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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, 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. Therefore, 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.

In 1994, Alcubierre introduced the so-called

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 [

Following Alcubierre’s seminal paper, many other studies appeared in the literature, either proposing alternatives to the original warp drive mechanism ([

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,

At the quantum level conformal gravity, as well as other theories with higher derivatives, was thought to be affected by the presence of “ghosts,” leading to possible instabilities of the quantum version of the theory. However, recent studies [

In view of a possible extension of Einstein’s general relativity into conformal gravity, in this paper we have reconsidered the Alcubierre WDM, basing it on CG rather than standard GR. In Section

In particular, we will show that for certain shaping functions, the Alcubierre metric in the context of conformal gravity does not violate the weak energy condition, as was the case of the original solution. This analysis continues in Section

Weyl in 1918 ([

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 the works done by Bach [

Bach [

Therefore, in conformal gravity, the stress-energy tensor is computed by combining together (

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

The original Alcubierre metric [

The original shaping function used by Alcubierre was

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

The top panels in Figure

Results for the two different shaping functions (left column ASF, right column HSF), computed with parameters

The expansion/contraction function

The weak energy condition ([

This violation of the WEC in GR is illustrated in Figure

The situation is different if we compute the energy density

However, the bottom right panel shows

The explicit expression of

In Figure

Energy density

This apparent “speed limit” at about

In any case, the results reported in Figure

In Figure

Stress-energy tensor components

The shapes of the other components of

We also want to point out that we set the spaceship motion in the positive direction of the

Figure

Energy density

In general, increasing the

In the previous section, we have discussed at length the weak energy condition (WEC) for 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 ([

Figure

Violation of the DEC in the case analyzed (CG with HSF and

On the contrary, our standard solution also verifies the strong energy condition (SEC) which states that

Finally, we want to estimate the energy necessary to establish our CG warp drive, under reasonable conditions. For this purpose, in Figure

Energy density

Figure

Therefore, we need to know the CG value for

The estimate in (

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.

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

We present here the expression for the energy density

In the main part of our work, we used the Hartle shaping function in (

By inserting these derivatives into (

This work was supported by a grant from the Frank R. Seaver College of Science and Engineering, Loyola Marymount University. The authors would like to acknowledge suggestions and clarifications by Dr. P. Mannheim and also thank the anonymous reviewers for the useful comments received.