# 阿庫別瑞引擎

## 數學形式

$ds^2 = g_{ij}dx^i dx^j = -\left(\alpha^2- \beta_i \beta^i\right)\,dt^2+2 \beta_i \,dx^i\, dt+ \gamma_{ij}\,dx^i\,dx^j$

$\alpha=1 ;\,$
$\beta^1=\beta^x=-v_s(t)f\left(r_s(t)\right), \beta^2=\beta^y=0, \beta^3=\beta^z=0 ;\,$
$\gamma_{ij}=\delta_{ij}= \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

$v_s(t)=\frac{dx_s(t)}{dt},$
$r_s(t)=\sqrt{(x-x_s(t))^2+y^2+z^2}$

$f(r_s)=\frac{\tanh(\sigma (r_s + R))-\tanh(\sigma (r_s - R))}{2 \tanh(\sigma R)}$

$R>0\,$$\sigma>0\,$為任意參數。

## 阿庫別瑞之後的發展

• Chris Van Den Broeck於一篇1999年的論文試圖闡述一些可能的課題，亦發表在《古典和量子引力》期刊。藉由將引擎運送的「曲速泡」的3+1維表面積收縮，同時擴張所包含的內部3維體積，Van Den Broeck能將運輸幾顆小原子所需的總能量減少到少於3個太陽質量。隨後，透過對Van Den Broeck度規稍微修改，Sergey Krasnikov可以將負能量總需求縮減到幾個毫克[9]
• González-Díaz的工作解決了二維量子不穩定性的問題。González-Díaz將結果發表在《物理評論D》62卷（Physical Review D Vol. 62），提議考慮封閉類時曲線的課題。這樣的改善允許了多連通空間（multiply-connected spaces），結束了短程線不完備性並滿足了量子不穩定性的要求。

## 注釋

1. ^ 1.0 1.1 Alcubierre, Miguel. The warp drive: hyper-fast travel within general relativity. Classical and Quantum Gravity. 1994, 11 (5): L73–L77. arXiv:gr-qc/0009013. Bibcode:1994CQGra..11L..73A. doi:10.1088/0264-9381/11/5/001.
2. ^ Thorne, Kip; Michael Morris; Ulvi Yurtsever. Wormholes, Time Machines, and the Weak Energy Condition. Physical Review Letters. 1988, 61 (13): 1446–1449. Bibcode:1988PhRvL..61.1446M. doi:10.1103/PhysRevLett.61.1446. PMID 10038800.
3. ^ See The Alcubierre Warp Drive by John G. Cramer, where Cramer notes that "Alcubierre, following the lead of wormhole theorists, argues that quantum field theory permits the existence of regions of negative energy density under special circumstances, and cites the Casimir effect as an example."
4. ^ Roundup. Lyndon B. Johnson Space Center. July 2012 [1 October 2013].
5. ^ Dr. Harold "Sonny" White. Warp Field Mechanics 101. NASA Johnson Space Center. 30 September 2011 [28 January 2013].
6. ^ Moskowitz, Clara. Warp Drive May Be More Feasible Than Thought, Scientists Say. Space.com. 17 September 2012 [10 January 2013].
7. ^ [1]
8. ^ 星際奇旅設定中，關於曲速引擎會運用到的「子空間/在科幻上」則與阿庫別瑞所提到的重力及空間扭曲無甚關連。
9. ^ [2]

## 文獻

• 原始論文：米給爾·阿庫別瑞(Alcubierre, Miguel). The Warp Drive: Hyper-Fast Travel Within General Relativity. Class. Quantum Grav. 1994年, 11: L73–L77. eprint
• 羅伯·洛(Low, Robert). Speed Limits in General Relativity. Class. Quantum Grav. 1999年, 16: 543–549. eprint