Physics Faculty Works Seaver College of Science and Engineering 2013 Conformal Gravity and the Alcubierre Warp Drive Metric Gabriele U. Varieschi Loyola Marymount University, [email protected] Zily Burstein Loyola Marymount University Follow this and additional works at: https://digitalcommons.lmu.edu/phys_fac Part of the Physics Commons Recommended Citation Gabriele U. Varieschi and Zily Burstein, “Conformal Gravity and the Alcubierre Warp Drive Metric,” ISRN Astronomy and Astrophysics, vol. 2013, Article ID 482734, 13 pages, 2013. doi: 10.1155/2013/482734 This Article is brought to you for free and open access by the Seaver College of Science and Engineering at Digital Commons @ Loyola Marymount University and Loyola Law School. It has been accepted for inclusion in Physics Faculty Works by an authorized administrator of Digital Commons@Loyola Marymount University and Loyola Law School. For more information, please contact [email protected]. Hindawi Publishing Corporation ISRN Astronomy and Astrophysics Volume 2013, Article ID 482734, 13 pages http://dx.doi.org/10.1155/2013/482734 Research Article Conformal Gravity and the Alcubierre Warp Drive Metric Gabriele U. Varieschi and Zily Burstein Department of Physics, Loyola Marymount University, Los Angeles, CA 90045, USA Correspondence should be addressed to Gabriele U. Varieschi; [email protected] Received 7 November 2012; Accepted 24 November 2012 Academic Editors: P. P. Avelino, R. N. Henriksen, and P. A. Hughes Copyright © 2013 G. U. Varieschi and Z. Burstein. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We present an analysis of the classic Alcubierre metric based on conformal gravity, rather than standard general relativity. e main characteristics of the resulting warp drive remain the same as in the original study by Alcubierre, that is, effective superluminal motion is a viable outcome of the metric. We show that for particular choices of the shaping function, the Alcubierre metric in the context of conformal gravity does not violate the weak energy condition, as was the case of the original solution. In particular, the resulting warp drive does not require the use of exotic matter. erefore, if conformal gravity is a correct extension of general relativity, superluminal motion via an Alcubierre metric might be a realistic solution, thus allowing faster-than-light interstellar travel. 1. Introduction type of super-luminal motion, unless large quantities of exotic matter (i.e., with negative energy density) can be cre- In 1994, Alcubierre introduced the so-called Warp Drive ated. Since our current knowledge of this type of exotic matter Metric (WDM), within the framework of general relativity is limited to some special effects in quantum �eld theory (GR), which allows in principle for super-luminal motion, (such as the Casimir effect), it is unlikely that the Alcubierre that is, faster-than-light travel [1]. is superluminal propul- WDM can be practically established within the framework of sion is achieved, respectively, by expanding and contracting general relativity. the space time behind and in front of a spaceship, while the Following Alcubierre’s seminal paper, many other studies spacecra� is le� inside a locally �at region of space time, appeared in the literature, either proposing alternatives to within the so-called warp bubble. the original warp drive mechanism ([5, 6]) or re�ning and In this way, the spaceship can travel at arbitrarily high analyzing in more detail the original idea ([7–19]). However, speeds, without violating the laws of special and general all these studies were conducted using standard GR and could relativity, or other known physical laws. Furthermore, the not avoid the violation of the WEC, meaning that some exotic spacecra� and its occupants would also be at rest in �at space matter would always be required for faster-than-light travel. time, thus immune from high accelerations and unaffected Similar issues also exist in other well-known GR solutions for by special relativistic effects, such as time dilation. Enormous super-luminal motion, such as space-time wormholes [20]. tidal forces would only be present near the edge of the warp Einstein’s general relativity and the related “Standard bubble, which can be made large enough to accommodate the Model” of cosmology have been highly successful in describ- volume occupied by the ship. ing our universe, from the solar system up to the largest However, Alcubierre [1]wasalsothe�rsttopointoutthat cosmological scales, but recently these theories have also led this hypothetical solution of Einstein’s equations of GR would to a profound crisis in our understanding of its ultimate