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
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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 following𝜆𝜆 metric: role. 𝐶𝐶 𝜇𝜇𝜇𝜇𝜇𝜇(𝑥𝑥푥 Since there is no evidence available yet as to the real nature of dark matter and dark energy, alternative gravi- 2𝛼𝛼𝛼𝛼𝛼 2 (2) tational and cosmological theories are being developed, in 𝑔𝑔𝜇𝜇𝜇𝜇 (𝑥𝑥) ⟶𝑔𝑔𝜇𝜇𝜇𝜇 (𝑥𝑥) =𝑒𝑒 𝑔𝑔𝜇𝜇𝜇𝜇 (𝑥𝑥) =Ω (𝑥𝑥) 𝑔𝑔𝜇𝜇𝜇𝜇 (𝑥𝑥) . 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, 𝛼𝛼𝛼𝛼𝛼 a theory invariant under all local stretchings of the space-time conformal gravity (CG) has emerged as a nonstandard exten- Ω(𝑥𝑥푥 𝑥𝑥𝑥 (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 (−, +, +, +) later we will use geometrized units ( , ), or c.g.s. renormalizable and unitary, thus providing a solution to the units when needed.): ghost problem. 𝑐𝑐 𝐺𝐺 In view of a possible extension of Einstein’s general rela- 𝑐𝑐𝑐𝐺𝐺𝐺 tivity into conformal gravity, in this paper we have recon- (3) sidered the Alcubierre WDM, basing it on CG rather than 4 1/2 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 𝑊𝑊 𝑔𝑔 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 standard GR. In Section 2, we review the fundamental prin- or on the following𝐼𝐼 = −𝛼𝛼 equivalent 𝑑𝑑 𝑥𝑥−𝑔𝑔 expression, 𝐶𝐶 𝐶𝐶 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 𝜇𝜇𝜇𝜇 2 In particular, we will show that for certain shaping func- 𝑊𝑊 𝑔𝑔 𝜇𝜇𝜇𝜇 1 where 𝐼𝐼 = −2𝛼𝛼 and𝑑𝑑 𝑥𝑥−𝑔𝑔is the gravitational𝑅𝑅 𝑅𝑅 − 𝑅𝑅 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 𝜇𝜇𝜇𝜇 𝑔𝑔 the case of the original solution. is analysis continues papers,𝑔𝑔is considered푔 푔푔푔�𝑔𝑔 ) 𝛼𝛼 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 of𝛼𝛼 an𝑔𝑔 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 transforms𝛼𝛼𝑔𝑔 as metric might be a realistic possibility, thus enabling faster- , while the conformal action than-light interstellar travel without requiring exotic matter. 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 is2𝛼𝛼 locally𝛼𝛼𝛼 conformally2 invariant, the only general𝐶𝐶 → coordinate𝐶𝐶 = 𝑒𝑒scalar𝐶𝐶 action𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 =Ω with such(𝑥𝑥푥𝑥𝑥𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 properties. 𝐼𝐼𝑊𝑊 Bach [34] introduced the gravitational �eld equations in 2. Conformal Gravity and the presence of a stress-energy tensor (We follow here the the Stress-Energy Tensor convention [21] of introducing the stress-energy tensor so that the quantity has the dimensions of an energy Weyl in 1918 ([23–25]) developed the “conformal” general- density.) as 𝑇𝑇𝜇𝜇𝜇𝜇 ization of Einstein’s relativity by introducing the conformal 𝑐𝑐𝑐𝑐00 (or Weyl) tensor, a special combination of the Riemann 𝑇𝑇𝜇𝜇𝜇𝜇 tensor , the Ricci tensor , and the curvature (5)
(or Ricci) scalar [33] as follows:𝜆𝜆 𝜇𝜇𝜇𝜇 1 𝜇𝜇𝜇𝜇 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 𝜇𝜇𝜇𝜇 𝜇𝜇𝜇𝜇𝜇𝜇 𝑊𝑊 = 𝑇𝑇 , 𝑅𝑅 𝜇𝜇 𝑅𝑅 =𝑅𝑅 as opposed to Einstein’s standard4𝛼𝛼𝑔𝑔 equations as 𝑅𝑅�𝑅𝑅 𝜇𝜇 (6) 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 1 𝜆𝜆𝜆𝜆 𝜇𝜇𝜇𝜇 𝜆𝜆𝜆𝜆 𝜇𝜇𝜇𝜇 𝜇𝜇𝜇𝜇 𝜆𝜆𝜆𝜆 𝜇𝜇𝜇𝜇 𝜆𝜆𝜆𝜆 𝐶𝐶 =𝑅𝑅 − 𝑔𝑔 𝑅𝑅 −𝑔𝑔 𝑅𝑅 −𝑔𝑔 𝑅𝑅 +𝑔𝑔 𝑅𝑅 1 8𝜋𝜋𝜋𝜋 2 where the “Bach𝐺𝐺𝜇𝜇𝜇𝜇 =𝑅𝑅 tensor”𝜇𝜇𝜇𝜇 − 𝑔𝑔𝜇𝜇𝜇𝜇[𝑅𝑅34𝑅] is the3 equivalent𝑇𝑇𝜇𝜇𝜇𝜇, in CG of 1 (1) the Einstein curvature tensor2 on the𝑐𝑐 le-hand side of (6). + 𝑅𝑅 𝑔𝑔𝜆𝜆𝜆𝜆𝑔𝑔𝜇𝜇𝜇𝜇 −𝑔𝑔𝜆𝜆𝜆𝜆𝑔𝑔𝜇𝜇𝜇𝜇 , 6 𝑊𝑊𝜇𝜇𝜇𝜇 𝐺𝐺𝜇𝜇𝜇𝜇 ISRN Astronomy and Astrophysics 3
has a very complex structure and can be de�ned in and is a “form function” or “shaping function” which a compact way as [39] needs to have values and , respectively, inside 𝑊𝑊𝜇𝜇𝜇𝜇 and outside𝑓𝑓𝑓𝑓�𝑠𝑠) the warp bubble, while it can have an arbitrary (7) shape in the transition𝑓𝑓𝑓 region of the𝑓𝑓� warp푓 bubble itself. 𝛼𝛼 𝛽𝛽 𝛼𝛼 𝛽𝛽 e original shaping function used by Alcubierre was or in an expanded𝑊𝑊𝜇𝜇𝜇𝜇 form= 2𝐶𝐶 as𝜇𝜇𝜇𝜇 ([37;𝛽𝛽,𝛽𝛽�40+𝐶𝐶]) 𝜇𝜇𝜇𝜇 𝑅𝑅𝛽𝛽𝛽𝛽, (12)
