added_by: arek

kind: direct

context_after: In a similar way one can show that the strong energy condition is also violated. We see then that, just as it happens with wormholes, one needs exotic matter to travel faster than the speed of light. However, even if one believes that exot

context_before: tive for all\/ observers. If one calculates the Einstein tensor from the metric (), and uses the fact that the four-velocity of the Eulerian observers is given by: then one can show that these observers will see an energy density given by:

section: Sec. 4 (energy conditions)

quote: The fact that this expression is everywhere negative implies that the weak and dominant energy conditions are violated.

status: EXACT

verified_by: werys


context_after: . Both the weak and the dominant energy conditions require the energy density to be positive for all\/ observers. If one calculates the Einstein tensor from the metric (), and uses the fact that the four-velocity of the Eulerian observers i

context_before: ortion of spacetime. A propulsion mechanism based on such a local distortion of spacetime just begs to be given the familiar name of the "warp drive" of science fiction. The metric I have just described has one important drawback, however:

quote: it violates all three energy conditions (weak, dominant and strong)


context_after: While warp drives are certainly interesting examples of speculative physics, the violation of the energy conditions, at least within the framework of standard general relativity, is unavoidable. Even in modified gravity, physically reasona

context_before: hows that the situation is actually much grimmer than advertised - within the framework adopted by those three papers all physically reasonable warp drives will certainly violate the WEC, and both the strong and dominant energy conditions.

section: Abstract

quote: Under plausible subsidiary conditions the null energy condition is also violated.


kind: contextual

context_after: Specifically, known results already include: -3pt Alcubierre warp drives have long been known to violate the WEC, and have more recently been shown to also violate the null energy condition (NEC). Nat\'ario zero-expansion warp drives have

context_before: ree recent articles have argued for the existence of "physically reasonable" positive-energy warp drive configurations, either satisfying the weak energy condition (WEC), or minimizing violations thereof. (See also the series of articles.)

section: Sec. 1 (Introduction)

quote: These claims are in sharp contrast to the 25-year-old consensus opinion that at least some energy condition violations are necessary for the generation of warp fields.


kind: definition

context_after: . It is certainly more than sufficient if for large distances, which is in turn certainly true if, which is in turn certainly true if the ADM mass is zero. Vanishing of the ADM mass is certainly the case for Alcubierre's warp drive, and als

context_before: y anything more precise about the energy conditions.) Fortunately, the average pressure is reasonably tractable. From equation (), Now, given that we want our warp field to be localizable, to not spread over and affect the whole spacetime,

section: Sec. 4 (generic Natario warp drives)

quote: it is natural to make the assumption that the flow field obeys some sort of fall-off conditions at spatial infinity


context_after: . Vanishing of the ADM mass is certainly the case for Alcubierre's warp drive, and also for Natario's zero-expansion warp drive, and is a first crude approximation for well-localized warp fields. (We will subsequently significantly weaken t

context_before: fect the whole spacetime, it is natural to make the assumption that the flow field obeys some sort of fall-off conditions at spatial infinity. It is certainly more than sufficient if for large distances, which is in turn certainly true if,

quote: which is in turn certainly true if the ADM mass is zero


context_after: Nat\'ario metric () in order to distinguish a warp drive from, for instance, the Painlev\'e-Gullstrand version of Schwarzschild spacetime. The line element () is sufficiently general and explicit to be physically interesting as a warp driv

context_before: e Christoffel symbols are C^1- at worst, that is, piecewise once differentiable with at worst step-function discontinuities. Then the Riemann tensor is C^0 at worst, piecewise continuous with at worst delta-function contributions. Finally,

quote: we emphasize the need for at least some subsidiary conditions to be applied to the generic


reason: the preceding 'assume that' is the authors stating their own localization assumption, which the theorem depends on; it is not a position the paper goes on to reject

reviewed_by: werys

context_after: "restore its asymptotics" after a warp drive passes through a particular point. A delicate point must be discussed before this proof is called closed and complete: The question regarding what happens to such a geodesic Eulerian observer wh

context_before: K increases as it returns to its original value.Since we still have to assume that any contribution to K_ij from the warp drive is sufficiently localized. This, however, is not allowed if the NEC is always satisfied, as given by equation.

