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Violations of the Weak Energy Condition for Lentz Warp Drives

Bill Celmaster, Steve Rubin

TL;DR

The paper investigates whether a Natário/Lentz-type warp drive can satisfy the weak energy condition (WEC) using classical matter sources. It first performs a direct calculation of the Eulerian energy density $E$ for Lentz's geometry, identifies a sign error in his derivation, and shows negative energy regions exist. It then generalizes to hyperbolic-potentials (HP) warp drives and analyzes modified rhomboidal and rectangular sources, finding that all tested configurations fail to guarantee $E\ge0$. The work also scrutinizes the applicability of known no-go theorems, arguing that their assumptions (differentiability, finite-energy, on-off behavior) constrain the viability of WEC-compliant warp drives, and that Lentz's construction does not provide a valid counterexample. Overall, the results reinforce the view that classical, non-exotic matter cannot source a WEC-compliant warp drive within these constructs, while clarifying the mistakes in Lentz's original claims and the conditions under which no-go theorems hold.

Abstract

Warp drive spacetimes capable of superluminal transportation, were first introduced in 1994 by Miguel Alcubierre and then generalized by others. These spacetimes violated the Weak Energy Condition (WEC). Lentz proposed a new type of warp drive in 2020. It was claimed that this warp spacetime has non-negative energy density and can therefore be sourced by a classical plasma. We demonstrate that Lentz's claim is incorrect. We begin with a direct calculation of the energy-momentum tensor of Lentz's warp drive in a Eulerian reference frame, and show that there are spacetime regions where the energy density is negative. We then examine the theoretical basis of Lentz's investigation and identify several derivation errors. The derivation errors can be somewhat ameliorated with a modified version of Lentz's geometry, which more closely respects the equalities and inequalities discussed by Lentz. Even so, we show that the modified geometry still violates the WEC even in the Eulerian reference frame. We conclude with a brief discussion of no-go theorems and their relationship to geometries of the kind proposed by Lentz.

Violations of the Weak Energy Condition for Lentz Warp Drives

TL;DR

The paper investigates whether a Natário/Lentz-type warp drive can satisfy the weak energy condition (WEC) using classical matter sources. It first performs a direct calculation of the Eulerian energy density for Lentz's geometry, identifies a sign error in his derivation, and shows negative energy regions exist. It then generalizes to hyperbolic-potentials (HP) warp drives and analyzes modified rhomboidal and rectangular sources, finding that all tested configurations fail to guarantee . The work also scrutinizes the applicability of known no-go theorems, arguing that their assumptions (differentiability, finite-energy, on-off behavior) constrain the viability of WEC-compliant warp drives, and that Lentz's construction does not provide a valid counterexample. Overall, the results reinforce the view that classical, non-exotic matter cannot source a WEC-compliant warp drive within these constructs, while clarifying the mistakes in Lentz's original claims and the conditions under which no-go theorems hold.

Abstract

Warp drive spacetimes capable of superluminal transportation, were first introduced in 1994 by Miguel Alcubierre and then generalized by others. These spacetimes violated the Weak Energy Condition (WEC). Lentz proposed a new type of warp drive in 2020. It was claimed that this warp spacetime has non-negative energy density and can therefore be sourced by a classical plasma. We demonstrate that Lentz's claim is incorrect. We begin with a direct calculation of the energy-momentum tensor of Lentz's warp drive in a Eulerian reference frame, and show that there are spacetime regions where the energy density is negative. We then examine the theoretical basis of Lentz's investigation and identify several derivation errors. The derivation errors can be somewhat ameliorated with a modified version of Lentz's geometry, which more closely respects the equalities and inequalities discussed by Lentz. Even so, we show that the modified geometry still violates the WEC even in the Eulerian reference frame. We conclude with a brief discussion of no-go theorems and their relationship to geometries of the kind proposed by Lentz.

Paper Structure

This paper contains 59 sections, 154 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Natário warp drives and the Eulerian energy density condition
  3. Lentz's original warp drive
  4. The warp drive hyperbolic differential equation
  5. Lentz's rhomboidal source
  6. Lentz's potential, ${ \if@compatibility \mathchar"011E {} \mathchar"011E }_{L}$
  7. Properties of hyperbolic potential (HP) warp drives
  8. HP warp drives almost satisfy the Lentz DE
  9. Energy density of HP warp drives
  10. Finiteness of the energy of HP warp drives
  11. The energy non-negativity condition and shift-derivative inequality
  12. The shift-derivative inequality
  13. The energy non-negativity condition
  14. A modified version of Lentz's geometry
  15. The modified potential, ${ \if@compatibility \mathchar"011E {} \mathchar"011E }_{\text{rh}}$
  16. ...and 44 more sections

Figures (19)

  • Figure 1: Unscaled rhomboidal source $\tilde{{ \if@compatibility \mathchar"011A {} \mathchar"011A }}_{\text{rhomb}}N_{z}(0,0,0)$ for $y=0$
  • Figure 2: $N_{z}^{L}$ for the rhomboidal source, evaluated on the slice y=0.
  • Figure 3: $N_{x}^{L}$ for the rhomboidal source, evaluated on the slice $y=0$.
  • Figure 4: Rhomboidal-source Eulerian energy density at $y=1.0\times10^{-6}.$
  • Figure 5: $\mathbf{{ \if@compatibility \mathchar"011A {} \mathchar"011A }_{\text{L}}\partial_{z}^{2}{ \if@compatibility \mathchar"011E {} \mathchar"011E }_{\text{L}}}$ for $y=10^{-6}$ (${ \if@compatibility \mathchar"010D {} \mathchar"010D }=0.43$)
  • ...and 14 more figures