Within the universe, the singularities Space-time singularities appear as cosmic anomalies. Examples of these are black holes or the Big Bang, points at which current models of physics break down and reality becomes an enigmatic puzzle. Their existence is expected, according to the mathematical predictions of the theorems that Roger Penrose enunciated in the 1960s and which were recognized with the Nobel Prize in 2020. However, little is known about their dynamics, that is, about the way in which space-time behaves near singularities.
Modern cosmology considers that a good approximation of the present-day universe on a large scale is given by the solution found by Alexander Friedmann in 1922 to the equations of Albert Einstein’s theory of relativity. According to Friedmann’s solution, space is not standing still, but is expanding over time – which was confirmed experimentally by Edwin Hubble in 1929. From the expansion of the universe, the existence of a singularity follows: if we turn back the clock and travel into the past, space will contract to a point (the Big Bang).
In his solution, Friedman assumed that space is homogeneous – that is, it behaves the same at all points – and isotropic – that it behaves the same in all directions. However, in the Big Bang this might not be true and, therefore, Friedmann’s solutions would not serve to explain what happens near this singularity.
Measurements of the cosmic background radiation indicate that shortly after the Big Bang, the universe expanded almost equally in all spatial directions. But A little asymmetry in isotropy could generate a different behavior close to the singularity marked by Friedmann’s solutions. In particular, unlike what happens in these, spatial dimensions could have different roles in shaping our destiny.
To answer these questions, in the 1970s Vladimir Belinsky, Isaak Khalatnikov, and Evgeny Lifshitz conjectured that shortly after the explosive birth of the universe, it went through a chaotic phase of development. Chaos, in this cosmic context, refers to fascinating complexity, rather than disorder. According to this hypothesis – called BKL–, chaotic fluctuations offer puzzling patterns and intricate mathematical structures that ultimately shaped our present-day cosmos. More than 50 years later, this mathematical question is still far from being answered.
The BKL conjecture suggests that singularities mostly have three characteristics. First, they are local, i.e. the particles are decouple from each other and each evolves independently towards the singularity. Thus, Einstein’s equations become ordinary differential equations.
Second, singularities are dominated by vacuum, meaning that for most types of matter, their effect on the dynamics of spacetime geometry is negligible near the singularity. In the words of John Wheeler, “matter doesn’t matter” near a singularity.
Finally, singularities are oscillatory and chaotic. At the same time, Charles Misner proposed a model to analyze these chaotic oscillations, which was coined with the term mixmaster –referring to an electric kitchen mixer for making dough–. This model describes a cosmological dancein which every spatial direction is converted into expansion and contraction, in the same way that pizza dough is made: the dough is kneaded, stretched and folded iteratively, changing direction slightly each time this procedure is repeated. In both cosmology and pizza making, a small modification of the initial conditions can lead to very complex and intricate results.
There are still many unanswered questions in this cosmic narrative about the birth of the universe. Acquiring experimental data and validating theories of gravity, particularly in the realm of extreme gravitational fields, remains very difficult. Thus, in the absence of direct observations, solid mathematical frameworks become crucial guides toward plausible and meaningful theories. The Penrose singularity theorems and the BKL conjecture reveal a rich picture of cosmic evolution that continues to provoke wonder and scientific curiosity, offering a different point of view on the intricate dynamics of the past, present, and future of the universe.
Phillipo Lappicy is a Marie Curie Fellow (Una4Career) at the Complutense University of Madrid
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
Editing and coordination: Agatha Timon Garcia-Longoria. She is the coordinator of the Mathematical Culture Unit of the Institute of Mathematical Sciences (ICMAT)
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