Can density be infinite?

First photo
of a black hole

Einstein’s general theory of relativity allows for the existence of objects with infinite density (singularities). There are two types:

1.      Black holes, accumulations of matter in a null volume, either at the center of a galaxy, or as the result of a supernova explosion.

2.      The universe, at its initial moment (the Big Bang).

A star like the sun is in equilibrium because the gravitational attraction, which tends to make it contract, is equal to the expansion caused by the nuclear reactions taking place inside the star. When a star much larger than the sun exhausts its nuclear fuel (first hydrogen, then helium, then other elements), as there are no longer nuclear reactions to stop the contraction, the star implodes. When the implosion rebounds, the star throws large quantities of matter into space: a supernova explosion, which for some time makes the star brighter than a whole galaxy. But there is always a remainder of matter, which gives rise to a new type of object.

Singularities

Hal 9000, from the film
2001, a space odyssey

A singularity is a mathematical concept applied to a function of a variable that reaches an infinite value for one or several finite values of its independent variable.

For example, the function y=1/x presents a singularity for x=0, since it is often said that 1 divided by zero is equal to infinity.

Black holes or gravastars?

As I explained in a previous post, our two fundamental physical theories, general relativity and quantum mechanics, predict infinities that physicists don’t like. General relativity does this in gravitational singularities: the Big Bang and black holes. Quantum mechanics, in vacuum energy and the quantities that must be renormalized in quantum field theory.

Until a little time ago, the theory of black holes, formulated by Subrahmanyan Chandrasekhar in 1930, stated the following: when a star 30 to 70 times more massive than the sun undergoes a supernova explosion, it expels most of its mass, but a part of it (at least 3.8 times more massive than the sun) collapses to such a point that it occupies zero volume, and so it will have an infinite density.

Do black holes have hair?

Black holes are strange objects. They are accumulations of extremely compact matter, which exerts such huge gravity that at less than a certain distance (the event horizon) nothing can escape their attraction, not even light. Hence their name.

The existence of black holes had been predicted in the 18th century by the English geologist John Michell and the French astronomer Laplace. At that time nobody paid attention, but from 1915, when Einstein formulated the theory of General Relativity, the interest in these mysterious objects grew. It was soon concluded that when a massive star exhausted its ability to produce nuclear fusion reactions, no force of nature would be able to overcome the gravitational pull of the remaining matter, resulting in a black hole. But for a long time there were doubts about their real existence, for the theory seemed to predict that the matter located inside a black hole would occupy a zero volume and therefore would have an infinite density. As physicists usually suspect that infinity is a mathematical concept that cannot happen in real life, there were two possibilities: either black holes do not exist, or Einstein's theory would have to be modified so that they wouldn’t have an infinite density.

Wormholes

Science fiction novels make it clear that, even if we were able to reach relativistic speeds (close to the speed of light), our need to personally explore the universe wouldn’t be satisfied. We’d like to travel to other stars with the same ease with which we cross the Atlantic today. We’d like to measure in days, if not hours, the time of a trip to the center of the galaxy (which probably contains a large black hole). Is there any chance of this happening?

To do this, we should discover in the future some property of the universe, now unknown, that would help us break the speed limit of light, which seems firmly established, and which would make us spend thousands of years on trips to most stars, except the nearest.

To solve the problem, science fiction authors have used essentially two different procedures:

Virtual particles

Werner Heisenberg

Heisenberg's uncertainty principle, one of the consequences of quantum mechanics, makes possible the birth of virtual particles in the void, apparently transgressing the principle of energy conservation, the most holy in physics. The reason is that Heisenberg’s principle can be expressed in several ways, one of which relates the uncertainty about the energy to the uncertainty about time:

DE.Dt≥ħ/2

This expression can be interpreted in the sense that a pair of objects, each with energy E, can appear spontaneously in the vacuum, provided that they lasts at most a time Dt<ħ/(2E). These pairs of objects are called virtual particles. One of these particles is always matter, the other antimatter, and their duration, according to this principle, is ridiculously small. A virtual electron, for instance, would last 1.3×10^-21 seconds (just above one sextillionth of a second). The higher the mass (energy) of the virtual particle, the less time it will last. After that time, the two particles will annihilate each other and disappear. Due to their short duration, the existence of virtual particles has not been experimentally verified.

Is it possible for these virtual particles to become real under certain circumstances? Yes it is, and it is believed that there are at least two situations (somewhat drastic, it is true) where this could happen.

Science or speculation?

Merging of two black holes

In December 2013 several media made reference to an article published in the journal Physical Review Letters, in the field called quantum gravity, a set of mutually incompatible theories that in the last 30 years have scarcely formulated a single testable prediction, although they are usually presented as the latest in physics and give rise to news broadly popularized by the general media. This scientific news was presented by some media with this title: quantum entanglement causes the appearance of wormholes. Some of the reviews contained important mistakes. I’ll cite two:

1.      The group showed that, by creating two black holes intertwined, later separated, a wormhole appeared, a “shortcut”' through the universe, connecting the two distant black holes to one another.

Comment: the group did not show that. It just found some equations that suggest that this might happen (or not, because mathematics is not the same as physics). Moreover, these theoretical speculations are based on string theory, which is just one among several alternative proposals of quantum gravity existing today.

The scientific work of Stephen Hawking

The scientific work of Stephen Hawking has been quite productive, although the media, influenced by his sad personal situation, tend to exaggerate its importance, putting him sometimes at the level of Einstein. His most renowned works are the following:

• The singularity theorems, published in 1970 in collaboration with Roger Penrose, proved that the application of the equations of Einstein’s General Relativity to the entire universe requires at least one singular point in that universe (a point where all the geodesics in the universe converge). As a consequence of this theorem, in the book The Large Scale Structure of Space-Time (1973, written with George Ellis), Hawking unequivocally embraced the theory that the universe began at a point of infinite density (the Big Bang).