The Schwarzschild Radius of a Black Hole

The Schwarzschild Radius is a mathematical relation that determines the necessary density (mass–volume ratio) that an object must possess to meet the conditions for an infinite gravitational collapse of the spacetime manifold to occur — the formation of a black hole.

The Schwarzschild radius is named after Karl Schwarzschild, a German-born Jewish physicist and astronomer of the early 20th century. He provided the first exact solution to Einstein’s Field Equations, writing three significant papers while fighting in WWI on both the eastern and western fronts (doing mathematical calculations for artillery).

While his solutions are abstractions, since they remove a great deal of the dynamics involved in actual systems (torque, coriolis effects, magnetism, and electric charge) — a typical and necessary step in the development of physics to mitigate the complexities involved at the forefront by temporarily reducing the problem to a simplified form — his solution nonetheless laid the foundations for our modern understanding of black holes, and the future of relativistic physics.

More importantly, Einstein’s Field Equation’s deal with the nature and behaviour of the spacetime manifold. Literally the foundation of reality. Therefore that singularities, event horizons, and black holes are scientifically central to the solution that describes reality was not just unexpected — it is among the nexus of discoveries that change everything about how we view the world.

The Schwarzchild Radius

The Schwarzschild radius describes the relationship between the mass and volume of an object. This relationship is expressed by the equation R_g = 2GM/c² where R_g is the Schwarzschild radius, G is the gravitational constant, M is the mass of the object in question, and c is the speed of light.

\[ R_g = \frac{2GM}{c^2} \]

In practice, there are really only two variables that concern us in this formula — the Schwarzschild radius (R_g) and the mass (M) — because the rest are universal constants, and are generally not subject to change. We can further reduce this to just the radius being significant, because when we are considering a specific object (such as the Sun, Milky Way, Mars, or a Neutron Star) the mass is not going to change either: it will be the mass of the object in question. To determine the Schwarzschild radius of an object, we simply input the mass and solve for R_g.

According to the Schwarzschild solution of Einstein’s Field Equations, there is a point during gravitational collapse (contraction, or compaction of an object) where a density is attained that exceeds a limit beyond which the mass of the object relative to its volume bends the spacetime manifold to such an extreme degree that it creates a gravity well. A region of space where spacetime collapses in on itself towards infinite contraction — the singularity.

An important concept about black holes, is that there is no specific radius condition to be met, because it is all about density. In other words, in nature there is no specific “Schwarzschild radius”. It is not a definite number that defines a black hole. Not a constant like the gravitational constant, G, or the speed of light, c. Rather, there is a specific Schwarzschild radius for every object which varies proportionally with the mass of the object.

The Schwarzschild radius will be greater for the Sun than the Earth because the Sun is more massive. It will be significantly less for an apple than either the Sun or Earth because an apple is significantly less massive.

To give you some numbers, the Schwarzschild radius for the Sun is about 2,954 meters. Right now the Sun has a radius of about 696 million meters, and you would have to compress that into about 3km (just under 2 miles) radius. Same mass, but a smaller volume.

Jupiter on the other hand you would have to compress to a radius of about 2.8 meters for it to become a black hole (with a starting radius of ~69 million meters). The Earth you would have to compress to about 8.9 mm (from a starting radius of ~6.4 million meters). Then once you get to the scale of a human being, if you were to turn me into a black hole at about 210lbs (95.3 kg) you would have to compress me to a sphere with a radius of about 1.415 × 10^-25m.

It is all about density, which is a derived quantity from SI units of volume and mass. While I mentioned radius and distance units a great deal, those numbers were determined from the original mass. Technically, or theoretically, provided the right conditions of density are met (the right mass compacted into a sufficiently small volume) we can have black holes of any size. Indeed, we might be able to manufacture black holes of any size.

We might want to do this since black holes release energy around their event horizon which, if harnessed, could turn mini black holes into portable power sources — literal free energy devices. These might exist in nature at every scale already, atomic-scale, microscopic, palm-sized, planetary, stellar and greater black holes floating eternally (presumably) through space.

The Schwarzschild Radius In Nature — Black Holes, Cosmology, and Gravitational Collapse

Not only are conditions that obey the Schwarzschild radius met throughout the universe — in the form of the objects we call black holes — the implications of this understanding are far-reaching. From this arise some interesting cosmological questions as well. For example, in the context of the Big Bang Theory the general outline of the theory states that the mass of the entire known universe was originally condensed into a single point.

