Conor Rowan

Date of Award

2019

First Advisor

Eric Kramer

Second Advisor

Mike Bergman

Abstract

The number of spatial dimensions clearly manifests in the laws of physics, and generalizing these laws to different dimensions gives hints as to why we live in 3 + 1 spacetime. Spacetime is a notion that emerged with relativity in the early 20th century and is a way of unifying space and time where a N + D spacetime is a world with N spatial dimensions and D temporal ones. For the sake of sanity, we assume throughout that D=1 and N is variable given that a world with multiple time dimensions is an even greater challenge to think and talk about. Dimensionality implicitly in uences the form of force laws for gravity and electromagnetism and changing the rate at which these forces vary with distance has far-reaching consequences. In different dimensions, the stability of circular orbits is no longer a safe assumption, nor is our \traditional" model of 3d quantum mechanics, which describes the behavior of atoms and their electrons. Assuming that a prospective dimensionality must be a positive whole number, we would expect it to be very large if picked at random. After all, a number chosen randomly between 1 and 1,000,000 is less than 100 only 0.01 percent of the time, and this is with an artificial upper limit. Intuitively, a world with this many dimensions seems chaotic and improbable{the Anthropic Cosmological Principle tells us that the world we observe must be conducive to the development of intelligent life, and perhaps our lowdimensional universe is exactly that. Thus, in addition to discussing a few interesting anecdotes and purely mathematical results, my thesis explores the various ways in which the undergraduate physics curriculum would differ in higher/lower dimensional space, offering insight into the nature of different-dimensional universes. Topics such as generalized waves and geometries are touched on, but the majority of the thesis is concerned with orbits, rocketry and quantum mechanics. It is shown that the theories of both Newton and Einstein predict the impossibility of stable orbits (circular or otherwise) in higher dimensions. The equation of a selfpowered two-dimensional rocket travelling in a variable gravitational field is formulated, and the feasibility of 2d rocketry is investigated with a computer simulation. In essence, this chapter analyzes the likelihood of space travel/exploration in a world where gravity is much stronger. Lastly, the arbitrary-dimensional atom is studied via the behavior of its electrons' orbitals and through generalized solutions to the governing equation of quantum mechanics, the Schrodinger equation. Many of these findings point to the convenience, if not necessity of 3 +1 dimensional spacetime for the evolution of advanced life.

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