Physics

Black Holes, explained

What they are, how they form, types & shapes, and how they die

Nicolus Rotich

Jan 6, 2020 — 9 min read

Fig. 1: An artist’s impression illustrating a supermassive black hole¹ (NASA)

Maybe you have never heard of a black hole, or you have heard someone mention, but somehow you still don’t know what to make of it. This article will give you a quick rundown on the most basic, yet essential concepts and fundamentals related to black holes. In particular, we will try to answer the questions: what they are, where they reside in our universe, how they form, as well as their classification by types and shapes they come in, and how they die.

So, What is a Black Hole?

A black hole is a region of extremely high density, formed when a massive object gets compressed into a small enough space. To put it in context, if our planet Earth, could be squeezed down into a 4.38 millilitre drop of water, it would result in a black hole. The corresponding radius of the volume so formed, is referred to as the Schwarzschild radius. This analogy is not based on a spherical volume shape, but a cylindrical shape of height h=2r . See the explanation for this in the black holes shapes — section.

As a concept, the black holes are primarily mathematical predictions. They are therefore the results that manifest as exact solutions to Einstein’s field equations. The first one of that kind was obtained by Karl Schwarzschild a German physicist, astronomer, and a 1st world war veteran in 1915 — the same year Einstein’s field equations were published. Hence the name Schwarzschild radius, which is presented with the following formula:

Where c is the speed of light in a vacuum, G is Newton’s universal gravitational constant, and M is the mass of the object.

Black holes are associated with a number of properties that explains how gravity works. Based on the principles of general relativity in which gravity is described as the bending of spacetime in presence of massive objects, black holes may also be said to be regions in spacetime that exhibit gravitational acceleration so strong that nothing can escape.

Before jumping to classification, there are a few more concepts to know about black holes a part from Schwarzschild radius, these are:

Black hole event horizon: this is a boundary while approaching a black hole, beyond which events of any kind cannot affect any outside observer. This refined definition was issued by David Finkelstein in 1958 based on general relativity. The earlier definition of the same concept by John Michell in 1784 meant that an event horizon was a region near a compact massive object, where gravity would increase to the extent that even light cannot escape. It is therefore noticeable that even though the concept hasn’t changed significantly, the definitions have been constantly refined to reflect its precise description. The last few decades have overseen the needful renaming of black hole horizon to apparent horizon — this came from the late Stephen Hawking. Hawking gave the justification that while cosmic events such as gravitational collapse may produce apparent horizons, they do not necessarily produce event horizons, and thus event horizons are the ultimate indicators of black hole existence. Hawking radiation was also a result of Stephen Hawking’s further development of black holes theories, in which he concluded that there exists black-body radiations emanating from black holes due to quantum fluctuation effects near the event horizon, and that would lead to eventual extinction of the black hole over a long period of time.
Black hole singularity: general relativity predicts that that at the centre of a black hole, there is a 1D — point containing the whole mass of the black hole in an infinitely small space. At this point, it is believed that the laws of physics as we know, no longer applies.
Black hole rotation: While Schwarzschild black holes are stationary, some black holes have been known to spin at an almost theoretical limit — meaning an angular speed which when converted to linear, approaches the speed of light in a vacuum. This was reported in early 2006 by Cosmos, an Australian science magazine³. Indeed GRS 1915+105, a black hole situated in the Milky Way, can rotate at an angular speed of 1150 cycles per second.

Black Holes Classification

The solutions of Einstein’s field equations are generally referred to as metrics, since they are usually presented in metric tensor notations. In mathematics, a metric tensor is a function that takes in pairs of tangent vectors as inputs, and returns real number scalars as output. Therefore as of today, there are mainly four known types of black holes based on their corresponding metrics allowing their existence. These are:

(i) Schwarzschild black holes

(ii) Reissner — Nordström black holes

(iii) Kerr black holes and

(iv) Kerr-Newman black holes

The above list is compiled on the basis of people who provided the exact solutions to Einstein’s field equations, describing the general theory of relativity. The most notable being the first one, Schwarzschild solution or metric. Schwarzschild’s effort is particularly commendable, as he accomplished this while serving in the German army during the first world war.

Another more precise classification albeit qualitative, is on the basis of rotation and charge of the black hole. Table 1 depicts this summary with the previously listed examples.

Table 1: Black hole classification matrix based on rotation and charge

As seen on Table 1, both Schwarzschild and Reissner — Nordström, are classified as non rotating black holes. On the other hand, Kerr and Kerr-Newman are some of the examples of rotating black holes. The most reliable metric for rotational dynamics of a black hole is the angular momentum J, which is presented as J = Mac. Where M is the mass of the black hole, usually represented in solar masses — black hole mass divided by that of the sun. Parameter a is a set of bounds restraining the actions of a black hole to its event horizon. In other words, if a is exceeded, the event horizon disappears.

Mechanics of Black Hole Formation

One of the well established ways in which black holes are formed, is as a result of gravitational collapse of interstellar bodies due to their own weight. There is an overall degeneracy in the stellar nucleosynthesis that continuously occurs during the lifetime of a star. This implies that a star’s rate of energy emission is not constant, but rather dependent on the internal energy of its constituents.

