A Closer Look at Natário’s Zero-Expansion Warp Drive

Abstract

We conduct a detailed analysis of Natário’s “zero-expansion” warp drive spacetime, focusing on scalar curvature invariants within the 3$+$1 formalism. This paper has four primary objectives: First, we establish the Petrov type classification of Natário’s spacetime, which has not been previously determined in the literature. We prove that Natário’s spacetime is Petrov type I, not fitting the Class B warped product spacetime definition. Second, we assess the relative magnitude of the Weyl scalar curvature invariant and compare it with the amplitudes of Einstein’s scalar and the Ricci quadratic and cubic invariants within the warp-bubble zone. Previous studies have focused on Ricci curvature and the energy-momentum tensor, neglecting the Weyl curvature, which we demonstrate plays a significant role due to the sharp localization of the form function near the warp-bubble radius. Third, we visualize several curvature invariants for Natário’s warp drive, as well as momentum density, which we show as the critical physical quantity governing the orientation of the warp drive trajectory, overshadowing space volume changes. Fourth, we critically examine claims that Natário’s warp drive is more realistic than Alcubierre’s. We demonstrate that Natário’s spacetime exhibits curvature invariant amplitudes 35 times greater than Alcubierre’s, given identical warp-bubble parameters, making Natário’s concept even less viable. Additionally, we address Mattingly et al.’s analysis, highlighting their underestimation of curvature invariant amplitudes by 21 orders of magnitude.

Appendix A: Curvature Invariants of the Natário Metric

In this appendix, we present the explicit formulas for the curvature invariants used in Section 4 to calculate and visualize them in Natário’s spacetime for different conditions. These curvature invariants are, namely, Einstein’s curvature invariant $G \equiv G_{\alpha }^{\alpha }= - R$ as per (2a), (3), the quadratic Ricci invariant, $r_{1} \equiv \widehat{R}_{\alpha }^{\,\,\,\beta } \widehat{R}_{\beta }^{\,\,\,\alpha } = \widehat{G}_{\alpha }^{\,\,\,\beta } \widehat{G}_{\beta }^{\,\,\,\alpha }$ as per (9a), (9b), Weyl’s curvature invariant, $I \equiv C_{\alpha \beta \gamma \delta } C^{\alpha \beta \gamma \delta }$ as per (8), and the cubic Ricci invariant, $r_{2} \equiv \widehat{R}_{\alpha }^{\,\,\,\beta } \widehat{R}_{\beta }^{\,\,\,\gamma } = \widehat{G}_{\alpha }^{\,\,\,\beta } \widehat{G}_{\beta }^{\,\,\,\gamma } \widehat{G}_{\gamma }^{\,\,\,\alpha }$ as per (11a), (11b).

$$\begin{aligned} G =-2\, v^2 \left( 3 \left( \frac{\partial f}{\partial r}\right) ^2 \, \cos ^2(\theta )+ \left( \frac{\partial f}{\partial r} + \frac{r}{2} \frac{\partial ^2 f}{\partial r^2}\right) ^2 \, \sin ^2(\theta ) \right) =2\, \varrho \; \kappa . \end{aligned}$$

(A1)

