Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology

Symmetry Integrability and Geometry-Methods and Applications

article

Yuri I. Manin¹  Matilde Marcolli² 

[1] Max-Planck-Institut für Mathematik;California Institute of Technology

关键词: Big Bang cosmology;    algebro-geometric blow-ups;    cyclic cosmology;    Mixmaster cosmologies;    modular curves;   

DOI  :  10.3842/SIGMA.2014.073

来源: National Academy of Science of Ukraine

【 摘 要 】

We introduce some algebraic geometric models in cosmology related to the ''boundaries'' of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point $x$. This creates a boundary which consists of the projective space of tangent directions to $x$ and possibly of the light cone of $x$. We argue that time on the boundary undergoes the Wick rotation and becomes purely imaginary. The Mixmaster (Bianchi IX) model of the early history of the universe is neatly explained in this picture by postulating that the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one. Penrose's idea to see the Big Bang as a sign of crossover from ''the end of previous aeon'' of the expanding and cooling Universe to the ''beginning of the next aeon'' is interpreted as an identification of a natural boundary of Minkowski space at infinity with the Big Bang boundary.

【 授权许可】

Unknown