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Cited by 15Cosmology and Gravitation TheoriesRelativity and Gravitational TheoryAdvanced Differential Geometry Research

Perfect fluid warp drive solutions with the cosmological constant

Open access
Santos-Pereira, Osvaldo L., Abreu, Everton M. C., Ribeiro, Marcelo B.·2021·The European Physical Journal Plus
DOI 10.1140/epjp/s13360-021-01899-7
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Abstract

The Alcubierre metric describes a spacetime geometry that allows a massive particle inside a spacetime distortion, called warp bubble, to travel with superluminal global velocities. In this work, we advance solutions of the Einstein equations with the cosmological constant for the Alcubierre warp drive metric having the perfect fluid as source. We also consider the particular dust case with the cosmological constant, which generalizes our previous dust solution (Santos-Pereira et al. 2020) and led to vacuum solutions connecting the warp drive with shock waves via the Burgers equation, as well as our perfect fluid solution without the cosmological constant (Santos-Pereira et al. 2021). All energy conditions are also analyzed. The results show that the shift vector in the direction of the warp bubble motion creates a coupling in the Einstein equations that requires off-diagonal terms in the energy–momentum source. Therefore, it seems that to achieve superluminal speeds by means of the Alcubierre warp drive spacetime geometry one may require a complex configuration and distribution of energy, matter and momentum as source in order to produce a warp drive bubble. In addition, warp speeds seem to require more complex forms of matter than dust for stable solutions and that negative matter may not be a strict requirement to achieve global superluminal speeds.

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Written from the paper’s full text.

TL;DR

The paper solves the Einstein equations with a cosmological constant for the Alcubierre warp-drive metric sourced by a perfect fluid, finding four solution classes and analyzing energy conditions. It concludes that warp speeds may require complex distributions of energy, matter, and momentum, and that negative matter density may not be strictly necessary.

Key findings

  • Four solution sets arise: Cases 1a and 2a have shift vector depending on \((t,y)\) or \((t,z)\), respectively, with equation of state \(\Lambda = 2\pi(3\mu - p)\) and conditions \(\Lambda - 8\pi\mu \ge 0\), \(\Lambda - 8\pi p \ge 0\).
  • Cases 1b and 2b require \(\Lambda = 8\pi\mu = 8\pi p = 0\), reduce to the earlier dust solution, and connect the warp drive to shock waves via the Burgers equation.
  • Energy conditions are summarized in Table I as weak \(\mu \ge 0\), strong \(\mu \ge 0\), dominant \(\mu + 3p \ge 0\), and null \(\mu + p \ge 0\); the text separately states the strong condition is satisfied if \(3p + \mu \ge 0\).
  • The paper conjectures that negative matter density may not be a strict requirement for global superluminal speeds, but stable warp-drive solutions may need more complex matter than dust.

Methods

  • The authors solve the Einstein equations with a cosmological constant using the Alcubierre warp-drive metric and a perfect-fluid energy-momentum tensor.
  • They calculate the covariant divergence of the energy-momentum tensor and impose null divergence, then analyze weak, dominant, strong, and null energy conditions.
  • Dust is treated as the special case of the perfect fluid with zero pressure, \(p=0\).

Limitations

  • The paper states that the dust EMT “seems to be not a stable source” for the warp drive, and that in Cases 1a and 2a the shift vector is not uniquely defined and can become complex when \(\Lambda=0\).
  • It also frames the broader proposal as a conjecture: warp drive may require a complex distribution of energy, matter, and momentum to stabilize the geometry.
  • Evident from the text: the work is a mathematical solution study with no numerical energy estimates, no stability analysis beyond source type, and no observational or experimental validation.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1140/epjp/s13360-021-01899-7

Cites

29 of 29 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 18 cited sources checked · 0 supported · 0 flagged

  • Findingown data
    § 1. INTRODUCTION · p. 3

    The perfect fluid with the cosmological constant as source for the Alcubierre warp drive results in four sets of differential equations.

