The Grandfather Paradox is the most common example used to illustrate causality problems in time travel. The scenario is as follows:
Someone who has the opportunity to travel back in time travels back in time to before his father was conceived and kills his grandfather there. The paradox in this situation arises from the fact that without the existence of his father, who is not born because of the death of his grandfather, the time traveler himself cannot be born and consequently could not have traveled back in time to his own grandfather to kill.
The grandfather paradox mostly serves to clarify the principle of causality and the impossibility of traveling into the past, which is justified with it. In the past, there is no cause for the time traveler to appear. If you consider the target system (e.g. the universe) as a closed system, the appearance of the time traveler in the past would also be a violation of the law of conservation of energy or the first two main laws of thermodynamics.
However, if one assumes in the context of a thought experiment that a person can “appear” due to a journey through time without having been born to date, then one would assume that the time traveler does not need a father and a grandfather because he is in his new time stepped into the world not from the womb but “finished” from the time machine. So he would not have to rely on biological ancestors.
Therefore, he could kill his “ancestors” without further consequences, especially since these “ancestors” were his ancestors only in the subjective memory of the time traveler. Like all memories, these memories are only certain states in the brain and, like the rest of the body, would have been generated by the time machine during this thought experiment.
Biological ancestors are either required for human existence or they are not required. However, the grandfather paradox assumes, on the one hand, that they are necessary, but at the same time inconsistently allows a person to emerge before they are conceived and thus without biological ancestors. It is through this lack of internal logic that the paradox arises.
Approaches to a solution are also worked out and discussed, in which one assumes the possibility of time travel.
One hypothesis is that there can be different strands of time and history side by side. Then you can say that when the time traveler arrives, the storyline splits into a story with time travelers and an “original” story without time travelers. While the time traveler is no longer born in the former, the time traveler is born in the latter and can therefore also organize time travel. The causality is thus guaranteed since there is no causal loop in any time thread. For obvious reasons, this solution is often used in connection with the many-worlds interpretation of quantum mechanics.
Another assumption is the so-called self-consistent universe: it is possible to travel in time, but not to produce causality violations in the process; any attempt to do so is guaranteed to fail. For example, the time traveler might simply not find his pistol, or he makes a mistake in programming the time machine and ends up somewhere completely different (or in a different time) and therefore cannot find his grandfather, or he confuses someone else with his grandfather (and now I know who committed the puzzling unsolved murder his grandfather once told him about), or he misses, or the pistol jams, or … This represents a restriction on the often invoked human free will – the implementation of human will (for example also the intention to fly on one’s own initiative) can be impaired by external circumstances (for example laws of nature).
The problem, however, is that if time travel is possible, then causal paradoxes could arise. A classic example is the grandfather paradox: if a time traveler kills his grandfather in the past before he fathered his father, he himself should never be born. The time traveler therefore did not even exist in his time and could not travel back either. Classical physics forbids such causal paradoxes.
The self-consistent universe can also be justified with quantum mechanics: In quantum mechanics, the state of the system is described by the quantum mechanical wave function. This has the special property that when two waves that have taken different paths are superimposed (normally, e.g., in the double-slit experiment, the left and right slit; in a time, loop, on the other hand, the part coming directly from the past and the part running through the time loop), these can annihilate each other, so that an event that can be reached in itself is no longer possible. It stands to reason that paradoxical strand of events would extinguish one another.