An old house. The facade is freshly painted, the stairs don't creak. A pretty picture. But try to move in heavy furniture, and the floorboards give way. The beauty of the facade says nothing about the strength of the floors.
It's much the same with black holes without singularities. For decades, physicists have drawn their static portraits — with a smooth, finite center instead of a collapse to a point. Such "regular" black holes are like a house with a perfect foundation on paper. But what happens when real matter (dust, gas, radiation) falls into that center? The authors tested not the picture but the theory itself: can it carry matter through the center without spawning a singularity on the way?
They considered the most general class of second-order gravity theories: those where the equations conserve energy and don't breed extra solutions. In such theories, under spherical symmetry, everything is described by two functions. And the "strength" condition turned out to be surprisingly simple: these two functions must change sign differently when crossing the center — one stays the same, the other mirrors. In terms of the mass attributed to the black hole, this means the mass must pass through zero oddly at the center and decrease there cubically fast, without a point mass.
And here's where it gets interesting. The researchers applied this test to three well-known models. The Hayward hole and the Dymnikova solution pass. Their internal structure is consistent: statics and dynamics don't contradict each other. But the Bardeen black hole fails. Its beautiful smooth center is just a facade. The theory behind it can't handle the simplest motion of matter. Smoothness in a static picture guarantees nothing.
The authors didn't stop at criticism. They showed how to build whole families of "strong" theories and described an infinite class of admissible models with a de Sitter core — the kind where the center resembles an inflating universe rather than a point. This doesn't guarantee that collapse always ends without a singularity, but it provides a necessary filter. Before spending years on calculations, you can check in a minute: will the theory pass an elementary stress test? The old house with a beautiful facade won't fool you anymore.
At the center of a regular Hayward black hole, the energy density is finite and comparable to that of an atomic nucleus — about $10^{44}$ kg/m³. That's unimaginably dense, but still not infinity.
In Greg Egan's novel "The Light of Distant Stars," black holes with quantum cores are explored, where matter doesn't collapse to a point but transforms — much like in theories with a "strong" center.