Black holes possesses three well known properties; the mass, charge, and the spin, all of which presents monumental difficulty to measure. The only measurable aspect of black holes is their effect in terms of gravitational waves as of the major breakthrough made in 2016⁴ by the LIGO and Virgo teams. Even then gravitational waves detection and measurement suffer immensely, from Newtonian noises and ambient ground motion, and thus a lot of the work on detection goes towards this stabilization. Black holes formation can be described by a number of evolutionary parameters (of the star before collapse) namely:

(i) Reduction in the original radius of the star over time

(ii) Increase in kinetic energy

(iii) Changes in thermodynamic parameters such as pressure, temperature, and accompanying chemistry of stellar nucleosynthesis with time

(iv) Increase in (quantum) gravitational potential, and finally

(v) Their extinction through black holes evaporation as predicted by Hawking radiation

Owing to the limitation of the scope of this article, we are going to briefly look at the first aspect in the above list.

Reduction in Radii Owing to Gravitational Collapse

This is the contraction of an astronomical object, due to the influence of its own gravity, and which tends to draw matter inward towards the center of gravity. This essentially causes a reduction in size, as the star losses its nucleosynthetic power. The timescales in which all the above occurs are infinitesimal. Quantum gravitational effects are particularly (usually) ignored during the formation and evolution of black holes. According to Hawking, the justification being the radius of curvature of event horizon becomes very large compared to Planck length — the lengths at which quantum fluctuations are expected to be noticeable.

If we were to assign a gravitational radius equivalent of λ to the star with respect to its gravitational strength. We can then track the changes by monitoring the reduction in the length scales, Δh over time as Δ h = λ-r, where r is the current radius at any given moment. For any celestial body, It can be shown that the initial value of gravitational radius before it becomes a black hole λ, is given by λ=2r²/rₛ. We would then be able to express the instantaneous changes in its radii with respect to Δh as follows:

Eq. (2) can be seen as a metric tensor to which a black hole event horizon can form if and only if 8Δh ≤ 3rₛ, yielding Schwarzschild radius when the value of Δ h=rₛ. Where rₛ is given by Eq. (1).

What is the Shape of a Black Hole?

While the mention of black holes is continually becoming a common talk in the streets, hardly does anyone asserts with significant confidence, whether they really know how black holes look like (in shape). In these cases therefore, mathematical descriptions take precedence, after all, it was only until the 1960s when black holes were thought of as purely mathematical constructs.

One of the very first conjectures was one done by Kip Thorne in 1972 on non-spherical black holes. In this short review, Thorne postulated that:

“Horizons form when and only when a mass m gets compacted onto a region whose circumference in every direction is less than or equal to 4πm’’ — Kip Thorne 1972

Even then he seemed to have contradicted himself, as he was describing non-spherical black holes. Nevertheless, his paper provided a comprehensive review, and research on black holes has since been receiving significant traction, especially in the field of mathematical physics.

Most of the literature describing black hole geometry, in one way or another also went as far as attempting to estimate other parameters such as thermodynamics, energy, and mechanics e.g. angular momenta. We will not concern ourselves with extra properties apart from the volume, and thus we will only attempt to characterize and predict the shapes of uncharged non-rotating black holes — also known as Schwarzschild black holes.

The antithesis of this section is that black holes characterized by Schwarzschild radius are spherical. At the end of this article, we should be convinced that Schwarzschild black holes take a specific volume pattern, that is non-spherical in nature. To show this, we start by subjecting a spherical object of volume, V=4π r³/3 to the formula in Eq. (1). We also introduce the mass density ρ, so we express rₛ as follows:

We then introduce Poisson’s gravitational potential ∇²ϕ=4πρG to the formula in Eq. (3), from ϕ(r) = GM/r — check out the conversion between Gauss and Poisson’s equations for gravity. Taking note that ∇² ϕ(r) = 2GM/r³, and comparing the resulting equation with the original Schwarzschild radius (Eq. 1), we find that:

We can therefore see that a spherical Schwarzschild black hole of above geometrical configuration would not exist. However, there are two possible geometries of black holes that Schwarzschild methodology predicts accurately: a cylinder of height h=2r, and a fraction of a four-dimensional sphere of volume V₄/π. For the two geometries V=2π r³, since that of a four sphere is V₄=2π²r³. We are going to subject these objects to a similar process as above:

Similarly, we notice that Poisson’s equation for gravity, 4πρG=∇²ϕ appears in the numerator of Eq. (5). Since, ∇²ϕ=2GM/r³, indeed replacing this in Eq. (5) results in Eq. (1) as expected.

Summary

In this article, we offered a more detailed explanation of what black holes are, before explaining their formation mechanics. We then took a look at the four known types of black holes based on their qualitative descriptions, such as rotational dynamics, charge, and based on the originators of their corresponding metric tensors. Finally, we discussed the expected shapes of Schwarzschild black holes, along with their intuitive mathematical descriptions. Based on this analysis, Schwarzschild black holes are primarily uncharged, non rotating, and takes cylindrical shapes. In some peculiar cases, they take the unique shape of a fraction of a four sphere, with their volumes expressible as V = 2π² r³.