section: Sec. 6 (asymptotics)

note: Hedge scan fired on 'assume that'; see hedge_review.

quote: This proves that the violation of the NEC is a necessary condition for the spacetime to


kind: equation

context_after: It is easy to understand the geometry of our spacetime from the previous expressions. First, from equation () we see that the 3-geometry of the hypersurfaces is always flat. Moreover, the fact that the lapse is given by

context_before: 1 f R > 0 and > 0 the function f(r)$ approaches very rapidly a "top hat" function: With the above definitions, the metric () can be rewritten as:

section: Sec. 3 (the metric)

note: Display equation 5 (\label{final_metric}), equation environment.


context_after: with R>0 and sigma>0 arbitrary parameters. Notice that for large sigma the function f(r) approaches very rapidly a "top hat" function: With the above definitions, the metric () can be rewritten as: It is easy to understa

context_before: We want to find a metric that will "push" the spaceship along a trajectory described by an arbitrary function of time x_s (t). A metric that has this property is given by (G = c = 1): where: and where f is the function:

section: Sec. 3 (shape function)

note: Display equation 3, equation environment.

quote: f ( r_s ) = \frac{\tanh ( \sigma ( r_s + R ) ) - \tanh ( \sigma ( r_s - R ) ) }{ 2 \tanh ( \sigma R )} ,


context_after: Figure shows a graph of theta as a function of x and, in the particular case when sigma = 8 and R =v_s = 1. The center of the perturbation corresponds to the spaceship's position. We clearly see how the volume elements a

context_before: not difficult to see that this expression reduces to: The expansion theta of the volume elements associated with the Eulerian observers is given in terms of K_ij as: From this expression it is not difficult to show that:

section: Sec. 3 (expansion)

note: Display equation 9 (\label{expansion}), equation environment.

quote: \theta = v_s \frac{x_s}{r_s} \frac{d f}{d r_s} .


context_after: The fact that this expression is everywhere negative implies that the weak and dominant energy conditions are violated. In a similar way one can show that the strong energy condition is also violated. We see then that, j

context_before: rvers. If one calculates the Einstein tensor from the metric (), and uses the fact that the four-velocity of the Eulerian observers is given by: then one can show that these observers will see an energy density given by:

note: Display equation 16, equation environment.


context_after: where is the Planck length. Thus, unless v_b is extremely large, the wall thickness cannot be much above the Planck scale. Typically, the walls of the warp bubble are so thin that the shape function could be considered a

context_before: r the sampling time is compared to the minimal radius of curvature. If we insert this into the quantum inequality and use we may neglect the term involving 1-f(rho) to find Now as an example, if we let alpha = 1/10, then

section: Sec. 3 (QI restrictions)

note: Display equation 22 (\label{eq:wall_thickness}), equation environment.

quote: \Delta \leq 10^2 v_b L_{Planck} ,


context_after: For a macroscopically useful warp drive, we want the radius of the bubble to be at least in the range of 100 meters so that we may fit a ship inside. It has been shown in the previous section that the wall thickness is c

context_before: approximation to the shape function given by Equation (). When one takes the derivative of this shape function, we find that the contributions to the energy come only from the bubble wall region, and we end up evaluating

section: Sec. 4 (total energy)

note: Display equation 26 (\label{eq:energy}), eqnarray environment.


context_after: Because a typical galaxy has a mass of approximately the energy required for a warp bubble is on the order of This is a fantastic amount of negative energy, roughly ten orders of magnitude greater than the total mass of

context_before: section that the wall thickness is constrained by (). If we use this constraint and let the bubble radius be equal to 100 meters, then we may neglect the second term on the right-hand-side of Equation (). It follows that

note: Display equation 27, equation environment.