;𝜆𝜆 ;𝜆𝜆 𝑠𝑠 𝑠𝑠 𝑠𝑠 tanh 𝜎𝜎 𝑟𝑟 +𝑅𝑅 − tanh 𝜎𝜎 𝑟𝑟 −𝑅𝑅 𝜇𝜇𝜇𝜇 1 𝜇𝜇𝜇𝜇 ;𝜆𝜆 2 ;𝜇𝜇�𝜇𝜇 𝜇𝜇𝜇𝜇 ;𝜆𝜆 𝑓𝑓 𝑟𝑟 = , 𝑊𝑊 =− 𝑔𝑔 𝑅𝑅 + 𝑅𝑅 +𝑅𝑅 (8) where basically indicates2 tanh ( the𝜎𝜎𝜎𝜎) radius of the spherical 6 3 warp bubble, while relates to the bubble thickness, 𝜆𝜆 𝜆𝜆 2 which decreases with increasing values of .Inthefollowing, −𝑅𝑅𝜇𝜇 ;𝜈𝜈𝜈𝜈� −𝑅𝑅𝜈𝜈 ;𝜇𝜇𝜇𝜇� + 𝑅𝑅𝑅𝑅𝜇𝜇𝜇𝜇 𝑅𝑅푅� 3 we will refer to the function𝜎𝜎𝜎 in (12) as the “Alcubierre shaping 𝜆𝜆 𝜆𝜆𝜆𝜆 2 function” (ASF). 1 1 𝜎𝜎 involving derivatives− 2𝑅𝑅𝜇𝜇 𝑅𝑅 up𝜆𝜆𝜆𝜆 to+ the𝑔𝑔 fourth𝜇𝜇𝜇𝜇 𝑅𝑅𝜆𝜆𝜆𝜆 order𝑅𝑅 − of𝑔𝑔 the𝜇𝜇𝜇𝜇𝑅𝑅 metric, with We will show that the particular form of the shaping respect to space-time coordinates.2 6 function can play an important role in the energy conditions erefore, in conformal gravity, the stress-energy tensor for the WDM. In our analysis we tested several different is computed by combining together (5) and (8) as functions obeying the general requirements for outlined above. In addition to the Alcubierre function above, in this paper we will also use the following: 𝑓𝑓 1 ;𝜆𝜆 2 ;𝜆𝜆 𝑇𝑇𝜇𝜇𝜇𝜇 = 4𝛼𝛼𝑔𝑔𝑊𝑊𝜇𝜇𝜇𝜇 = 4𝛼𝛼𝑔𝑔 − 𝑔𝑔𝜇𝜇𝜇𝜇𝑅𝑅 ;𝜆𝜆 + 𝑅𝑅;𝜇𝜇�𝜇𝜇 +𝑅𝑅𝜇𝜇𝜇𝜇 ;𝜆𝜆 6 3 𝑚𝑚 (13) 𝜆𝜆 𝜆𝜆 𝑠𝑠 𝜇𝜇 𝜈𝜈 ;𝜇𝜇𝜇𝜇� 2 𝜇𝜇𝜇𝜇 𝑟𝑟 𝑠𝑠 −𝑅𝑅 ;𝜈𝜈𝜈𝜈� −𝑅𝑅 + 𝑅𝑅𝑅𝑅 𝑠𝑠 1− ;𝑟𝑟< 𝑅𝑅� 3 𝑓𝑓 𝑟𝑟 = 𝑅𝑅 where is a positive integer.0; Since this𝑟𝑟𝑠𝑠 particular> 𝑅𝑅� function 𝜆𝜆 1 𝜆𝜆𝜆𝜆 − 2𝑅𝑅𝜇𝜇 𝑅𝑅𝜆𝜆𝜆𝜆 + 𝑔𝑔𝜇𝜇𝜇𝜇 𝑅𝑅𝜆𝜆𝜆𝜆𝑅𝑅 for isusedbyHartletoillustratethewarpdrivein 2 his textbook𝑚𝑚 [20], we will refer to the function in (13)asthe 2 (9) “Hartle shaping function” (HSF). 1 𝜇𝜇𝜇𝜇 𝑚𝑚𝑚 − 𝑔𝑔 𝑅𝑅 . e top panels in Figure 1 illustrate the differences 6 is form of the tensor will be used in the following sections, between the Alcubierre shaping function (top le panel, for in connection to the Alcubierre metric, to compute the as used originally by Alcubierre) and the Hartle shaping energy density and other relevant quantities. function (top right panel, for ). All functions in this For this purpose, we have developed a special Mathe- 𝜎𝜎�gure�휎 are computed for a �xed value of andattime matica program which enables us to compute all the tensor , when the spaceship is located𝑚𝑚 at𝑚 the origin. All quantities quantities of both GR and CG, for any given metric. In shown in the different panels are plotted𝑅𝑅 as𝑅 a function of the𝑡𝑡� particular, this program can compute the conformal tensor 0 coordinate of the spaceship motion and of the transverse in (1), the Bach tensor in (8), or the stress-energy tensor in (9) by performing all the necessary covariant cylindrical coordinate . Similar coordinates will 𝑥𝑥 𝐶𝐶derivatives.𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 Given the complexity𝑊𝑊𝜇𝜇𝜇𝜇 of these types of compu- also be used in the other �gures.2 (e2 cylindrical coordinate tations,𝑇𝑇 we𝜇𝜇𝜇𝜇 have tested extensively our program against the should be𝜌𝜌� considered𝑦𝑦 +𝑧𝑧 as non-negative and all results for and computed by Kazanas and quantities2 in2 the �gures plotted only for . However, for Mannheim in [38] for different metrics, obtaining a perfect illustrative𝜌𝜌�𝑦𝑦 + purposes𝑧𝑧 and also to follow similar �gures in the agreement. 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 𝜇𝜇𝜇𝜇 𝐶𝐶 𝑊𝑊 literature (such as those in [1, 11, 12]), we𝜌𝜌 decided휌� to let run on negative values in all �gures, except in the last one, where 3. Alcubierre Metric, Shaping Functions, and we restrict for a correct energy calculation.) 