Summary of Key Points
In the 18th century, British clergyman John Michell used Newtonian mechanics to calculate the radius formula for a “black hole,” which turned out to be exactly the same as the Schwarzschild radius formula derived from Einstein’s general theory of relativity more than a hundred years later. However, Newtonian mechanics is “incorrect” in the context of strong gravitational fields like those around black holes (due to flawed physical assumptions). The fact that this “incorrect theory” led to the correct answer is essentially a mathematical coincidence combined with the historical limitations of those assumptions, revealing some interesting logic in scientific research.
Detailed Explanation
1. How did the clergyman use Newtonian mechanics to calculate the “black hole”?
John Michell based his calculations on two concepts from the Newtonian era:
- That light consists of particles with mass (the then-popular particle theory of light);
- That objects must achieve a certain “escape velocity” to escape the gravitational pull of a celestial body (for example, a rocket needs to reach 7.9 km/s to leave Earth).
He applied Newton’s formula for escape velocity: Escape velocity \(v = \sqrt{\frac{2GM}{r}}\) (where \(G\) is the gravitational constant, \(M\) is the mass of the celestial body, and \(r\) is its radius). If a celestial body’s gravity is so strong that the speed of light (\(c\)) equals the escape velocity, then no light can escape, and the body would become an “invisible dark star”—what we now call a black hole. Substituting \(c\) for \(v\) in Michell’s formula yields \(r = \frac{2GM}{c^2}\), which is identical to the black hole radius calculated by Karl Schwarzschild using general relativity in 1916.
2. Why is Newtonian mechanics considered “incorrect” in this context?
The core assumptions of Newtonian mechanics are that space is flat, time is uniform, and gravity is a force between objects. However, black holes represent regions of extreme gravitational strength, where general relativity is necessary to explain phenomena (for example, considering space-time as a sheet of fabric that is bent by a massive object, causing light to follow the curved path and unable to escape).
Furthermore, later experiments proved that light has no rest mass, debunking Michell’s assumption that light consists of particles with mass. Therefore, the foundation of his calculations (Newtonian gravity + the idea that light has mass) was fundamentally incorrect.
3. Why did an incorrect theory lead to the correct formula?
This is a matter of mathematical coincidence. Although Michell’s physical assumptions were wrong, the escape velocity formula he used happened to match the mathematical result derived from general relativity. For instance, when general relativity is applied in a “weak gravitational approximation” (similar to the range where Newtonian mechanics is valid), the formulas yield the same result. However, the physical interpretations of these two theories are vastly different: Michell believed that light was being pulled back by the gravity, while general relativity suggests that light is simply deflected by the curvature of space-time.
4. Why didn’t this discovery gain much attention at the time?
There were two main reasons:
- Technical limitations: Telescopes in the 18th century were not capable of detecting black holes (the first image of a black hole was captured only in 2019), so Michell’s theoretical prediction remained unverifiable;
- Theoretical evolution: The wave theory of light later replaced the particle theory (as demonstrated by experiments like the double-slit interference experiment), proving that light has no mass, rendering Michell’s assumptions obsolete. As a result, his theory was forgotten for over a century.
5. What can we learn from this?
This example highlights an important lesson in science and finance: Just because the outcome is correct does not mean the method used is also valid. In investing, someone might make a profit by guessing market trends correctly but not due to sound analytical methods; similar to this, in scientific research, we cannot rely solely on the results but must also evaluate the correctness of the underlying assumptions and logic. A correct result derived from an incorrect method may be a temporary coincidence that cannot be replicated.
Final Note
This story illustrates that scientific progress often involves trial and error and continuous iteration. Incorrect theories can still provide valuable insights, while correct theories are typically built upon the work of predecessors with refined assumptions. In finance, old models may be replaced by newer ones, but some of the mathematical tools or perspectives from those older models can still be useful.