    “We found that the perfect fluid with the cosmological constant as source for the Alcubierre warp drive results in four sets of differential equations”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § 1. INTRODUCTION · p. 3

    Considering that zero pressure reduces the perfect fluid to the dust EMT, the solution for this case is identical to the one found in Ref. [26], namely, the vacuum solution of the Einstein equations connecting the warp drive to shock waves.

    “Considering that the zero pressure reduces the perfect fluid to the dust EMT, the solution for this case is identical to the one we found in Ref. [26], namely, the vacuum solution of the Einstein equations connecting the warp drive to shock waves.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [26] O.L. Santos-Pereira, E.M.C. Abreu, M.B. Ribeiro, Dust content content solutions for the Alcubierre warp drive spacetime. Eur. Phys. J. C 80, 786 (2020) source

      Not checked in the free run.

  • Findingown data
    § 6. CONCLUSIONS AND FINAL REMARKS · p. 12

    For Cases 1a and 2a, the equation of state relating the cosmological constant Λ, matter density μ, and fluid pressure p is Λ = 2π(3μ - p).

    “having the following equation of state relating the cosmological constant Λ, the matter density µ and the fluid pressure p : Λ = 2 π (3 µ -p ) .”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § 6. CONCLUSIONS AND FINAL REMARKS · p. 12

    The presence of the cosmological constant allows the shift vector to be a real valued function for Cases 1a and 2a, as seen from Eqs. (3.11)-(3.12) and (3.16)-(3.17), namely Λ - 8πμ ≥ 0 and Λ - 8πp ≥ 0.

    “The presence of the cosmological constant allows the shift vector to be a real valued function as can be seen from Eqs. (3.11) and (3.12) for Case 1a in Table II, and Eqs. (3.16) and (3.17) for Case 2a, namely, Λ -8 πµ ≥ 0 and Λ -8 πp ≥ 0 .”
    Quote in the paperSubstantial claim
  • Findingnone given
    § 6. CONCLUSIONS AND FINAL REMARKS · p. 12

    The energy conditions are all satisfied for the perfect fluid if the conditions in Table I are satisfied.

    “The energy conditions are all satisfied for the perfect fluid if the conditions in Table I are satisfied.”
    Quote in the paperNot a substantial claim
  • May combine two claims. 9
  • Classified as probably not a substantial claim. 2

  • May combine two claims.

    Findingown data
    § 1. INTRODUCTION · p. 3

    Two of the four sets of differential equations are very similar and raise the possibility for the shift vector to be a complex function in one case depending on (t,y) coordinates and in another case depending on (t,z) coordinates.

    “two of them are very similar and raise the possibility for the shift vector to be a complex function in one case, depending on the ( t, y ) coordinates, and in another case depending on the ( t, z ) coordinates.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data
    § 1. INTRODUCTION · p. 3

    The other two sets of solutions are identical to each other and similar to the solution found in Ref. [26], except that now there is a cosmological constant coupled with the Burgers equation, and the warp drive is connected to shock wave solutions.

    “The other two sets of solutions are identical to each other and similar to the solution we found in Ref. [26], except that now there is a cosmological constant coupled with the Burgers equation and, again, the warp drive is connected to shock waves solutions.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [26] O.L. Santos-Pereira, E.M.C. Abreu, M.B. Ribeiro, Dust content content solutions for the Alcubierre warp drive spacetime. Eur. Phys. J. C 80, 786 (2020) source

      Not checked in the free run.

  • May combine two claims.

    Findingreasoning
    § 5. DUST AS A PARTICULAR CASE FROM THE PERFECT FLUID · p. 12

    In the case of dust EMT, there is no longer a set of equations 1a and 2a to be solved, only 1b and 2b, which are identical to the ones appearing in Ref. [26].

