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Schwarzschild Radius: The Edge of Darkness

Teodo on 21 Oct 2024 (Edited on 4 Nov 2024)
Latest activity Reply by Teodo on 17 Nov 2024 at 16:29

Inspired by the suggestion of Mr. Chen Lin (MathWorks), I am writing this post with a humble and friendly intent to share some fascinating insights and knowledge about the Schwarzschild radius. My entry, which is related to this post, is named: 'Into the Abyss - Schwarzschild Radius (a time lapse)'.
MATLAB Shorts Mini Hack: Schwarzschild Radius
The Schwarzschild radius (or gravitational radius) defines the radius of the event horizon of a black hole, which is the boundary beyond which nothing, not even light, can escape the gravitational pull of the black hole. This concept comes from the Schwarzschild solution to Einstein’s field equations in general relativity. Black holes are regions of spacetime where gravitational collapse has caused matter to be concentrated within such a small volume that the escape velocity exceeds the speed of light.
This is a rudimentary scientific post, as the matter of Schwarzschild radius - it's true meaning and function, is a much, much, much-more complex "thing" (not known to us entierly, by the third degre of epistemological explanation(s)).
And, very important is to mention: I am NOT an expert - by any means, on this topic, just a very curious guy, in almost anything, that has to do with science.
Schwarzschild Radius (Gravitational Radius)
The Schwarzschild radius (Rₛ) is the critical radius at which an object of mass must be compressed to form a black hole, specifically, a non-rotating, uncharged black hole, known as a Schwarzschild black hole. The Schwarzschild radius is given by the formula: .
Where: .
Key Characteristics are, that for any mass, if that mass is compressed within a sphere with radius equal to , the gravitational field is so strong that not even light can escape, thus forming a black hole. The Schwarzschild radius is proportional to the mass. Larger masses have larger Schwarzschild radii.
Example:
For the Sun : .
So, if the Sun were compressed into a sphere with a radius of ~3 km, it would become a black hole!
Stellar-mass Black Holes form from the collapse of massive stars (roughly ). Their Schwarzschild radius ranges from a few kilometers to tens of kilometers.
Supermassive Black Holes found at the centers of galaxies, such as Sagittarius A in the Milky Way (), their Schwarzschild radii span from a few million to billions of kilometers!
Primordial or Micro Black Holes, are the hypothetical small black holes with masses much smaller than stellar masses, where the Schwarzschild radius could be extremely tiny.
A black hole, in general, is a solution to Einstein’s general theory of relativity where spacetime is curved to such an extent that nothing within a certain region, called the event horizon, can escape.
MATLAB Shorts Mini Hack: Schwarzschild Radius
Types of Black Holes:
1. Schwarzschild (Non-rotating, Uncharged):
- This is the simplest type of black hole, described by the Schwarzschild solution.
- Its key feature is the singularity at the center, where the curvature of spacetime becomes infinite.
- No charge, no angular momentum (spin), and spherical symmetry.
2. Kerr (Rotating):
- Describes rotating black holes.
- Involves an additional parameter called angular momentum.
- Has an event horizon and an inner boundary, known as the ergosphere, where spacetime is dragged around by the black hole's rotation.
3. Reissner–Nordström (Charged, Non-rotating):
- A black hole with electric charge.
- A charged black hole has two event horizons (inner and outer) and a central singularity.
4. Kerr–Newman (Rotating and Charged):
- The most general solution, describing a black hole that has both charge and angular momentum.
Relationship Between Schwarzschild Radius and Black Holes
Formation of Black Holes: When a massive star exhausts its nuclear fuel, gravitational collapse can compress the core beyond the Schwarzschild radius, creating a black hole.
Event Horizon: The Schwarzschild radius marks the event horizon for a non-rotating black hole. This is the boundary beyond which no information or matter can escape the black hole.
Curvature of Spacetime: At distances closer than the Schwarzschild radius, spacetime curvature becomes so extreme that all paths, even those of light, are bent towards the black hole’s singularity.
BTW, the term singularity, scientificaly 😊, means that: we do not have a clue what is really happening right there...
Detailed Properties of Black Holes:
a. Singularity:
