Before filming even began on Ridley Scott's summer blockbuster "Prometheus" screenwriter Damon Lindelof (creator of the hit show "Lost") suggested exploiting the power of social media to help answer audiences' mounting questions about the mysterious plot of the film. The problem: 20th Century Fox was insisting the sci-fi epic bore absolutely no relation to Mr. Scott's "Alien" franchise, but "our position was that this was not entirely the case," says Mr. Lindelof, who has about 200,000 Twitter followers and tweets several times a day.

To...