Conformal Gravity and the Alcubierre Warp Drive Metric

We present an analysis of the classic Alcubierre metric based on conformal gravity, rather than standard general relativity. The main characteristics of the resulting warp drive remain the same as in the original study by Alcubierre, namely that effective super-luminal 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. Therefore, if conformal gravity is a correct extension of general relativity, super-luminal motion via an Alcubierre metric might be a realistic solution, thus allowing faster-than-light interstellar travel.

i.e., faster-than-light travel [1]. This super-luminal propulsion is achieved by respectively expanding and contracting the space-time behind and in front of a spaceship, while the spacecraft is left inside a locally flat region of space-time, within the so-called warp bubble.
In this way the spaceship can travel at arbitrarily high speeds, without violating the laws of special and general relativity, or other known physical laws. Furthermore, the spacecraft and its occupants would also be at rest in flat space-time, thus immune from high accelerations and unaffected by special relativistic effects, such as time dilation. Enormous tidal forces would only be present near the edge of the warp bubble, which can be made large enough to accommodate the volume occupied by the ship.
However, Alcubierre [1] was also the first to point out that this hypothetical solution of Einstein's equations of GR would violate all three standard energy conditions (weak, dominant and strong; see [2], [3], and [4] for definitions). In particular, the violation of the weak energy condition (WEC) implies that negative energy density is required to establish the Alcubierre WDM, thus making it practically impossible to achieve this type of superluminal motion, unless large quantities of exotic matter (i.e., with negative energy density) can be created. Since our current knowledge of this type of exotic matter is limited to some special effects in quantum field theory (such as the Casimir effect), it is unlikely that the Alcubierre WDM can be practically established within the framework of General Relativity.
Einstein's General Relativity and the related "Standard Model" of Cosmology have been highly successful in describing our Universe, from the Solar System up to the largest cosmological scales, but recently these theories have also led to a profound crisis in our understanding of its ultimate composition. From the original discovery of the expansion of the Universe, which resulted in standard Big Bang Cosmology, scientists have progressed a long way towards our current picture, in which the contents of the Universe are today described in terms of two main components, dark matter (DM) and dark energy (DE), accounting for most of the observed Universe, with ordinary matter just playing a minor role.
Since there is no evidence available yet as to the real nature of dark matter and dark energy, alternative gravitational and cosmological theories are being developed, in addition to standard explanations of dark matter/dark energy invoking the existence of exotic new particles also yet to be discovered. In line with these possible new theoretical ideas, Conformal Gravity (CG) has emerged as a non-standard extension of Einstein's GR, based on a possible symmetry of the Universe: the conformal symmetry, i.e., the invariability of the space-time fabric under local "stretching" of the metric (for reviews see [21], [22]). This alternative theory has been re-introduced in recent years (following the original work by H.

II. CONFORMAL GRAVITY AND THE STRESS-ENERGY TENSOR
H. Weyl in 1918 ([23], [24], [25]) developed the "conformal" generalization of Einstein's relativity by introducing the conformal (or Weyl) tensor, a special combination of the Riemann tensor R λµνκ , the Ricci tensor R µν = R λ µλν and the curvature (or Ricci) scalar R = R µ µ [33]: where C λ µλν (x) is invariant under the local transformation of the metric: The factor Ω(x) = e α(x) determines the amount of local "stretching" of the geometry, hence the name "conformal" for a theory invariant under all local stretchings of the space-time (see [22] and references therein for more details).
This conformally invariant generalization of GR was found to be a fourth-order theory, as opposed to the standard second-order General Relativity, since the field equations originating from a conformally invariant Lagrangian contain derivatives up to the fourth order of the metric, with respect to the space-time coordinates. Following work done by R. Bach [34], C. Lanczos [35] and others, CG was ultimately based on the Weyl or conformal action: 1 1 In this paper we adopt a metric signature (-,+,+,+) and we follow the sign conventions of Weinberg [33].
or on the following equivalent expression, differing from the previous one only by a topological invariant: where g ≡ det(g µν ) and α g is the gravitational coupling constant of Conformal Gravity (see [36], [37], [38], [39]). 2 Under the conformal transformation in Eq. (2), the Weyl tensor transforms as C λµνκ → C λµνκ = e 2α(x) C λµνκ = Ω 2 (x)C λµνκ , while the conformal action I W is locally conformally invariant, the only general coordinate scalar action with such properties.
R. Bach [34] introduced the gravitational field equations in the presence of a stress-energy tensor 3 T µν : as opposed to Einstein's standard equations, where the "Bach tensor" W µν [34] is the equivalent in CG of the Einstein curvature tensor G µν on the left-hand side of Eq. (6).
W µν has a very complex structure and can be defined in a compact way as [40]: or in an expanded form as ( [38], [41]): In this section we will leave fundamental constants, such as c and G, in all equations, but later we will use geometrized units (c = 1, G = 1), or c.g.s. units when needed. 2 In these cited papers, α g is considered a dimensionless constant by using natural units. Working with c.g.s. units, we can assign dimensions of an action to the constant α g so that the dimensionality of Eq. (5) will be correct. 3 We follow here the convention [36] of introducing the stress-energy tensor T µν so that the quantity cT 00 has the dimensions of an energy density.
involving derivatives up to the fourth order of the metric with respect to space-time coordinates.
Therefore, in Conformal Gravity, the stress-energy tensor is computed by combining together Eqs. (5) and (8): This form of the tensor will be used in the following sections, in connection to the Alcubierre metric, to compute the energy density and other relevant quantities.
For this purpose, we have developed a special Mathematica program which enables us to compute all the tensor quantities of both GR and CG, for any given metric. In particular, this program can compute the conformal tensor C λµνκ in Eq. [39] for different metrics, obtaining a perfect agreement.

