same way by stars of any size. The statement that neutron stars have a maximum mass is therefore not Buchdahl’s theorem: naming a maximum mass requires an equation of state, and the equation of state at those densities is not known. What Buchdahl gives is a compactness that no equation of state can exceed. Paper 17 asked for an address, and saying that there is a limit requires saying a limit on what: a mass, a radius, or a ratio. The three are different statements.
Fifth, the corpus now holds three bounds demanding three different amounts of input. The Chandrasekhar bound (Paper 141) demands an equation of state and bounds a mass; Buchdahl’s bound demands no equation of state and bounds a ratio; the horizon demands no matter at all and bounds a length. The less demanded, the less said: Chandrasekhar names a definite mass at the price of demanding microphysics, Buchdahl demands almost nothing at the price of naming only a ratio, and the horizon demands nothing and names only a length. One word, limit, naming three different statements. Demand and conclusion are in balance, and without seeing that balance one cannot tell which bound supports which claim.
IMPORTANT (scope): a census of established standard material (Layer 2). Every work cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that Buchdahl’s theorem is in error (it is not) / that it gives a maximum mass for neutron stars (it does not; that requires an equation of state). What is added: making explicit that all three TOV factors exceed one, locating the bound as the divergence of the uniform-density solution, writing that what is bounded is a ratio rather than a mass, and setting three bounds side by side by the input they demand. What remains: in the Newtonian world, if a thing was about to collapse one raised the pressure; in general relativity that move tightens the noose, because what was added to support also joins the pull. A bound is the point at which the support becomes its own enemy.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Pressure Pulls Too, and That Is Why There Is a Bound [Paper 145]. Zenodo.
DOI: 10.5281/zenodo.22082693
ORCID: 0009-0006-8587-3450