Yet Hawking also acknowledged that once his mathematical depiction of the geometry of space is transformed back into the real domain with a real-time variable—the domain of mathematics that does apply to our universe—the singularity reappears. As he noted, “When one goes back to the real time in which we live, however, there will still appear to be singularities. . . . Only if [we] lived in imaginary time would [we] encounter no singularities. . . . In real time, the universe has a beginning and an end at singularities that form a boundary to space-time and at which the laws of science break
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