𝜌𝜌 the Weak Energy Condition e expansion/contraction function of the volume elements behind/in𝜌𝜌휌� front of the spaceship was also computed e original Alcubierre metric [1] considered a spaceship by Alcubierre as [1] 𝜃𝜃 traveling along the -axis, with motion described by a function and spaceship velocity . Using (14) the 3+1 formalism of GR,𝑥𝑥 the metric was written in Cartesian 𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑥𝑥�𝑥𝑥 (𝑡𝑡) 𝑑𝑑𝑑𝑑 𝑟𝑟 coordinates𝑥𝑥 (𝑡𝑡푡 as ( ) 𝑣𝑣 (𝑡𝑡푡 � 𝑡𝑡𝑡𝑡 (𝑡𝑡푡푡𝑡𝑡𝑡𝑡 𝜃𝜃𝜃𝜃�𝑠𝑠 (𝑡𝑡) and is illustrated for (in𝑟𝑟 geometrized𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠 units, i.e., (10) in traditional units) in Figure 1(b), for the two different 𝑐𝑐𝑐 shaping functions. Again,𝑠𝑠 the choice of the parameters𝑠𝑠 , where 2 is the2 distance from the spaceship2 position2 2 as 𝑣𝑣 =1 𝑣𝑣 =𝑐𝑐 𝑑𝑑𝑑𝑑 = −𝑑𝑑𝑑𝑑 + 𝑑𝑑𝑑𝑑𝑑𝑑�𝑠𝑠(𝑡𝑡푡𝑡𝑡�𝑡𝑡𝑠𝑠)𝑑𝑑𝑑𝑑 + 𝑑𝑑𝑑𝑑 + 𝑑𝑑𝑑𝑑 , , and isthesameasinthetoppanelsinthe�gure.e expansion fortheASFisthesameasFigure1in[1], while𝑅𝑅 𝑟𝑟𝑠𝑠 (11) 𝜎𝜎the corresponding𝑚𝑚 for the HSF is slightly different but still 2 2 2 𝜃𝜃 𝑠𝑠 𝑠𝑠 𝑟𝑟 (𝑡𝑡) = 𝑥𝑥�𝑥𝑥(𝑡𝑡푡 +𝑦𝑦 +𝑧𝑧 , 𝜃𝜃 4 ISRN Astronomy and Astrophysics
Alcubierre shaping function Hartle shaping function 1 1 0 0 −1 −1
1 1
0.5 0.5
0 0 −1 −1 0 0 1 1
(a) Expansion of the volume elements Expansion of the volume elements 1 1 0 0 −1 −1 4 4
2 2
0 0
−2 −2
−4 −4 −1 −1 0 0 1 1
(b) Energy density in GR Energy density in GR
1 0 1 −1 0 −1
0 0
− 0.1 − 0.1
− 0.2 − 0.2 −1 −1 0 0 1 1
(c) Energy density in CG Energy density in CG 1 1 0 0 −1 −1 4000 20000 3000
0 2000
1000 − 20000 0 −1 −1 0 0 1 1
(d)
F 1: Results for the two different shaping functions (le column ASF, right column HSF), computed with parameters , , , , , . (a) Alcubierre and Hartle shaping functions. (b) Expansion of the volume elements . (c) Energy density 𝑠𝑠 computed with general relativity. (d) Energy density computed with conformal gravity. �e �E� is veri�ed in the case𝑣𝑣 shown=1𝑅𝑅 in𝑅 the00 𝑔𝑔 bottom𝜎𝜎�휎𝑚𝑚 right�푚 panel.𝛼𝛼 =1𝑡𝑡�푡 00 𝜃𝜃 𝑇𝑇 𝑇𝑇 ISRN Astronomy and Astrophysics 5 shows expansion of the normal volume elements behind the inthetwoplotsatthebottomofthe�gurecanbeattributed spaceship and contraction in front of it. to the different shaping functions and their derivatives up e weak energy condition ([2–4]) requires that to the fourth order. All these derivatives enter the complex for all timelike vectors . Alcubierre has also expression of in CG, as in the master equation (9), and shown𝜇𝜇 𝜈𝜈 that for the Eulerian observers𝜇𝜇 in the warp drive their interplay ultimately𝜇𝜇𝜇𝜇 determines the shape of , or of 𝜇𝜇𝜇𝜇 𝑇𝑇metric,𝑡𝑡 𝑡𝑡 and≥0 for their 4-velocity ,𝑡𝑡 the following relation the other components,𝑇𝑇 in a way which is hard to00 predict holds [1]: before the actual computation is performed. 𝑇𝑇 𝑛𝑛𝜇𝜇 In Figure 2, we analyze the dependence of the energy (15) 2 2 2 density on the spaceship velocity . In this case we 𝜇𝜇𝜇𝜇 00 𝑠𝑠 consider only00 the Hartle shaping function for and 𝜇𝜇 𝜈𝜈 1 𝑣𝑣 𝜌𝜌2 𝑑𝑑𝑑𝑑 which implies𝑇𝑇 that𝑛𝑛 𝑛𝑛 the=𝑇𝑇 energy=− density𝑠𝑠 is𝑠𝑠 negative , every- 𝑠𝑠 8𝜋𝜋 4𝑟𝑟 𝑑𝑑𝑑𝑑 ,𝑇𝑇 and we compute inCG(𝑣𝑣 ) for speeds where for any choice of the shaping function00 and, therefore, ranging from the subluminal00 to the super-luminal𝑚𝑚𝑚 the WEC is violated (also the dominant𝑇𝑇 energy condition, 𝑅𝑅𝑅 (for 𝑇𝑇 we obtain𝛼𝛼𝑔𝑔 the=1 same function DEC, and the strong energy condition, SEC,𝑓𝑓 are violated in asinthebottomrightpanelof𝑣𝑣𝑠𝑠 =Figure 0.25𝑐𝑐 1). e shape of the the analysis based on GR [1]). 𝑣𝑣energy𝑠𝑠 = 3.00𝑐𝑐 density function𝑣𝑣𝑠𝑠 = 1.00𝑐𝑐 is about the same for speeds up to is violation of the WEC in GR is illustrated in , although the function values increase with speed. Figure 1(c), where is calculated using (15) for both For higher velocities, the function develops two “downward shaping functions. Although00 the results in the two panels lobes”𝑣𝑣𝑠𝑠 = 1.50𝑐𝑐 which eventually become negative for speeds are slightly different,𝑇𝑇 they obviously show negative energy . is implies that the WEC is veri�ed for speeds up to densities and therefore a complete violation of the WEC. , while at higher velocities exotic matter would𝑣𝑣𝑠𝑠 be≳ e situation is different if we compute the energy density 2.50𝑐𝑐required to sustain the warp drive. in the framework of CG, following (9), setting 𝑣𝑣𝑠𝑠 ≈is 2.50𝑐𝑐 apparent “speed limit” at about might for00 simplicity, and using the completely contravariant form be raised or overcome completely by adopting a different 𝑇𝑇of the stress-energy tensor, instead of the covariant𝛼𝛼 one.