ERGY CONDITION
The original Alcubierre metric [1] considered a spaceship traveling along the x-axis, with motion described by a function x s (t) and spaceship velocity v s (t) = dxs(t) dt . Using the 3+1 formalism of GR, the metric was written in Cartesian coordinates as (c = 1): where r s is the distance from the spaceship position: and f (r s ) is a "form function" or "shaping function" which needs to have values f = 1 and f = 0 respectively inside and outside the warp bubble, while it can have an arbitrary shape in the transition region of the warp bubble itself.
The original shaping function used by Alcubierre was: where R > 0 basically indicates the radius of the spherical warp bubble, while σ > 0 relates to the bubble thickness, which decreases with increasing values of σ. In the following, we will refer to the function in Eq. (12) as the "Alcubierre shaping function" (ASF).
We will show that the particular form of the shaping function can play an important role in the energy conditions for the WDM. In our analysis we tested several different functions obeying the general requirements for f outlined above. In addition to the Alcubierre function above, in this paper we will also use the following: where m is a positive integer. Since this particular function for m = 4 is used by J. Hartle to illustrate the warp drive in his textbook [20], we will refer to the function in Eq. (13) as the "Hartle shaping function" (HSF).
The top panels in Fig. 1 illustrate the differences between the Alcubierre shaping function The cylindrical coordinate ρ = y 2 + z 2 should be considered as non-negative and all quantities in the figures plotted only for ρ ≥ 0. However, for illustrative purposes and also to follow similar figures in the literature (such as those in [1], [11], [12], etc.), we decided to let ρ run on negative values in all figures, except in the last one, where we restrict ρ ≥ 0 for a correct energy calculation.
The expansion/contraction function θ of the volume elements behind/in front of the spaceship was also computed by Alcubierre as [1]: and is illustrated for v s = 1 (in geometrized units, i.e., v s = c in traditional units) in the second row of Fig. 1, for the two different shaping functions. Again, the choice of the parameters R, σ, and m is the same as in the top panels in the figure. The expansion θ for the ASF is the same as Fig. 1 in Ref. [1], while the corresponding θ for the HSF is slightly different, but still shows expansion of the normal volume elements behind the spaceship and contraction in front of it.
The weak energy condition ( [2], [3], [4]) requires that T µν t µ t ν ≥ 0 for all timelike vectors t µ . Alcubierre has also shown that for the Eulerian observers in the warp drive metric, and for their 4-velocity n µ , the following relation holds [1]: which implies that the energy density T 00 is negative everywhere for any choice of the shaping function f and, therefore, the WEC is violated (also the dominant energy condition -DECand the strong energy condition -SEC-are violated in the analysis based on GR [1]).
This violation of the WEC in GR is illustrated in the third row of Fig. 1, where T 00 is calculated using Eq. (15) for both shaping functions. Although the results in the two panels are slightly different, they obviously show negative energy densities and therefore a complete violation of the WEC.
The situation is different if we compute the energy density T 00 in the framework of CG, following Eq. (9), setting α g = 1 for simplicity, and using the completely contravariant form of the stress-energy tensor, instead of the covariant one. As seen in the bottom row of This apparent "speed limit" at about v s ≈ 2.50 c might be raised or overcome completely by adopting a different shaping function, instead of the HSF used here, but this analysis would go beyond the scope of this work. In any case, the results reported in Fig. 2 show that a warp drive in CG with positive energy density is possible for a wide range of spaceship velocities; therefore, if CG is the correct extension of GR, the Alcubierre warp drive might be a viable mechanism for super-luminal travel.
In Fig. 3 we present the other components of the stress-energy tensor. These were computed with the same Mathematica program, following Eq. (9) with α g = 1, leading to even more complex expressions than the one for T 00 (we will omit to report these expressions for brevity). To simplify the computation, we used the covariant components T µν and adopted cylindrical coordinates around the x-axis, (t, x, ρ, φ) ≡ (0, 1 Rather, Fig. 3 shows that a "bias" towards one of the two possible directions is induced by some components of T µν . Fig. 4 illustrates one last dependence of the energy density T 00 on the parameters used.
In this case, we set R = 1, v s = 1.00 c and consider the Hartle shaping function as in Eq. We also checked that changing the value of m does not have a strong effect on the "speed limit" of v s ≈ 2.50 c, reported above for the case m = 4. Thus, this value of the parameter seems to be the most adequate for this type of solutions.

