Quantum effects in the Alcubierre warp drive spacetime

The expectation value of the stress-energy tensor of a free conformally invariant scalar field is computed in a two-dimensional reduction of the Alcubierre “warp drive” spacetime. The stress-energy is found to diverge if the apparent velocity of the spaceship exceeds the speed of light. If such behavior occurs in four dimensions, then it appears implausible that “warp drive” behavior in a spacetime could be engineered, even by an arbitrarily advanced civilization. e-mail: [email protected] 1 Alcubierre [1] has described a spacetime which has several of the properties associated with the “warp drive” of science fiction. By causing the spacetime to contract in front of a spaceship, and expand behind, the Alcubierre warp drive spacetime allows a spaceship to have an apparent speed relative to distant objects much greater than the speed of light. The stress-energy needed to have a spacetime of this sort is known to require matter which violates the weak, strong, and dominant energy conditions [1]. While quantized fields can locally violate the energy conditions, Pfenning and Ford [2] have recently demonstrated that the configuration of exotic matter needed to generate the warp “bubble” around the spaceship is quite implausible. In this letter, a different issue involving quantum effects and the warp drive spacetime is examined. The curved spacetime associated with the warp drive will create a nonzero expectation value for the stress-energy of a quantized field in that spacetime. This field is assumed to be a spectator in the spacetime, not responsible for the stress-energy which supports the exotic warp drive metric. While calculating the expectation value of the stressenergy of a quantized field in a spacetime is generally an extremely difficult task, the work involved is greatly reduced if one confines attention to a two-dimensional spacetime. The warp drive spacetime admits a natural two-dimensional reduction containing the worldline of the spaceship. A coordinate transformation then renders the two-dimensional metric into a static form. For a conformally invariant massless quantized scalar field, the stress-energy is then completely determined by the trace anomaly, conservation, and the values of two integration constants which are determined by the state of the field [3,4]. The resulting expressions for 〈Tμ〉 are found to be everywhere regular so long as the ship does not exceed the speed of light, v < 1. However, for apparent ship velocities exceeding the speed of light, the stress-energy diverges at a particular distance from the ship dependent upon the speed. This divergence is associated with an event horizon which forms in the twodimensional spacetime. If the instability is not an artifact of working in two dimensions, then the spaceship would presumably be precluded from attaining apparent velocities greater than light due to metric backreaction effects. 2 The warp drive metric proposed by Alcubierre may be written as ds = −dt + (dx− vf(r)dt) + dy + dz, (1) where v is the apparent velocity of the spaceship,