Visualization and analysis of the curvature invariants in the Alcubierre warp-drive spacetime

Abstract

In the Alcubierre warp-drive spacetime, we investigate the following scalar curvature invariants: the scalar I, derived from a quadratic contraction of the Weyl tensor, the trace R of the Ricci tensor, and the quadratic r1 and cubic r2 invariants from the trace-adjusted Ricci tensor. In four-dimensional spacetime the trace-adjusted Einstein and Ricci tensors are identical, and their unadjusted traces are oppositely signed yet equal in absolute value. This allows us to express these Ricci invariants using Einstein‘s curvature tensor, facilitating a direct interpretation of the energy-momentum tensor. We present detailed plots illustrating the distribution of these invariants. Our findings underscore the requirement for four distinct layers of an anisotropic stress-energy tensor to create the warp bubble. Additionally, we delve into the Kretschmann quadratic invariant decomposition. We provide a critical analysis of the work by Mattingly et al., particularly their underrepresentation of curvature invariants in their plots by 8 to 16 orders of magnitude. A comparison is made between the spacetime curvature of the Alcubierre warp-drive and that of a Schwarzschild black hole with a mass equivalent to the planet Saturn. The paper addresses potential misconceptions about the Alcubierre warp-drive due to inaccuracies in representing spacetime curvature changes and clarifies the classification of the Alcubierre spacetime, emphasizing its distinction from class B warped product spacetimes.

Appendix A: Curvature invariants of the Alcubierre metric

$$\begin{aligned} f_r= & {} \frac{\tanh {[\sigma (\sqrt{(x-x_{0}(t))^2+y^2+z^2}+\rho )}]-\tanh {[\sigma (\sqrt{(x-x_{0}(t))^2+y^2+z^2}-\rho )}]}{2\tanh {[\sigma \rho ]}} \nonumber \\ \end{aligned}$$

(A1)

$$\begin{aligned} v_s= & {} \frac{\partial x_{0}(t)}{\partial t} \end{aligned}$$

(A2)

$$\begin{aligned} G= & {} -\frac{1}{2} v_s \left( v_s \left( \frac{\partial f_r}{\partial z}^2+\frac{\partial f_r}{\partial y}^2+4 \left( \frac{\partial f_r}{\partial x}^2+f_r \frac{\partial ^2 f_r}{\partial x^2}\right) \right) +4 \frac{\partial ^2 f_r}{\partial t \partial x}\right) \end{aligned}$$

(A3)

$$\begin{aligned} r_1= & {} \frac{1}{16} v_s^2 \left( 11 v_s^2 \frac{\partial f_r}{\partial z}^4+11 v_s^2 \frac{\partial f_r}{\partial y}^4+8 \left( 2 v_s^2 \frac{\partial f_r}{\partial x}^4+v_s^2 f_r^2 \left( \frac{\partial ^2 f_r}{\partial x \partial z}^2\right. \right. \right. \nonumber \\{} & {} \left. +\frac{\partial ^2 f_r}{\partial x \partial y}^2+ 2 \frac{\partial ^2 f_r}{\partial x^2}^2\right) +4 v_s \frac{\partial f_r}{\partial x}^2 \left( v_s f_r \frac{\partial ^2 f_r}{\partial x^2}+\frac{\partial ^2 f_r}{\partial t \partial x}\right) \nonumber \\{} & {} +2 v_s f_r \left( \frac{\partial ^2 f_r}{\partial x \partial z} \frac{\partial ^2 f_r}{\partial t \partial z}+\frac{\partial ^2 f_r}{\partial x \partial y} \frac{\partial ^2 f_r}{\partial t \partial y}+2 \frac{\partial ^2 f_r}{\partial x^2} \frac{\partial ^2 f_r}{\partial t \partial x}\right) \nonumber \\{} & {} \left. -\frac{\partial ^2 f_r}{\partial x \partial z}^2-\frac{\partial ^2 f_r}{\partial x \partial y}^2+\frac{\partial ^2 f_r}{\partial t \partial z}^2+\frac{\partial ^2 f_r}{\partial t \partial y}^2+2 \frac{\partial ^2 f_r}{\partial t \partial x}^2\right) \\{} & {} + 2 v_s \frac{\partial f_r}{\partial z}^2 \left( 11 v_s \frac{\partial f_r}{\partial y}^2-4 \left( -3 v_s \frac{\partial f_r}{\partial x}^2+v_s f_r \frac{\partial ^2 f_r}{\partial x^2}+\frac{\partial ^2 f_r}{\partial t \partial x}\right) \right) \nonumber \\{} & {} +32 v_s \frac{\partial f_r}{\partial x} \frac{\partial f_r}{\partial z} \left( v_s f_r \frac{\partial ^2 f_r}{\partial x \partial z}+\frac{\partial ^2 f_r}{\partial t \partial z}\right) + 32 v_s \frac{\partial f_r}{\partial y} \frac{\partial f_r}{\partial x} \left( v_s f_r \frac{\partial ^2 f_r}{\partial x \partial y}+\frac{\partial ^2 f_r}{\partial t \partial y}\right) \nonumber \\{} & {} \left. -8 v_s \frac{\partial f_r}{\partial y}^2 \left( -3 v_s \frac{\partial f_r}{\partial x}^2+v_s f_r \frac{\partial ^2 f_r}{\partial x^2}+\frac{\partial ^2 f_r}{\partial t \partial x}\right) -8 \left( \frac{\partial ^2 f_r}{\partial z^2}+\frac{\partial ^2 f_r}{\partial y^2}\right) ^2\right) \nonumber \end{aligned}$$

