Ordered structures that never repeat can exist in the merged space and time of Einstein’s relativity
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The stunning materials called quasicrystals have an ordered structure, but without a regular repeating pattern. They were found in meteorites and the debris of the first atomic bomb test. Scientists have now discovered that they can theoretically inhabit an even stranger realm: spacetime, the mixture of time and space from Einstein’s theory of special relativity.
Instead of existing in two or three spatial dimensions, the structures of these quasicrystals connect space and timephysicists report in an article submitted January 12 to arXiv.org. Although quasicrystals are theoretical, researchers suggest that such space-time quasicrystals could appear in nature, perhaps even underlying the structure of the universe.
A crystal is a repeating structure. If you make a copy of a crystal and drag it around itself, you can find places where the patterns match perfectly. You can imagine doing the same thing with your bathroom floor tiles or wallpaper patterns. But quasicrystals, despite their seemingly ordered structure, do not repeat as regularly.
Crystals and quasicrystals are mathematical concepts that also appear in the real world, usually in two- or three-dimensional materials. It was not obvious that space-time quasicrystals could exist. “I had a feeling that it probably wouldn’t be possible to create a real space-time quasicrystal,” says theoretical physicist Felix Flicker of the University of Bristol in England. But, he says, that’s exactly what the researchers seem to have done. “The things they came up with are… the most elegant things one could have in space-time as a combined entity.”
Despite the lack of repetition of quasicrystals, their order means that their general characteristics are similar in different locations. An ant sitting atop part of a quasicrystal would see a structure similar to that seen by an ant in a different location. But the different space-time domains are another matter.
Space-time obeys a rule known as Lorentz symmetry. Lorentz symmetry means that something remains unchanged whether you are sitting still or moving at close to the speed of light. For example, the laws of physics respect Lorentz symmetry: they do not change for rapidly moving observers. Lorentz symmetry also does not apply to previously known quasicrystals, nor to normal crystals: a stationary ant would observe a different structure from that of an ant close to the speed of light. In relativity, observers traveling at high speed observe an apparent shortening of objects, which distorts the structure of materials.
But the new space-time quasicrystals obey Lorentz symmetry. They would appear the same to a stationary ant as to an ant on a high-speed rocket. The researchers mathematically formulated their quasicrystals by taking a four-dimensional slice through a grid of higher-dimensional points and projecting those points onto the slice. The slice has a slope which is an irrational number – a number which cannot be written as a fraction of two integers, like pi. The irrational slope means that the slice never directly intersects the grid points, which helps produce a structure that never repeats.
Quasicrystals are a mathematical concept that appears in the structure of real materials, but this concept could appear elsewhere. “The space-time we live in could be a quasi-crystal,” says Sotiris Mygdalas of the Perimeter Institute in Waterloo, Canada, co-author of the study.
Space-time quasicrystals could be relevant to some theories of quantum gravity that propose that, on very small scales, space-time is divided into individual points, Mygdalas says. The structure of quasicrystals could provide a framework for breaking up space-time while respecting Lorentz symmetry.
Researchers are also investigating potential applications for string theorywhich describes fundamental particles as tiny vibrating strings and suggests that the universe may have 10 dimensions. Since the universe we experience has only three dimensions of space and one of time, proponents of string theory generally suggest that the extra dimensions are so small that we cannot interact with them. Alternatively, quasicrystals suggest a way in which all 10 dimensions could be curled up, while still allowing the seemingly infinite space and time we experience to exist. This endless space and time could be constructed from curved space if one took an irrationally tilted slice of it, the same way researchers designed their mathematical quasicrystals.
There is still work to be done to see if these ideas come to fruition. The authors call them “admittedly half-baked” in their article.
The allure of a space-time quasicrystal, however, exists regardless. “It’s beautiful mathematics,” says theoretical physicist Gregory Moore of Rutgers in New Brunswick, New Jersey, who was not involved in the work. “Physics is very highly speculative.”