$$\begin{aligned} r_1&=\frac{1}{16 r^2} v^2 \Biggl [4 r^2 v^2 (-4 \cos (2 \theta )+3 \cos (4 \theta )+109) \frac{\partial f}{\partial r}^4 \nonumber \\&\quad +r^2 \Biggl (8 r^2 \frac{\partial ^3 f}{\partial r^3}^2 \sin ^2(\theta ) \left( 4 v^2 f^2 \cos ^2(\theta )-1\right) \nonumber \\&\quad + \frac{\partial ^2 f}{\partial r^2}^2 \Biggl (v^2 \Biggl (11 r^4 \sin ^4(\theta ) \frac{\partial ^2 f}{\partial r^2}^2+16 r^2 f \sin ^2(\theta ) (5 \cos (2 \theta )+1) \frac{\partial ^2 f}{\partial r^2} \nonumber \\&\quad +32 f^2 (4 \cos (2 \theta )+3 \cos (4 \theta )+5)\Biggr )+32 (4 \cos (2 \theta )-5)\Biggr ) \nonumber \\&\quad +32 r \frac{\partial ^3 f}{\partial r^3} \frac{\partial ^2 f}{\partial r^2} \left( v^2 f^2 \sin ^2(2 \theta )-3 \sin ^2(\theta )\right) \Biggr )\nonumber \\&\quad +32 r v^2 \sin ^2(\theta ) \frac{\partial f}{\partial r}^3 \left( r^2 (4 \cos (2 \theta )+21) \frac{\partial ^2 f}{\partial r^2}-2 f (7 \cos (2 \theta )+9)\right) \nonumber \\&\quad +8 r \frac{\partial f}{\partial r} \Biggl ( \frac{\partial ^2 f}{\partial r^2} \Biggl (v^2 \sin ^2(\theta ) \Biggl (11 r^4 \sin ^2(\theta ) \frac{\partial ^2 f}{\partial r^2}^2+2 r^2 f (13 \cos (2 \theta )-5) \frac{\partial ^2 f}{\partial r^2}\nonumber \\&\quad +8 f^2 (3 \cos (2 \theta )+5)\Biggr )+8 (\cos (2 \theta )-5)\Biggr )\nonumber \\&\quad -4 r \frac{\partial ^3 f}{\partial r^3} \left( v^2 f \sin ^2(2 \theta ) \left( r^2 \frac{\partial ^2 f}{\partial r^2}-2 f\right) +2 \sin ^2(\theta )\right) \Biggr )\nonumber \\&\quad +8 \frac{\partial f}{\partial r}^2 \Biggl (v^2 \Biggl (\sin ^2(\theta ) \Biggl (r^4 (32-7 \cos (2 \theta )) \frac{\partial ^2 f}{\partial r^2}^2+4 r^2 f (5 \cos (2 \theta )-13) \frac{\partial ^2 f}{\partial r^2}\nonumber \\&\quad +8 f^2 (3 \cos (2 \theta )+5)\Biggr )-8 r^3 f \frac{\partial ^3 f}{\partial r^3} \sin ^2(2 \theta )\Biggr )-8 (3 \cos (2 \theta )+5)\Biggr )\Biggr ]. \end{aligned}$$

(A2)

$$\begin{aligned} I&= \frac{1}{6 r^2}\ v^2 \Biggl [-6 r^4 \frac{\partial ^3 f}{\partial r^3}^2 \left( \sin ^2(\theta )-v^2 f^2 \sin ^2(2 \theta )\right) \nonumber \\&\quad +8 r^2 \frac{\partial ^2 f}{\partial r^2}^2 \Biggl (v^2 \Biggl (r^4 \sin ^4(\theta ) \frac{\partial ^2 f}{\partial r^2}^2+3 r^2 f \sin ^2(\theta ) (3 \cos (2 \theta )+1) \frac{\partial ^2 f}{\partial r^2}\nonumber \\&\quad +3 f^2 (4 \cos (2 \theta )+3 \cos (4 \theta )+5)\Biggr )-12 \sin ^2(\theta )\Biggr )\nonumber \\&\quad +r^2 v^2 (-52 \cos (2 \theta )-11 \cos (4 \theta )+351) \frac{\partial f}{\partial r}^4\nonumber \\&\quad +24 r^3 \frac{\partial ^3 f}{\partial r^3} \frac{\partial ^2 f}{\partial r^2} \left( v^2 f^2 \sin ^2(2 \theta )-2 \sin ^2(\theta )\right) \nonumber \\&\quad +16 r v^2 \frac{\partial f}{\partial r}^3 \left( r^2 \sin ^2(\theta ) (\cos (2 \theta )+29) \frac{\partial ^2 f}{\partial r^2}-6 f \sin ^2(2 \theta )\right) \nonumber \\&\quad +4 r \frac{\partial f}{\partial r} \Biggl (4 \sin ^2(\theta ) \frac{\partial ^2 f}{\partial r^2} \Biggl (v^2 \Biggl (4 r^4 \sin ^2(\theta ) \frac{\partial ^2 f}{\partial r^2}^2+3 r^2 f (5 \cos (2 \theta )+1) \frac{\partial ^2 f}{\partial r^2}\nonumber \\&\quad +3 f^2 (3 \cos (2 \theta )+5)\Biggr )+12\Biggr )\nonumber \\&\quad +3 r \frac{\partial ^3 f}{\partial r^3} \left( v^2 f \sin ^2(2 \theta ) \left( 4 f-r^2 \frac{\partial ^2 f}{\partial r^2}\right) +4 \sin ^2(\theta )\right) \Biggr )\nonumber \\&\quad +6 \frac{\partial f}{\partial r}^2 \Biggl (2 \sin ^2(\theta ) \left( -5 r^4 v^2 (\cos (2 \theta )-3) \frac{\partial ^2 f}{\partial r^2}^2-8\right) \nonumber \\&\quad +r^2 v^2 f \left( (76 \cos (2 \theta )+5 \cos (4 \theta )+15) \frac{\partial ^2 f}{\partial r^2}-4 r \frac{\partial ^3 f}{\partial r^3} \sin ^2(2 \theta )\right) \nonumber \\&\quad +8 v^2 f^2 \sin ^2(\theta ) (3 \cos (2 \theta )+5)\Biggr )\Biggr ]. \end{aligned}$$

