Superluminal travel requires negative energies

Abstract

I investigate the relationship between faster-than-light travel and weak-energy-condition violation, i.e., negative energy densities. In a general spacetime it is difficult to define faster-than-light travel, and I give an example of a metric which appears to allow superluminal travel, but in fact is just flat space. To avoid such difficulties, I propose a definition of superluminal travel which requires that the path to be traveled reach a destination surface at an earlier time than any neighboring path. With this definition (and assuming the generic condition) I prove that superluminal travel requires weak-energy-condition violation.

Figures (5)

• Figure 1: A null geodesic in the metric of Eq. (\ref{['eqn:metric']}). It appears that one can reach arbitrary distances before $t = 1$.
• Figure 2: Superluminal travel is produced by modifying the shaded region of Minkowski space. The modification is localized between $x_1$ and $x_2$ and after $t_0$. Because of this modification, there is a causal path $P$ connecting $(t_1, x_1)$ to $(t_2, x_2)$, even though $x_2-x_1 > t_2-t_1$.
• Figure 3: A superluminal travel arrangement. The metric has been so arranged so that a causal path (solid line) exists between $A$ and $B$ but there are no other causal paths (such a possibility is shown dashed) that connect the 2-surfaces $\Sigma_A$ and $\Sigma_B$.
• Figure 4: Congruence of null geodesics from $\lambda (s)$ followed into the future until they reach points near $B$ with $t = 0$ at a curve $\lambda' (s)$ with tangent vector $Z$. At points near $B$, $\lambda' (s)$ must have negative $z$ coordinate.
• Figure 5: Circular conducting plates give rise to a negative pressure and energy density, and a consequent advancement of the time of arrival of a null ray from $A$ to $B$.