    “In the case of dust EMT there is no longer a set of equations 1a and 2a to be solved, only 1b and 2b, that are identical to the ones appearing in Ref. [26].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [26] O.L. Santos-Pereira, E.M.C. Abreu, M.B. Ribeiro, Dust content content solutions for the Alcubierre warp drive spacetime. Eur. Phys. J. C 80, 786 (2020) source

      Not checked in the free run.

  • May combine two claims.

    Findingreasoning
    § 3.2. Solving the Einstein equations with Λ for the perfect fluid · p. 8

    For Case 1a, the shift vector β is not uniquely defined and is a complex valued function that depends only on (t,y) spacetime coordinates.

    “The set of these last equations implies that the shift vector β is not uniquely defined. It is a complex valued function that depends only on ( t, y ) spacetime coordinates.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingcitation
    § 3.2. Solving the Einstein equations with Λ for the perfect fluid · p. 8

    For Case 1b, Eq. (3.14) is the Burgers equation that connects the warp drive to shock waves, and the cosmological constant, fluid pressure and matter density are all equal to zero, making the warp drive metric a vacuum solution for the Einstein equations.

    “Eq. (3.14) is the Burgers equation that connects the warp drive to shock waves, as discussed in Ref. [26]. The cosmological constant, fluid pressure and matter density are all equal to zero and the warp drive metric (2.6) is a vacuum solution for the Einstein equations.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [26] O.L. Santos-Pereira, E.M.C. Abreu, M.B. Ribeiro, Dust content content solutions for the Alcubierre warp drive spacetime. Eur. Phys. J. C 80, 786 (2020) source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 1. INTRODUCTION · p. 2

    The original Alcubierre warp drive metric was established without solving the Einstein equations.

    “In the original paper by M. Alcubierre the warp drive metric was established without solving the Einstein equations [1].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [1] M. Alcubierre, The warp drive: hyper-fast travel within general relativity. Class. Quantum Gravity 11, L73 (1994) source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § 1. INTRODUCTION · p. 3

    In the authors' second paper, solutions for the warp drive metric with perfect fluid and a special case of anisotropic fluid with heat flux, both with zero cosmological constant, indicate that positive matter densities are possibly capable of generating superluminal speeds.

    “In our second paper [27] we investigated solutions for the warp drive metric having the perfect fluid and a special case of anisotropic fluid with heat flux, but both with zero cosmological constant in the Einstein equations. The resulting solutions indicate that positive matter densities are possibly capable of generating superluminal speeds.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [27] O.L. Santos-Pereira, E.M.C. Abreu, M.B. Ribeiro, Fluid dynamics in the warp drive spacetime geometry. Eur. Phys. J. C 81, 133 (2021) source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § 1. INTRODUCTION · p. 2

    The Einstein tensor components for the Alcubierre warp drive metric imply that negative energy density would be required to create the warp bubble, violating the weak and dominant energy conditions.

    “The Einstein tensor components were calculated and it was noticed that negative energy density would be required to create the warp bubble, thus violating the weak and dominant energy conditions.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [1] M. Alcubierre, The warp drive: hyper-fast travel within general relativity. Class. Quantum Gravity 11, L73 (1994) source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § 1. INTRODUCTION · p. 3

    In the authors' third paper, charged dust as source EMT for the Alcubierre metric with cosmological constant yielded solutions connecting electric energy density with the cosmological constant, and some solutions had positive matter density and satisfied energy conditions.

    “In our third paper [28] a charged dust was used as source EMT for the Alcubierre metric and the Einstein equations which included the cosmological constant. We obtained solutions connecting the electric energy density with the cosmological constant and, again, some solutions were found having positive matter density and satisfying the energy conditions.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [28] O.L. Santos-Pereira, E.M.C. Abreu, M.B. Ribeiro, Charged dust solutions for the warp drive spacetime. Gen. Rel. Gravit. 53, 23 (2021) source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § 1. INTRODUCTION · p. 3

    Solutions of the Einstein equations for the Alcubierre warp drive metric having dust as source connect the warp drive geometry to the Burgers equation in a particular case, and shock waves appear as vacuum solutions.

