Why did the ghost appear when Gertrude and hamlet were alone Act 3?

In Act 3 of Shakespeare's "Hamlet," the ghost appears when Gertrude and Hamlet are alone to emphasize Hamlet's emotional turmoil and the weight of his unresolved feelings toward his father's murder. The ghost serves as a reminder of Hamlet's duty to avenge his father's death, heightening the tension of the moment. Additionally, it symbolizes Hamlet's internal struggle between his desire for action and his feelings of betrayal and disgust toward his mother. The ghost's appearance also signifies the thin line between reality and the supernatural in Hamlet's world.