𝑔𝑔 =1 As shaping function, instead of the HSF used𝑣𝑣𝑠𝑠 ≈ here, 2.50𝑐𝑐 but this seen in the bottom row of Figure 1, the energy density in analysis would go beyond the scope of this work. We do CG is completely different from the one calculated within the not ascribe any physical signi�cance to this new speed limit, framework of GR. In the bottom le panel is computed which is probably just an arti�cial product of the shaping with the ASF and the resulting function is in part00 positive and functions used in this analysis. in part negative, thus still violating the WEC.𝑇𝑇 In any case, the results reported in Figure 2 show that a However,thebottomrightpanelshows computed warp drive in CG with positive energy density is possible for a wide range of spaceship velocities; therefore, if CG is the with the HSF and in this case the energy density is00 completely correct extension of GR, the Alcubierre warp drive might be non-negative, showing that the WEC is veri�ed𝑇𝑇 and no exotic matter is needed to establish the warp drive. (In a viable mechanism for super-luminal travel. this panel of Figure 1 (and also in the other �gures) we In Figure 3, we present the other components of the show a step discontinuity at the edge of the warp bubble, stress-energy tensor. ese were computed with the same common to all our CG solutions. ese discontinuities can Mathematica program, following (9) with , leading be mathematically replaced by appropriate delta functions, as to even more complex expressions than the one for (we 𝑔𝑔 described by the equations used in the Appendix to model will omit to report these expressions for brevity).𝛼𝛼 =1 To simplify00 the Hartle shaping function. However, these delta functions the computation, we used the covariant components𝑇𝑇 do not need to be included in the physical solutions for and adopted cylindrical coordinates around the -axis, 𝜇𝜇𝜇𝜇 in CG shown in the �gures. erefore, these energy densities00 , instead of the Cartesian coordinates𝑇𝑇 are completely non-negative and they fully determine𝑇𝑇 the of the original Alcubierre metric. e results in the different𝑥𝑥 expansion of the volume elements required for the warp drive (𝑡𝑡panels𝑡𝑡𝑡𝑡𝑡𝑡� 𝑡𝑡푡 are labeled 푡 𝑡𝑡𝑡𝑡𝑡� accordingly.푡 We recall that the stress- effect.) is non-negative energy density plot in the bottom energy tensor is symmetric, , and only the nonzero right panel of Figure 1 isthemainresultofourpaperasit components are illustrated in this �gure, under similar shows that—if CG is the correct extension of GR—it might conditions used before (CG𝑇𝑇 with𝜇𝜇𝜇𝜇 =𝑇𝑇 HSF𝜈𝜈𝜈𝜈 and parameters , be possible to establish a warp drive without having to use , ). negative energy (mass), thus overcoming the main difficulty e shapes of the other components of are𝑚𝑚 more𝑚 𝑠𝑠 of the warp drive mechanism. complex𝑅𝑅𝑅 𝑣𝑣 than= 1.00𝑐𝑐 the one of , but they can all be𝜇𝜇𝜇𝜇 determined e explicit expression of in CG, computed with analytically, either in covariant00 or contravariant𝑇𝑇 form, using our Mathematica program, is rather00 cumbersome and is our program. If this exact𝑇𝑇 form of the stress-energy tensor reproduced in (A.1) of the Appendix.