    “Our results showed that solutions of the Einstein equations for the Alcubierre warp drive metric having dust as source connect them in a particular case the warp drive geometry to the well-known Burgers equation, which describes the dynamics of the waves moving through an inviscid fluid. Hence, shock waves appear to be vacuum solutions of the Einstein equations endowing the warp drive metric [26].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [26] O.L. Santos-Pereira, E.M.C. Abreu, M.B. Ribeiro, Dust content content solutions for the Alcubierre warp drive spacetime. Eur. Phys. J. C 80, 786 (2020) source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § 6. CONCLUSIONS AND FINAL REMARKS · p. 12

    In Ref. [26], the authors found an intrinsic relationship between the warp drive and shock waves by solving Einstein equations for the warp drive metric and dust particle EMT, but concluded there is an impossibility of coupling the dust as a source in this case.

    “In Ref. [26] we found this intrinsic relationship between the warp drive and shock waves by solving Einstein equations for the warp drive metric and the dust particle EMT, but we concluded that there is an impossibility of coupling the dust as a source in this case.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [26] O.L. Santos-Pereira, E.M.C. Abreu, M.B. Ribeiro, Dust content content solutions for the Alcubierre warp drive spacetime. Eur. Phys. J. C 80, 786 (2020) source

      Not checked in the free run.

11 of 11

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The perfect fluid warp drives (dust, non-zero pressure, non-zero pressure with cosmological constant) in [4-8] are just Riemann-flat Minkowski space in disguise.

    “Their perfect fluid warp drives (dust, non-zero pressure, non-zero pressure with cosmological constant) are just Riemann-flat Minkowski space in disguise.”

    From Generic warp drives violate the null energy condition (2022) · cites this paper as [7]

    Section 3.2 (p. 7) says that with a cosmological constant, a dust-case warp-drive solution is no longer a vacuum solution; only Case 1b has zero cosmological constant, pressure, and density and is described as vacuum. The Conclusions (p. 12) also describe other perfect-fluid solutions with an equation of state relating the cosmological constant, density, and pressure. The paper does not identify its solutions as Riemann-flat Minkowski space, and its distinction between vacuum and non-vacuum cases contradicts the blanket claim. (LLM, conf=0.88)

    Checked by a reasoning model — may be wrong.

    Experimental — accuracy not yet measured

  • In [7], the attempt at including a cosmological constant is at best a redefinition of terms in the stress-energy tensor.

    “For the 'cosmological' solution of [7] the attempt at including a cosmological constant is at best a redefinition of terms in the stress-energy tensor.”

    From Generic warp drives violate the null energy condition (2022) · cites this paper as [7]

    The paper does include a cosmological constant, but it does not characterize that inclusion as merely a redefinition of stress-energy terms. In Conclusions and Final Remarks (p. 12), it says the constant changes the solutions: in Cases 1a and 2a it permits a real-valued shift vector that would otherwise be complex. (LLM, conf=0.77)

    Checked by a reasoning model — may be wrong.

    Experimental — accuracy not yet measured

  • In all situations [4-8], one is working within the framework of the generic Alcubierre metric, so energy condition violations must be present.

    “In all of these situations [4-8], one is working within the framework of the generic Alcubierre metric, so we know from first principles that energy condition violations must be present.”

    From Generic warp drives violate the null energy condition (2022) · cites this paper as [7]

    Section 2.3 (p. 6) says that, for the Alcubierre warp-drive metric considered here, the weak and dominant energy conditions may be satisfied if the cosmological constant is positive and large enough. The Conclusions (p. 12) also state that all energy conditions are satisfied for the perfect fluid under the conditions in Table I. Thus the paper does not support the claim that using this metric necessarily entails energy-condition violations. (LLM, conf=0.89)

    Checked by a reasoning model — may be wrong.

    Experimental — accuracy not yet measured

  • The 'cosmological' warp drive in [7] is not cosmological.

