**Chapters of 120 page dissertation on Casimir effect**

1. we review the nature of the quantum vacuum and discuss the diff erent contributions to the vacuum energy density arising

from diff erent sectors of the standard model.

2. we discuss cosmology and the introduction of the cosmological constant into Einstein’s field equations.

3. we explore the Casimir e ffect and study a number of mathematical techniques used to obtain a fi nite physical result for the Casimir energy. We also review the experiments that have verifi ed the Casimir force.

4. we discuss the introduction of extra dimensions into physics. We begin by reviewing Kaluza Klein theory, and then discuss three popular higher dimensional models: bosonic string theory, large extra dimensions and warped extra dimensions.

5. devoted to an original derivation of the Casimir energy we derived for the scenario of a higher dimensional vector fi eld coupled to a scalar field in thef fifth dimension.

6. we explore a range of vacuum scenarios and discuss research we have performed regarding moduli stability

7. explores a novel approach to spacecraft propulsion we have proposed based on the idea of manipulating the extra dimensions of string/M theory.

8. we discuss some issues in heterotic string phenomenology derived from the free fermionic approach

We have investigated the possibility of moduli stability using the Casimir effect in RS1. We have calculated the one loop corrections arising from a massive scalar fi eld with periodic boundary conditions in the compactifi ed extra dimension by applying the Schwinger proper time technique and exploiting the Jacobi Theta function. We have populated the bulk with numerous fields in an attempt to uncover stabilization scenarios with an emphasis on phenomenologically motivated fi eld content. Extending on our work in Chapter 5, we have explored the implications of the existence of a fi ve-dimensional vector field with a VEV in the compact dimension coupling to one of the scalar fi elds, and noted its relevance as a tuning parameter.

We have demonstrated that the Casimir energy of the SM fi elds, in conjunction with the Higgs eld, cannot provide the necessary potential to stabilize the extra dimension. The fermionic nature of the SM fi elds contribute a positive Casimir energy which is not balanced by the Higgs fi eld when we include the three lepton generations (and their anti-particles, excepting the left-handed antineutrinos) and the six quarks with three color degrees of freedom (and their anti-particles). We have also investigated the possibility of stability for a range of Higgs masses based on experimental lower limits and theoretical upper limits, and found the same result.

We have investigated the possibility of adding an exotic massive anti-periodic fermionic field to this fi eld setup, and discovered that a light field (m=0.02 in normalized units) is su fficient to generate a stable minimum of the potential. The minimum is located at a negative energy density, which corresponds to an AdS spacetime. Reduction of the mass of the exotic fi eld causes the minimum to become deeper. Our justi cation for the addition of this field is the possibility of the existence of a light sterile bulk neutrino.

Next, we built on on previous work by considering the e ects on the Casimir energy of a scalar field coupled to a Lorentz violating vector field. This f ield, which is completely charactered by the parameter , allows for fi ne tuning of the Casimir energy and the stabilization radius. Fine tuning is a generic feature of stabilization schemes, and in this model the simple addition of a vector field in only the fifth dimension creates additional freedom for the stabilization schemes. We discovered that no stable minimum of the potential can be found with either a single periodic or antiperiodic massless scalar field coupled to a Lorentz violating vector in the case of a Higgs vacuum. However, when the SM fi elds are included a periodic massless scalar fi eld coupled to a Lorentz violating vector can lead to a stable minimum, but this is dependent on the parameter . Our motivations for studying the phenomenology of Lorentz violating fields stem from the recent surge of activity regarding the possibility of Lorentz invariance violations and the potential role of Lorentz violating

fi elds in cosmology.

We also outlined a higher dimensional field con guration, which creates a positive energy density minimum of the Casimir energy. We find that at least two additional exotic fi elds are required. Our example highlights a possible connection

between dark energy, the heirachy problem, and additional bulk fi elds. The capability of this model to explain so many apparantly unrelated phenomenon under the common framework of extra dimensional boundary conditions makes this model particularly appealing.

Since releasing our results, we have received interest from a group at Fermilab performing similar research. They have discovered that if standard model fields (without the Higgs boson) populate the bulk, the fermion condensation can also

stabilize the two branes in the RS model. From recent correspondence we have learned that they felt that our work was a more natural way to stabilize the extra dimension, and has given them new thoughts with regards to the issue.

There has also been recent interest in the literature relating dark energy to Casimir energy, and for this reason we have briefly reviewed cosmological aspects of extra dimensions by considering an anisotropic cosmology. Using simple arguments, we have found the relation between the Casimir energy density and the pressure in both large and compact dimensions necessary for accelerated cosmological expansion.