quote: E \leq -6.2\times10^{70} v_b L_{Planck} \sim -6.2\times10^{65} v_b {\rm grams}.


context_after: This is a fantastic amount of negative energy, roughly ten orders of magnitude greater than the total mass of the entire visible universe. If one can violate the quantum inequality restrictions and make a bubble with a w

context_before: qual to 100 meters, then we may neglect the second term on the right-hand-side of Equation (). It follows that Because a typical galaxy has a mass of approximately the energy required for a warp bubble is on the order of

note: Display equation 29, equation environment.

quote: E \leq - 3 \times 10^{20} M_{galaxy} v_b .


context_after: . This puts the warp drive in the mass scale of large traversable wormholes. The new geometry satisfies the quantum inequality concerning WEC violations and has the same advantages as the original Alcubierre spacetime. width 5cm 2mm ^dagger

context_before: elgium 4cm Abstract I show how a minor modification of the Alcubierre geometry can dramatically improve the total energy requirements for a 'warp bubble' that can be used to transport macroscopic objects. A spacetime is presented for which

quote: the total negative mass needed is of the order of a few solar masses, accompanied by a comparable amount of positive energy


context_after: . The most natural way to do this is the following: ds^2 = - dt^2 + B^2(r_s) [(dx - v_s(t) f(r_s) dt)^2 + dy^2 + dz^2]. For simplicity, the velocity v_s will be taken constant. B(r_s) is a twice differentiable function such that, for some a

context_before: me drawbacks of the new geometry are discussed. Throughout this note, we will use units such that c=G=hbar=1, except when stated otherwise. A modification of the Alcubierre geometry We will solve the problem of the large negative energy by

section: Sec. 2 (the modification)

quote: keeping the surface area of the warp bubble itself microscopically small, while at the same time expanding the spatial volume inside the bubble


context_after: and has the same advantages as the original Alcubierre spacetime. width 5cm 2mm ^dagger chris.vandenbroeck@fys.kuleuven.ac.be Introduction In recent years, ways of effective superluminal travel (EST) within general relativity have generate

context_before: cts. A spacetime is presented for which the total negative mass needed is of the order of a few solar masses, accompanied by a comparable amount of positive energy. This puts the warp drive in the mass scale of large traversable wormholes.

quote: The new geometry satisfies the quantum inequality concerning WEC violations


context_after: . width 5cm 2mm ^ast vdbroeck@starlab.net Introduction Since Alcubierre published his 'warp drive' spacetime, the proposal has been criticized from various viewpoints by a number of authors. One of the problems, concerning the amount of exo

context_before: n the (im)possibility of warp bubbles 1.5cm Chris Van Den Broeck ^ast Physics Division, Starlab Research Boulevard St.-Michel 47, 1040 Brussels, Belgium 4cm Abstract Various objections against Alcubierre's warp drive geometry are reviewed.

quote: Superluminal warp bubbles seem an unlikely possibility within the framework of general relativity and quantum field theory, although subluminal bubbles may still be possible


context_after: . Due to the absence of horizons, potential problems due to diverging vacuum fluctuations will not arise, and there will be no tachyonic motion of exotic matter. Possibly the geometry can be chosen in such a way that the necessary negative

context_before: ved without retaining some unphysical features or introducing new ones, such as high energy densities, curvature radii of the order of the Planck length, and naked singularities. We have limited the discussion to superluminal warp bubbles.

