In physics, we sometimes say that an idea “breaks down”. What do we mean by that?
When a theory “breaks down”, we mean that it stops being accurate. Newton’s theory of gravity is excellent most of the time, but for objects under strong enough gravity or high enough speed its predictions stop matching reality and a new theory (relativity) is needed. This is the sense in which we say that Newtonian gravity breaks down for the orbit of mercury, or breaks down much more severely in the area around a black hole.
When a symmetry is “broken”, we mean that it stops holding true. Most of physics looks the same when you flip it in a mirror, a property called parity symmetry. Take a pile of electric and magnetic fields, currents and wires, and you’ll find their mirror reflection is also a perfectly reasonable pile of electric and magnetic fields, currents and wires. This isn’t true for all of physics, though: the weak nuclear force isn’t the same when you flip it in a mirror. We say that the weak force breaks parity symmetry.
What about when a more general “idea” breaks down? What about space-time?
In order for space-time to break down, there needs to be a good reason to abandon the idea. And depending on how stubborn you are about it, that reason can come at different times.
You might think of space-time as just Einstein’s theory of general relativity. In that case, you could say that space-time breaks down as soon as the world deviates from that theory. In that view, any modification to general relativity, no matter how small, corresponds to space-time breaking down. You can think of this as the “least stubborn” option, the one with barely any stubbornness at all, that will let space-time break down with a tiny nudge.
But if general relativity breaks down, a slightly more stubborn person could insist that space-time is still fine. You can still describe things as located at specific places and times, moving across curved space-time. They just obey extra forces, on top of those built into the space-time.
Such a person would be happy as long as general relativity was a good approximation of what was going on, but they might admit space-time has broken down when general relativity becomes a bad approximation. If there are only small corrections on top of the usual space-time picture, then space-time would be fine, but if those corrections got so big that they overwhelmed the original predictions of general relativity then that’s quite a different situation. In that situation, space-time may have stopped being a useful description, and it may be much better to describe the world in another way.
But we could imagine an even more stubborn person who still insists that space-time is fine. Ultimately, our predictions about the world are mathematical formulas. No matter how complicated they are, we can always subtract a piece off of those formulas corresponding to the predictions of general relativity, and call the rest an extra effect. That may be a totally useless thing to do that doesn’t help you calculate anything, but someone could still do it, and thus insist that space-time still hasn’t broken down.
To convince such a person, space-time would need to break down in a way that made some important concept behind it invalid. There are various ways this could happen, corresponding to different concepts. For example, one unusual proposal is that space-time is non-commutative. If that were true then, in addition to the usual Heisenberg uncertainty principle between position and momentum, there would be an uncertainty principle between different directions in space-time. That would mean that you can’t define the position of something in all directions at once, which many people would agree is an important part of having a space-time!
Ultimately, physics is concerned with practicality. We want our concepts not just to be definable, but to do useful work in helping us understand the world. Our stubbornness should depend on whether a concept, like space-time, is still useful. If it is, we keep it. But if the situation changes, and another concept is more useful, then we can confidently say that space-time has broken down.
4 gravitons 
On a tangential note, I have a basic question.
Why does string theory require AdS space and why is our observed universe a dS space.
Follow-up questions:
Can string theory be modified to support dS space
Can different parts of the universe exist in different kinds (dS or AdS) space or is it required to be a constant property everywhere ?
Does space-time curvature have any additional effects on general relativity equation that become significant in cases where curvature is high ?
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So far, the idea that string theory requires AdS space is just a conjecture. People know some ways to get AdS space out of string theory, and while there are some proposals for how to get dS space it’s much more controversial whether they actually work. But we don’t know one way or the other, it goes beyond the contexts in which string theory has a precise definition so far.
As to why our observed universe is a dS space, do you just mean how that was observed? Essentially, the observation is that the expansion of the universe is accelerating.
dS and AdS are statements about how the universe behaves overall (for example, an AdS space has a boundary at a finite distance away, dS space does not). You can have different parts of the universe that look like they’re AdS or dS though. One of the proposals string theorists have for making string theory consistent with the appearance of a dS universe is of this sort, they use something called a quintessence field to make the universe temporarily behave like it’s dS.
