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.

If the mass of the remainder of the star is within certain limits, the implosion stops because the strong nuclear interaction causes a repulsion between the elementary particles (neutrons) that ultimately make up the star. The result is a neutron star, a pulsar. But if the mass of the remainder is even greater (more than 3 or 4 times greater than the mass of the sun), the attraction of gravity would overcome the strong nuclear interaction, the implosion cannot be stopped, and the result is a black hole, a singularity, with all the mass of the remainder of the star concentrated in a null volume. Therefore, although its mass is always finite, the density (mass/volume) would be infinite.

Karl Schwarzchild

At a certain distance from the center of a black hole (the Schwarzschild radius) the escape velocity (the speed that a body must to escape the attraction of a gravitational mass) is equal to the speed of light. Since neither matter nor energy can move faster, that means that anything closer than that distance to the center of the black hole cannot escape from it. Not even light can escape from the inside of a black hole. That is precisely why John Wheeler proposed the name black hole for these objects in 1967, which was reportedly suggested to him by one of his students. The spherical surface whose center is the center of the black hole and whose radius is the Schwarzschild radius is called the event horizon of the black hole.

We know that physicists always try to avoid infinities whenever they appear in their equations. Einstein, for example, was against the idea that black holes (they were not called that at the time, but Schwarzschild singularities) have infinite density. Various theories have subsequently been proposed, some indistinguishable from general relativity under normal conditions, to try to eliminate singularities. I talked about one of them in another post.

Roger Penrose

One of the ways to try and solve the infinite density problem was by trying to find solutions to Einstein’s equations under conditions that would not lead to singularities; for example, by making the black hole rotate, or have an electric charge, or by trying to show that under the most general possible conditions the singularity would not appear. But in the late 1960s, Roger Penrose and Stephen Hawking showed that singularities do appear, whatever the initial conditions. In turn, their theorems proved that Einstein’s equation leads without any possible escape to the fact that the universe began at the Big Bang. In 2020, Penrose received the Nobel Prize in physics for this reason (Hawking had already died).

Is there a way to escape infinite density? Yes. We know (I have said it several times in these posts) that general relativity can only be applied since 5×10^-44 seconds after the Big Bang. Before that time (the Planck time) the universe was so small that the other great physical theory, quantum theory, must also be applied. This theory is usually applied to very small objects. The problem is that we do not have a theory of quantum gravity, which combines general relativity with quantum theory. There have been several attempts to construct such a theory, but so far no one has succeeded. Until we have it, we do not know what could have happened in the first fraction of a second after the Big Bang, and we also do not know what exactly happens at the center of a black hole, where the same theory would have to be applied.

Meanwhile, some say that it is not true that black holes have infinite density. What they do is divide the mass inside the black hole (remember that mass is always finite) by the volume of the sphere contained inside the event horizon, the radius of which is the distance to the center of the black hole from the point where the escape velocity equals the speed of light: the distance at which not even light can escape from the black hole. That volume is never infinite either. The quotient of a finite number between another finite number different from zero is always finite. Consequently, they claim that the gigantic supermassive black holes at the center of galaxies have a density lower than that of water. But this is cheating, because it assumes that the density inside the black hole is constant, which no theory allows us to assert. We should not confuse the average density of the sphere contained inside the event horizon with the density of the compressed matter inside the black hole. For now, we cannot avoid the possibility that this matter occupies a volume equal to zero, as predicted by Einstein’s equations.

The same post in Spanish

Thematic Thread about Standard Cosmology: Previous Next

Manuel Alfonseca