Robert Brandenberger, a physicist at McGill University who was not involved in the research, said the novel paper “sets a new standard for the rigor of analysis” of the mathematics of the beginning of time. In some cases, what at first glance appears to be a singularity – a point in space-time where mathematical descriptions lose their meaning – may actually be an illusion.
Taxonomy of singularities
The main question faced by Geshnizjani, Ling and Quintin is whether there is a point before inflation at which the laws of gravity break down into a singularity. The simplest example of a mathematical singularity is what happens to the function 1/X How X is approaching zero. The function takes a number X as input and outputs another number. How X becomes smaller and smaller, 1/X becomes larger and larger, approaching infinity. If X is zero, the function is no longer well defined: it cannot be relied upon as a description of reality.
Sometimes, however, mathematicians can bypass a singularity. For example, consider the Prime Meridian, which passes through Greenwich, England at zero longitude. If you had a 1/longitude function, you would go berserk in Greenwich. But there’s actually nothing physically special about the outskirts of London: you could easily redefine zero longitude to pass through another place on Earth, and then your function would behave completely normally as it approached the Royal Observatory, Greenwich.
Something similar happens at the boundary of mathematical models of black holes. The equations describing spherical, non-rotating black holes, developed by physicist Karl Schwarzschild in 1916, contain a term whose denominator tends to zero at the black hole’s event horizon – the surface surrounding the black hole beyond which nothing can escape. This has led physicists to believe that the event horizon is a physical singularity. However, eight years later, astronomer Arthur Eddington showed that if a different set of coordinates was used, the singularity disappeared. Like the prime meridian, the event horizon is an illusion: a mathematical artifact called a coordinate singularity that arises solely from the choice of coordinates.
In contrast, at the center of a black hole, density and curvature approach infinity in a way that cannot be eliminated by another coordinate system. The laws of general relativity start spewing gibberish. This is called the curvature singularity. This means that something is happening that cannot be described by current physical and mathematical theories.