So, I have been watching some science videos regarding Einstein's theory on general relativity and until today the predictions based on his equations have been proven to stand.
My question would be: what happens in the scientific community if one experiment proves it wrong (not only Einstein, but even other laws and theories of physics? Do we automatically get rid of these centuries-old theories, or do they get rewritten to fit the new experiment, or do they stay the same, but with exceptions?
My question would be, what happens in the scientific community if one experiment proves it wrong
We have already seen what happens in this circumstance by looking at what happened to Newtonian gravity.
First, well before the development of general relativity there were observations that did not fit with Newtonian gravity. For example, Uranus’ orbit did not match Newtonian predictions. It was found that by modifying the predictions by including an unobserved source of gravity, the data could be coerced into fitting the observations. Subsequent observations confirmed the planet Neptune. As another example, Mercury’s orbit also did not fit, and a similar additional planet named Vulcan was proposed. The planet Vulcan was never observed through other means.
Now, afterward general relativity was developed. It explained the orbit of Mercury without requiring Vulcan. In addition, many other phenomena were predicted and discovered. Many of these phenomena were not predicted by Newtonian gravity or the wrong value was predicted. Through the course of these observations Newtonian gravity was explicitly falsified.
However, after Newtonian gravity was falsified it still continued to be taught in schools. The Apollo space program and other spacecraft successfully reached their destinations using the falsified Newtonian gravity theory.
The thing is that although the theory was falsified it had also been verified for centuries and none of that verification was removed by the falsification. Newtonian gravity continued to accurately predict all of the phenomena that it had ever been shown to accurately predict. If you were only interested in those previously verified phenomena then you could continue to use Newtonian gravity with confidence, and there is a strong incentive to do so because it is computationally far simpler than general relativity.
So, at some point when an experiment falsifies general relativity then new sources will be sought and if they cannot be found then that will place limits on its domain of validity, but it will not reverse any of the evidence that validates it within its domain of validity. Furthermore, just as general relativity needed to reduce to Newtonian gravity in the appropriate domain, so any future theory will need to reduce to general relativity in the appropriate domain.
If the future theory is computationally more difficult than general relativity, then we would continue to use general relativity just as we have continued to use Newtonian gravity. Thus, we would fully expect future students to learn general relativity just as current students still learn Newtonian gravity. General relativity will not go away, even after such an experiment
Words like "proven" and "wrong" have to be used carefully in this context. It is more meaningful to talk about "accuracy" and "limits". If an experiment was conducted tomorrow that contradicted general relativity it would by no means make general relativity a useless theory, nor would we get rid of it.
The purpose of theories in physics isn't to "prove" anything about the real world. This isn't even something they are capable of doing. Their purpose is to as accurately as possible predict the outcome of experiments.
Of course if there comes a time when a prediction of general relativity is not found in nature the likely reaction will be to attempt to reconcile the theory with the experiment, rather than throwing it out and starting from square 1.
Every currently accepted physical theory will likely be eventually found "wrong" at some scale. But all their predictions have already been experimentally verified a zillion-and-one times at the scales currently accessible to apparatus we know how to construct. So all these theories will forever be valid approximations to "the truth" at these scales, i.e., any more foundationally correct theory will necessarily reduce to our currently accepted approximate theories at these scales.
Take Newtonian mechanics, for example. Go too fast and it fails, requiring the use of special relativity. But $\sqrt{1-v^2/c^2}$ reduces to $1$ when $v\ll c$, so special relativity reduces to newtonian. Or study very small masses and it again fails, requiring quantum mechanics. Or study very large masses and it fails yet again, requiring general relativity.
And since you bring it up, maybe even general relativity will eventually fail. For example, maybe $G$ varies over cosmological times, which would require some kind of modification to general relativity. Oh, and maybe it might be this kind of modification -- https://pubs.giss.nasa.gov/abs/ca00010g.html But even if Canuto, et al, are right, general relativity will remain a valid approximation at the time scales accessible to our currently available apparatus.
