Conformal Gravity and the Alcubierre Warp Drive Metric

Conformal Gravity and the Alcubierre Warp Drive Metric

Gabriele U. Varieschi and Zily Burstein, Department of Physics, Loyola Marymount University, Los Angeles, California.

Received 7 November 2012; accepted 24 November 2012.

Abstract

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.

The tensor equations of this paper are given below as named results in words, with the article's own equation numbers; the full expressions and the six figures are at the source.

1. Introduction

In 1994, Alcubierre introduced the so-called Warp Drive Metric (WDM), within the framework of general relativity (GR), which allows in principle for super-luminal motion, that is, faster-than-light travel. This superluminal propulsion is achieved, respectively, by 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 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. 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 super-luminal motion, unless large quantities of exotic matter, that is, 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.

Following Alcubierre's seminal paper, many other studies appeared in the literature, either proposing alternatives to the original warp drive mechanism or refining and analyzing in more detail the original idea. However, all these studies were conducted using standard GR and could not avoid the violation of the WEC, meaning that some exotic matter would always be required for faster-than-light travel. Similar issues also exist in other well-known GR solutions for super-luminal motion, such as space-time wormholes.

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 and dark energy, 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 and 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 nonstandard extension of Einstein's GR, based on a possible symmetry of the universe: the conformal symmetry, that is, the invariability of the space-time fabric under local "stretching" of the metric. This alternative theory has been reintroduced in recent years, following the original work by Weyl, leading to cosmological models which do not require the existence of dark matter and dark energy.

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 have shown that CG as a quantum theory is both renormalizable and unitary, thus providing a solution to the ghost problem.

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 2, we review the fundamental principles of CG and the calculation of the stress-energy tensor in this gravitational theory. In Section 3, we consider the Alcubierre metric in CG and compute the energy density for different shaping functions of the metric.

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 4, where we study other energy conditions and estimate the total energy required for this CG warp drive. Finally, in Section 5, we conclude that if CG is a correct extension of GR, super-luminal motion via an Alcubierre metric might be a realistic possibility, thus enabling faster-than-light interstellar travel without requiring exotic matter.

2. Conformal Gravity and the Stress-Energy Tensor

Weyl in 1918 developed the "conformal" generalization of Einstein's relativity by introducing the conformal, or Weyl, tensor: equation (1) builds it as a special combination of the Riemann tensor, the Ricci tensor and the curvature scalar, subtracting the metric-weighted Ricci terms at one half and adding the metric-weighted curvature scalar at one sixth. The Weyl tensor with one index raised is invariant under the local transformation of the metric given in equation (2), in which the metric is multiplied by the square of a position-dependent conformal factor, the exponential of twice a scalar function of position. That factor 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.

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, Lanczos, and others, CG was ultimately based on the Weyl or conformal action of equation (3), the integral over space-time of the square root of minus the metric determinant times the full contraction of the Weyl tensor with itself, weighted by minus the gravitational coupling constant of conformal gravity; or on the equivalent expression of equation (4), differing from the previous one only by a topological invariant, in which the contraction of the Weyl tensor with itself is replaced by the contraction of the Ricci tensor with itself minus one third of the squared curvature scalar. In this paper we adopt a metric signature with one minus and three pluses and follow the sign conventions of Weinberg; in this section we leave fundamental constants in all equations, but later we use geometrized units, or centimetre-gram-second units when needed. Under the conformal transformation, the Weyl tensor picks up the same conformal factor, while the conformal action is locally conformally invariant — the only general coordinate scalar action with such properties.

Bach introduced the gravitational field equations in the presence of a stress-energy tensor, equation (5): the Bach tensor equals the stress-energy tensor divided by four times the gravitational coupling constant. This is to be compared with Einstein's standard equations, equation (6), where the Einstein curvature tensor equals eight pi times Newton's constant over the cube of the speed of light, times the stress-energy tensor. The "Bach tensor" is the equivalent in CG of the Einstein curvature tensor on the left-hand side of the Einstein equations. We follow the convention of introducing the stress-energy tensor so that the speed of light times its time-time component has the dimensions of an energy density.

The Bach tensor has a very complex structure and can be defined in a compact way as equation (7), twice the second covariant derivative of the Weyl tensor plus the Weyl tensor contracted with the Ricci tensor, or in the expanded form of equation (8), which involves 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 equations (5) and (8) into the master equation (9). 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 derivatives. Given the complexity of these types of computations, we have tested extensively our program against results published by Mannheim for different metrics, obtaining a perfect agreement.