IV. OTHER ENERGY CONDITIONS AND WARP DRIVE ENERGY ESTIMATE
In the previous section we have discussed at length the weak energy condition -WECfor the Conformal Gravity Alcubierre warp drive. We have seen that, if the Hartle shaping function is used, this condition is not violated for a wide range of spaceship velocities, including super-luminal speeds. In this section we will briefly analyze the other main energy conditions and estimate the energy necessary to establish the warp drive in CG.
The dominant energy condition -DEC-is reported in the literature ( [2], [3], [4]) as T 00 ≥ |T µν | for any µ, ν, or equivalently as assuming the WEC plus the additional condition that T µν t µ is a non-spacelike vector, i.e., T µν T ν λ t µ t λ ≤ 0. It is easy to see that using as a vector t µ the 4-velocity n µ of the Eulerian observers [1], the previous condition for the DEC becomes T 0 λ T 0λ ≤ 0. Figure 5 illustrates the violation of the DEC for our standard solution (AWD with HSF and m = 4, R = 1, v s = 1.00 c). The plotted function T 0 λ T 0λ is not negative everywhere, as required by the DEC, but shows a violation for the central portion of the warp bubble. Even if this energy condition appears to be violated, this does not notably affect the feasibility of our CG warp drive. We recall that the DEC is usually related to the standard perfect fluid stress-energy tensor, T µν = (ρ + p)U µ U ν + pg µν , where here ρ and p are the fluid density and pressure, while U µ is the fluid 4-velocity. In this context the DEC requires ρ ≥ |p|, but this condition is not required in general by all classical forms of matter [3]; therefore, its violation in our case is not particularly significant.
On the contrary, our standard solution also verifies the strong energy condition -SECwhich states that T µν t µ t ν ≥ 1 2 T λ λ t σ t σ for all timelike vectors t µ . Again, using the Eulerian 4-velocity vector in place of t µ , the previous condition is equivalent to T 00 + 1 2 T λ λ ≥ 0. Since the scalar T λ λ is identically zero for all our solutions, as checked using our Mathematica program, the SEC is equivalent to T 00 ≥ 0, which is the WEC already verified in Sect. III.
Finally, we want to estimate the energy necessary to establish our CG warp drive, under reasonable conditions. For this purpose, in Fig. 6 we computed once again the energy density T 00 for our AWD with the Hartle shaping function (α g = 1, m = 4, v s = 1.00 c), but this time for R = 10000 cm = 100 m, a reasonable radius for a warp bubble enclosing our spaceship. Figure 6 illustrates this solution, plotted only for ρ = y 2 + z 2 ≥ 0, as this is the correct interval for the transverse coordinate ρ. The cylindrical symmetry of this solution can also be better appreciated in this type of plot. We then followed the procedure outlined in Refs. [8], [11], to integrate the local energy density over the proper volume, in cylindrical coordinates at time t = 0 over all space, obtaining the total energy E: where g = Det |g ij | is the determinant of the spatial metric on the constant time hypersurface. Since we assume that the spaceship is traveling at constant velocity, v s = 1.00 c, the total energy is also constant with time. In the last equation, we reinstated a factor of c to obtain the correct dimensions (see footnote before Eq. (5)) and also inserted an overall multiplicative factor α g , which corresponds to the conformal gravity coupling constant in Eq. (9). This factor is necessary since our computation of T 00 in Fig. 6 was done assuming α g = 1.
Therefore, we need to know the CG value for α g in order to complete our energy estimation. Unfortunately, the value of this coupling constant is not well determined yet. The only value in the literature is reported by P. Mannheim [30]: since this coupling constant has the dimensions of action. Inserting this value for α g in Eq.
The estimate in Eq. (18) would imply that an enormous amount of (standard) massenergy is needed to establish our warp drive at a velocity equal to the speed of light, with a reasonable size for the warp bubble. However, the CG value of α g is not well-established, since the number in Eq. (17)  Therefore, our estimate could be reduced by many orders of magnitude. Moreover, the energy necessary to establish the 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.

V. CONCLUSIONS
In this paper we have analyzed in detail the Alcubierre warp drive mechanism within the framework of Conformal Gravity. We have seen that a particular choice of the shaping function (Hartle shaping function, instead of the original Alcubierre one) can overcome the main limitation of the AWD in standard General Relativity, namely the violation of the weak energy condition.
In fact, we have shown that for a wide range of spaceship velocities, the CG solutions do not violate the WEC, and, therefore, the AWD mechanism might be viable, if CG is the correct extension of the current gravitational theories. All the components of the stress-energy tensor can be analytically calculated, using a Mathematica program based on Conformal Gravity. Thus, 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 verified, while the DEC is violated, at least in the case we considered. Finally, we estimated the energy needed to establish a reasonable warp drive at the speed of light. This energy depends critically on the value of α g , the CG coupling constant, which is not well known. Therefore, this estimate will need to be refined in future studies.      Integrating this local energy density over all space, we obtain an estimate for the total energy E required to establish the warp drive.