section: Conclusion

quote: Subluminal bubbles are still an open possibility, and it is not inconceivable that microscopic ones might even occur naturally


context_after: . There are other problematic issues with both of these models, but they are physically different from the Nat\'ario class, (unit lapse, spatially flat 3-geometry), and must be directly addressed using different techniques. Some defective w

context_before: measure zero.) While the transformations made in appendices A.1 and A.2 of reference satisfy the determinant condition, they fail the more basic condition; they are simply not coordinate transformations. For our purposes then it means that

section: Sec. 3

quote: the non-unit-lapse and van den Broeck warp drive spacetimes cannot simply be dismissed out of hand


context_after: . A modest numerical analysis is carried out on a set of example configurations, finding total energy requirements four orders of magnitude smaller than the solar mass. Extraordinarily, the example configurations are generated by purely pos

context_before: tic energy distributions. With this new interpretation, it becomes a relatively simple matter to generate solitonic configurations, within a certain subclass, that respect the positive energy constraint. Using this newfound interpretation,

quote: a superluminal solitonic spacetime is presented that possesses positive semi-definite energy


context_after: . It follows from the Lorentz-invariant eigenvalue equation that the Eulerian energy, Eq., is the first eigenvalue (whose eigenvector is the normal vector) and thus satisfies the first constraint enforced by the WEC for a type-I stress-ener

context_before: in hand, the full WEC can now easily be investigated to study the nature of the energy distribution in other frames of reference. By analyzing the Lorentz-invariant eigenvalues of the stress-energy tensor calculated using Eq., it turns out

quote: the WEC is violated in compact regions within the energy distribution


reason: the reopening sentence that follows (E-0030) is now quoted in full and carried verbatim by the corrected claim C-0021; the span itself remains the paper's own words

context_after: However, the WEC is not violated everywhere and if the configuration in indeed satisfies the WEC, as claimed, then it may still be possible to satisfy the WEC in the presented configurations too, given sufficient modifications. The ansatz

context_before: for a type-I stress-energy tensor, which the configuration in Fig. () appears to be. However, the principle momenta constraints is where the violations occur, namely, rho + p_i^p<0 for some i in 123 in compact regions in the distribution.

quote: No amount of modification to the configuration could get rid of these WEC-violating regions.


context_after: : steering a warp drive is a problem of energy budget, not negative energy. Introduction Alcubierre's warp drive (reviewed in) showed that general relativity permits a compact region (a "bubble") to be transported through an ambient spaceti

context_before: nt class: the 3 is the collimation penalty of dipole exhaust over a floor of 1. The radiating equilibrium is linearly unstable (flux anti-damps the radial mode), yet growth sim luminosity is bounded over any finite burn by the fuel budget.

quote: The drive is causal, subluminal, and a reaction drive needing no exotic matter in the thin-shell idealization


context_after: The barrier is the size of the budget, the sense in which warp propulsion is here an engineering problem. [Local in retarded time.] The solution is a finite maneuver on a u -slab; global, eternal existence is not claimed, and mass(u)>0 on

context_before: light (Deltaetatoinfty) sends mass_f/mass_0to0. A steering history that changes direction has (the integrated rapidity arc exceeds the net rapidity, as boosts do not add as vectors), so a turn costs strictly more than the collinear figure.

section: Limitations and scope

quote: The cost is steep, but positive and finite: no negative energy and no quantum-inequality violation appear at any point.


context_after: ; the one certified positive case, the Fuchs shell (Table), evades the theorem's hypotheses through its Schwarzschild exterior rather than contradicting it. We offer as an independent verifier for the warp-drive community: given any metric,

context_before: ructure-preserving symplectic integrator and reported with an on-cone witness, and a Ford-Roman comparison preserve this ordering, with Rodal the mildest by one to two orders of magnitude yet still violating. Across the four matched drives

quote: the invariant NEC margin is negative wherever the wall is Type I, and the wall is Type IV elsewhere, an explicit, all-observer, all-velocity realization of the Santiago-Schuster-Visser theorem


context_after: Now as an example, if we let alpha = 1/10, then where is the Planck length. Thus, unless v_b is extremely large, the wall thickness cannot be much above the Planck scale. Typically, the walls of the warp bubble are so th

context_before: ecified parameter that describes how much smaller the sampling time is compared to the minimal radius of curvature. If we insert this into the quantum inequality and use we may neglect the term involving 1-f(rho) to find

note: Display equation 21, equation environment.

