Why; if ONLY the non-Christian is there at this 'judgement'?
Look back at that verse which speaks about those who were brought before the Great White Throne Judgment. They were ALL cast into Hell after their judgment, no one passed to Heaven at that judgment. They were judged according to their deeds - which some people think speaks of degrees of punishment in Hell. The Book of Life is there to prove that their names are NOT written in it. It is also called the Lamb's Book of Life. The question then is what is this Book of Life, how are names put in and when do names get blotted out/removed?
There are a few schools of thought on this. The idea of a book that, from the beginning of time, has all the names of all the humans that will ever be conceived. Numerous times in the Psalms, for example, David asked God to "blot of his enemies names". The book of Revelation speaks about this Book of Life from which those who worshiped the Antichrist and took his "mark", had their names blotted out of the book:
Rev. 13:8 "And all who dwell on the earth will worship him, every one whose name has not been written from the foundation of the world in the Book of Life of the Lamb Who has been slain." (speaking of the Antichrist)
Rev. 17:8 "The beast, which you saw, once was, now is not, and will come up out of the Abyss and go to his destruction. The inhabitants of the earth whose names have not been written in the Book of Life from the creation of the world will be astonished when they see the beast, because he once was, now is not, and yet will come."
When you get to Revelation 20, it is speaking of all those who were the unsaved from all time being in the final judgment. Their names have been blotted out of the Book of Life. The above verses also speak of those who WILL take the Mark of the Beast, the Antichrist, and seems to say that there are some names that NEVER get written in the book. The Book of Life in the Revelation 20 passage is brought to the final judgment as evidence that the name of those judged is blotted out forever from it. This is a reference that explores this idea further here.