𝑇𝑇 Here we present just could be established in the region surrounding the spacecra, the graphical computation of in Figure 1,orintheother warp drive motion would be feasible within the framework of �gures in this paper. We have00 tested the validity of these conformal gravity. results by running the program𝑇𝑇 in several different ways, Wealso want to point out that we set the spaceship motion including computing the stress-energy tensor for a simpli�ed in the positive direction of the -axis (setting ), three-dimensional Alcubierre metric (coordinates , , and and this is re�ected in the symmetry, or lack thereof, of the only), always obtaining consistent results. e differences components of . While some𝑥𝑥 components,𝑣𝑣𝑠𝑠 such= +1.00𝑐𝑐 as 𝑥𝑥 𝑦𝑦 00 𝜇𝜇𝜇𝜇 𝑡𝑡 𝑇𝑇 𝑇𝑇 6 ISRN Astronomy and Astrophysics
1 1 1 0 0 0 −1 −1 −1 300 1000 2000 200 500 1000 100
0 0 0 −1 −−11 −1 0 0 0 1 1 1
(a)
1 1 0 1 0 0 −1 −1 −1 4000 10000 6000 3000 4000 2000 5000
1000 2000
0 0 0 −1 −1 −1 0 0 0 1 1 1
(b)
1 1 1 0 0 0 −1 −1 −1 30000 15000 20000
15000 20000 10000 10000 10000 5000 5000 0 0 0 −1 −1 −1 0 0 0 1 1 1
(c)
1 1 1 0 0 0 −1 −1 −1 60000 40000 60000 40000 40000 20000 20000 20000 0 0 0 − 20000 −1 −1 −1 0 0 0 1 1 1
(d)
F 2: Energy density computed with conformal gravity and the Hartle shaping function. Parameters used are , , , , and variable 00 . e energy density becomes in part negative for ,sothe�E�isveri�edforspeedsupto 𝑔𝑔 about . 𝑇𝑇 𝑅𝑅푅푅 𝑚𝑚�푚 𝛼𝛼 =1 𝑠𝑠 𝑠𝑠 𝑡𝑡𝑡 𝑣𝑣 = 0.25𝑐𝑐 푐𝑐�𝑐𝑐 𝑣𝑣 ≳ 2.50𝑐𝑐 𝑠𝑠 𝑣𝑣 ≃ 2.50𝑐𝑐 ISRN Astronomy and Astrophysics 7
Stress-energy tensor: component Stress-energy tensor: component 1 1 0 0 −1 −1 4000
2000 3000
2000 0
1000 − 2000
0 0 −1 −1 0 0 1 1
(a) Stress-energy tensor: component Stress-energy tensor: component 1 1 0 0 −1 −1 2000 2000
1500
1000 0
500 − 2000 0 −1 −1 0 0 1 1
(b) Stress-energy tensor: component Stress-energy tensor: component
1 1 0 0 −1 −1 1000
1000
500 0
− 1000
0 −1 −1 0 0 1 1
(c)
F 3: Stress-energy tensor components , in cylindrical coordinates , computed with conformal gravity and the Hartle shaping function. Parameters used are , , , , and . 𝜇𝜇𝜇𝜇 𝑇𝑇 (𝑡𝑡푡 𝑡𝑡푡 𝑡𝑡푡푡 �푡푡 푡푡 푡푡 푡푡 𝑠𝑠 𝑔𝑔 𝑣𝑣 =1 𝑅𝑅𝑅 𝑚𝑚�푚 𝛼𝛼 =1 𝑡𝑡�푡 8 ISRN Astronomy and Astrophysics
(or ), , , , , appear to be symmetric observers [1], the previous condition for the DEC becomes under the exchange , the other components, . 00 01 10 11 22 33 𝑇𝑇and 𝑇𝑇 =𝑇𝑇, are𝑇𝑇 not𝑇𝑇 symmetric𝑇𝑇 under this exchange 0 Figure0𝜆𝜆 5 illustrates the violation of the DEC for our 02 and, therefore, these𝑥𝑥 components 푥푥𝑥𝑥 must contain information𝑇𝑇 = 𝑇𝑇standard𝜆𝜆𝑇𝑇 ≤0 solution (AWD with HSF and , , 20 12 21 about𝑇𝑇 the𝑇𝑇 spaceship=𝑇𝑇 direction of motion. is argument over- ). e plotted function is not negative comes the objection addressed in [12] that since is everywhere, as required by the DEC,0 but0𝜆𝜆𝑚𝑚 shows�푚 a violation𝑅𝑅𝑅 symmetric about the plane, there is uncertainty00 in for𝑣𝑣𝑠𝑠 the=1.00𝑐𝑐 central portion of the warp bubble.𝑇𝑇 𝜆𝜆𝑇𝑇 Even if this energy where the space-time is expanded/contracted, thus making𝑇𝑇 condition appears to be violated, this does not notably affect it impossible for the spaceship𝑥𝑥𝑠𝑠 =0 to know in which direction thefeasibilityofourCGwarpdrive.WerecallthattheDEC of the -axis, positive or negative, to move. Rather, Figure 3 is usually related to the standard perfect �uid stress-energy shows that a “bias” towards one of the two possible directions tensor, , where here and are is induced𝑥𝑥 by some components of . the �uid density and pressure, while is the �uid 4-velocity. Figure 4 illustrates one last dependence of the energy In this context,𝑇𝑇𝜇𝜇𝜇𝜇 = (𝜌𝜌 the DEC 휌 𝜌𝜌휌𝜌𝜌𝜇𝜇𝑈𝑈 requires𝜈𝜈 휌 𝜌𝜌𝑝𝑝𝜇𝜇𝜇𝜇 , but this𝜌𝜌 condition𝑝𝑝 𝜇𝜇𝜇𝜇 density ontheparametersused.Inthiscase,weset𝑇𝑇 , is not required in general by all classical𝑈𝑈𝜇𝜇 forms of matter 00 and consider the Hartle shaping function as in [3]; therefore, its violation in our𝜌𝜌 𝜌� case𝜌𝜌� is not particularly (13), while𝑇𝑇 varying the integer parameter . In addition𝑅𝑅𝑅 to signi�cant. our𝑣𝑣𝑠𝑠 = standard 1.00𝑐𝑐 value, , we have also tried values from �n the contrary, our standard solution also veri�es the to , as shown in the �gure. 