Why the Past Does not Wait for Us
The grandfather paradox is a well-known problem of a time travel. According to this paradox, a time traveller goes back in time and kills his grandfather before he even meets his grandmother. As a result of this, the time traveller can never be born. However, if he was never born, then he is not able to travel through time and kill his grandfather, which subsequently means that the traveller would be born after all.
We must realise that a totally necessary precondition to this paradox is that the past remains absolutely unchanged forever, waiting for our time traveller. Therefore, the almost 14 billion-year history of the whole universe must be stored somewhere and being fixed up to a quantum level. What is more, this past must exist without any smallest change and be fully flexible at the same time, reacting to the actions of time travellers. Nothing else but our time traveller can change it, so it is almost a "personal" world, and this indicates that such past is only in our fantasy.
If a time traveller could not change history, meaning he could not interact with his surroundings in the past, it would not be time travelling in the usual sense of the term. This would be a mere historic 3D "movie". So, should time travel be possible, we ought to consider a changeable past. If it is changeable, the past will look (very) different when reached by the time traveller. Hence, the past is currently not what it used to be when we were there. (In an extreme case, there would be no past to return to.)
Speaking metaphorically, let us imagine that the time traveller is a fish born from a fish egg in a mountain river, carried away to a valley. He can later swim back up to the mountains to the exact place of his birth, but he will not find the fish egg from which he was born, neither his mother delivering the egg. Moreover, we can even imagine that the mountains are very dry, without any water source, and he was born during a flood wave. Should the time traveller return back, he will not find anything remotely similar to his original water world.
Now let me offer you a more advanced visual metaphor inspired by the 4D Minkowski space-time model as well as some cosmological models. Our 3D universe is a 3D hyper-surface of a 4D hyper-bubble which expands. This expansion is time. For a better understanding, we can go by one dimension lower and picture only a 3D bubble with a 2D surface. Now we can imagine that the bubble is full of "liquid" which the surface is composed of as well, or alternatively there is only a surface membrane and the bubble is empty. There is also a third possibility: there are only parts of the past inside the bubble, some drops or trickles or similar. In all cases, the past is either dynamic or non-existent, but never static and unchanged.
I personally prefer the empty bubble, which is a result of the fact that we cannot see the past from our presence, so it looks like we live inside a 3D hyper-membrane only. Of course, a concrete physical mechanism of our time must be discovered and proved by physics. I certainly do not insist on the 4D bubble or any particular solution. Although, the 4D Minkowski space-time model with its specific hyperbolic attributes offers us a good clue. Instead, I am merely suggesting that there are solutions which remove the grandfather paradox and all time travel paradoxes, and these solutions contain changes to the past or even physical non-existence of the past. The question whether the dynamics of the past is very slow, immediate and disappearing, or like a wave after a boat, is an area for further investigation of natural sciences.
Because the case of the empty bubble seems to be simple, we will investigate further the bubble full of "liquid", which is in fact "liquid of past". One possibility is that the time traveller would not meet his grandfather, because the granddad would have disappeared and would not be there anymore. But if he was still there, the death of the granddad (by the hand of the time traveller) and its consequences would be delayed behind the actual position/motion of the 3D surface, never reaching it because the velocity of the impacts would be the same as the velocity of the 3D surface. (The velocity of the light cone movement alongside the time axis is c within the Minkowski space-time model, and mutual velocity of two objects with c velocity is again c.) Returning back to the original time, the time traveller would not see any consequences of his action and would not notice any changes to history. Our time would not wait for impacts of changed history, and would simply move farther. By the way, the traveller would not be able to go back to that original moment of time, from which he departed.
Furthermore, in the case of "full bubble", there will not be any 3D time membrane; there will be only a 4D hyper-space which we cannot see by light rays. (We know this from our 3D world where we cannot see 4D space.) Hence, being in history, we would not observe any world similar to ours, and our way of interaction with surroundings would be very physically different. We would not be able to see the history (via light), to touch objects there, neither we would be able to speak to people who in fact would not be people like in our world.
Therefore, in this bubble model, there is no way of considering time travelling precisely in a traditional meaning of the term. Let me offer you one possibility a bit similar to the conventional understanding of the term. Imagine, that we can dent the bubble and move a local part of the membrane to the past or to the future. It would be something different than a common extraction of the time traveller from our time. Instead, it would be a local continuous deformation of our space-time where the time traveller goes back in time together with us and the entire environment. In this case, the grandfather paradox is out of the question and practically impossible. When the time goes back before our birth, there could be our granddad, but certainly not us. And so, the causality that would otherwise be heavily damaged by the grandfather paradox, is saved.
There is one very probable conclusion resulting from this thinking: Considering our time as a motion of our universe (3D hyper-surface), we are looking at it from a hyper-space and thinking in hyper-time. In this case, we perceive our time only as one motion in hyper-time. Based on the assumption of the hyper-time, there is no problem regarding the beginning of our time in the big bang as a creation of our universe from the multiverse. Hyper-time would be the time of multiverse. This would also explain the origin of time relativity.
Ultimately, the grandfather paradox does not suggest that time travelling is impossible, however, it shows us what kind of time travelling is impossible (fixed waiting past). Therefore, it offers us detailed information on how the nature of time travel should look like.