What does it mean for the laws of physics to "break down" at a singularity? [duplicate]

Question

When the statement says:

As you get to the center of a singularity the laws of physics "break down".

What exactly does that mean?

Answer 1

The "laws of physics" are mathematical models that describe the behavior of things in the universe. We use the laws to determine things like how matter moves and how fields interact by doing math.

A singularity in general relativity is a place where the curvature of spacetime is equal to infinity. The curvature is a representation of the gravitational field, so another way to think about it is that the gravitational strength is infinity. We can't meaningfully do calculations with this infinity, so we can't use the rules of general relativity to predict the behavior of anything at a singularity.

There's also good reason to believe that infinities in nature aren't real. To have infinite density, like at a singularity, you would need to have infinite energy or zero size, since density is (energy)/(volume). And those conditions can't really be met in nature. This is evidence that general relativity does not correctly describe the behavior of the universe at singularities.

General relativity is an excellent mathematical model to describe the universe, but it is not the whole story. The known laws of physics (general relativity) give nonsensical answers (infinity) at singularities. At a singularity the known laws of physics don't work.

What we need is a better law of physics. For example, people are working on theories of quantum gravity, where there are no singularities in the center of black holes. Instead there is a finite sized, finite density quantum mechanical object. This way there are no infinities, and you would get sensible answers about how the universe works. This is a very challenging problem, so we don't have a complete theory of quantum gravity yet.

Answer 2

Here is an easy way to understand this.

Imagine you have a law of physics which states that the strength of some interaction decreases as the interacting things move farther apart, so the interaction scales as ~ 1/(distance^2). Now for small distances, the interaction becomes strong; for teeny distances the interaction strength becomes enormous, and as distance goes to zero the interaction strength becomes infinite.

Since there are no infinities in nature, that "1/(distance^2)" law is said to break down as distance goes to zero because it then furnishes a nonsense result.

Answer 3

Note that this is not very different from saying that the laws of plane geometry break down as a surface becomes curved. The so-called "laws" are a description of what happens under a specific set of conditions, though ideally a very wide range of conditions. When you get outside that assumed range, the simplified everyday forms may become less accurate and more complicated descriptions may be needed, and we may not yet know what those complicated forms are. Black holes are just an extreme case of that.

Answer 4

Ok, so allow me to indulge you in a slightly different aspect of the story, which would imply that a singularity is not a place where physics "breaks down", but instead prevents physics from breaking down.

The idea of a singularity theorem is that there are some conditions on how geodesics behave (in the Penrose theorem this is the null energy condition), how physically things happen (say the existence of a trapped surface where light cannot escape in any direction), a causal condition (global hyperbolicity and no closed timelike curves) and how something physically interesting happens (a non-compact Cauchy surface). If you put all these together, at least in a certain way you get that there must be a light ray that has to stop abruptly somewhere. Not because physics "breaks down" (in the conventional sense), but because a non-compact surface cannot become a compact Cauchy surface.

Now, more physically, there is this thing called the Bousso bound or the covariant entropy bound, which states that the most entropy that can go through a compact surface is related to the area of that surface. So, this becomes $S=\frac{\text{Area of } I}{4G}$, where $I$ is the compact surface with a converging pair of light rays (making the surface marginally trapped). Now, this is an upper bound -- so, if this was violated, the compact lightsheet -- the surface formed by these converging pair of light rays till they converge -- would have more entropy flux than can go through the surface! So you say that at least one of these light rays must abruptly end somewhere. This is a null geodesic incompleteness -- a singularity, granted not exactly a curvature singularity but one that can be attributed to one for a number of reasons -- and so the existence of a singularity saves the violation of the Bousso bound!

So, my point is that the laws breaking down is a very tongue-in-cheek statement. It is true that we don't understand how these things can resolve or how they can affect the causal structure in a certain way. But, the fact is that physics near singularities is more important -- and well understood -- than physics at a singularity (which is not a very meaningful question at certain scales), and is the one that is physically more relevant.