At the center of a black hole, within the Schwarzschild radius, lies the singularity, a point (or ring in the case of rotating black holes) where gravitational forces compress matter to infinite density and spacetime curvature becomes infinite. General relativity breaks down at the singularity, and a quantum theory of gravity is required for a complete understanding.
b. Event Horizon:
The event horizon is not a physical surface but a boundary where the escape velocity equals the speed of light. For an outside observer, objects falling into a black hole appear to slow down and fade away near the event horizon due to gravitational time dilation, a prediction of general relativity. From the perspective of the infalling object, however, it crosses the event horizon in finite time without noticing anything special at the moment of crossing.
c. Hawking Radiation: (In the post, I told that there is no radiation - to make it simple, although, there is a relatively newly-found (theoretically) radiation. Truth to be said, some physicists are still chalenging this notion, in some of it's parts...)
Quantum mechanical effects near the event horizon predict that black holes can emit radiation (Hawking radiation), a process through which black holes can lose mass and, over very long timescales, potentially evaporate completely. This process has a temperature inversely proportional to the black hole's mass, making large black holes emit extremely weak radiation. (Very trivialy speaking: the concept supposes that an anti-particle is drawn from the vakum and is anihilated with the black's hole matter (particle), and in the process, the black hole looses mass gradually and proportionally to the released energy - very slowly(!)).
This radiation is significant only for small black holes.
Gravitational Time Dilation (here, as well, things become 'super-weird'...)
Near the Schwarzschild radius, the intense gravitational field leads to time dilation. For an external observer far from the black hole, time appears to slow down for an object moving toward the event horizon. As it approaches the Schwarzschild radius, time dilation becomes so extreme that the object appears frozen in time at the horizon.
The time dilation factor is given by:
Eg. Approaching the Schwarzschild radius and theoretically remaining just outside of it for a few hours would correspond to the passage of approximately several decades on Earth due to relativistic time dilation.
Using relativistic equations, it's estimated that near the event horizon 2 hours (120 minutes) near the black hole Sagittarius A* (as already mentioned ~ 4 million ) - in the center of our galaxy Milky Way, could correspond to 83 years passing on Earth! However, this varies based on the precise distance from the event horizon (give or take, a decade 😬).
Information Paradox (definte answer on this question, 'hold's the keys of the universe' 😊, maybe...)
The black hole information paradox arises from the seeming contradiction between general relativity and quantum mechanics.
According to quantum mechanics, information cannot be destroyed, yet anything falling into a black hole seems to be lost beyond the event horizon. Hawking radiation, which allows a black hole to evaporate, does not appear to carry information about the matter that fell into the black hole, leading to ongoing debates and research into how information is preserved in the context of black holes, or not...!
Schwarzschild Radius is the key parameter defining the size of the event horizon of a non-rotating black hole. Black Holes are regions where the Schwarzschild radius constrains all physical phenomena due to extreme gravitational forces, forming event horizons and singularities. The interaction between general relativity and quantum mechanics in the context of black holes (e.g., Hawking radiation and the information paradox) remains one of the most intriguing areas in modern theoretical physics.
I hope you will find this post, and information provided, interesting.
Giampiero Campa
Giampiero Campa on 12 Nov 2024 at 21:18
Teodo
Teodo on 17 Nov 2024 at 16:29
Exactly as I stated in the Article.
Rena Berman
Rena Berman on 5 Nov 2024 at 21:55
Such a nice article!
Adam Danz
Adam Danz on 4 Nov 2024
When I view some of the minihack code with the intention of understanding each line, I feel like my code-comprehension skills reach the Schwarzschild radius.
Teodo
Teodo on 4 Nov 2024 at 22:59
Respectfully, Adam, you’re far too modest. Your insights have taught me a great deal, and your contributions remain some of my favorites - particularly your work on rain effects.
In fact, this entry is directly inspired by your work and code.
Chen Lin
Chen Lin on 4 Nov 2024 at 20:32
Use AI for help!
Julie
Julie on 22 Oct 2024
Great!
Chen Lin
Chen Lin on 22 Oct 2024

Perfect. Thanks for sharing the knowledge with the community!