If someone traveled through space-time and encountered their past self, proving their identity would indeed be a challenging task. Time travel scenarios often involve complex paradoxes and logical conundrums, and this situation is no exception.
In a strict, self-consistent model of time travel, it would be impossible to prove one's identity beyond a reasonable doubt because any proof could potentially create a paradox. Interacting with one's past self could alter the events that led to the future self's existence, creating a paradoxical situation. This is known as the "grandfather paradox," where someone travels back in time and prevents their own birth by killing their grandfather before they have children.
To maintain consistency, some time travel theories suggest that events would unfold in a predetermined manner, preventing any changes to the timeline. In this case, the future self's presence and actions would have been a part of the past self's experiences all along. The past self may not require any proof since they would have memories of encountering their future self.
Alternatively, if we consider a more flexible model of time travel, such as the multiple timelines theory, proving identity becomes less straightforward. In this theory, time travel creates separate branching timelines, where each decision or action leads to a different reality. In such a scenario, the future self could provide personal information or knowledge that only they would know, but it would be impossible to prove the authenticity of this information to the past self with absolute certainty.
Ultimately, the concept of time travel and its implications are still hypothetical and subject to scientific and philosophical debate. As of now, there is no scientific consensus on the possibility, nature, or consequences of time travel, so proving one's identity in a time travel scenario remains an intriguing but unresolved question.