𝑚𝑚 strong energy condition (SEC) which states that In general, increasing the value increases the internal 𝑚𝑚�푚 for all timelike vectors . Again, using the𝜇𝜇 Eule-𝜈𝜈 𝑚𝑚volume𝑚 of𝑚𝑚 the𝑚𝑚 warp bubble, where space time is �at, and 𝜇𝜇𝜇𝜇 rian 4-velocity𝜆𝜆 𝜎𝜎 vector in place of ,𝜇𝜇 the previous𝑇𝑇 condition𝑡𝑡 𝑡𝑡 ≥ therefore decreases the thickness𝑚𝑚 of the bubble wall where 𝜆𝜆 𝜎𝜎 the space-time distortion takes place, in a way similar to that (1/2is equivalent)𝑇𝑇 𝑡𝑡 𝑡𝑡 to 𝜇𝜇𝑡𝑡 . Since the scalar of the parameter in the original Alcubierre warp drive. is identically zero00 for all our solutions,𝜆𝜆 𝑡𝑡 as checked using our𝜆𝜆 𝜆𝜆 𝜆𝜆 However, increasing also increases the energy required to Mathematica program,𝑇𝑇 + (1/2 the) SEC𝑇𝑇 is≥0 equivalent to 𝑇𝑇 , establish𝜎𝜎 the warp drive (as we checked by integrating the which is the WEC already veri�ed in Section 3. 00 functions in Figure 4).𝑚𝑚 erefore, it appears to be convenient Finally, we want to estimate the energy necessary𝑇𝑇 ≥0 to to use a low value for this parameter. As shown in the different establish our CG warp drive, under reasonable conditions. panels of the �gure, the �rst value, , does not work For this purpose, in Figure 6 we computed once again the since it does not leave a �at space-time volume inside the energy density for our AWD with the Hartle shaping bubble, while seems to create𝑚𝑚 a𝑚 very small volume function ( 00, , ), but this time for inside the bubble. erefore, intermediate values such as = cm 𝑇𝑇 m, a reasonable radius for a warp bubble 𝑔𝑔 𝑠𝑠 4–6 would appear𝑚𝑚𝑚 to be more adequate to establish the warp enclosing our𝛼𝛼 spaceship.=1𝑚𝑚�푚𝑣𝑣 = 1.00𝑐𝑐 drive. Meanwhile, the solution for would require𝑚𝑚 𝑅𝑅 𝑅𝑅𝑅�Figure 6 illustrates= 100 this solution, plotted only for much more energy and would not give any advantage, except , as this is the correct interval for the transverse reducing the thickness of the warp bubble. We also checked 𝑚𝑚 𝑚𝑚 coordinate . e cylindrical symmetry of this solution𝜌𝜌 can� thatchangingthevalueof does not have a strong effect on 2 2 also𝑦𝑦 + be𝑧𝑧 better≥0 appreciated in this type of plot. We then the “speed limit” of ,reportedaboveforthecase followed the procedure outlined in [8, 11], to integrate the . us, this value of the parameter seems to be the most 𝜌𝜌 𝑚𝑚 local energy density over the proper volume, in cylindrical adequate for this type𝑠𝑠 of solutions. 𝑣𝑣 ≈ 2.50𝑐𝑐 coordinates at time over all space, obtaining the total 𝑚𝑚�푚 energy as 𝑡𝑡𝑡 erg (16) 4. Other Energy Conditions and Warp Drive 𝐸𝐸 3 00 10 Energy Estimate where𝐸𝐸𝐸𝐸�𝑔𝑔𝑐𝑐𝑐𝑐𝑐𝑐Det 𝑥𝑥is𝑔𝑔 the𝑇𝑇 determinant= 1.86 × 10 of𝛼𝛼 the𝑔𝑔 spatial, metric on the constant time hypersurface. Since we assume that the In the previous section, we have discussed at length the spaceship𝑔𝑔� is traveling|𝑔𝑔𝑖𝑖𝑖𝑖| at constant velocity, , the weak energy condition (WEC) for the conformal gravity total energy is also constant with time. In the last equation, we Alcubierre warp drive. We have seen that, if the Hartle reinstated a factor of to obtain the correct𝑣𝑣 dimensions𝑠𝑠 = 1.00𝑐𝑐 (see shaping function is used, this condition is not violated for a Section 2) and also inserted an overall multiplicative factor wide range of spaceship velocities, including super-luminal , which corresponds𝑐𝑐 to the conformal gravity coupling speeds. In this section, we will brie�y analyze the other constant in (9). is factor is necessary since our computation main energy conditions and estimate the energy necessary to of𝛼𝛼𝑔𝑔 in Figure 6 was done assuming . establish the warp drive in CG. erefore,00 we need to know the CG value for in order e dominant energy condition (DEC) is reported in the to𝑇𝑇 complete our energy estimation. Unfortunately,𝛼𝛼𝑔𝑔 =1 the value literature ([2–4]) as for any , , or equivalently as of this coupling constant is not well determined yet.