(A4)

$$\begin{aligned} I{} & {} =\frac{1}{3} v_s^2 \left( 4 v_s^2 \frac{\partial f_r}{\partial z}^4+4 v_s^2 \frac{\partial f_r}{\partial y}^4+4 v_s^2 \frac{\partial f_r}{\partial x}^4+3 v_s^2 f_r^2 \frac{\partial ^2 f_r}{\partial x \partial z}^2+3 v_s^2 f_r^2 \frac{\partial ^2 f_r}{\partial x \partial y}^2\right. \nonumber \\{} & {} \quad + 4 v_s^2 f_r^2 \frac{\partial ^2 f_r}{\partial x^2}^2+8 v_s^2 f_r \frac{\partial f_r}{\partial x}^2 \frac{\partial ^2 f_r}{\partial x^2}\nonumber \\{} & {} \quad +4 v_s \frac{\partial f_r}{\partial z}^2 \left( 2 v_s \left( \frac{\partial f_r}{\partial y}^2+ \frac{\partial f_r}{\partial x}^2\right) -v_s f_r \frac{\partial ^2 f_r}{\partial x^2}-\frac{\partial ^2 f_r}{\partial t \partial x}\right) \nonumber \\{} & {} \quad +12 v_s \frac{\partial f_r}{\partial x} \frac{\partial f_r}{\partial z} \left( v_s f_r \frac{\partial ^2 f_r}{\partial x \partial z}+\frac{\partial ^2 f_r}{\partial t \partial z}\right) +6 v_s f_r \frac{\partial ^2 f_r}{\partial x \partial z} \frac{\partial ^2 f_r}{\partial t \partial z}+6 v_s f_r \frac{\partial ^2 f_r}{\partial x \partial y} \frac{\partial ^2 f_r}{\partial t \partial y}\nonumber \\{} & {} \quad + 12 v_s \frac{\partial f_r}{\partial y} \frac{\partial f_r}{\partial x} \left( v_s f_r \frac{\partial ^2 f_r}{\partial x \partial y}+\frac{\partial ^2 f_r}{\partial t \partial y}\right) \nonumber \\{} & {} \quad +4 v_s \frac{\partial f_r}{\partial y}^2 \left( 2 v_s \frac{\partial f_r}{\partial x}^2-v_s f_r \frac{\partial ^2 f_r}{\partial x^2}-\frac{\partial ^2 f_r}{\partial t \partial x}\right) \nonumber \\{} & {} \quad + 8 v_s \left( \frac{\partial f_r}{\partial x}^2+f_r \frac{\partial ^2 f_r}{\partial x^2}\right) \frac{\partial ^2 f_r}{\partial t \partial x}+3 \frac{\partial ^2 f_r}{\partial t \partial z}^2+3 \frac{\partial ^2 f_r}{\partial t \partial y}^2+4 \frac{\partial ^2 f_r}{\partial t \partial x}^2\nonumber \\{} & {} \quad \left. -3 \left( 4 \frac{\partial ^2 f_r}{\partial y \partial z}^2+\left( \frac{\partial ^2 f_r}{\partial z^2}-\frac{\partial ^2 f_r}{\partial y^2}\right) ^2\right) -3 \left( \frac{\partial ^2 f_r}{\partial x \partial z}^2+\frac{\partial ^2 f_r}{\partial x \partial y}^2\right) \right) \end{aligned}$$

