The Wiki says, "Physicists are undecided whether the prediction of singularities means that they actually exist". Is this really true? originally appeared on Quora, the place to gain and share knowledge, empowering people to learn from others and better understand the world. You can follow Quora on Twitter, Facebook, and Google Plus.

The Wiki says, "Physicists are undecided whether the prediction of singularities means that they actually exist". Is this really true?

It looks like a short, sweet summary of a wide-ranging issue which belongs more to the philosophy of physics than to physics.

Before starting, it’s worth drawing a rather obvious distinction. That is, between a mathematical representation of a physical something, and the physical something itself. Yes, this is straightforward, but my goodness, the problems when you get the two confused…

With that out of the way, it’s pretty uncontroversial that singularities can exist in many perfectly good physical theories (that is, in the mathematical representations of physical things). In fact, we can go stronger: singularities are often useful in physical theories. Or stronger still, singularities are often essential in physical theories.

But despite this, it not at all clear that there are singularities existing in the things represented (i.e., in the physical quantities existing out there in the world).

To try to tease this apart, it’s useful to make a switch from the question’s implied subject, that is, the singularities at the center of black holes as arising in the General Relativity. Because here, intuition is rubbish — at least mine is. So think of a kettle of boiling water (intuition is better).

Our best theories of matter use singularities to represent phase transitions (that is transitions like boiling). In both statistical mechanics and thermodynamics, phase transitions are characterized by singularities in the equations of state (more precisely, by singularities in some derivative in the expression for the free energy). The nature of the phase transition and its physical properties can be analysed and categorized by which derivative it appears in (hence first-order vs second order phase transitions).

There is something weird about this though. In a finite system, the free energy of a finite system — say — a kettle of water can be written as a finite-but-very-long sum of infinitely differential expressions. And such a finite sum can be proven to have no singularities anywhere. It is only when we take the infinite limit (known as the thermodynamic limit) that the expression can gain singularities and the points of phase transition start to be represented mathematically.1

But we are doing this while conscious it is wrong — we know that the kettle of water is finite, yet it obviously boils. Effectively, we have to lie a bit (make it infinite) then read off the nature of the singularities to analyse the phase transition, then draw our conclusions while trying to forget that we lied. And this is the way to get the most effective representation of the system that we know of.

In a sense, these singularities act both as essential to the modelling of the system, but simultaneously an indication that the representation we are using is overidealized in some way. There is no sense in which the “actual system” is singular (however that might be cashed out).

Right, so moving back to the question. First, General Relativity does seem to predict singularities as arising under some very general conditions of gravitational collapse, as shown by the Penrose-Hawking singularity theorems.

So here, we have some genuine singularities in the representation of gravitational collapse.

But what does that mean? My answer, drawing from examples in more familiar circumstances would be that something very interesting, very physical and very significant will be happening at the points represented by the singularities. And also that we should take seriously the predictions of GR for the “black hole” regions surrounding them, just as we take seriously statistical mechanics as a representation of states on their approach towards boiling and freezing.

But, based on the existence of a singularity in the mathematics of GR: our best theoretical representation of gravity, should we say that there is a singularity actually “out there” in the world? No, not based on the examples we have a handle on. Whether some kind of ultimate “finite bittiness” of spacetime comes to our rescue (as a direct analogy with kettles might suggest), or something else, we do not know. But we do have enough examples of places in which we use an idealization to produce a theoretically-useful-but-not-actually-out-there-singularity to be cautious.

So, to the wiki which states that physicists are “undecided about the existence of actual singularities” (which I assume means singularities in the actual object represented, not in the theoretical representation), then I’d say — yes. This is a good, if rather cryptic summary of the appropriate attitude to take.

1 And you can’t just fix it by going for “steepness” instead. You can get steep changes in the free energy that don’t become singularities in the infinite limit; these do not represent phase transitions.

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