Black Holes are where God divided by zero, so the saying goes. As short sayings go, that’s not a bad description of a black hole’s singularity, and it gives one a good idea why singularities are so problematic in physics.

In my last post I wrote about the cosmic censorship conjecture, and how it might be violated in hypothetical 5-dimensional black holes. I didn’t delve too deeply into the conjecture itself because there are actually multiple versions of the conjecture. The weak cosmic censorship conjecture is basically as I stated, that a singularity must be enclosed by an event horizon. It’s a bit more subtle than that, since an event horizon isn’t a local object in space, but rather a global structure in spacetime. So the formal definition is that a singularity can’t be seen by an observer sitting far away from a black hole (what is called null infinity). If you were close to a black hole you might catch a glimpse of a singularity, but the structure of spacetime is such that you couldn’t tell someone far from a black hole what you saw.

The upshot of all this is that anybody reasonably distant from a black hole could never see a singularity or interact with it. Since it can’t effect things outside a black hole, we don’t have to care about how it would affect things. As long as Pandora’s box stays closed, there’s no need to worry. As I mentioned in the earlier post, there are theoretical examples in general relativity where a singularity can be seen by a distant observer, but none of them seem likely to occur in a real situation.

The strong cosmic censorship conjecture takes a different approach. This is where the “dividing by zero” idea comes in. If you were to divide a number by zero, you might think the answer is infinity. After all, zero can go into a number like 1 an infinite number of times. But the actual answer is undefined. The formal reason has to do with the subtleties of mathematics, but for our purposes suppose we first divided by a very small number. For example, if we divide 1 by 0.01, the answer would be 100. If we divided 1 by 0.0001 we get 10,000. If we kept dividing by an ever smaller number, our answer would get bigger and bigger. This seems to say that 1/0 is infinity, but suppose instead we divided 1 by -0.01. In that case the answer would be -100. Dividing by -0.0001 we get -10,000. That would make 1/0 negative infinity. Starting with a small negative number or a small positive number gets us to the same 1/0, so is the answer positive or negative? The ratio 1/0 is meaningless without knowing how we approached zero.

A singularity is similar to this in that it is indeterministic. If all you have is a singularity, you have no idea how it became a singularity. Likewise if a singularity were to interact with other objects in the universe, the outcome would be unpredictable. So the strong cosmic censorship conjecture proposes that general relativity must be deterministic. As a result, singularities must be excluded from interaction with the rest of the universe.

It turns out there are solutions to Einstein’s field equations that satisfy the weak conjecture but not the strong conjecture, and vice versa. By itself general relativity is not bound by either censorship conjecture. But general relativity is also perfectly fine with warp drive and time travel as well, and they don’t seem to be physically possible for reasons beyond relativity. So it’s likely that some physical process prevents these kinds of weird singularities from occurring.

Then again, we’ve been wrong before.

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Did the big bang really happen? Yes, despite recent claims to the contrary.  A new paper in Physical Letters B has the popular press wondering if there was no big bang, but the actual paper claims no such thing.

The big bang is often presented as some kind of explosion from an initial point, but actually the big bang model simply posits that the universe was extremely hot and dense when the universe was young. The model makes certain predictions, such as the existence of a thermal cosmic background, that the universe is expanding, the abundance of elements, etc. All of these have matched observation with great precision. The big bang is a robust scientific theory that isn’t going away, and this new paper does nothing to question its legitimacy.

That doesn’t mean there aren’t unanswered questions about the big bang. For example, simple big bang models show that if you go back in time far enough, there is time when the entire universe was an infinitely dense singularity. This singularity would mark time zero for the cosmos. As many of you know, singularities are problematic, and they tend to stir up lots of debate. That’s where this paper comes in.

