In the eyes of astronomy enthusiasts, a black hole is a sphere with a radius of \frac{2GM}{c^2}. This is the result for a Schwarzschild black hole in General Relativity, and it can also be derived from classical mechanics. Although using classical mechanics is technically incorrect, for most astronomy enthusiasts (including the author), it is currently the only feasible way to understand it (the complex derivations of General Relativity can be quite overwhelming). Of course, in reality, a black hole is not a sphere; it is a point of extreme density. As for how high the density is, the generally accepted view is that it is infinite. However, rigorous physics does not accept the concept of infinity, or rather, physics does not accept any claim of infinity. Therefore, quantum gravity theories are being actively developed to unify relativity and quantum mechanics, but that is another story. \frac{2GM}{c^2} is merely the event horizon of the black hole; inside the horizon, we know nothing. This article mainly explores the changes in the shape of the event horizons during the merger of two black holes from the perspective of classical mechanics. Readers will find that the shapes of these horizons are quite interesting.
In classical mechanics, a black hole is defined as follows: the escape velocity at the surface of a celestial body exceeds the speed of light, so even light cannot escape, making this “hole” very black. That is to say, the total energy of a photon (the sum of gravitational potential energy and kinetic energy, in the classical sense) must be negative, indicating it is bound. Mathematically, this is expressed as:
\frac{1}{2}mc^2 - \frac{GM_1 m}{r_1} - \frac{GM_2 m}{r_2} - \dots - \frac{GM_n m}{r_n} \leq 0
where -\frac{GMm}{r} represents the gravitational potential energy of the photon. In other words, if there are n point masses, the event horizon created by them is determined by the following equation:
\frac{M_1}{r_1} + \frac{M_2}{r_2} + \dots + \frac{M_n}{r_n} \geq \frac{c^2}{2G}
By establishing a coordinate system, we can describe the shapes of these surfaces. Let us first look at the case of two black holes of equal mass. Before viewing the simulation results below, readers might want to close their eyes and imagine what these surfaces might look like. This is a very important exercise; it helps you build a series of physical intuitions, which assist in verifying the correctness of the final results.
During the merger of two equal-mass black holes, the equation for the event horizon is:
\frac{1}{r_1} + \frac{1}{r_2} \geq \frac{c^2}{2GM}
By establishing an appropriate Cartesian coordinate system, the equation can be expressed as:
\frac{1}{\sqrt{x^2+y^2+z^2+d^2+2dx}} + \frac{1}{\sqrt{x^2+y^2+z^2+d^2-2dx}} \geq \frac{c^2}{2GM}
where 2d represents the distance between the two bodies. During the merger process, d continuously decreases until it reaches 0. Below is a series of simulation images:
When the distance is far, the event horizons are two spheres:
As they gradually approach, the event horizon area expands due to the superposition of gravitational forces:
The horizons begin to intersect:
The horizons slowly merge:
It looks quite like a peanut:
It is very close to a sphere:
The final result, becoming a single sphere:
Its surface area is larger than the sum of the two original ones.
This is a small animation I made (a simple frame-by-frame overlay):
In fact, this shape is essentially the same as the Roche lobe in astrophysics. I found that this process is somewhat like the reverse of cell division!
What if the masses are not equal? Please see:
The merger of multiple black holes is even more interesting:
I have been studying theoretical physics recently. Although it is very interesting, the mathematics is quite complex. In order not to lose the physical essence in the complex ocean of mathematics, I occasionally look for some interesting things to play with to stimulate my thinking. This article is a product of such a context. Although the classical mechanical analysis is not entirely correct, it still allows us to catch a glimpse of the phenomenon, which is sufficient. Regardless, this is just for physical intuition and geometric appreciation. Of course, it also exercised my skills in using mathematical software; these images were all generated using Mathematica 8.
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