You cannot prove a theory wrong, because a theory is like a world-view. Claiming that a theory can be proven wrong is like claiming that the concept of "color" or "speed" is wrong. Theories can of course be inconsistent, but they would then be rejected instantly.
Now, physics theories are still connected to "real life" with what is called a model. To understand the difference between a theory and a model, a good analogy is to think of a theory as the rules of a game, and of a model as a very game (as in, a concrete game that is happening with all of its specificities, choices, etc). The very game is constrained by the rules of the game, and it is what is tangible or "measurable". You're never "measuring" the rules of a game, as you never "measure" a theory. At best, you can "measure" a model of a theory.
Obviously, some properties of a theory are universally true, in the sense that they are true in every model, but it is never easy to conclude that what you measured is what you think you did.
Why is it so? Quite simply, because there is a big difference between the abstract model you write on your piece of paper in a given theoretical framework (say, a model of black hole in general relativity), and a "real life" phenomenon.
First, to identify the "real life phenomenon" as a very model you wrote explicitely on your piece of paper is not choice-free. A choice is involved, and by choice I mean that someone used of his subjectivity to actually claim that what he measures "in real life" is actually identified to this abstract model written on a piece of paper.
Already here, you understand that people can easily object that this identification is legitimate. Maybe the predictions made by your model are not wrong, but the measured "real life" phenomenon is actually not represented faithfully by your abstract model. I call this the downward modelization problem.
Secondly, it is always possible to complexify an abstract model so that it predicts what you measure with good enough accuracy. Well, that's kind of trivial, as you can always add as many free parameters as you want, and do a fit of the curve of your "real life" phenomenon. The question of finding a simple, canonical enough abstract model that predicts your real life phenomenon is at least as hard as the problem above. I call this one the upward modelization problem.
So, what would happen now if the prediction of an abstract model are not fitting the measures of a "real life" phenomenon? If the measures are contradicting a universal claim, then assuming there is no mistake in your measurement apparatus, that would kind of "prove" the theory wrong. Smart physicists are never going to assume that though (stupid ones are going to publish thousands of articles about why it is so tho, think of neutrino going faster than c years ago). If anything, when such a universal claim is "rejected" by a measure, the first thing that comes to mind to smart physicists, is that the experiment was poorly done. The same comes to mind to experimentalists by the way. They need a way to be convinced that their apparatus is good, and they precisely conclude that it is so when their measure of a universal claim (or say, or physics constant) fits what is accepted in the theory.
What if the rejected claim is non-universal? Well, most people will now assume that you did a mistake in the downward modelization, as in, you "forgot to add some effects to your model". They will then try to add these "missing effects" into the abstract model to find a "cheated" solution to the upward modelization problem. The solution is "cheated" because it is rare that it is going to be canonical or legitimate, and somehow, the newly created cheated abstract model of your theory is able to predict with a good enough accuracy your real life phenomenon.
So, what do you do to actually reject a theory, or even to realize that a phenomenon you measured is actually rejecting a theory rather than a model (like, say, dark matter)? There is no choice-free, algorithmic way to do it. You're forced to use your brain, which is apparently hard for most physicists in 2020.
In 2011, the OPERA experiment claimed to have detected neutrinos that traveled faster than the speed of light. Einstein's theory of relativity doesn't allow particles with mass to travel faster than the speed of light, so this experiment violated his theory of relativity.
What happened subsequently, is described in this truly excellent summary, which gives a timeline of all the discussion and further experiments that were done afterwards.
I'm not going to repeat that article here, but here's the gist of it:
Some steps in the above sequence may vary, depending on the type of experiment, its scale, and the theory being violated.
I'm a bit late, but I think there's still stuff to contribute in here. To make things a bit easier, I'll give an example not with General Relativity, but with Newtonian gravity instead (just because the calculations are easy enough for me to do them in here).
Suppose you are an engineer in 1900, before Relativity was discovered. You trust Newtonian gravity pretty well and you know that the force between two masses is given by the law $$F = \frac{G M m}{r^2}, \tag{1}$$ where $G$ is Newton's constant, $M$ and $m$ are the objects' masses, and $r$ is the distance between their centers of mass.