The paper presents a big bang model without an initial singularity. It does this by looking at a result derived from general relativity known as the Raychaudhuri equation. Basically his equation describes how a volume of matter changes over time, so its a great way of finding where physical singularities exist in your model. But rather than using the classical Raychaudhuri equation, the authors use a variation with a few quantum tweaks. This approach is often called semi-classical, because it uses some aspects of quantum theory, but isn’t a complete quantum gravity model (which we don’t have).

You can have a big bang without a beginning. Credit: Ethan Siegel.

What the authors show is that their modified Raychaudhuri model eliminates the initial singularity of the big bang. It also predicts a cosmological constant, which is a proposed mechanism for dark energy. Their model is really basic, but this first result shows that this type of approach could work. The catch is that by eliminating the singularity, the model predicts that the universe had no beginning. It existed forever as a kind of quantum potential before “collapsing” into the hot dense state we call the big bang. Unfortunately many articles confuse “no singularity” with “no big bang.”

While this is an interesting model, it should be noted that it’s very basic. More of a proof of concept than anything else. It should also be noted that replacing the big bang singularity with an eternal history isn’t a new idea. Many inflation models, for example, make similar predictions. But none of these ideas eliminate the big bang, which is an established scientific fact.

Paper: Ahmed Farag Alia & Saurya Das. Cosmology from quantum potential. Phys. Let. B. 741, 276–279. (2015)

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Yesterday’s post on black holes stirred up things a bit, particularly among those who truly think black holes don’t exist. Some of the counter-arguments centered on the strangeness of event horizons, but most of it centered on the fact that black holes have singularities, which clearly defy all things science. Ergo, black holes don’t exist. Here we go again.

A (too) simple black hole model. Credit: Northern Arizona University

To begin with, let’s talk a little bit about what a singularity is and what it has to do with black holes. In simple terms, as the weight of an object tries to cause it to collapse, the pressure of the material pushes back until an equilibrium is reached. In the case of a star, it is the pressure of heated plasma that balances the weight, while for a white dwarf or neutron star it is the quantum degeneracy pressure of electrons or neutrons respectively. But at ever higher densities, the weight becomes harder to overcome. What’s worse, since mass and energy are related in relativity, there comes a point where the pressure of a material actually increases the weight, so the very thing used to counter gravity becomes part of the problem. It’s kind of like putting a fire out with water. Pour water on a small fire and it quickly goes out. Pour water on the Sun, and the weight of the water actually increases the Sun’s mass, which causes the Sun to heat up.

So the basic idea is that for things more dense than a neutron star, you get to the point where things are too dense to reach equilibrium. Even trying to create enough pressure to counter gravity would only help gravity, and so collapse is inevitable. The material of a star would simple collapse until there is literally no smaller volume to occupy. It would become a point of infinite density and zero volume, which is known as a singularity. At this point some of you are probably thinking “see, this is the kind of model-dependent nonsense I’m talking about.” Clearly singularities are nonsense, so clearly black holes don’t exist.

A galaxy with a black hole jet. Credit: NASA/JPL-Caltech

Interestingly, this is exactly the type of argument many astrophysicists made when black holes were first proposed. Even Einstein doubted black holes were possible. So you’re in good company if you doubt black holes, but you’re also about a century behind the times. Despite the way black holes are often presented, it wasn’t the model that convinced astrophysicists of black holes, it was the evidence.

The first “black hole” solution to Einstein’s general relativity equations was found by Karl Schwarzschild in 1916. At the time Schwarzschild himself showed that strange nonsensical things happened in the solution, which we now call the event horizon and the singularity. Interestingly, it was the event horizon that was seen as more problematic, because the “singularity” was just a mathematical concept, just like treating objects as point masses in Newtonian gravity. But soon more sophisticated models showed that matter within the event horizon of a black hole would most certainly collapse into a singularity. So for decades it was thought that black holes simply wouldn’t form. Surely the dynamics of material would prevent anything that dense from actually happening.