(A5)

$$\begin{aligned} r_2= & {} -\frac{3}{64} v_s^3 \left( 3 v_s^3 \frac{\partial f_r}{\partial z}^6+v_s^2 \left( 9 v_s \frac{\partial f_r}{\partial y}^2+4 v_s \frac{\partial f_r}{\partial x}^2-12 \left( v_s f_r \frac{\partial ^2 f_r}{\partial x^2}+\frac{\partial ^2 f_r}{\partial t \partial x}\right) \right) \frac{\partial f_r}{\partial z}^4\right. \\{} & {} + 16 v_s^2 \frac{\partial f_r}{\partial x} \left( v_s f_r \frac{\partial ^2 f_r}{\partial x \partial z}+\frac{\partial ^2 f_r}{\partial t \partial z}\right) \frac{\partial f_r}{\partial z}^3+v_s \left( 9 v_s^2 \frac{\partial f_r}{\partial y}^4+8 v_s \left( v_s \frac{\partial f_r}{\partial x}^2\right. \right. \\{} & {} \left. - 3 \left( v_s f_r \frac{\partial ^2 f_r}{\partial x^2}+\frac{\partial ^2 f_r}{\partial t \partial x}\right) \right) \frac{\partial f_r}{\partial y}^2+16 v_s \frac{\partial f_r}{\partial x} (v_s f_r \frac{\partial ^2 f_r}{\partial x \partial y}+\frac{\partial ^2 f_r}{\partial t \partial y}) \frac{\partial f_r}{\partial y}\\{} & {} - 4 \left( \frac{\partial ^2 f_r}{\partial z^2}+\frac{\partial ^2 f_r}{\partial y^2}\right) ^2+4 \left( (v_s^2 f_r^2+3) \frac{\partial ^2 f_r}{\partial x \partial z}^2 +2 v_s f_r \frac{\partial ^2 f_r}{\partial t \partial z} \frac{\partial ^2 f_r}{\partial x \partial z}+\frac{\partial ^2 f_r}{\partial x \partial y}^2+\frac{\partial ^2 f_r}{\partial t \partial z}^2\right. \\{} & {} +\left. \left. 3 \left( v_s f_r \frac{\partial ^2 f_r}{\partial x \partial y}+\frac{\partial ^2 f_r}{\partial t \partial y}\right) ^2\right) \right) \frac{\partial f_r}{\partial z}^2+16 v_s \left( -v_s^2 \frac{\partial f_r}{\partial y} \frac{\partial ^2 f_r}{\partial x \partial z} \frac{\partial ^2 f_r}{\partial x \partial y} f_r^2\right. \\{} & {} + v_s \frac{\partial f_r}{\partial y} \left( \frac{\partial ^2 f_r}{\partial x \partial z} \left( v_s \frac{\partial f_r}{\partial y} \frac{\partial f_r}{\partial x}-\frac{\partial ^2 f_r}{\partial t \partial y}\right) -\frac{\partial ^2 f_r}{\partial x \partial y} \frac{\partial ^2 f_r}{\partial t \partial z}\right) f_r-2 \frac{\partial ^2 f_r}{\partial z^2} \frac{\partial f_r}{\partial x} \frac{\partial ^2 f_r}{\partial x \partial z}\\{} & {} \left. -2 \frac{\partial ^2 f_r}{\partial y^2} \frac{\partial f_r}{\partial x} \frac{\partial ^2 f_r}{\partial x \partial z}+ \frac{\partial f_r}{\partial y} \frac{\partial ^2 f_r}{\partial x \partial z} \frac{\partial ^2 f_r}{\partial x \partial y}+v_s \frac{\partial f_r}{\partial y}^2 \frac{\partial f_r}{\partial x} \frac{\partial ^2 f_r}{\partial t \partial z}- \frac{\partial f_r}{\partial y} \frac{\partial ^2 f_r}{\partial t \partial z} \frac{\partial ^2 f_r}{\partial t \partial y}\right) \frac{\partial f_r}{\partial z}\\{} & {} +3 v_s^3 \frac{\partial f_r}{\partial y}^6-4 v_s \frac{\partial f_r}{\partial y}^2 \frac{\partial ^2 f_r}{\partial y^2}^2+4 v_s^3 \frac{\partial f_r}{\partial y}^4 \frac{\partial f_r}{\partial x}^2+16 v_s \frac{\partial ^2 f_r}{\partial y^2}^2 \frac{\partial f_r}{\partial x}^2\\{} & {} + 12 v_s^3 f_r^2 \frac{\partial f_r}{\partial y}^2 \frac{\partial ^2 f_r}{\partial x \partial z}^2+4 v_s \frac{\partial f_r}{\partial y}^2 \frac{\partial ^2 f_r}{\partial x \partial z}^2-16 v_s f_r \frac{\partial ^2 f_r}{\partial y^2} \frac{\partial ^2 f_r}{\partial x \partial z}^2\\{} & {} +4 v_s^3 f_r^2 \frac{\partial f_r}{\partial y}^2 \frac{\partial ^2 f_r}{\partial x \partial y}^2 +12 v_s \frac{\partial f_r}{\partial y}^2 \frac{\partial ^2 f_r}{\partial x \partial y}^2-16 v_s f_r \frac{\partial ^2 f_r}{\partial y^2} \frac{\partial ^2 f_r}{\partial x \partial y}^2\\{} & {} +12 v_s \frac{\partial f_r}{\partial y}^2 \frac{\partial ^2 f_r}{\partial t \partial z}^2+4 v_s \frac{\partial f_r}{\partial y}^2 \frac{\partial ^2 f_r}{\partial t \partial y}^2\\{} & {} +16 v_s^3 f_r \frac{\partial f_r}{\partial y}^3 \frac{\partial f_r}{\partial x} \frac{\partial ^2 f_r}{\partial x \partial y}-32 v_s \frac{\partial f_r}{\partial y} \frac{\partial ^2 f_r}{\partial y^2} \frac{\partial f_r}{\partial x} \frac{\partial ^2 f_r}{\partial x \partial y}-12 v_s^3 f_r \frac{\partial f_r}{\partial y}^4 \frac{\partial ^2 f_r}{\partial x^2}\\{} & {} + 16 v_s f_r \frac{\partial ^2 f_r}{\partial y^2}^2 \frac{\partial ^2 f_r}{\partial x^2}+24 v_s^2 f_r \frac{\partial f_r}{\partial y}^2 \frac{\partial ^2 f_r}{\partial x \partial z} \frac{\partial ^2 f_r}{\partial t \partial z}-16 \frac{\partial ^2 f_r}{\partial y^2} \frac{\partial ^2 f_r}{\partial x \partial z} \frac{\partial ^2 f_r}{\partial t \partial z}\\ \end{aligned}$$