(A3)

$$\begin{aligned} r_2&= \frac{1}{128 \ r^3}3 v^4 \Biggl [64 \ v^2 f^3 \left( r \frac{\partial ^2 f}{\partial r^2} \cos ^2(\theta )+\sin ^2(\theta ) \frac{\partial f}{\partial r}\right) \Biggl (\sin ^2(2 \theta ) \ \frac{\partial ^3 f}{\partial r^3}^2 r^4\nonumber \\&\quad +2 (4 \cos (2 \theta )+5 \cos (4 \theta )+7) \frac{\partial ^2 f}{\partial r^2}^2 r^2+4 \ \sin ^2(2 \theta ) \left( 2 \frac{\partial f}{\partial r}+r \frac{\partial ^2 f}{\partial r^2}\right) \frac{\partial ^3 f}{\partial r^3} r^2\nonumber \\&\quad +8 (\cos (2 \theta )+3) \sin ^2(\theta ) \ \frac{\partial f}{\partial r} \frac{\partial ^2 f}{\partial r^2} r+16 \sin ^2(2 \theta ) \frac{\partial f}{\partial r}^2\Biggr )\nonumber \\&\quad -4 r v^2 \Biggl (2 \sin ^2(\theta ) \ \frac{\partial ^2 f}{\partial r^2}^2 \Biggl (r^2 \frac{\partial ^3 \ f}{\partial r^3}^2 \sin ^2(2 \theta )+4 r \frac{\partial ^2 \ f}{\partial r^2} \frac{\partial ^3 f}{\partial r^3} \sin ^2(2 \ \theta )\nonumber \\&\quad -8 (4 \cos (2 \theta )+\cos (4 \theta )+1) \ \frac{\partial ^2 f}{\partial r^2}^2\Biggr ) r^4\nonumber \\&\quad +4 \frac{\partial \ f}{\partial r} \frac{\partial ^2 f}{\partial r^2} \Biggl (8 r^2 \cos ^2(\theta ) \frac{\partial ^3 f}{\partial r^3}^2 \sin ^4(\theta )\nonumber \\&\quad +(-20 \cos (2 \theta )+5 \cos (4 \theta )+47) \ \frac{\partial ^2 f}{\partial r^2}^2 \sin ^2(\theta )\nonumber \\&\quad +2 r (\cos \ (2 \theta )+7) \sin ^2(2 \theta ) \frac{\partial ^2 f}{\partial \ r^2} \frac{\partial ^3 f}{\partial r^3} \Biggr ) \ r^3+\frac{\partial f}{\partial r}^2 \Biggl (96 r^2 \sin ^2(\theta \ ) \frac{\partial ^3 f}{\partial r^3}^2 \cos ^4(\theta )\nonumber \\&\quad +(139 \ \cos (2 \theta )+226 \cos (4 \theta )+69 \cos (6 \theta )+718) \ \frac{\partial ^2 f}{\partial r^2}^2\nonumber \\&\quad +16 r (3 \cos (2 \theta )+11) \ \sin ^2(2 \theta ) \frac{\partial ^2 f}{\partial r^2} \ \frac{\partial ^3 f}{\partial r^3}\Biggr ) r^2\nonumber \\&\quad +16 \frac{\partial f}{\partial r}^3 \Biggl ((72 \cos (2 \theta )+19 (\cos (4 \theta \ )+7)) \frac{\partial ^2 f}{\partial r^2} \sin ^2(\theta ) \nonumber \\&\quad +2 r \ (\cos (2 \theta )+5) \sin ^2(2 \theta ) \frac{\partial ^3 \ f}{\partial r^3}\Biggr ) r\nonumber \\&\quad -32 (-14 \cos (2 \theta )+\cos (4 \ \theta )-35) \sin ^2(\theta ) \frac{\partial