Eon by Greg Bear.

Above our planet hangs a hollow Stone, vast as the imagination of Man. The inner dimensions are at odds with the outer: there are different chambers to be breached, some even containing deserted cities. The furthest chamber contains the greatest mystery ever to confront the Stone's scientists. But tombstone or milestone, the Stone is not an alien structure: it comes from the future of our humanity. And the war that breaks out on Earth seems to bear witness to the Stone's prowess as oracle . . .