A few months ago, David Fincher was having a problem with his new movie. This in itself wasn’t especially surprising, as Fincher’s productions seem to attract crises of the cosmic-joke variety, be they midshoot injuries (Se7en), last-minute casting switcheroos (Panic Room), or on-the-fly script rewrites (Alien³). Despite the director’s meticulous planning—he can spend years preparing for a film—something usually goes awry. He’s used to it. “All movies are a trial,” he says matter-of-factly. “It’s war.”

But...