I’m not sure what you’re asking in your last question. The general relativity equations already include the effects of space-time curvature. However, if you have a theory that goes beyond general relativity, like a theory of quantum gravity, then you will also get new effects that become more significant when the curvature is high. Is that what you were asking about?
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Thanks for the detailed response.
Yes, for the last part I was referring to any new effects that become significant when curvature is high.
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Matt
A small correction:
The Ads timelike boundary is at an infinite proper distance away from any point in the bulk. It takes only a finite time for a light ray to reach it (and bounce back), though.
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What are the metric equations for a ds space and an ads space?
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You can find some in the relevant Wikipedia pages.
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The breakdown of an idea or a concept is a very interesting topic with many nuances and subtleties. Your post is mostly focused on the concept of spacetime ( that is related with classical GR and its domain of applicability).
Almost everybody expects GR ( and perhaps spacetime itself) to break down when mathematical absurdities occur ( like infinite densities at the Big Bang or unbounded tidal deformations inside black holes etc).
In spacelike singularities there is no extension in the future ( or in the past) classically.
Even for realistic rotating black holes, GR predicts an instability in the vicinity of the inner horizon ( mass inflation) that develops a “weak” null singularity ( in the mildest cases ). That singularity may be extendible, although this extension might not be “smooth”. Another indication that familiar concepts ( smooth manifolds, differentiability ) may not be adequate near strong curvature regions.
Violations of strong and weak cosmic censorship are also related to this.
Not only for the strictly classical theory , but also when QFTs are involved.
For example, what really happens when a black hole evaporates, at the final stage?
In a sense, if it disappears completely and leaves behind, locally, flat spacetime, that’s kind of probing arbitrarily small distances and strong curvature regimes from the external perspective.
Weak CC is violated in that case and we have the information loss problem.
Semi-classical physics does not give a prediction for this final stage ( we don’t know even if the hole disappears completely or it leaves a remnant behind).
From all the above it’s not clear, though, if it is the concept of spacetime that becomes obsolete ( or at least a useful ” macroscopic” approximation ) and has to be replaced by something more abstract, or if there are different, or complementary physical descriptions of the same reality.
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4gravitons,
This is slightly off-topic, my question is about quantum electrodynamics, but, depending on your answer, I might add something about the breaking down issue.
I have no experience with this theory. I am a chemist and I’ve only learned standard QM for atomic spectra, orbitals, etc. I imagined that there should be no problem to use the full relativistic theory (the best, most precise theory of mankind, as it is usually presented) to describe an atom. Yet, from what I can find in literature this is not the case. Even if computation power is not a problem you can’t just enter the QED equations and get a nice simulation of a hydrogen atom on your computer screen. You need to make some assumptions (like treat the proton’s field classically, etc.).
Is my understanding of this correct? Thanks!
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So, certainly the particular assumption you mention is unnecessary: people use QED to calculate properties of positronium, where the two particles have the same mass and thus you can’t assume that either is classical.
In general, people use approximations whenever they can get away with them, so for a problem like the hydrogen atom where there are so many convenient small parameters (ratio of electron to proton mass, electron velocity over the speed of light, fine structure constant), basically everything you find in the literature will be making some approximation or other. That doesn’t mean that you can’t in principle calculate whatever you’d like to know without those approximations.
(One approximation is inevitable, the approximation of a small fine structure constant. That’s because the theory is expected to genuinely break down outside of that approximation, due to something called the Landau pole.)
Outside of the above, I don’t know of any reason why you couldn’t calculate anything you’d want to know about a hydrogen atom with QED, or even a more complicated atom (though the last would in many cases require more computer power than anybody’s got). It might not look like the “nice simulation on your computer screen” you’re imagining, because you’re likely imagining something with wavefunctions for energy levels and wavefunctions are a feature of the QM approximation. But yeah any actual observable should still be calculable.
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