f}{\partial \ r}^4 \Biggr ) f^2\nonumber \\&\quad +8 \Biggl (16 r^2 v^2 (38 \cos (2 \theta )+9 \cos \ (4 \theta )+49) \sin ^2(\theta ) \frac{\partial f}{\partial \ r}^5\nonumber \\&\quad +4 r^3 v^2 \sin ^2(\theta ) \left( 96 r \frac{\partial ^3 f}{\ \partial r^3} \cos ^4(\theta )+(48 \cos (2 \theta )-25 \cos (4 \ \theta )+201) \frac{\partial ^2 f}{\partial r^2}\right) \ \frac{\partial f}{\partial r}^4\nonumber \\&\quad +2 \sin ^2(\theta ) \Biggl (r^4 \ v^2 \frac{\partial ^2 f}{\partial r^2} \Biggl (16 r (\cos (2 \ \theta )+5) \frac{\partial ^3 f}{\partial r^3} \cos ^2(\theta \ ) \nonumber \\&\quad +(-56 \cos (2 \theta )-25 \cos (4 \theta )+241) \ \frac{\partial ^2 f}{\partial r^2}\Biggr )-32 (7 \cos (2 \theta \ )+9)\Biggr ) \frac{\partial f}{\partial r}^3\nonumber \\&\quad +r \Biggl (v^2 (-148 \ \cos (2 \theta )-5 \cos (4 \theta )+137) \sin ^2(\theta ) \ \frac{\partial ^2 f}{\partial r^2}^3 r^4\nonumber \\&\quad +32 \sin ^2(\theta ) \ \left( 3 v^2 \cos ^2(\theta ) \sin ^2(\theta ) \ \frac{\partial ^2 f}{\partial r^2}^2 r^4-7 \cos (2 \theta \ )-9\right) \frac{\partial ^3 f}{\partial r^3} r\nonumber \\&\quad +16 (-16 \cos (2 \ \theta )+\cos (4 \theta )-49) \frac{\partial ^2 f}{\partial \ r^2}\Biggr ) \frac{\partial f}{\partial r}^2\nonumber \\&\quad +4 r^2 \Biggl (-r^2 (7 \ \cos (2 \theta )+9) \frac{\partial ^3 f}{\partial r^3}^2 \sin \ ^2(\theta )\nonumber \\&\quad +4 r \frac{\partial ^2 f}{\partial r^2} \left( v^2 \ \cos ^2(\theta ) \sin ^2(\theta ) \frac{\partial ^2 \ f}{\partial r^2}^2 r^4-13 \cos (2 \theta )-23\right) \ \frac{\partial ^3 f}{\partial r^3} \sin ^2(\theta )\nonumber \\&\quad + \ \frac{\partial ^2 f}{\partial r^2}^2 \left( 19 r^4 v^2 \ \frac{\partial ^2 f}{\partial r^2}^2 \sin ^6(\theta )+20 \cos (2 \ \theta )-17 \cos (4 \theta )-131\right) \Biggr ) \frac{\partial \ f}{\partial r}\nonumber \\&\quad \!+\!2 r^3 \frac{\partial ^2 f}{\partial r^2} \ \Biggl (r^4 v^2 (1\!-\!3 \cos (2 \theta )) \sin ^4(\theta ) \ \frac{\partial ^2 f}{\partial r^2}^4\!+\!(56 \cos (2 \theta )\!-\!30 \ \cos (4 \theta )\!