The density of the Bing Bang Singularity far exceeds the conditions required by the Schwarzschild radius for an infinite gravitational collapse. So according to the math alone, the universe began as a black hole. Perhaps it is a black hole still, in a way that we don’t yet fully understand. There is a lot we don’t know, such as how the uncontrolled gravitational collapse transitioned to the “explosion” and expansion we have seen since. What condition sparked the first moment of time in our universe?

(While I don’t believe the Big Bang event 13.7 billion years ago was the beginning of time within infinite space, an event did occur. Many physicists and cosmologists consider this to be the beginning of time in the universe.)

One theory is that black holes play a role in the formation of stars. We are fairly certain that when massive stars die, they can form black holes if they are of significant mass. (The Sun, for reference, is not a massive star. It is an average mass and temperature star, even on the low end.) There are some physicists who suggest that black holes might play a role in the formation of stars also.

For example, while the Sun itself is not a black hole — it doesn’t obey the mass–volume–density conditions that meet the requirement for a black hole — it is possible that there are very small black holes at the center of stars which played a role in their formation through the gravitational collapse of nebulae. In some cases initiating it, or being involved in the initial collapse in some way.

Indeed, it is another mass-volume-density relationship known as Jean’s mass which determines the collapse of nebulae (giant space clouds made of mostly hydrogen, some helium, trace heavier elements, and ice) which form into protostars once they collapse to a sufficient density. If a nebulae is massive enough, and once a specific density is met in a region of said cloud, it collapses gravitationally inwards on itself, forming the first phase of life of a star: a protostar — essentially a star-seed.

The interesting thing is that Jean’s Mass and the Schwarzschild radius are two different stages of gravitational collapse. More like two different scales of gravitational collapse determined by the scale of masses involved — with different results because of the differences in mass. The same dynamic, fundamentally, but in one the mass is sufficient to form a black hole. In another, a star. If the mass involved is not great enough to form a star, then a planet like a gas giant, brown dwarf, or a moon is formed.

These are all just stages of gravitational collapse that we see in the universe.

The Schwarzschild Radius in the Holographic Fractal Universe Model

The significance of the Schwarzschild Radius in the Holographic Fractal Universe Model is multi-faceted. First, Schwarzschild’s solution to Einstein’s Field Equations indicated not only the existence of singularities and black holes, but demonstrated that they are central to our understanding of reality and the very dynamics of the universe. After all, solving the equations for the field — the fabric of spacetime, the spacetime manifold, the gravitational and electromagnetic fields of reality — produced a black hole singularity.

This forced us to rethink everything, or to look at everything anew, with eyes untainted by prejudice. Second, it also leads us to consider the masses, energies, and densities in the universe more assiduously. Looking more carefully at space, at the dynamics of the universe at all scales, right down to what is probably the most important universal dynamic of all (at least from our scalar resolution of the universe): the atom.

Although the Schwarzschild Radius and Schwarzschild Metric are approximations (because Schwarzschild excluded basically all dynamic quantities in his explanation that would otherwise be required to accurately describe basically any real-world system in nature) it leads us to look at the atom in a completely different way. More specifically, at the nucleus in the center of the atom, and the proton within it.

A nearly infinite energy exists in the nucleus of all atoms that is not explained or dealt with (yet) in modern physics. We do not understand what it is, where it comes from, how it exists, or the mechanisms by which it does so. These are the Nuclear Forces — the Strong Nuclear Force and the Weak Nuclear Force — which are the strongest of the Four Fundamental Forces. Significantly stronger than both the Gravitational Interaction and the Electromagnetic Interaction.

When we consider these forces in the context of Relativistic Physics, the Mass-Energy Equivalency states that mass and energy are the same thing. When we consider the energy involved in holding the positively charged protons in the nucleus together eternally, as well as quarks (which is what the Strong Force and Weak Force do, respectively) this is a constant force exerted, which is energy, which is mass — changing the mass profile of atoms.

In addition to the nuclear forces, we also have to consider the energy density of the vacuum structure within atoms and the atomic nuclei. When we factor all of these forces, energies, and equivalent mass together, the nucleus of the atom appears to obey the conditions for a black hole.

In the Holographic Fractal Universe Model of the Unified Field Theory, that is exactly what exists in the nucleus of all atoms: an atomic-scale black hole that provides the mechanism by which protons are held together in the nucleus in spite of (relatively) massive repulsive electrostatic forces pulling them apart.

This is covered completely in the next section: [link].