Physics allows for black holes far smaller than the enormous objects typically found at the centers of galaxies or created when massive stars collapse. Under the right conditions, microscopic black holes could also form when spacetime enters an unusual critical state and organizes into a repeating, crystal-like pattern.
Researchers from Goethe University Frankfurt and TU Wien have now found a way to describe this process mathematically. For the first time, they derived an exact formula for the phenomenon by using an unconventional mathematical approach.
Microscopic Black Holes and Critical Collapse
Most familiar black holes are associated with violent cosmic events, including the deaths of massive stars. In theory, however, there is no strict lower size limit. Extremely small black holes could arise from special critical states in which even a tiny addition of energy can determine whether the system disperses or collapses.
Conditions like these may have existed in the early universe, shortly after the Big Bang, when matter and energy were packed into an intensely chaotic environment. Such circumstances could potentially have produced primordial black holes.
Computer simulations had already shown that these unusual critical structures should be possible. The challenge was finding a mathematical description that could reproduce what the simulations revealed. Researchers at Goethe University Frankfurt and TU Wien have now done that with a formula that can be worked out analytically, using what the scientists describe as essentially paper and pencil.
How a Tiny Change Can Trigger Collapse
“Sometimes a tiny, seemingly insignificant cause is enough to trigger a huge and dramatic change,” says Prof. Daniel Grumiller from TU Wien. “Take liquid water at zero degrees Celsius, for example. 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.”
Einstein’s theory of relativity suggests that spacetime can undergo something conceptually similar. Matter influences the geometry of spacetime, and changes in the distribution of particles can therefore alter its curvature.
“We say that spacetime is curved by mass,” explains Christian Ecker from the Institute for Theoretical Physics at Goethe University Frankfurt. “Large objects such as stars curve spacetime strongly — for example, we can observe this when light rays are deflected by massive stars. But smaller masses also produce spacetime curvature, just to a lesser extent.”
Under critical conditions, that curvature can arrange itself into a repeating pattern across space and time. The researchers describe the resulting structure as a kind of “spacetime crystal.” The process that produces this finely balanced state is known as critical collapse.
“This spacetime crystal is a very peculiar and fascinating object,” says Grumiller. “It is a kind of intermediate state, an unstable point that can evolve in two different directions. It may simply dissolve again, leaving behind ordinary spacetime filled with freely moving particles. But if a tiny amount of energy is added, the evolution takes a completely different path: the inconspicuous spacetime crystal turns into a black hole.”
A Black Hole Prediction Dating Back to 1993
Computer simulations first indicated in 1993 that black holes could form spontaneously through this kind of critical behavior. Physicists have spent decades trying to reproduce the process with mathematical equations, but deriving the necessary formulas proved exceptionally difficult.
The Vienna and Frankfurt researchers found a way around the problem by changing the number of dimensions in which they performed the calculations.
“Our universe has four dimensions — three dimensions of space and one dimension of time,” explains Christian Ecker. “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.”
It might seem that adding dimensions would make an already difficult problem even more complicated. Surprisingly, the opposite can happen. The researchers found that certain complex calculations become much easier when the number of dimensions approaches infinity.
Solving Four-Dimensional Physics With Infinite Dimensions
The team first analyzes the problem in this hypothetical setting with infinitely many dimensions. Researchers can then investigate whether the resulting solution can be translated back toward a universe with fewer dimensions, including the four-dimensional spacetime in which we live.
This mathematical detour allowed the scientists to extract information about critical collapse that had previously been extremely difficult to obtain analytically.
“Our technique turns out to be remarkably stable. Depending on the desired precision, we can systematically improve our formulas using additional approximation methods,” says Florian Ecker from TU Wien. “This gives us a new method for studying black-hole-related phenomena that could previously not be analyzed analytically.”
The approach could give physicists a new way to study black hole formation and other extreme behaviors of spacetime without relying entirely on numerical computer simulations.
