While the possibility of time travel in physics is still debated, the explosive growth of virtual-reality simulations opens up new possibilities to rigorously explore such time travel and its consequences in the digital domain. Here we provide a computational model of time travel and a computer program that allows exploring digital time travel. In order to explain our method we formalize a simplified version of the famous grandfather paradox, show how the system can allow the participant to go back in time, try to kill their ancestors before they were born, and experience the consequences. The system has even come up with scenarios that can be considered consistent "solutions" of the grandfather paradox. We discuss the conditions for digital time travel, which indicate that it has a large number of practical applications.