But in the 1960s neutron stars were discovered. It became clear that you could have several solar masses compressed into the volume the size of a small city. Neutron stars are fairly close to the critical density of a black hole, so it wasn’t unreasonable to imagine a collision or accretion of mass from another star triggering the formation a black hole. By the 1970s and 1980s there was growing evidence of black holes, both from x-ray binaries and the like for stellar mass black holes, and quasars and galaxy jets for supermassive black holes. In our own galaxy we now have orbital evidence from stars that show a supermassive black hole in our own galaxy. It is now decidedly clear that black holes exist.

But what about that pesky singularity? That’s actually a matter of some debate. Some argue that Hawking radiation will prevent singularities from forming. Some argue that things like dark energy might prevent their formation. Others argue that “singularities” might actually be a mechanism for forming other universes. It’s all pretty speculative, and they should all be considered a bit speculative.

But none of that disputes the existence of black holes. They are just discussions about one strange aspect of black holes. Models are a good way to understand things, but it’s the evidence that wins the day.

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The basic model of a black hole can be summed up as follows: gravity wins. The root cause of all black holes—be they tiny primordial black holes, solar mass black holes, or supermassive galactic black holes—is gravity. Squeeze enough mass into a small enough volume and gravity does the rest.

The problem (at least according to general relativity) is that gravity does its job too well. Once matter enters a black hole, it simply cannot resist the pull of gravity. As a result all the matter within a black hole is squeezed down to a point of zero volume and infinite density, known as a singularity.

Black hole singularities have long been the bugaboo of gravitational physics. They cause several problems, not the least of which is that the laws of physics as we understand them break down near the singularity. As a result there has been a lot of research on trying to eliminate singularities from our theoretical models.

One way to get around singularities is by simply ignoring them. When black holes form, they become enclosed by an event horizon. A black hole’s event horizon is a kind of cosmic roach motel. Matter and energy can fall into a black hole, but anything that crosses the event horizon is forever stuck inside. Since the singularity always resides within the event horizon, we can never observe it from the outside. The singularity remains safely hidden from view. Roger Penrose even went so far as to propose the cosmic censorship hypothesis, which argues that every singularity must be enclosed by an event horizon, which hides it from the universe.

But this hypothesis has never been proven in general, and several theoretical counter-examples (known as naked singularities) have been found. It would seem then that simply hiding singularities doesn’t solve the problem. Another approach to the problem is to find a mechanism by which the singularity never forms in the first place.

Most of the effort in this area has been to find patchwork solutions to Einstein’s equations for general relativity. That is, patch together a black hole solution outside the event horizon (the exterior solution) with a non-singular solution inside the event horizon (the interior solution). By mathematically sewing the two solutions together, one gets a non-singular black hole. These solutions demonstrate that non-singular black holes are theoretically possible, but they say nothing about how such a black hole might form.

Recently, however, Mbonye and Kazanas have found an exact solution to Einstein’s equations which contains no central singularity. Mbonye and Kazanas arrived at their solution by assuming a black hole contains exotic matter.

Exotic matter is a theoretical material which has a negative energy density instead of the usual positive energy density. This negative energy density means that exotic matter can hold its own against gravity. When ordinary matter is squeezed by gravity, its energy density goes up. This higher energy density means gravity squeezes even more strongly, which means an even higher energy density, and so on. It is this feedback loop which means that gravity wins in the end. The harder gravity squeezes, the harder it can squeeze, until all that remains is the singularity.

But exotic matter works differently. When gravity squeezes on exotic matter, its energy density goes down. This means gravity can’t win no matter how hard it tries, and it is impossible to form a singularity. What Mbonye and Kazanas have shown is that exotic matter allows for the creation of non-singular black holes.

Mbonye and Kazanas don’t specify what this exotic matter is, but their formulation implies one possible candidate: dark energy. We don’t know what dark energy is, but we do know two things: it cannot be regular matter, and it has a negative energy density. Mbonye and Kazanas haven’t proven that their exotic matter is dark energy, but their work points to the idea that maybe, just maybe, one of newest mysteries of gravitational astrophysics might just solve one of the oldest ones.

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