com- violate all three standard energy conditions (weak, dominant, position. From the original discovery of the expansion of the and strong; see [2–4] for de�nitions). In particular, the universe, which resulted in standard big bang cosmology, violation of the weak energy condition (WEC) implies that scientists have progressed a long way towards our current negative energy density is required to establish the Alcubierre picture, in which the contents of the universe are today WDM, thus making it practically impossible to achieve this described in terms of two main components, dark matter 2 ISRN Astronomy and Astrophysics (DM) and dark energy (DE), accounting for most of the where is invariant under the local transformation of observed universe, with ordinary matter just playing a minor the following metric: role. (푥 Since there is no evidence available yet as to the real nature of dark matter and dark energy, alternative gravi- 2(푥 2 (2) tational and cosmological theories are being developed, in (푥 ⟶ (푥 = (푥 =Ω (푥 (푥 . addition to standard explanations of dark matter/dark energy e factor determines the amount of local invoking the existence of exotic new particles also yet to be “stretching” of the geometry, hence the name “conformal” for discovered. In line with these possible new theoretical ideas, (푥 a theory invariant under all local stretchings of the space-time conformal gravity (CG) has emerged as a nonstandard exten- Ω(푥 (see [22] and references therein for more details). sion of Einstein’s GR, based on a possible symmetry of the is conformally invariant generalization of GR was universe: the conformal symmetry, that is, the invariability of found to be a fourth-order theory, as opposed to the standard the space-time fabric under local “stretching” of the metric second-order general relativity, since the �eld equations (for reviews see [21, 22]). is alternative theory has been originating from a conformally invariant Lagrangian contain reintroduced in recent years (following the original work by derivatives up to the fourth order of the metric, with respect Weyl [23–25]), leading to cosmological models which do not to the space-time coordinates. Following the works done by require the existence of DM and DE ([26–32]). Bach [34], Lanczos [35], and others, CG was ultimately based At the quantum level conformal gravity, as well as other on the following Weyl or conformal action (In this paper we theories with higher derivatives, was thought to be affected adopt a metric signature and we follow the sign by the presence of “ghosts,” leading to possible instabilities of conventions of Weinberg [33]. In this section we will leave the quantum version of the theory. However, recent studies fundamental constants, such as and ,inallequations,but [26]haveshownthatCGasaquantumtheoryisboth (−, +, +, +푥 later we will use geometrized units ( , ), or c.g.s. renormalizable and unitary, thus providing a solution to the units when needed.): ghost problem. In view of a possible extension of Einstein’s general rela- =1 tivity into conformal gravity, in this paper we have recon- (3) sidered the Alcubierre WDM, basing it on CG rather than 4 1/2 standard GR. In Section 2, we review the fundamental prin- or on the following = − equivalent − expression, differing, from the ciples of CG and the calculation of the stress-energy tensor previous one only by a topological invariant: in this gravitational theory. In Section 3, we consider the Alcubierre metric in CG and compute the energy density for (4) different shaping functions of the metric. 4 1/2 2 In particular, we will show that for certain shaping func- 1 where = −2 and −is the gravitational − coupling , con- tions, the Alcubierre metric in the context of conformal 3 gravity does not violate the weak energy condition, as was stant of conformal gravity (see [21, 36–38].) (In these cited the case of the original solution. is analysis continues papers,is considered푔 푔푔푔( 푥 a dimensionless constant by using in Section 4, where we study other energy conditions and natural units. Working with c.g.s. units, we can assign dimen- estimate the total energy required for this CG warp drive. sions of an action to the constant so that the dimensional- Finally, in Section 5, we conclude that if CG is a correct ity of (5) will be correct.) Under the conformal transforma- extension of GR, super-luminal motion via an Alcubierre tion in (2), the Weyl tensor transforms as metric might be a realistic possibility, thus enabling faster- , while the conformal action than-light interstellar travel without requiring exotic matter. is2 locally(푥 conformally2 invariant, the only general → coordinate = scalar action =Ω