quote: \Delta \leq {3\over 4}\sqrt{3\over\pi} {v_b \over \alpha^2} .


context_after: The ansatz Eq. contains an interesting property related to an O(3) operation. The dynamical equations Eq. are not invariant under the total spacial inversion. Additionally, the expansion is not invariant either, since under this operation,

context_before: , the principle momenta constraints is where the violations occur, namely, rho + p_i^p<0 for some i in 123 in compact regions in the distribution. No amount of modification to the configuration could get rid of these WEC-violating regions.

section: Sec. 3.2 (immediately after E-0025)

quote: However, the WEC is not violated everywhere and if the configuration in indeed satisfies the WEC, as claimed, then it may still be possible to satisfy the WEC in the presented configurations too, given sufficient modifications.


context_after: at v_sapproxvdbTransitionVS, with a stress-energy that has no rest frame. The Type-IV labels are certified physical, not numerical, by a three-solver and 50 -digit cross-check. The geometric control of the algebraic type. The vorticity of

context_before: the residual non-Type I points. A velocity-resolved Hawking-Ellis type map across the luminal transition. Sweeping v_s through 1, we find a clean dichotomy (Section): the Rodal irrotational geometry is globally Type I at all speeds, while

section: Abstract/results (transition value 0.36 stripped by canonicaliser; in main.tex:29 as \vdbTransitionVS)

quote: the Alcubierre and Nat\'ario walls are Type-IV dominated at all speeds, and Van den Broeck above its Type-I to Type-IV transition


context_after: with and n sufficiently large. As an example, let us choose n=80. Then one can check that will be negative for 0 leq w leq 0.981 and positive for w>0.981. It has a strong negative peak at w=0.349, where it reaches the va

context_before: at = (1,0,0,0) t=0 when r_s=r=(x^2+y^2+z^2)^ 1/2 B r= R + B r= R + $. In addition, we will demand that a large number of derivatives vanish at this point. A choice that meets our requirements is

section: Sec. 3 (the interpolating profile, labelled B)

note: Display equation 12 (\label{B}), equation environment.


context_after: To explicitly see a specific example of this behaviour, let us work in an orthonormal basis. Take rho_0>0 and Gamma>1, and consider: Then in the natural rest frame we have. But a moving observer, with 4-velocity, where n^i is any unit spat

context_before: nergy conditions is typically the ANEC (averaged null energy condition) where one averages the NEC along timelike geodesics using an affine parameterization. As a cautionary example regarding application of the energy conditions, note that

section: Sec. on energy conditions (the collective rebuttal)

quote: all three of the recent Lentz, Bobrick-Martire, and Fell-Heisenberg articles merely assert the existence of one sub-class of timelike observers for which the energy density is positive. This is not enough to show that the WEC is satisfied.


context_after: . The solitons are also shown to be capable of being sourced from the stress-energy of a conducting plasma and classical electromagnetic fields. This is the first example of hyper-fast solitons resulting from known and familiar sources, reo

context_before: ral relativity. The negative-energy sources required for these solitons must be created through energy-intensive uncertainty principle processes as no such classical source is known in particle physics. This paper overcomes this barrier by

section: Abstract

quote: constructing a class of soliton solutions that are capable of superluminal motion and sourced by purely positive energy densities


context_after: . We show that a class of subluminal, spherically symmetric warp drive spacetimes, at least in principle, can be constructed based on the physical principles known to humanity today. Introduction The classical-relativistic Alcubierre drive

context_before: ity and the rate of time can be chosen in a controlled manner. Conceptually, we demonstrate that any warp drive, including the Alcubierre drive, is a shell of regular or exotic material moving inertially with a certain velocity. Therefore,

section: Abstract/Introduction

quote: any warp drive requires propulsion


context_after: ; construct superluminal warp-drive solutions which satisfy quantum inequalities; provide optimizations for the Alcubierre metric that decrease the negative energy requirements by two orders of magnitude; and introduce a warp drive spacetim

context_before: l. In this study, we develop a model of a general warp drive spacetime in classical relativity that encloses all existing warp drive definitions and allows for new metrics without the most serious issues present in the Alcubierre solution.

