Black Holes, Quantum Cosmology and Negative Dimensions

Multidimensional spaces are widely used in mathematics and physics. This talk will discuss the emergence of negative dimensions in black hole thermodynamics and quantum cosmology. Most black holes violate the third law of thermodynamics. The entropy of a Schwarzschild black hole is inversely proportional to the square of the temperature, and tends to infinity rather than zero when the temperature goes to zero. We are looking for quantum statistical models with such exotic thermodynamic behavior. It is shown that a Schwarzschild black hole in $D=4$ spacetime dimensions corresponds to a Bose gas in space with $d=-4$ negative spatial dimensions. The Riemann zeta function is used to determine the entropy of a Bose gas in negative dimensions. If we assume that the creation and evolution of the universe obey the second law of thermodynamics, and the "pre-Big Bang" state of the universe was a sphere of negative dimension, then in the semiclassical approximation to the Wheeler-DeWitt equation, the entropy reaches its maximum value at the de Sitter spacetime with the observed dimension $D = 4$ for de Sitter radius $l=0.57$.