-\!58) \frac{\partial ^2 f}{\partial r^2}^2 \nonumber \\&\quad +16 r (\ \cos (2 \theta )-2) \sin ^2(\theta ) \frac{\partial ^3 \ f}{\partial r^3} \frac{\partial ^2 f}{\partial r^2}-4 r^2 \sin \ ^2(\theta ) \frac{\partial ^3 f}{\partial r^3}^2\Biggr )\Biggr ) \ f\nonumber \\&\quad +2 r \Biggl (48 r^2 v^2 \cos ^2(\theta ) (38 \cos (2 \theta )+3 \ \cos (4 \theta )-5) \frac{\partial f}{\partial r}^6 \nonumber \\&\quad +8 r^3 v^2 \ (-44 \cos (2 \theta )+9 \cos (4 \theta )-133) \sin ^2(\theta ) \ \frac{\partial ^2 f}{\partial r^2} \frac{\partial f}{\partial \ r}^5\nonumber \\&\quad +2 \Biggl (r^4 v^2 (132 \cos (2 \theta )+41 \cos (4 \theta \ )-341) \sin ^2(\theta ) \frac{\partial ^2 f}{\partial r^2}^2\nonumber \\&\quad -16 \ (56 \cos (2 \theta )+3 \cos (4 \theta )+37)\Biggr ) \ \frac{\partial f}{\partial r}^4\nonumber \\&\quad +32 r \Biggl (-r^4 v^2 (\cos (2 \ \theta )+18) \frac{\partial ^2 f}{\partial r^2}^3 \sin \ ^4(\theta )+2 r (3 \cos (2 \theta )+7) \frac{\partial ^3 \ f}{\partial r^3} \sin ^2(\theta )\nonumber \\&\quad -4 (13 \cos (2 \theta )+\cos \ (4 \theta )-2) \frac{\partial ^2 f}{\partial r^2}\Biggr ) \frac{\ \partial f}{\partial r}^3\nonumber \\&\quad +4 r^2 \Biggl (r^4 v^2 (14 \cos (2 \theta \ )-33) \sin ^4(\theta ) \frac{\partial ^2 f}{\partial r^2}^4\nonumber \\&\quad +16 \ (-17 \cos (2 \theta )+2 \cos (4 \theta )+12) \frac{\partial ^2 \ f}{\partial r^2}^2+8 r (11-5 \cos (2 \theta )) \sin ^2(\theta ) \ \frac{\partial ^3 f}{\partial r^3} \frac{\partial ^2 f}{\partial \ r^2}\nonumber \\&\quad -2 r^2 (5 \cos (2 \theta )+1) \sin ^2(\theta ) \ \frac{\partial ^3 f}{\partial r^3}^2\Biggr ) \frac{\partial f}{\partial r}^2-4 r^3 \sin ^2(\theta ) \frac{\partial ^2 f}{\partial r^2} \Biggl (9 r^4 v^2 \sin ^4(\theta ) \ \frac{\partial ^2 f}{\partial r^2}^4\nonumber \\&\quad +24 (3 \cos (2 \theta )-5) \ \frac{\partial ^2 f}{\partial r^2}^2+4 r (3 \cos (2 \theta )-11) \ \frac{\partial ^3 f}{\partial r^3} \frac{\partial ^2 f}{\partial \ r^2}-4 r^2 \sin ^2(\theta ) \frac{\partial ^3 f}{\partial r^3}^2\ \Biggr ) \frac{\partial f}{\partial r}\nonumber \\&\quad +r^4 \sin ^2(\theta ) \ \frac{\partial ^2 f}{\partial r^2}^2 \Biggl (-3 r^4 v^2 \sin \ ^4(\theta ) \frac{\partial ^2 f}{\partial r^2}^4+(48-96 \cos (2 \ \theta )) \frac{\partial ^2 f}{\partial r^2}^2 \nonumber \\&\quad +48 r \sin ^2(\theta ) \frac{\partial ^3 f}{\partial r^3} \ \frac{\partial ^2 f}{\partial r^2}+4 r^2 \sin ^2(\theta ) \ \frac{\partial ^3 f}{\partial r^3}^2\Biggr )\Biggr )\Biggr )\Biggr ]. \end{aligned}$$

(A4)