Department of Mathematics

9-21-2018 – Photos of Jared Speck, Assoc. Professor in the dept of Mathematics. (Vanderbilt University / Steve Green)

Prof. Jared Speck, along with Grigorios Fournodavlos (University of Crete) and Igor Rodnianski (Princeton University) recently published a breakthrough paper about the stability of the big bang theory. Their paper answers a 50-year-old conjecture made by physics.  It was published in the highly-prestigious Journal of the American Mathematical Society, which is considered as one of the top few general-interest mathematical journals.

The theory of the big bang has become part of the canon of cosmology. One limitation of the theory is that it was built upon solutions that have symmetry. For example, most textbook solutions describe a big bang-containing universe that has no spatial variation at all, meaning that at each fixed time, the universe looks the same at every point and in every direction. Such symmetric solutions are relatively easy to write down an analyze, but they are at best an approximation to our physical universe.

In their paper, Prof. Speck and his collaborators derived a rigorous mathematical theory of the big bang that does not require any symmetry assumptions. Their main result is a mathematical theorem showing that the big bang is not an artifact of symmetry, but rather can occur in physically realistic solutions that do not have any symmetry properties.

More precisely, using tools from partial differential equations and geometry, they proved that the big bang is a dynamically stable prediction of general relativity. This means that if you have a single solution to Einstein’s equations with a big bang in the past, then all nearby solutions – regardless of whether they have symmetry – also must have a big bang in the past.

The proof given by Speck and his collaborators works for solutions that satisfy a certain property, called sub-criticality, which in the mid-1980’s was conjectured by physicists to imply the dynamic stability of the big bang. Their work also has implications beyond general relativity because it develops general mathematical tools that allow one to study equations in which the terms are becoming infinite.

In 1915, Einstein formulated his grand theory of gravity, general relativity. In 1927-1931, the Belgian Catholic priest Georges Lemaître and other leading scientists at the time found solutions to Einstein’s equations that describe a universe that is expanding, consistent with astronomer Edwin Hubble’s 1929 observation of the expansion of the universe. In a profound leap of logic, Lemaître posited that an expanding universe could have originated from one that was initially a single point of diameter zero. Such an initial configuration of vanishing size has come to be known as the “big bang,” where the “bang” refers to a fundamental prediction of general relativity: the curvature of space-time is infinite at the initial point. Lemaître is thus informally known as the “father of the big bang.”