$$\begin{aligned} \begin{aligned}&\quad +16 v_s^2 \frac{\partial f_r}{\partial y}^3 \frac{\partial f_r}{\partial x} \frac{\partial ^2 f_r}{\partial t \partial y}+8 v_s^2 f_r \frac{\partial f_r}{\partial y}^2 \frac{\partial ^2 f_r}{\partial x \partial y} \frac{\partial ^2 f_r}{\partial t \partial y}-16 \frac{\partial ^2 f_r}{\partial y^2} \frac{\partial ^2 f_r}{\partial x \partial y} \frac{\partial ^2 f_r}{\partial t \partial y}\\&\quad + 4 \left( 4 \frac{\partial ^2 f_r}{\partial y^2}^2-3 v_s^2 \frac{\partial f_r}{\partial y}^4\right) \frac{\partial ^2 f_r}{\partial t \partial x}\\&\quad +4 \frac{\partial ^2 f_r}{\partial z^2}^2 \left( 4 \left( v_s \frac{\partial f_r}{\partial x}^2+v_s f_r \frac{\partial ^2 f_r}{\partial x^2}+\frac{\partial ^2 f_r}{\partial t \partial x}\right) -v_s \frac{\partial f_r}{\partial y}^2\right) \\&\quad -8 \frac{\partial ^2 f_r}{\partial z^2} \left( v_s \frac{\partial ^2 f_r}{\partial y^2} \frac{\partial f_r}{\partial y}^2+4 v_s \frac{\partial f_r}{\partial x} \frac{\partial ^2 f_r}{\partial x \partial y} \frac{\partial f_r}{\partial y}+2 \left( v_s f_r \left( \frac{\partial ^2 f_r}{\partial x \partial z}^2+\frac{\partial ^2 f_r}{\partial x \partial y}^2\right) \right. \right. \\&\quad +\left. \left. \left. \frac{\partial ^2 f_r}{\partial x \partial z} \frac{\partial ^2 f_r}{\partial t \partial z}+\frac{\partial ^2 f_r}{\partial x \partial y} \frac{\partial ^2 f_r}{\partial t \partial y}-2 \frac{\partial ^2 f_r}{\partial y^2} \left( v_s \frac{\partial f_r}{\partial x}^2+v_s f_r \frac{\partial ^2 f_r}{\partial x^2}+\frac{\partial ^2 f_r}{\partial t \partial x}\right) \right) \right) \right) \end{aligned} \end{aligned}$$

(A6)