As I was researching about escape velocities in my previous post, I found an interesting connection to black holes. Since astronomy has been a long time passion of mine, I decided to conduct some further research in today's post. I was surprised to find that the seemingly simple connection to the escape velocity was actually a flaw in reasoning!
The Schwarzschild radius, also known as the event horizon, is the distance from the centre of the black hole (the singularity) at which no light or radiation can escape once entered.
Check out this article to learn more about the fascinating nature of singularities and what could possibly be lurking at the centre of black holes.
The Schwarzschild radius $R_s$ is given by $R_s=\frac{2GM}{c^2}$ where $G$ is the gravitational constant, $M$ is the mass of the object and $c$ is the speed of light. For example, the Schwarzschild radius of the Sun with $M=1.9\cdot 10^{30}kg$ is calculated as follows: $$R_s=\frac{2\cdot 1.9\cdot 10^{30} \cdot 6.67430 \cdot 10^{-11}}{{(3\cdot 10^8)}^2}$$ $$=2.8km$$This means that if all of the Sun's mass could be reduced in a sphere of radius 2.8km, the object would then be considered a black hole.
If you have seen the equation for the Schwarzschild radius before, you might be tempted to instantly use the escape velocity formula above to "derive" the equation. If we let the speed of light $c$ be the escape velocity $v_e$, then the following miracle happens:
$$\frac{1}{2}mc^2=\frac{GmM}{R_s}$$ $$\bbox[border: 2px solid red]{R_s=\frac{2GM}{c^2}}$$I call this a "miracle" precisely because it is: as many others have stated on forums, as beautiful as it may seem, the fact that this "derivation" works is merely a coincidence. This is because the Newtonian notion of escape velocity is incompatible with the relativistic notion of light's inability to escape the event horizon. See this discussion forum for more.
In fact, in 1783, English natural philosopher John Mitchell (1724 - 1793) published this exact "derivation". Although the formula ended up being correct, this was purely coincidental and his reasoning was invalid.
A proper derivation requires the use of mathematics and physics which is beyond the scope of this post. However, a key mathematical idea is something called a metric - a function that defines the distances between each pair of elements in a set.
Below is the Pythagorean theorem stated as differentials, where $ds$ represents displacement, the change in distance from your original point. $$ds^2=dx^2+dy^2$$ Adding the third dimension, $$ds^2=dx^2+dy^2+dz^2$$ Adding the fourth dimension of time, $$ds^2=dt^2+dx^2+dy^2+dz^2$$... no! Actually, according to relativity, it is actually $$ds^2=-c^2dt^2+dx^2+dy^2+dz^2$$ The $c^2$ term helps keep the units consistent. Indeed, $-c^2dt^2$ has units of $m^2s^{-2}\cdot s^2=m^2$, like the others. This equation, called the Minkowski metric, serves as the foundation of special relativity and the proof of the formula for the Schwarzschild radius.
As I was conducting further research, although I was not able to comprehend the works completely, I was intrigued to see that some approaches, such as this thesis and this Wikipedia article involved the use of such topics as geodesics and the Euler-Lagrange equation, two of the topics I initially brainstormed for my IB Extended Essay. I will soon post about my EE, so stay tuned!
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