Robert Brandenberger, a physicist at McGill University who was not involved in the study, said the new research “sets a new standard of accuracy in analyzing” the mathematics of the beginning of time. In some cases, what initially appears to be a singularity – a point in spacetime where mathematical descriptions lose their meaning – may actually be just an illusion.

## Classification of singularities

The central issue facing Jeshnjani, Ling and Quentin is whether there is a point before inflation at which the laws of gravity break down in the singularity. The simplest example of a mathematical singularity is what happens to the function 1/*s* like *s* Zero approach: The function takes a number *s* As an input, another number outputs. like *s* Getting smaller and smaller 1/*s* It gets bigger and bigger, approaching infinity. if *s* If it is zero, the function is no longer well defined: it cannot be relied upon as a description of reality.

However, sometimes, mathematicians can get around the singularity. For example, consider the Prime Meridian, which passes through Greenwich, England, at zero longitude. If you had a function of 1/longitude, it would get wild at Greenwich. But there’s actually nothing physically special about the outskirts of London: you could easily redefine zero longitude to pass through somewhere else on Earth, and then your function would behave quite normally when approaching the Royal Observatory at Greenwich.

Something similar happens at the limits of mathematical models of black holes. The equations describing non-rotating spherical black holes, developed by physicist Karl Schwarzschild in 1916, contain a term whose denominator reaches zero at the black hole’s event horizon, the surface surrounding the black hole beyond which nothing can escape. This led physicists to believe that the event horizon was a physical singularity. But eight years later, astronomer Arthur Eddington showed that if a different set of coordinates is used, the singularity disappears. Like the prime meridian, the event horizon is an illusion: a mathematical artifact called a coordinate singularity, which arises only because of the choice of coordinates.

In contrast, at the center of a black hole, the density and curvature reach infinity in a way that cannot be eliminated using a different coordinate system. The laws of general relativity are starting to spew nonsense. This is called the curvature singularity. It means that something is happening that is beyond the ability of current physical and mathematical theories to describe.

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