3. Alcubierre Metric, Shaping Functions, and the Weak Energy Condition

The original Alcubierre metric considered a spaceship travelling along the x-axis with an arbitrary velocity. In the three-plus-one formalism of GR, the metric was written in Cartesian coordinates as equation (10), with the ship's trajectory and the transverse distance from it defined in equations (11) and (12). The metric contains a shaping function of that distance, which must equal one at the ship and fall to zero outside the warp bubble, while it can have an arbitrary shape in the transition region of the warp bubble itself. We will refer to the function in equation (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 shaping functions. In addition to the Alcubierre function above, in this paper we will also use equation (13), a piecewise function built from the transverse distance and the bubble radius, raised to an even power set by a positive integer. Since this particular function for that integer equal to two is used by Hartle to illustrate the warp drive in his textbook, we will refer to the function in equation (13) as the "Hartle shaping function" (HSF).

Figure 1. Top panels: the Alcubierre shaping function and the Hartle shaping function, plotted over the plane containing the ship at the origin. Second row: the expansion of the volume elements for the two profiles, showing expansion of the normal volume elements behind the spaceship and contraction in front of it. Third row: the energy density in general relativity, negative for both profiles. Bottom row: the energy density in conformal gravity — part positive and part negative for the Alcubierre profile, and completely non-negative for the Hartle profile.

The top panels in Figure 1 illustrate the differences between the Alcubierre shaping function and the Hartle shaping function. All quantities shown in the different panels are plotted as a function of the coordinates with the spaceship located at the origin. The expansion of the volume elements behind and in front of the spaceship was also computed by Alcubierre as equation (14), and is shown in the second row of the figure; it shows expansion of the normal volume elements behind the spaceship and contraction in front of it.

The weak energy condition requires that the energy density measured by any observer be non-negative. Alcubierre showed that for the Eulerian observers in the warp drive the relation of equation (15) holds — the energy density is proportional to minus the square of the ship's velocity times the squared transverse gradient of the shaping function, and is therefore negative everywhere it does not vanish — so the WEC is violated; the dominant and strong energy conditions are violated as well in the analysis based on GR.

This violation of the WEC in GR is illustrated in the third row of Figure 1 for the two 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 in conformal gravity, setting the coupling constant to one 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 Figure 1, the energy density in CG is completely different from the one calculated within general relativity. The bottom left panel is computed with the ASF and the resulting function is in part positive and in part negative, thus still violating the WEC. The bottom right panel is computed with the HSF and in this case the energy density is completely non-negative, showing that the WEC is verified and no exotic matter is needed to establish the warp drive.

In this panel of Figure 1, and also in the other figures, we show a step discontinuity at the edge of the warp bubble, common to all our CG solutions. These discontinuities can be mathematically replaced by appropriate delta functions, as described by the equations used in the Appendix to model the Hartle shaping function; however, these delta functions do not alter the non-negative character of the energy densities in CG shown in the figures. Therefore, these energy densities are completely non-negative and they fully determine the expansion of the volume elements required for the warp drive effect.

This non-negative energy density plot in the bottom right panel of Figure 1 is the main result of our paper as it shows that — if CG is the correct extension of GR — it might be possible to establish a warp drive without having to use negative energy or mass, thus overcoming the main difficulty of the warp drive mechanism.

The complete analytical expression of the energy density, obtained with our Mathematica program, is rather cumbersome and is reproduced in the Appendix. We have tested the validity of these results by running the program in several different ways, including computing the stress-energy tensor for a simplified case. The difference between the two plots at the bottom of the figure can be attributed to the different shaping functions and their derivatives up to the fourth order. All these derivatives enter the complex expression of the stress-energy tensor in CG, as in the master equation (9), and their interplay ultimately determines the shape of the energy density, or of the other components, in a way which is hard to predict before the actual computation is performed.

Figure 2. The conformal-gravity energy density for the Hartle shaping function, with the integer parameter equal to two and the bubble radius equal to one, at ship velocities from a quarter of the speed of light to three times the speed of light. The energy density becomes in part negative above about two and a half times the speed of light, so the weak energy condition is verified for speeds up to about that value.

In Figure 2, we analyze the dependence of the energy density on the spaceship velocity. In this case we consider only the Hartle shaping function with the integer parameter equal to two and the bubble radius equal to one, and we compute the energy density in CG, with the coupling constant set to one, for speeds ranging from a quarter of the speed of light to three times the speed of light; at exactly the speed of light we obtain the same function as in the bottom right panel of Figure 1. The shape of the energy density function is about the same for speeds up to one and a half times the speed of light, although the function values increase with speed. For higher velocities, the function develops two "downward lobes" which eventually become negative for speeds above about two and a half times the speed of light. This implies that the WEC is verified for speeds up to about two and a half times the speed of light, while at higher velocities exotic matter would be required to sustain the warp drive.

This apparent "speed limit" 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. We do not ascribe any physical significance to this new speed limit, which is probably just an artificial product of the shaping functions used in this analysis.

