Historical research guide · 0984 · 2009
Investigation into Compactified Dimensions: Casimir Energies and Phenomenological Aspects
Richard K. Obousy
In one minute
Richard Obousy’s Baylor doctoral thesis takes one number seriously: the energy of empty space depends on how big the universe’s curled-up extra dimensions are, and it depends on it steeply — as the inverse fourth power of their radius. Most of the thesis is careful housekeeping. He reviews what the quantum vacuum is, how the cosmological constant entered Einstein’s equations, how the Casimir force is calculated and how it was measured, and what extra dimensions are for. Then he does his own calculation: what Casimir energy a scalar field acquires when it is coupled to a vector field pointing along the fifth dimension, and whether the resulting energy can hold that dimension at a fixed size. Chapter seven is where it turns into hardware. If the radius of an extra dimension sets the local expansion rate of space, then a technology able to shrink or grow it locally would expand space behind a ship and contract it ahead — a warp bubble, arrived at from string theory rather than from exotic matter.
Connection
Chapter 2 gets its cleanest statement of the vacuum as an adjustable quantity rather than a fixed background: change one geometric parameter and the energy of empty space changes as its fourth power. Chapter 4 gets the mechanism that follows — a warp bubble produced by locally tuning the size of a compact dimension, with an energy budget, a speed ceiling and a named next calculation, which is the same programme the Defense Intelligence Agency later commissioned from this author.
From the source
The Casimir energy of a field living in a compactified fifth dimension goes as the inverse fourth power of that dimension’s radius, and for a massless scalar with periodic boundary conditions the vacuum energy density is minus three times the Riemann zeta function at five, over sixty-four pi squared times the radius to the fourth — so the vacuum energy density and the cosmological constant are tied to a single geometric number.Chapter 7, Equations 7.3.1 to 7.3.3; the same expression derived in Chapter 5, Equation 5.1.26
The thesis’s own derivation adds a Lorentz-violating vector field with a non-zero expectation value only along the fifth dimension, coupled to a scalar; this splits the Kaluza-Klein tower unevenly and multiplies the Casimir energy by a single tuning parameter, but the contribution stays attractive and shrinks the extra dimension, so scalars coupled this way cannot stabilise it on their own.Chapter 5, Sections 5.1.1 to 5.1.3; Equations 5.1.1, 5.1.2 and 5.1.26
Populating the bulk with the Standard Model fields and the Higgs fails to stabilise the extra dimension across the whole experimentally allowed range of Higgs masses, because the fermions’ positive Casimir energy is not balanced; adding one light anti-periodic exotic fermion — justified as a possible sterile bulk neutrino — does produce a stable minimum, sitting at negative energy density in an anti-de Sitter spacetime, and a minimum at positive energy density needs at least two additional exotic fields.Chapter 6, Sections 6.1.3 to 6.1.7; Section 6.5, Discussion
Solving the vacuum Einstein equations for a five-dimensional toy universe with empty space and no cosmological constant gives the Hubble rate of the three large dimensions equal to minus the Hubble rate of the compact one — the contraction of the extra dimension inflates the others, so even in pure general relativity the physics of the compact space governs the expansion rate of the space we live in.Chapter 7, Section 7.3.1, Equations 7.3.6 to 7.3.11
A sufficiently advanced technology able to increase or decrease the radius of the extra dimension locally would thereby adjust the local expansion and contraction of spacetime, creating a warp bubble; the craft always stays inside its own light cone, using expansion behind it to recede from an object and contraction ahead to approach one, at any desired speed.Chapter 7, Section 7.3.1, Equation 7.3.5 and the paragraph following it
The thesis’s own back-of-the-envelope budget: making space expand locally at light speed needs the Hubble rate raised by ten to the twenty-six, the extra-dimensional radius shrunk by a factor of ten trillion, a local vacuum energy density near ten to the forty-two joules per cubic metre and about ten to the forty-five joules for a thousand cubic metre craft — Jupiter’s mass-energy in antimatter, dropping dramatically for a thin shell — while the Planck-length floor on the radius sets a ceiling near ten to the thirty-two times light speed; how to manipulate an extra dimension locally, perhaps by locally changing string tension or countering winding modes, is named as the vital next piece of research.Chapter 7, Sections 7.3.2 to 7.3.4, Equations 7.3.12 to 7.3.21
The way in
https://arxiv.org/abs/0901.3640How to cite it
Richard K. Obousy (2009) Investigation into Compactified Dimensions: Casimir Energies and Phenomenological Aspects arXiv:0901.3640