𝛼𝛼𝑔𝑔 e only assuming the WEC plus00 the𝜇𝜇𝜇𝜇 additional condition that value in the literature is reported by Mannheim [30] as is a nonspacelike vector,𝑇𝑇 ≥ that |𝑇𝑇 is,| 𝜇𝜇 𝜈𝜈 .Itiseasyto𝜇𝜇𝜇𝜇 𝜇𝜇 erg s (17) see that using as a vector the 4-velocity𝜈𝜈 𝜇𝜇 𝜆𝜆 of the Eulerian𝑇𝑇 𝑡𝑡 𝜇𝜇𝜇𝜇 𝜆𝜆 82 𝑇𝑇 𝑇𝑇 𝑡𝑡 𝑡𝑡 ≤0 𝑔𝑔 𝑡𝑡𝜇𝜇 𝑛𝑛𝜇𝜇 𝛼𝛼 = 3.29 × 10 , ISRN Astronomy and Astrophysics 9
1 1 0 0 −1 −1 200 1500
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0 0 −1 −1 0 0 1 1
(a)
1 1 0 0 −1 −1 4000 10000
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(b)
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(c)
F 4: Energy density computed with conformal gravity and the Hartle shaping function. Parameters used are , , , , and variable � ��1�.00 In all cases, the �EC is veri�ed. 𝑠𝑠 𝑔𝑔 𝑇𝑇 𝑣𝑣 =1 𝑅𝑅𝑅 𝛼𝛼 =1 𝑡𝑡𝑡 𝑚𝑚 since this coupling constant has the dimensions of action. where we also converted this energy into equivalent mass and Inserting this value for in (16), we obtain the energy comparedourresultwiththesolarmass g. estimate as e estimate in (18) would imply that an enormous33 𝑔𝑔 𝛼𝛼 amount of (standard) mass or energy is𝑀𝑀 needed⊙ = 1.99 to establish× 10 erg our warp drive at a velocity equal to the speed of light, with 92 71 (18) a reasonable size for the warp bubble. However, the CG 𝐸𝐸 � 퐸퐸�퐸 퐸 𝐸 = 6.81 × 10 𝑔𝑔 38 = 3.4퐸 퐸 10 𝑀𝑀⊙, 10 ISRN Astronomy and Astrophysics
DEC in CG ( ), or possibly the number of baryons in the universe 1 ( 68)[41]. 0 𝑁𝑁erefore, 푁 𝑁80 our estimate could be reduced by many orders −1 of𝑁𝑁 magnitude. 푁 𝑁 Moreover, the energy necessary to establish the 2000 warp drive might also be decreased by using a more efficient shaping function, an analysis which we leave for a future study on the subject. 1000 5. Conclusions 0 In this paper, we have analyzed in detail the Alcubierre warp drive mechanism within the framework of conformal − 1000 gravity. We have seen that a particular choice of the shaping function (Hartle shaping function, instead of the original − 2000 Alcubierre one) can overcome the main limitation of the −1 AWD in standard general relativity, namely, the violation of 0 the weak energy condition. 1 In fact, we have shown that for a wide range of spaceship velocities, the CG solutions do not violate the WEC, and, F 5: Violation of the DEC in the case analyzed (CG with HSF therefore, the AWD mechanism might be viable, if CG is the and , , , , ). e plotted function correct extension of the current gravitational theories. All the components of the stress-energy tensor can be analytically is not negative𝑠𝑠 everywhere,𝑔𝑔 as required by the DEC, but shows0 𝑚𝑚푚푚0𝜆𝜆 a violation𝑅𝑅�푅 for𝑣𝑣 the푚 푅 central.00𝑐𝑐 𝛼𝛼 portion=1 𝑡𝑡 of𝑡 the warp bubble. calculated, using a Mathematica program based on conformal 𝜆𝜆 𝑇𝑇 𝑇𝑇 gravity. us, a warp drive can, at least in principle, be fully established following our computations. We have also checked two other main energy conditions: the SEC is always veri�ed, while the DEC is violated, at least 15000 10000 in the case we considered. Finally, we estimated the energy 5000 needed to establish a reasonable warp drive at the speed of × 10− 13 0 light. is energy depends critically on the value of , the 4 CG coupling constant, which is not well known. erefore, this estimate will need to be re�ned in future studies. 