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Introduction
Over the last decade there has been theoretical interest in curiosities dubbed 'warp-drives' initiated by the 1994 paper by M. Alcubierre [Alcubierre, 1994]. These warp drives are constructs that allow some object (a spacecraft) to travel at superluminal velocities by manipulating space-time in a way such that the spacecraft never locally exceeds the speed of light however in a manner identical to the inflationary stage of the universe the spacecraft does have a relative speed, defined as change of proper spatial distance over proper spatial time, faster than the speed of light.
Interest in warp drives has not been solely confined to the realm of theoretical speculation as shown by the formation of the NASA Breakthrough Propulsion Program and the British Aerospace Project Greenglow, both of whose purpose has been to investigate the realization of these ideas.
In the spirit of the Morris, Thorne and Yurtsever paper (Morris et al, 1988) these warp drives, though highly speculative in nature, provide an unique and inspiring opportunity to ask the question `what constraints do the laws of physics place on the abilities of an arbitrarily advanced civilization'. In this paper a new and innovative mechanism to generate the necessary `Alcubierre warp bubble' is proposed.
To appreciate this new mechanism there are several areas of physics that must first be reviewed. These will be discussed in the following section.
Background
Cosmological Constant
Current observations of distant supernova (Garner, 1998) indicate that the universe is expanding at an increasing rate. This expansion is realized in Einstein’s equations as the cosmological constant term. It is, in essence, a ‘fix’ in Einstein’s Field equations that match theory with observations. The Lambda term provides a necessary addition to the gravitational field to correlate with what we see in the cosmos.
The nature of this Lambda term is a mystery. Physicists are not certain what creates the additional component to the gravitational field; we simply know that it is there. Several ideas exist as to the nature of this field, dark energy, for example, is a popular contemporary phrase for the lambda term. Efforts have been made to explain the cosmological constant using the more modern quantum field theory, created after Einstein and his gravitational equations.
Quantum Field Theory
Quantum field theory (QFT) is widely regarded as the most successful physical theory of all time. Its predictions are precise to many decimal places and no experiment has ever contradicted the predictions of the theory. With this said, it would seem only natural to try to account for the cosmological constant using QFT. The calculations are based on summing all the zero point oscillations with a Planck scale cut-off which would give us an estimate to the overall vacuum energy density, which is equivalent to calculating the cosmological constant.
The energy density predicted using standard equations is of order 1071 GeV4, which conflicts with the observed value of 10-48 GeV4. This is by far the worst predication of theoretical physics and is off by a factor of 10119. This failure of quantum theory has recently been reexamined using brane world scenarios born of string theory.
Brane World Models
QFT and Einstein’s General Relativity (GR) have long been known to be completely incompatible. When one attempts to describe gravity using QFT one obtains meaningless results. Similarly one cannot use the equations of GR to perform quantum calculations.
String theory is a modern and valiant attempt to create a quantum theory of gravity, in essence, to unify GR and QFT under one simple framework. The fundamental ‘object’ from which all matter and energy arises is a vibrating string, a strand of energy, which can acquire varying vibrational modes and patterns which manifest themselves as the bosons and fermions we are familiar with.
One highly popularized by product of this unification is the prediction that we do not inhabit a universe of three spatial and one temporal dimensions, but that there are, in fact, extra dimensions, too small to probe with today’s technology.
A particularly modern approach is to utilize the idea that a ‘brane’ is the more fundamental object than the string. In brane world models, our universe is a ‘3 brane’ which exists in a higher dimensional ‘bulk’.
Utilizing brane world scenarios it is possible to decrease the vacuum energy which was found to give a prediction of the cosmological constant far in excess of that observed. To achieve this decrease the final ingredient is something called supersymmetry.
Supersymmetry
Supersymmetry (SUSY) is a recent theoretical development of particle physics which postulates that every boson and fermion has a ‘partner’ particle, called its ‘superpartner’. The theory developed in the 1970s and was motivated mainly by a need to suppress the quadratic divergence (infinity) associated with calculating the mass squared of the Higgs boson. To date, no superpartner has been discovered. This implies that supersymmetry is somehow ‘broken’ in our universe. Consider, for example, beta decay where a single electron is ejected from a nucleus. If supersymmetry were ‘preserved’ there would be an equal chance of us seeing the supersymmetric partner of the electron, the ‘selectron’ being ejected from the nucleus. As this is clearly not the case, it is presumed that supersymmetric partners are far more massive than the standard particles that we are familiar with, and require more energy to create. This is why we say the supersymmetry is broken.
