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3. Krasnikov tube

Krasnikov tube

The Krasnikov tube is an attempt to modify the Alcubierre warp drive metric in a way that does not violate the Flux Energy Condition (the matter used to support it does not go faster than light itself).

1. History

2. Topology

The Krasnikov tube has the topology $\mathbb{R}^n$.

3. Metrics and coordinates

The two-dimensional case

\begin{eqnarray} ds^2 &=& - (dt - dx) (dt + k(x,t) dx)\\ &=& -dt^2 + (1 - k(x,t)) dx dt + k(x,t) dx^2 \end{eqnarray} \begin{equation} k(x,t) = 1 - (2 - \delta) \theta_\varepsilon(t-x) [\theta_\varepsilon(x) - \theta_\varepsilon(x + \varepsilon - D)] \end{equation} \begin{equation} \theta_\varepsilon(x) = \begin{cases} 1 & x > \varepsilon \\ 0 & x < 0 \end{cases} \end{equation}

The four-dimensional case

\begin{equation} ds^2 = -dt^2 + (1 - k(x,t, \rho)) dx dt + k(x,t, \rho) dx^2 + d\rho^2 + \rho^2 d\phi^2 \end{equation} \begin{equation} k(x,t) = 1 - (2 - \delta) \theta_\varepsilon(\rho_{max} - \rho) \theta_\varepsilon(t-x - \rho) [\theta_\varepsilon(x) - \theta_\varepsilon(x + \varepsilon - D)] \end{equation}

4. Tensor quantities

5. Symmetries

6. Stress-energy tensor

7. Curves

8. Equations

9. Causal structure

10. Asymptotic structure

11. Energy conditions

12. Limits and related spacetimes

13. Misc.

Bibliography