Physicists from Goethe University Frankfurt and TU Wien have derived the first exact mathematical formula describing how a structured “spacetime crystal” can undergo critical collapse and transform into a microscopic black hole following a tiny energy shift, according to research published in Physical Review Letters.
The Infinite-Dimensional Breakthrough in Gravity
For over three decades, computer simulations have hinted at a precise mathematical pattern underlying the birth of microscopic black holes. Back in 1993, computational models showed that when spacetime reaches a critical threshold, black hole formation follows strict rules. Yet, analytical proof remained elusive because the equations in our familiar four dimensions of space and time are notoriously intractable.
Escaping Four-Dimensional Complexity
The Vienna and Frankfurt research team bypassed this limitation using an unexpected geometric workaround. Instead of tackling the problem in four dimensions, they solved it in infinitely many dimensions. In the limit of infinite dimensions, the mathematical complexities that typically block analytical progress simply vanish. Once the exact formula was derived in this infinite space, the team translated the solution back to our physical universe.
“Our technique turns out to be remarkably stable. This gives us a new method for studying black hole related phenomena that could previously not be analysed analytically,” noted Florian Ecker from TU Wien.
Phase Transitions on a Knife-Edge
The concept mirrors everyday thermodynamic phase transitions, such as water freezing into ice. Prof. Daniel Grumiller from TU Wien explained the mechanics of the transition, stating, “Take liquid water at zero degrees Celsius. A very small change is enough to make the water freeze. The water molecules then spontaneously arrange themselves into a regular pattern and form an ice crystal.”
According to Einstein’s theory of relativity, matter and energy dictate the curvature of spacetime. While massive stars create extreme gravitational distortions, smaller masses generate localized effects. Under specific conditions, these distortions organize into a repeating pattern—a spacetime crystal.
From Crystal to Microscopic Black Hole
This structure sits at a knife-edge. Left undisturbed, the spacetime crystal dissolves back into ordinary spacetime filled with freely moving particles. However, adding an infinitesimal amount of energy triggers critical collapse. “The inconspicuous spacetime crystal turns into a black hole,” Grumiller outlined.
Unlocking Primordial Physics
While stellar-mass black holes form via the catastrophic core collapse of dying stars, these theoretical microscopic black holes emerge from delicate critical states. Christian Ecker of Goethe University Frankfurt emphasized the scale of these gravitational formations, noting that smaller masses produce minor spacetime curvature that can compound under precise critical thresholds.
By securing an analytical framework for critical collapse, researchers now possess a robust tool for investigating primordial and microscopic black holes without relying solely on numerical approximations. The breakthrough bridges a thirty-year gap between early computational observations and rigorous theoretical physics.
