Gravitational building codes: why space sets a limit on force ✛

Original: "Upper bounds on the force function in spatially regular self-gravitating matter configurations"
· Shahar Hod
arXiv:2607.23661v1 · 2026-07-26 · CC BY 4.0 · 1 min
Einstein's theory limits the maximum force in nature — and new evidence shows that this limit is even stricter than previously thought.

Building codes prevent overloading a house's foundation — otherwise it cracks. Einstein's field equations also have their own 'building codes': they set a limit on the force of gravity that no star or planet can exceed. At the core is the famous number c⁴/G, the Planck force, about 10⁴⁴ newtons.

In 2002, physicists suggested that this is the maximum possible force in nature. However, it was long impossible to prove this for real objects without black holes. Recently, Shahar Hod derived strict bounds directly from the field equations. He considered spherical bodies (stars, white dwarfs, neutron stars) and introduced a force function — the product of pressure and surface area of a sphere. It turned out that depending on the properties of matter, this function cannot exceed 2, 1, or even 0.5. The strictest limit — 0.5 — applies to most known substances where the sum of energy and pressure is non-positive. In other words, the real maximum force is about half of c⁴/G.

The force c⁴/G is comparable to Mount Everest, compressed to atomic size, pressing on a surface. Even in the densest neutron stars, the pressure is weaker.

These numbers are exact consequences of the theory. They serve as cosmic building codes: if future observations of supernovae or gravitational waves show a deviation, it would indicate new physics. Just as an engineer checks a foundation against codes, astrophysicists can now test the most extreme objects in the universe.

c⁴/G ≈ 1.2×10⁴⁴ N is the force that would be required to squeeze Mount Everest into a microscopic volume and make it press on a surface. Even in the most extreme neutron stars, the pressure does not reach this limit, leaving a remarkable margin.

F = 4\pi r^2 p(r)
The product of pressure and the area of a sphere of radius r
Scientists
Bernhard RiemannJoseph WeberKarl SchwarzschildKip ThorneRainer WeissAlbert Einstein
Concepts
mass limit gravitational constant fr gravity Einstein field equations upper limit Birkhoff's theorem Law of Universal Gravitation equivalence principle gravity gravitational field mass grand unified theory free fall Bekenstein bound singularity dimensionality degrees of freedom Witten's positive energy theorem supergravity Friedmann equations