**Emerging Possibilities in Spacecraft Propulsion**

In Chapter 7, we take an excursion from pure research and explore some of the exciting possibilities that are opened up when we introduce higher dimensions into physics. Speci cally, we discuss the concept of field propulsion, which is a hypothetical system of propelling spacecraft beyond the conventional rocket technology. Recently we published a paper outlining the physics of such a device, and performed calculations regarding the energy requirements to create a warping of spacetime, and also calculations redefining maximum obtainable speeds. We also published a layman’s version of this idea, which subsequently received an reasonable amount of attention; this reflects a public interest in this area. We feel that pursuing such exciting areas is an excellent way to attract fresh young minds to the fi eld of physics, and also to explore how physics places ultimate limitations on technological developments.

**Warp Drive**

An alternative to the wormhole idea is the warp drive, which involves creating an asymmetric bubble of locally contracting/expanding spacetime around a spacecraft, eff ectively stretching and compressing space itself. Over the last decade, there has been a respectable level of scientifi c interest regarding the concept of the `warp drive’. Recently, we explored a novel approach to generate the asymmetric bubble of spacetime required to create such a warp drive, which we will summarize in this section.

As discussed throughout this dissertation, certain classes of higher dimensional models suggest that the Casimir eff ect is a candidate for the cosmological constant. We demonstrate that a su fficiently advanced civilization could, in principal, manipulate the radius of the extra dimension to locally adjust the value of the cosmological constant. This adjustment could be tuned to generate an expansion/contraction of spacetime around a spacecraft, creating an exotic form of fi eld-propulsion. Due to the fact that spacetime expansion itself is not restricted by relativity, a faster-than-light `warp drive’ could be created. Calculations of the energy requirements of such a drive are performed and an `ultimate’ speed limit, based on the Planckian limits on the size of the extra dimensions, is found.

**The Physics of Warp Drives**

Numerous papers discussing the idea of warp drives have emerged in the literature in recent years. The basic idea is to formulate a solution to Einstein’s equations whereby a warp bubble is driven by a local expansion of spacetime behind

the bubble and a contraction ahead of the bubble. One common feature of these papers is that their physical foundation is the GR. An element missing from all the papers is that there is little or no suggestion as to how such a warp bubble may be

created.

The aim of this section is not to discuss the plausibility of warp drive, the questions associated with violation of the null energy condition, or issues regarding causality. The aim of this paper is to suggest that a warp bubble could be generated using ideas and mathematics from quantum fi eld theory, and to hypothesize how such a bubble could be created by a su fficiently advanced technology. By associating the cosmological constant with the Casimir Energy due to the KK modes of vacuum fluctuations in higher dimensions, especially in the context of M-theory derived or inspired models, it is possible to form a relationship between and the radius of the compact extra dimension.

Result indicates that a suffi ciently advanced technology with the ability to locally increase or decrease the radius of the extra dimension would be able to locally adjust the expansion and contraction of spacetime, creating the hypothetical warp bubble discussed earlier. A spacecraft with the ability to create such a bubble will always move inside its own local light-cone. However, the ship could utilize the expansion of spacetime behind the ship to move away from some object at any desired speed, or equivalently, to contract the space-time in front of the ship to approach any object. The possibility of the size of the compact geometry vary depending on the location in four dimensional spacetime has been explored in the context of string theory, but never from the perspective of propulsion technology.

**Energy Requirements**

In this section, we perform some elementary calculations to determine how much energy would be required to reach superluminal speeds. We also determine an absolute speed limit based on fundamental physical limitations.

The currently accepted value for the Hubble constant is 70 km/sec/Mpsc. A straightforward conversion into SI units gives H = 2.17 X 10^18 m/s/m. This tells us that one meter of space would expand to two meters of space if one were prepared to wait two billion billion seconds, or sixty five billion years. The fundamental idea behind the warp drive presented here is to increase Hubble’s constant locally around the spaceship, such that space no longer expands at such a sedentary rate, but locally expands at an arbitrarily fast velocity. For example, if we want space to locally expand at the speed of light, a simple calculation shows us by what factor we would need to increase H.

10^26 times

Assuming some arbitrarily advanced civilization was able to create such an e ffect, we might postulate that this civilization could utilize the most e fficient method of energy production : matter-antimatter annihilation. (a jupiter mass converted into energy) This would drop dramatically if we assumed a thin-shell of modfii ed spacetime instead of bubble encompassing the volume of the craft.

Max speed = 10^32 times light speed

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Brian Wang is a Futurist Thought Leader and a popular Science blogger with 1 million readers per month. His blog Nextbigfuture.com is ranked #1 Science News Blog. It covers many disruptive technology and trends including Space, Robotics, Artificial Intelligence, Medicine, Anti-aging Biotechnology, and Nanotechnology.

Known for identifying cutting edge technologies, he is currently a Co-Founder of a startup and fundraiser for high potential early-stage companies. He is the Head of Research for Allocations for deep technology investments and an Angel Investor at Space Angels.

A frequent speaker at corporations, he has been a TEDx speaker, a Singularity University speaker and guest at numerous interviews for radio and podcasts. He is open to public speaking and advising engagements.