The space-time oddities of modern physics may have just been taken to a new level of odd, as researchers have revealed that space and time can be used to form a variety of structures that may then be able to become tiny black holes.
The unusual discovery, reported by researchers from Vienna and Frankfurt, presents a new formula for this unusual effect, which they claim can be used to create a crystal-like structure resulting from spacetime self-organization due to a process physicists call critical collapse.
The findings, now reported in Physical Review Letters, reveal the first successful description of this bizarre phenomenon using a novel mathematical trick, which allowed researchers to derive a precise formula for the phenomenon.
Baby Black Holes
Although black holes are often envisaged as large physical structures that result from the powerful conditions involving stellar deaths, not all of them are so monstrous.
In theory, tiny black holes can also exist, emerging from very minuscule critical states where only the smallest amount of energy is introduced. These states, according to physicists, are believed to have once existed immediately after the genesis of our universe, known as the Big Bang, at which time a disorderly blend of particles persisted in the newborn cosmos—conditions that would have been ripe for the creation of what are known as primordial black holes.
These structures are already theoretically verified through computer simulations, although in their recent research, the Goethe University Frankfurt and TU Wien collaboration has now taken the study of these tiny cosmic monsters to a new level by deriving a mathematical formula to confirm longstanding theories about these tiny black holes.
Curving Spacetime at Smaller Scale
According to Professor Daniel Grumiller, a researcher at TU Wien, even the smallest events can sometimes trigger major changes.
“Take liquid water at zero degrees Celsius, for example,” Grumiller recently said in a statement. “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,” he says.
Why is this significant? A primary reason involves Einstein’s revolutionary ideas about gravity, in which a similar effect occurs, albeit involving space and time. Specifically, Einstein’s theory holds that particles that change locations can cause changes to the surrounding spacetime.
Christian Ecker of the Institute for Theoretical Physics at Goethe University Frankfurt observes that spacetime is warped more strongly in proportion to the size of objects (in other words, those possessing greater mass).
“Large objects such as stars curve spacetime strongly,” Ecker notes. “For example, we can observe this when light rays are deflected by massive stars.” However, massive celestial objects aren’t the only ones that can curve spacetime.
“Smaller masses also produce spacetime curvature, just to a lesser extent,” Ecker explains.
Patterns in Space and Time
According to the researchers, repeating patterns emerge in space and time because of spacetime curvature, in which spacetime can self-organize into a regular, repetitive structure.
This structural form, which they liken to being a sort of “spacetime crystal,” results from a process known as critical collapse.
Grumiller calls the resulting spacetime “crystal,” a “very peculiar and fascinating object,” which he says can be thought of as “a kind of intermediate state, an unstable point that can evolve in two different directions.” Following its formation, Grumiller says that the crystal may then simply dissipate, “leaving behind ordinary spacetime filled with freely moving particles.”
That is, unless an energy input is introduced.
“If a tiny amount of energy is added, the evolution takes a completely different path,” Grumiller says, whereby “the inconspicuous spacetime crystal turns into a black hole.”
Simulating Primordial Black Holes
According to Grumiller and his colleague, Christian Ecker, deriving accurate formulas for such phenomena has proven especially difficult over the years. However, Ecker says they were able to overcome this challenge by instituting a novel trick of mathematics.
“Our universe has four dimensions—three dimensions of space and one dimension of time,” Ecker recently said. “But in principle, nothing prevents us from writing down physical equations for a larger number of dimensions—five dimensions, forty-two dimensions, or even infinitely many.”
Despite the expectation that such conditions might cause theoretical interpretations to become very complicated, the team was able to show that the opposite can be the case, with some questions physicists would normally deem to be extremely complex actually being reduced to relatively simple outcomes.
The team says they hope to explore the possibility that their mathematical formula might be reinterpreted for contexts involving fewer dimensions, which would allow the current models, which relate to the possibility of an infinite number of dimensions, to be scaled back to four-dimensional applications.
So far, doing so has allowed the team to explore four-dimensional universal qualities by taking what one might liken to being a shortcut through a sort of theoretical universe consisting of many dimensions. However, for now, the team’s findings are already proving very promising.
“Our technique turns out to be remarkably stable,” according to Florian Ecker, also with TU Wien.
“Depending on the desired precision, we can systematically improve our formulas using additional approximation methods,” Ecker added. “This gives us a new method for studying black-hole-related phenomena that could previously not be analyzed analytically.”
The team’s recent paper, “Analytic Discrete Self-Similar Solutions of Einstein-Klein-Gordon at Large 𝐷,” appeared in Physical Review Letters on May 12, 2026.
Micah Hanks is the Editor-in-Chief and Co-Founder of The Debrief. A longtime reporter on science, defense, and technology with a focus on space and astronomy, he can be reached at micah@thedebrief.org. Follow him on X @MicahHanks, and at micahhanks.com.