It is in Chapter Twelve that Arthur Dimmesdale, standing on the scaffold with Hester and Pearl at night, sees the meteor that he interprets as a sign regarding his own position and, in particular, his guilt. The narrator talks at length about the way such signs that appear in the night sky are interpreted, and it is clear that the narrator believes Dimmesdale is looking at this comet through his own guilty situation and feelings and sees it as a sign that speaks to him alone. Note how the narrator comments upon what Arthur Dimmesdale saw in the sky:
We impute it, therefore, solely to the disease in his own eye and heart, that hte minister, looking upward to the zenith, behld there the appearance of an immense letter--the letter "A"--marked out in lines of dull red light. Not but the meteor may have shown itself at that point, burning duskily through a veil of cloud; but with no such shape as his guilty imagination gave it; or, at least, with so little definiteness, that another's guilt might have seen another symbol in it.
It is clear, therefore, that Arthur Dimmesdale interprets the comet that he sees in the night sky as being about himself and his own situation, reflecting his own guilt at not wearing the same scarlet "A" that Hester has been forced to wear and because of which she has suffered so much. What he has tried to keep secret, the heavens themselves have chosen to reveal, from his perspective.