In any case, the results reported in Figure 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 Figure 3, we present the other components of the stress-energy tensor. These were computed with the same Mathematica program, following the master equation with the coupling constant set to one, leading to even more complex expressions than the one for the energy density, which we omit for brevity. To simplify the computation, we used the covariant components and adopted cylindrical coordinates around the x-axis, instead of the Cartesian coordinates of the original Alcubierre metric. The results in the different panels are labeled accordingly. We recall that the stress-energy tensor is symmetric, and only the non-zero components are shown. All the components can be computed analytically, either in covariant or contravariant form, using our program. If this exact form of the stress-energy tensor could be established in the region surrounding the spacecraft, warp drive motion would become possible.

4. Other Energy Conditions and Warp Drive Energy Estimate

In the previous section, we have discussed at length the weak energy condition 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.

Figure 5. Violation of the dominant energy condition in the case analyzed, conformal gravity with the Hartle shaping function: the plotted quantity is non-negative over most of the region but shows a violation for the central portion of the warp bubble.

The dominant energy condition is reported in the standard references; for the Eulerian observers of the warp drive the condition takes the reduced form used here. Figure 5 illustrates the violation of the DEC for our solution: the relevant quantity is non-negative almost 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 and is not required in general by all classical forms of matter; therefore, its violation in our case is not particularly significant.

On the contrary, our standard solution also verifies the strong energy condition. The SEC is equivalent to the energy density plus one half of the trace of the stress-energy tensor being non-negative. Since that trace is identically zero for all our solutions, as checked using our Mathematica program, the SEC is equivalent to the energy density being non-negative, which is the WEC already verified in Section 3.

Finally, we want to estimate the energy necessary to establish our CG warp drive, under reasonable conditions. For this purpose, in Figure 6 we computed once again the energy density for our warp drive with the Hartle shaping function, at unit coupling constant, integer parameter two and a ship velocity of exactly the speed of light, but this time for a bubble radius of one hundred metres — a reasonable radius for a warp bubble enclosing our spaceship.

Figure 6 illustrates this solution, plotted only for non-negative values of the transverse cylindrical radius, as this is the correct interval for that coordinate. The cylindrical symmetry of this solution can also be better appreciated in this type of plot. We then followed the procedure outlined in the literature, to integrate the local energy density over the proper volume, in cylindrical coordinates at a fixed time over all space, obtaining the total energy as equation (16): the integral of the square root of the spatial metric determinant times the energy density, weighted by the coupling constant and the speed of light, which evaluates to 1.86 × 10¹⁰ times the coupling constant, in ergs. Since we assume that the spaceship is traveling at constant velocity, the total energy is also constant with time. In the last equation, we reinstated a factor of the speed of light to obtain the correct dimensions and also inserted an overall multiplicative factor of the conformal gravity coupling constant, which is necessary since our computation of the energy density in Figure 6 was done assuming that constant equal to one.

Figure 6. The conformal-gravity energy density for a warp bubble of radius one hundred metres at a ship velocity equal to the speed of light, plotted against the longitudinal coordinate and the transverse cylindrical radius. Integrating this local energy density over all space gives the estimate for the total energy required to establish the warp drive.

Therefore, we need to know the CG value for the coupling constant 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 Mannheim, equation (17), as 3.29 × 10⁸² erg seconds — the constant has the dimensions of action. Using it, we obtain the estimate of equation (18): a total energy of about 6 × 10⁹² erg, equal to 6.81 × 10⁷¹ grams, which is 3.42 × 10³⁸ solar masses, where we converted the energy into equivalent mass and compared our result with the solar mass of 1.99 × 10³³ grams.

The estimate in equation (18) would imply that an enormous amount of standard mass or energy 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 value quoted in equation (17) represents only an estimate of the macroscopic value of this coupling constant. This does not need to be the same as the microscopic value associated with the fundamental theory, which could be reduced by a factor of N, where N could be the number of occupied baryonic states in a galaxy, about 10⁶⁸, or possibly the number of baryons in the universe, about 10⁸⁰. 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.

5. 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 — the Hartle shaping function, instead of the original Alcubierre one — can overcome the main limitation of the Alcubierre warp drive 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 warp drive 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 estimate depends on the CG coupling constant, which is not well known. Therefore, this estimate will need to be refined in future studies.

Appendix. Energy Density Expression in Conformal Gravity

The complete analytical expression of the energy density in conformal gravity, computed using our Mathematica program, is given in equation (A.1). In the main part of our work, we used the Hartle shaping function, written more explicitly in equation (A.2) in terms of the Heaviside step function, where the bracketed subscript indicates the positive part of the function. The derivatives of the Hartle shaping function, up to the fourth order, were computed in terms of the Heaviside step function and the Dirac delta function, equation (A.3), also using the standard relations for the derivatives of those distributions. By inserting these derivatives into equation (A.1), we obtained the energy density plots of the bottom right panel of Figure 1 and of Figures 2, 4 and 6.

Acknowledgment

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.

(The reference list of the published article is omitted here; the complete text is at the source.)