quote: We present the first general model for subluminal positive-energy, spherically symmetric warp drives


context_after: . Introduction Warp drive spacetimes, first introduced by Alcubierre and later by others, offer several unique transportation properties for timelike observers. These properties include the possibility of accelerating through geodesic motio

context_before: s well-known warp drive solutions such as the Alcubierre metric. We generate the spacetime metric numerically, evaluate the energy conditions, and confirm that the shift vector distribution cannot be reduced to a coordinate transformation.

quote: This study demonstrates that classic warp drive spacetimes can be made to satisfy the energy conditions by adding a regular matter shell with a positive ADM mass


context_after: : at points where, for example, this happens when the second derivative terms outweigh contributions proportional to phi^2. Violation of Energy Conditions in QFT Quantization and positivity do not mix well. For instance, the prototypical ex

context_before: ewise, DEC also holds for this reason, and the same is true for the electromagnetic field. However, we also see that so the SEC can fail even for this model if m>0. The nonminimally coupled field, with coupling xi, has stress-energy tensor

section: Sec. on non-minimal coupling

quote: As the additional terms are not of the sum of squares form, even NEC can be violated


context_after: (certainly above 2 -dimensions) or, as already mentioned, along null curves. The argument above, and the analogous argument for the energy density, relies on 'classical positivity' of the quantity in question. This permits a number of rela

context_before: curve, e.g., contracted against a null vector or possibly differing future-pointing causal vector fields, to obtain QNEI, QDEI etc. Variants exist for averages over suitable Lorentzian submanifolds, instead of timelike curves (see, e.g.).

section: Sec. on QEI variants

quote: One may show that no such bounds exist for smearings over spacelike surfaces


context_after: As the additional terms are not of the sum of squares form, even NEC can be violated: at points where, for example, this happens when the second derivative terms outweigh contributions proportional to phi^2. Violation of Energy Conditions

context_before: _0 = u. Then so this theory obeys WEC due to the 'sum of squares' form. Likewise, DEC also holds for this reason, and the same is true for the electromagnetic field. However, we also see that so the SEC can fail even for this model if m>0.

quote: The nonminimally coupled field, with coupling xi, has stress-energy tensor


context_after: "warp drive" spacetime of Ref.. It is the goal of this paper to analyze a spacetime recently proposed by Krasnikov which, although differing from that of Ref. in several key respects, shares with it the property of allowing superluminal tr

context_before: ry. However, Ford and Roman have proven inequalities which limit the magnitude and duration of negative energy density. These "quantum inequalities" (QIs) strongly suggest that it is unlikely that stable Lorentzian wormholes can exist, and

section: Introduction

quote: similar conclusions have been drawn by Pfenning and Ford with regard to the


context_after: (mass), thus overcoming the main difficulty of the warp drive mechanism. The explicit expression of T^00 in CG, computed with our Mathematica program, is rather cumbersome and is reproduced in Eq. () of the Appendix. Here we present just t

context_before: pletely non-negative, showing that the WEC is verified and no exotic matter is needed to establish the warp drive. This non-negative energy density plot in the bottom right panel of Fig. 1 is the main result of our paper as it shows that -

section: Sec. 3 (main result)

quote: if CG is the correct extension of GR - it might be possible to establish a warp drive without having to use negative energy


context_after: , 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 Alcubie

context_before: ather 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

quote: for particular choices of the shaping function, the Alcubierre metric in the context of conformal gravity does not violate the weak energy condition


context_after: . 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 Uni

context_before: 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,

quote: 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