Now let us say you want to build, on Earth, a $10 \text{ m}$ tall building. What is the gravitational force a brick in the building will feel?
Noticing that $r$ can be written as $r = R + h$, where $R$ is the radius of the Earth and $h$ is the height of the brick with respect to the ground, you write the force as $$F = \frac{G M m}{(R+h)^2}.$$
However, you notice that $h \sim 10 \text{ m}$ is much smaller than $R \sim 6000 \text{ km}$. Hence, you decide to write the force as a Taylor series in $h$. In other words, you do a mathematical trick you learned back in Calculus I to write the force in the form $$F = \frac{G M m}{R^2}\left(1 - 2 \frac{h}{R} + 3 \frac{h^2}{R^2} - 4\frac{h^3}{R^3} + \cdots\right),$$ where the dots represent terms proportional to $h^4$ or higher. The sum goes on forever, but if you compute it you'll get back to the original expression. For simplicity, you write $g = \frac{GM}{R^2}$ so you get to $$F = mg\left(1 - 2 \frac{h}{R} + 3 \frac{h^2}{R^2} - 4\frac{h^3}{R^3} + \cdots\right).$$ Plugging in the experimental values for $G$, $M$, and $R$ you'll find roughly $g \approx 9.8 \text{ m}/\text{s}^2$.
You then decide to plug in the values $h = 10 \text{ m}$ and $R = 6000 \text{ km}$ to get an idea of how much each term contributes. You'll then notice that you get $$F = mg\left(1 - 0.00000333\ldots + 0.000000000008333\ldots - 0.0000000000000000185185185\ldots + \cdots\right),$$ i.e., you notice the contributions proportional to $h$ and higher are really tiny. You think for a while and notice that the building you're constructing is not sensible to forces as tiny as $\frac{mgh}{R} \approx 0.000032666\ldots \text{ m}/\text{s}^2 \times m$. Since these corrections are so small, you choose to drop them and work with $$F \approx mg.$$ After all, your building won't change if you used the more precise Newtonian theory.
The upshot here is: Physics is an experimental science. Many times we can work with better theories, but choose not to, because old theories are often easier to make computations and are really reliable for some experiments. If your experiment is not sensible enough to notice the difference, there is no reason to use the more complicated theory. Hence, even though the engineer knows the "better theory" to be given by Eq. (1) on the top of this answer, they choose to use $F = mg$ because the building is not sensible enough to notice the difference. Of course, things would change if the engineer was building a satellite. In this case, one might have $\frac{h}{R} \sim 1$, meaning the Newtonian corrections would be extremely relevant. Furthermore, if the engineer was designing a GPS system, the Newtonian theory wouldn't be enough, and they would need to work with General Relativity.
In modern Physics, we see all currently known and well-tested theories as effective theories: they are effective in describing the phenomena we are interested up to a certain scale, and after that they have no compromise of working. However, even if they fail for extreme cases, they work really well in a certain range of validity, and we can get quite good predictions and understanding with them.
If GR was falsified by an experiment, we would continue to use it in the situations we know it works, but search for something new in the cases it doesn't. In some sense, that's what we are already doing. As far as I know, few physicists believe General Relativity to be completely right. Most of us think quantum effects must come into play at very small distances and we'll need a theory of Quantum Gravity to deal with that stuff, which would be close to the Big Bang or within black holes, for example. However, in most situations, General Relativity works amazingly well.
I think it's safe to say most scientists hope for well-trusted theories to be falsified. When something goes wrong between theory and experiment, there is the opportunity to learn more. We get a clue on how the Universe really works and get to be astonished once more with whatever we learn. Few, if any, physicists want things to work perfectly, for that would be quite boring.
In short, if GR was falsified, we'd need a new theory of gravity for those phenomena which GR can't describe, but we'd still trust it for the phenomena we know it describes well.