𝛼𝛼𝑔𝑔 3 Appendix 2 Energy Density Expression in Conformal Gravity 1 We present here the expression for the energy density 0 in conformal gravity, computed using our Mathematica00 − 10000 program. is is the general form of for any shaping𝑇𝑇 0 function and its derivatives, up to the00 fourth order. e 10000 energy density is a function of coordinates𝑇𝑇 , , , 𝑠𝑠 and, therefore,𝑓𝑓푓𝑓𝑓 ] it has a cylindrical symmetry around the - F 6: Energy density computed with conformal gravity and 2 2 axis. e distance is de�ned in (11), 𝑡𝑡 𝑥𝑥is𝜌𝜌 the� spaceship𝑦𝑦 +𝑧𝑧 the Hartle shaping function,00 plotted for . Parameters used are , , , , and . Integrating this velocity, and is the conformal gravity coupling constant:𝑥𝑥 𝑇𝑇 𝑠𝑠 𝑠𝑠 local energy density over all space, we obtain𝜌𝜌휌 an� estimate for the total 𝑟𝑟 𝑣𝑣 𝑠𝑠 𝑔𝑔 𝑔𝑔 𝑣𝑣energy=1𝑅𝑅required� 푅𝑅𝑅 to establish𝑚𝑚푚푚𝛼𝛼 the=1 warp drive.𝑡𝑡𝑡 𝛼𝛼 2 00 𝑠𝑠 2 2 2 2 𝐸𝐸 𝑣𝑣 𝑇𝑇 =𝛼𝛼𝑔𝑔 6 −4𝑟𝑟𝑠𝑠𝑣𝑣𝑠𝑠 𝜌𝜌 6𝑥𝑥𝑥𝑥�𝑠𝑠𝑡𝑡 + 5𝜌𝜌 −1+𝑓𝑓𝑟𝑟𝑠𝑠 3𝑟𝑟𝑠𝑠 value of is not well established, since the number in ′ 3 4 2 2 2 (17) represents only an estimate of the macroscopic value of ×𝑓𝑓𝑟𝑟𝑠𝑠 + 4𝑟𝑟𝑠𝑠 𝑣𝑣𝑠𝑠 𝑥𝑥𝑥𝑥�𝑠𝑠𝑡𝑡 + 3𝜌𝜌 𝑔𝑔 this coupling𝛼𝛼 constant. is does not need to be the same ′ 4 ′ 2 as the microscopic value associated with the fundamental ×𝑓𝑓𝑟𝑟𝑠𝑠 +𝑓𝑓𝑟𝑟𝑠𝑠 theory, which could be reduced by a factor of , where 4 2 could be the number of occupied baryonic states in a galaxy × −24𝑥𝑥𝑥𝑥�𝑠𝑠𝑡𝑡 +31+5𝑣𝑣𝑠𝑠 𝑁𝑁 𝑁𝑁 2 2 2 4 × 𝑥𝑥𝑥𝑥�𝑠𝑠𝑡𝑡 𝜌𝜌 + 27 + 10𝑣𝑣𝑠𝑠 𝜌𝜌 ISRN Astronomy and Astrophysics 11
2 2 2 2 ′′ 2 +𝑣𝑣𝑠𝑠 5𝜌𝜌 3𝑥𝑥푥𝑥𝑥𝑠𝑠𝑡𝑡 + 2𝜌𝜌 ×𝑓𝑓 𝑟𝑟𝑠𝑠 + 8𝑟𝑟𝑠𝑠 −1+𝑓𝑓𝑟𝑟𝑠𝑠 2 2 (3) 𝑠𝑠 × −2 + 𝑓𝑓 𝑟𝑟 × 2𝑥𝑥푥𝑥𝑥𝑠𝑠𝑡𝑡 +𝜌𝜌 𝑓𝑓 𝑟𝑟𝑠𝑠 2 2 𝑠𝑠 𝑠𝑠 𝑠𝑠 2 (4) (A.1) ×𝑓𝑓𝑟𝑟 − 4𝑟𝑟 𝑥𝑥푥𝑥𝑥𝑡𝑡 +𝑟𝑟𝑠𝑠𝑥𝑥푥𝑥𝑥𝑠𝑠𝑡𝑡 𝑓𝑓 𝑟𝑟𝑠𝑠 . 2 2 × 4𝑥𝑥푥𝑥𝑥𝑠𝑠𝑡𝑡 + 3𝜌𝜌 Inthemainpartofourwork,weusedtheHartleshaping ′′ function in (13), or more explicitly as × −1+𝑓𝑓𝑟𝑟𝑠𝑠 𝑓𝑓 𝑟𝑟𝑠𝑠 ′ 2 2 𝑠𝑠 𝑠𝑠 𝑠𝑠 − 2𝑟𝑟 𝑓𝑓 𝑟𝑟 16𝑥𝑥푥𝑥𝑥𝑡𝑡 − 8𝜌𝜌 + 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑠𝑠 𝑠𝑠 4 𝑠𝑠 𝑠𝑠 𝑟𝑟 1−𝑟𝑟 /𝑅𝑅 + 1−𝑟𝑟 /𝑅𝑅(A.2) 𝑠𝑠 𝑓𝑓 𝑟𝑟 = 1− ≡ , + −60𝑥𝑥푥𝑥𝑥𝑡𝑡 𝑅𝑅 2 2 2 2 where indicates the positive part of the function. e 𝑠𝑠 𝑠𝑠 + −97 + 5𝑣𝑣 𝑥𝑥푥𝑥𝑥𝑡𝑡 𝜌𝜌 derivatives+ of the HSF, up to the fourth order, were computed 2 4 ′′ in terms[⋯] of the Heaviside step function and the Dirac + −37 + 3𝑣𝑣𝑠𝑠 𝜌𝜌 𝑓𝑓 𝑟𝑟𝑠𝑠 delta function , also using the relations for the derivatives 2 of these special functions 𝐻𝐻𝐻𝐻𝐻 ; 𝑠𝑠 𝑠𝑠 +𝑟𝑟 4 −4+𝑣𝑣 𝛿𝛿𝛿𝛿𝛿; as follows: 4 2 𝑛𝑛 𝑛𝑛 𝑑𝑑푑𝑑𝑑푑�𝑑𝑑𝑛𝑛𝑑𝑑𝑑 푑푑푑𝑛𝑛 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑠𝑠 𝑠𝑠 × 𝑥𝑥푥𝑥𝑥𝑡𝑡 +3−9+𝑣𝑣 𝑑𝑑𝑑𝑑𝑑𝑑𝑑�𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑� 𝑥𝑥 𝑑𝑑 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿� = (−1) 𝑛𝑛푛 𝑛𝑛𝑛𝑛𝑛 2 2 4 𝑚𝑚𝑚 𝑚𝑚 𝑠𝑠 ′ 𝑠𝑠 𝑠𝑠 ×𝑥𝑥푥𝑥𝑥𝑡𝑡 𝜌𝜌 − 11𝜌𝜌 𝑠𝑠 𝑚𝑚 𝑟𝑟 𝑟𝑟 (3) 2 2 𝑓𝑓 𝑟𝑟 =− 𝐻𝐻 1− 𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑅𝑅 𝑅𝑅 𝑅𝑅 ×𝑓𝑓 𝑟𝑟 +𝑣𝑣 𝑓𝑓𝑟𝑟 𝑚𝑚𝑚 𝑚𝑚 ′′ 𝑠𝑠 𝑠𝑠 2 2 2 𝑠𝑠 𝑚𝑚 (𝑚𝑚𝑚) 𝑟𝑟 𝑟𝑟 𝑠𝑠 𝑓𝑓 𝑟𝑟 =− 2 𝐻𝐻 1− × 𝜌𝜌 5𝑥𝑥푥𝑥𝑥𝑡𝑡 + 3𝜌𝜌 𝑅𝑅 𝑅𝑅 𝑅𝑅 ′′ 2 2 2𝑚𝑚𝑚 𝑚𝑚 𝑠𝑠 𝑠𝑠 ×𝑓𝑓 𝑟𝑟𝑠𝑠 +𝑟𝑟𝑠𝑠𝑥𝑥푥𝑥𝑥𝑠𝑠𝑡𝑡 𝑚𝑚 𝑟𝑟 𝑟𝑟 + 2 𝛿𝛿 1− 2 2 𝑅𝑅 𝑅𝑅 𝑠𝑠 𝑅𝑅 × 4𝑥𝑥푥𝑥𝑥𝑡𝑡 + 3𝜌𝜌 𝑚𝑚𝑚 𝑚𝑚 (3) 𝑠𝑠 𝑠𝑠 (3) 𝑠𝑠 𝑚𝑚 (𝑚𝑚𝑚)(3 𝑚𝑚𝑚) 𝑟𝑟 𝑟𝑟 𝑠𝑠 𝑠𝑠 𝑓𝑓 𝑟𝑟 =− 𝐻𝐻 1− ×𝑓𝑓 𝑟𝑟 + 2𝑓𝑓 𝑟𝑟 𝑅𝑅 𝑅𝑅 𝑅𝑅 2 2 2 2 2𝑚𝑚𝑚 𝑚𝑚 𝑠𝑠 𝑠𝑠 × −8 𝑥𝑥푥𝑥𝑥𝑠𝑠𝑡𝑡 + 4𝜌𝜌 +𝑣𝑣𝑠𝑠 3𝑚𝑚 (𝑚𝑚𝑚) 𝑟𝑟 𝑟𝑟 + 3 𝛿𝛿 1− 2 2 2 𝑅𝑅 𝑅𝑅 𝑅𝑅 𝑠𝑠 𝑚𝑚 × −𝜌𝜌 5𝑥𝑥푥𝑥𝑥𝑡𝑡 + 3𝜌𝜌 3 3𝑚𝑚𝑚 𝑠𝑠 𝑠𝑠 2 𝛿𝛿 1−𝑟𝑟 /𝑅𝑅 ′′ 𝑚𝑚3 𝑟𝑟 𝑚𝑚 𝑠𝑠 𝑠𝑠 𝑠𝑠 + ×𝑓𝑓 𝑟𝑟 −𝑟𝑟𝑥𝑥푥𝑥𝑥𝑡𝑡 𝑅𝑅 𝑅𝑅 1−𝑟𝑟𝑠𝑠/𝑅𝑅 2 2 𝑚𝑚𝑚 × 4𝑥𝑥푥𝑥𝑥𝑠𝑠𝑡𝑡 + 3𝜌𝜌 (4) 𝑠𝑠 𝑠𝑠 𝑚𝑚 (𝑚𝑚𝑚)(𝑚𝑚4𝑚)(𝑚𝑚𝑚) 𝑟𝑟 (3) 𝑓𝑓 𝑟𝑟 =− 𝑠𝑠 𝑅𝑅𝑚𝑚 𝑅𝑅 ×𝑓𝑓 𝑟𝑟 𝑟𝑟𝑠𝑠 2 2 2 ×𝐻𝐻1− 𝑠𝑠 𝑠𝑠 𝑅𝑅 +𝑟𝑟 −16 2𝑥𝑥푥𝑥𝑥𝑡𝑡 −𝜌𝜌 2 2𝑚𝑚𝑚 ′′ 𝑚𝑚 (𝑚𝑚𝑚)(7𝑚𝑚 𝑚𝑚) 𝑟𝑟𝑠𝑠 𝑠𝑠 𝑠𝑠 + 4 × −1+𝑓𝑓𝑟𝑟 𝑓𝑓 𝑟𝑟 𝑅𝑅 𝑅𝑅 2 4 𝑚𝑚 𝑠𝑠 + 4 6+𝑣𝑣𝑠𝑠 𝑥𝑥푥𝑥𝑥𝑠𝑠𝑡𝑡 𝑟𝑟 ×𝛿𝛿1− 2 2 2 𝑅𝑅 𝑠𝑠 𝑠𝑠 𝑚𝑚 + 43 + 3𝑣𝑣 𝑥𝑥푥𝑥𝑥𝑡𝑡 𝜌𝜌 3 3𝑚𝑚𝑚 4 2 2 𝑠𝑠 𝛿𝛿 1−𝑟𝑟𝑠𝑠/𝑅𝑅 𝑠𝑠 𝑠𝑠 6𝑚𝑚 (𝑚𝑚4 𝑚) 𝑟𝑟 𝑚𝑚 + 19𝜌𝜌 +𝑣𝑣 𝑥𝑥푥𝑥𝑥𝑡𝑡 + 𝑠𝑠 2 2 𝑅𝑅 𝑅𝑅 1−𝑟𝑟 /𝑅𝑅 𝑠𝑠 4 𝑚𝑚 × 4𝑥𝑥푥𝑥𝑥𝑡𝑡 + 3𝜌𝜌 4𝑚𝑚𝑚 𝑠𝑠 2𝑚𝑚 𝑟𝑟𝑠𝑠 𝛿𝛿 1−𝑟𝑟 /𝑅𝑅 4 𝑚𝑚 2 × −2+𝑓𝑓𝑟𝑟𝑠𝑠 𝑓𝑓 𝑟𝑟𝑠𝑠 + . (A.3) 𝑅𝑅 𝑅𝑅 1−𝑟𝑟𝑠𝑠/𝑅𝑅 12 ISRN Astronomy and Astrophysics
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