If time in physics is static and just one of the dimensions, what exactly makes it different from space in physical models?
The key differences is that one can predict across time, but not across space. From a photo of a room one can easily see what happens in one hour. Most things will be at the same place, the shadows from a sun will slightly move, a cat sleeping in a chair will not be there. But that is it.
Now suppose one is gets a one-hour video of a wall in the room and the question is to describe the whole room at the end of that hour. The answer is that one cannot. As with the story of Plato’s Cave at best one can speculate from a shadows on the wall. But the speculation will be not much better than random guesses.
Physical equations reflects that. Speaking mathematically the equations are hyperbolic or parabolic in nature, but this is just precise expression in the language of math of the notion of predictability across time. Another way to say this is that each particle follows its worldline and one in principle can calculate the worldline for each particle from initial conditions both into future and past.
But this predictability is lost in General Relativity at singularities. It follows from the equations that in the center of black holes or at the Big Bang it is not possible to predict. One cannot draw the worldline through singularity. There literally anything goes.
Which explains why the singularities of General Relativity are considered bad. Our experience tells us that predictability across time is always there. Thus those singularities must be an artifact of the model and should be corrected with a better theory.
Yet General Relativity survived more and more stringent tests and now we can even detect gravitational waves from the merge of black holes. So maybe past and future are not always predictable.
Russian philosopher Lev Shestov (Лев Исаакович Шестов) observed 100 years ago in his books and articles that the notion that the past was set in stone and could not be altered was pretty much an axiom of western philosophy. Not much changed since then.
Most of discussions about changing the past are about time-travelling. And even in those discussions it is typically assumed that the past of time-travelers do not change and they do remember how the word was before their time-travel affected the history.
Yet the idea of the past that changes allows to resolve the paradox of free will. Our experience tells us that everything is predetermined and follows from the initial conditions. On the other hand most people believe in the free will. Why it is so?
One way to address it is to dismiss it and state the free will is an illusion. There is no free will and the feeling that there is one is just a consequence of a particular initial conditions of the Universe. This is a valid solution to the paradox but it is not only one.
Another solution is to assume that the free will is very real, but its act changes not only the future but also the past. The timeline becomes such that the action naturally follows from the altered past. So everything is still predetermined and follows from the initial conditions, it is just the conditions are different.
It is interesting how minuscule such changes can be. Nassim Taleb in his book “Fooled by Randomness” gave an example that while a good billiard player could predict where the ball go after second or even second reflection from the pool walls, to predict the result after 100th collision one needed to account to gravitational influence off all particles in the universe. So the act of the free will can just slightly altered a path of a photon at the Big Bang and then everything naturally follows from that to the new reality.
The Stanford Encyclopedia of Philosophy in its entry on the free will does not mention this simple solution to the paradox while listing in details a fascinating example by Harry Frankfurt how one can be morally responsible for one’s choice in the deterministic world. Which brings back the observation by Lev Shestov that unchangeable past is too sacred to depart from it.
Physics does not describe the flow of time. The time coordinate in physical equations is no different from a frame number in a movie. And as with movies, a time flow and a motion in physical models only appear when one watches a simulation of the model. Without that the model is a static construction. Essentially equations of the motion should be called equations of particle paths. They do not describe the motion. Rather the equations describe the whole path of particles statically without giving any clue why we perceive the particles as moving.
It is easy to understand why it is so. Physics are based on what we observe. But an observation is always about the past as it is based on memory. When I see at a watch that shows 9:33 I do not perceive that immediately. I need some reflection to understand that. And when I got it, the result of the observation is already about a past moment. So time in equations is similar to perception of time in memory. As with memory, one can go investigate any arbitrary moment in the past and even play the time in the opposite direction. But memory does not tell what is now, so is equations in physics.
Is it possible to describe perception of now at all? Obviously one can refer in words about that perception, but it seem to go beyond such reference and try to describe what happens is impossible. Understanding of a language comes from the memory, so as with physical equations, the language cannot describe now.
It could that it was such considerations that lead to Parmenides declaring 2500 years ago that a movement was an illusion. Based on experience the past is static and fixed. Interpolating that gives that the whole universe is a static construction that does not change with no place for now. It is just one is going with a flow and perceiving an illusion of change. As I understand, Buddhism also has this notion that the time flow is an illusion . (It is interesting that both Parmenides and Buddha lived in about the same time.)
However, the notion that perception of now is illusion tells nothing. It does not describe how this illusion arises or what it means to avoid it.
Completely opposite approach was taken by St. Augustine, who declared 1600 years ago that only now existed and the past and the future did not. It follows then it is the notion of static past and following from it predetermined future is illusion. But then again, if the static past is an illusion, then the truth should be that the past is changeable, and experience tells that it is not.
There is no movement, the wise man said.
Another was silent and walked before him.
He could not object more strongly;
The Russian poet Alexander Pushkin in a concise form gave a usual interpretation of surviving fragments of the poem by Parmenides. But these fragments do not say that there is no movement. They only assert that the movement is an illusion, and the gods and Parmenides know the real truth.
To answer to the silent wise man, modern Parmenides can take out a smartphone and record a video. And it will be clear that there was no walking, but just a film. One can watch it arbitrary number of times at any speed, in any direction, with any special effects and an arbitrary number of pauses. That we are watching this movie only once with no access to the remote control is our lot as mortals.