Einstein's General Relativity was born in 1916 as a complete theory of gravity for that era. Mass, energy, radiation, and pressure are unified in a single geometric language: the curvature of spacetime.
The field equations
The heart of the theory is contained in the field equations:
Gμν = (8πG/c4) Tμν
where the energy-momentum tensor contains crucial information: pressure is also a source of gravity.
This apparently technical observation becomes devastating when applied to gravitational collapse. During the collapse of a massive star, matter is compressed, density increases, internal pressure rises. But in General Relativity, greater pressure means more gravitational curvature, and therefore even more collapse.
A feedback loop is created: compression increases gravity, gravity increases compression.
The singularity paradox
If we take the theory in its "pure" form, without other physical ingredients, the result is clear: a star that exceeds a certain mass limit has no way to stop. Contraction continues, density diverges, the radius tends to zero, and the geometry of spacetime collapses into a singularity.
Yet the sky tells a different story. The black holes we observe do not instantly collapse until they disappear; they exist as stable objects, persist for billions of years, grow through orderly accretion, merge in binary systems producing gravitational waves with regular and repeatable waveforms.
Images of regions near the horizon show geometrically well-defined structures, bright rings, coherent shadows. In other words: the universe is populated by black holes that behave as stationary structures, not as objects in terminal collapse.
The missing ingredients
The thesis we will develop is opposite to the traditional one: the paradox does not arise from an intrinsic defect of General Relativity, but from the fact that Einstein could not know, in 1916, two fundamental ingredients:
- The Pauli Exclusion Principle (1925): two identical fermions cannot occupy the same quantum state
- The neutron (1932, Chadwick): a massive, electrically neutral fermion that can be compressed to very high densities
In 1916, physics had an incomplete view of matter. Atoms consisted of protons and electrons; the atomic nucleus, in its internal complexity, was still largely unexplored territory.
Degeneracy pressure
The Pauli Principle has implications far beyond spectroscopy. If we try to compress a system of fermions, we reach a point where all the "low" quantum states are occupied, and particles can no longer be pushed to lower energy levels.
Matter opposes a purely quantum resistance to compression: degeneracy pressure.
pdeg ∝ ρ5/3
At nuclear densities (ρnuc ~ 1017 kg/m3), neutron degeneracy pressure reaches almost inconceivable values:
pdeg ~ 1033 Pa
Every further attempt at compression produces an increasingly steep pressure increase, building a true quantum wall against collapse.
The arrest radius
When neutron degeneracy pressure becomes comparable to average gravitational pressure, physics changes sign: the term that previously fueled collapse becomes what stops it.
The result is the emergence of an arrest radius: a value of the stellar radius at which degeneracy pressure balances gravity. For a star of a few solar masses, this radius is on the order of a few kilometers.
And here the surprise emerges: the quantum arrest radius turns out to be comparable to the Schwarzschild radius:
Rs = 2GM/c2
This is no trivial coincidence. It is a signature of the unity of physics: nature has "fitted" the fundamental constants so that the quantum mechanism that stops collapse comes into play exactly where light can no longer escape.
The dark star
The gravitational collapse scenario changes face. The star does not continue to contract until it disappears into a singularity. It stops when matter reaches a state of maximum compression compatible with the Exclusion Principle.
The final state is not a mathematical point of infinite density, but a physical structure: a configuration of quantum matter at nuclear density, supported by neutron degeneracy pressure.
From the outside, this structure is completely obscured by the event horizon. But inside there is no singularity: there is a dark star, a stable structure of quantum matter frozen in a state of maximum compression.
Profound consequences
This reinterpretation has profound consequences:
- Resolves the singularity paradox: instead of a "hole" in physics, we have a structure of real matter
- Addresses the information paradox: information remains stored in the degrees of freedom of matter at the inner surface
- Is compatible with observations: LIGO/Virgo mergers, James Webb supermassive black holes, Event Horizon Telescope images
The lesson
Einstein's General Relativity is not "wrong": it is a correct theory, but formulated at a time when the quantum structure of matter had not yet been revealed.
Applied alone, without the Exclusion Principle and without the neutron, it inevitably leads to the idea of singularities. Integrated with quantum mechanics and fermion physics, it shows another possibility: collapse stops and what we call a "black hole" is, in reality, a dark star obscured by gravity.
The singularity is not a necessity of nature, it is the product of a correct theory applied without a crucial piece of knowledge.