    “Their 'cosmological' warp drive is not cosmological.”

    From Generic warp drives violate the null energy condition (2022) · cites this paper as [7]

    The Introduction (pp. 2–3) describes a warp drive as a localized spacetime distortion and says this study adds a cosmological constant to the Einstein equations for that drive. This supports distinguishing the presence of a cosmological constant from a cosmological model, but the paper does not itself state that its warp drive is ‘not cosmological.’ (LLM, conf=0.75)

    Experimental — accuracy not yet measured

  • In [7], the spacetime is asymptotically flat; it does not match (anti)de Sitter at large distances, so it is not a 'cosmological' warp bubble.

    “Furthermore, given that the entire calculation is taking place within the framework of the generic Alcubierre metric, we know the spacetime is asymptotically flat; it does not match (anti)de Sitter at large distances, so it is not a 'cosmological' warp bubble.”

    From Generic warp drives violate the null energy condition (2022) · cites this paper as [7]

    Section 2.1 (p. 4) uses the Alcubierre warp-drive metric, supporting that part of the claim. Our copy does not state that this spacetime is asymptotically flat, fails to match (anti) de Sitter at large distances, or is therefore not a cosmological warp bubble. The Introduction (p. 3) says the paper includes a cosmological constant in the Einstein equations, but does not resolve those asymptotic claims. (LLM, conf=0.78)

    Experimental — accuracy not yet measured

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Dust Case 1a both exists and is excluded

    Major

    Section 3.2 describes a nonvacuum dust solution arising from Case 1a with a cosmological constant. Section 5 instead says that imposing zero pressure leaves only Cases 1b and 2b, which are vacuum solutions.

  • Burgers-equation cases both include and exclude the cosmological constant

    Minor

    The introduction says the two identical shock-wave solutions have a cosmological constant coupled to the Burgers equation. Case 1b, one of those solutions, explicitly requires Λ = 0; the paper says Case 2b is the same case.

  • Dominant energy condition differs between text and Table I

    Minor

    Section 4.2 says the dominant energy condition is satisfied for μ > 0, while Table I gives μ + 3p ≥ 0. The table's condition depends on pressure; the stated condition does not.

  • Strong energy condition differs between text and Table I

    Minor

    Section 4.3 says the strong energy condition requires 3p + μ ≥ 0, while Table I lists μ ≥ 0. These conditions differ when pressure is nonzero.

4 of 4

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. F.S.N. Lobo, P. Crawford, Weak energy condition violation and superluminal travel. Lect. Notes Phys. 617, 277 (2003) · DOI · Paper page
  2. H.G. White, Warp field mechanics 101. J. Brit. Interplanet. Soc. 66, 242 (2011)

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Related

Authors
Santos-Pereira, O. L., Abreu, E. M. C., & Ribeiro, M. B.
Affiliations
Universidade Federal do Rio de Janeiro (BR)
Corresponding author
Ribeiro, Marcelo B.
ORCID
Venue
The European Physical Journal Plus
Publisher
Springer Science and Business Media LLC
Published
2021-09
Volume
136
Issue
9
ISSN
2190-5444 (ISSN-L)
DOI
10.1140/epjp/s13360-021-01899-7
OpenAlex
W3195841321
Semantic Scholar
83e5bba34bd6955e548270d76cc2b9ce1ecc73ed
Cited by
15
References
29
Citations by year
2026: 1 · 2025: 2 · 2024: 2 · 2023: 8 · 2022: 2
FWCI
1.61
Type
journal article
Language
en
Open access
Licence
Locations
  • The European Physical Journal Plus (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University) (cc-by)
Funders
conselho nacional de desenvolvimento científico e tecnológico (406894/2018-3)
Topics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
Keywords
Alcubierre metric, warp drive, perfect fluid, cosmological constant, dust solutions, superluminal travel, energy conditions, shift vector, shock waves, Burgers equation, warp bubble

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