The Chen and Gu approach
To summarize what has been discussed so far, there exists a cosmological constant whose value cannot be predicted with any accuracy using quantum field theory. The Chen and Gu approach ‘tames’ the QFT calculation by an almost exact cancellation of fields in our universe by fields in the bulk (Chen et al, 2004). The basic approach is that supersymmetric contributions to the vacuum energy actually subtract from the contributions in our 3 brane, thereby reducing the overall vacuum energy calculations to a value that more closely agrees with observation. Review the fundamental elements, the Chen and Gu approach supposes that:
- we exists in a 3 brane in which supersymmetry is broken
- our 3 brane exists in a higher dimensional bulk in which supersymmetry is not broken
- our 3 brane has an inherent ‘thickness’ that ‘bulges’ into the bulk
- supersymmetric fields in the bulk almost perfectly cancel fields in the 3 brane.
The new approach to ‘warp drive’, which this paper exploits, is that the vacuum energy is, in fact a function of the size (radius) of the extra dimensions. Recall that the vacuum energy is what is also referred to as the cosmological constant in General Relativity. This vacuum energy directly effects space and is responsible for its current expansion. The main theme of this paper is that if one could locally manipulate the size of an extra dimension then, in theory, one could expand/contract space-time at will.
Technical Setup
In the context of brane-world models our universe is a (3+1) brane residing in some higher dimensional bulk [Brax et al, 2004]. It is known phenomenologically that supersymmetry is broken on our 3-brane, however is has been suggested that it may not be broken on the bulk [Giudice, 1999].
Unbroken SUSY decrees that the components of the Chiral or Gauge Multiplets share equal masses in the bulk and have the same interaction strength. However on the 3-brane SUSY breaking induces a mass square difference between them. Motivated by string theory the 3-brane has an effective thickness characterized by the string length ls. As a result the Casimir energy is non-trivial in the extra dimensional volume that encompasses the brane. This energy has the necessary features to account for the cosmological constant.
For simplicity assume an M4(Tn) manifold with extra dimensional radius a. SUSY breaking around the brane alters the Casimir energy which leads to a mass shift of the bulk fields. It is the aim of this paper to demonstrate that the mass shift is directly related to the radius of the extra dimension and, as such, a local change in the radius of the extra dimension will have the effect of altering the mass shift and thus the Casimir Energy which locally effects the value of the cosmological constant in the region effectively creating a 'bubble' of inflation/contraction.
A spacecraft with the ability to create such a bubble will always move inside its own local light-cone. However the ship can utilize the expansion of space-time 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.
Conclusions
This paper has introduced a novel method of spacecraft propulsion that could potentially revolutionize space exploration. The idea is, however extremely theoretical in that we have not yet discovered:
- Supersymmetric particles
- Extra dimensions
- Branes
These ideas are certainly at the forefront of theoretical physics and evidence for supersymmetry and extra dimensions may be revealed in the next generation of particle accelerator to open at CERN in 2007.
In the meantime, there are certainly avenues of theoretical research, based on this theory that can be followed. First and foremost, the most important aspect of this novel propulsion is the idea that a compact dimension can be manipulated, i.e. its radius can be modified by some technological means. String theory suggests that dimensions are held compact by strings wrapping around them [Cleaver et al, 1994]. If this is, indeed the case, then it may be possible to increase or decrease the string tension, or even annihilate string winding modes. This would achieve the desired effect of changing the size of the extra dimensions which would lead to propulsion under this model. It would thus be prudent to research this area further and perform calculations as to the energies required to affect an extra dimension and to try and relate this energy to the acceleration a spacecraft might experience. It is the ambition of the author to expand on these ideas in future papers.
nomenclature
a = radius of extra dimension
Evac = vacuum energy density
g = measure
G = gravitational constant
k = wave vector
m = mass of field
S = Action
R = Ricci scalar
y = location of 3 brane within bulk
M4 = Minkowski space
Tn = toroidal space
Rmn = Ricci tensor
Tmn = stress energy tensor
j = field
d = brane thickness
ru = vacuum energy density
kn = constant
G = Gamma function
dr = shift in vacuum energy density
gmn = metric tensor
L = cosmological constant
References
Alcubierre, M., The Warp Drive: Hyperfast Travel Within General Relativity, Class.Quant.Grav.11:L73-L77,1994
Brax, P., and Van de Bruck, C., andDavis, A., Brane World Cosmology, Rept.Prog.Phys.67:2183-2232,2004
Chen, P., and Gu, J., Casimir effct in a supersymmetry-breaking brane world as dark energy. By SLAC- PUB-10772, Sep 2004.
Cherednikov, I., On Casimir Energy Contribution to Observable Value of the Cosmological Constant, Acta Phys. Polon. B33:1973-1977, 2002
Cleaver, G and Rosenthal P., String Cosmology and the Dimension of Space-time, Nucl.Phys.B457:621-642,1995
Garnavich, P., Observational Evidence From Supernovae for an Accelerating Universe and a Cosmological Constant,
Astron.J.116:1009-1038,1998
Giudice, G., and Rattazzi, R., Theories with gauge mediated supersymmetry breaking,Phys. Rept. 322, 419 (1999).
Morris, M., and Thorne, K,. and Yurtsever, U., Worm-holes, Time Machines, and the Weak Energy Conditions, Phys. Rev. Lett., 61:1446-1449, 1988
