Black holes are very dense objects living in the universe. Their density is about 10¹⁰ times the density of our sun. Such a density means that close to their core, under a given radius, nothing can escape, not even light.
A lot of scientists imagined such ‘holes’ where nothing can escape. Einstein was the first to actually find a good scientific theory to model them, one century ago. However, we have only been recently able to observe them directly (the famous photo of a black hole, April 2019).
In this article, we’ll discuss how a black hole can still emit a lot of energy (in the form of light). This paradoxical phenomenon shows that black holes are really strange and fascinating objects !
Galileo, Newton or Einstein ?
A bit of history
Ancient civilisations (egyptians, greeks, chinese) already had a rather good understanding of the law of physics, and especially gravity, since we can observe its effect every single second in our lives. Copernic and then Galileo advocated that objects movements in a gravitational field do not depend on their mass. As a consequence, every object thrown from a tower will arrive at the bottom at exactly the same time (+- epsilon because of air). Also, Earth is turning around the Sun, as every other planets, and not the inverse. In return for this extremely powerful idea (which is the core idea of general relativity), they got fire and death by the catholic church, but that’s another story.
Newton then arrived a few centuries later, with a practical theory for forces, and dynamics induced by them, including gravity. His theory was predicting that Copernic and Galileo where right: the movement of an object in a gravitational field does not depend on its mass. It was the standard for many years since Newton’s theory explained 99.9% cases we could observe. Except one.
One thing sounded strange to scientists for a long time. Why is Mercury’s orbit not stable ? The angle of the ellipse is increasing by a small amount every year. Even after adding the influence of other planets, sun eruptions, sun wind, hypothetical alien vessel, we still have this 43 arc seconds increase every century. Just imagine how small that is: 0.012 degree every century, so only 0.00012 degree per year. But still, we observe this slight increase. And nobody can explain it. This is bad for science !
Einstein was very interested in how light works. What are these rays, coming from the sun ? Are they particles ? What are their mass ? Why can’t I touch them ? Does light have a speed ? He was not very good at maths and physics, hardly passing his exams to enter university. But he had very creative ideas, and thought physics in the right way: first imagine the situation, simulate it in your head, then come up with the solution and test it in the real world. Einstein developed special relativity (which was really a good physical interpretation of others’ work, Lorentz especially), and then began to think about gravity and its link with light, if any. He started to develop general relativity (next section) but he knew Mercury’s precession problem and thought it was the perfect spot to prove his theory. If he predicted the 43 arcseconds per century, it was the proof to the world that his ideas were right !
First let’s start with the simple version of general relativity, the first theory of Einstein. This theory is founded on two axioms: nothing can go faster than the speed of light in void (c = 299354234 m/s) and law of physics are the same in every Galilean referential. These two axioms are absolutely fundamental for the rest of the theory. A Galilean referential is a referential where an object initially idle won’t move as time passes.
Now imagine you’re in a train going at the speed of light. I am on the side watching you. You take a ball in your hand and throw it in the air at, say, 1 meter, and it takes 3 seconds for the whole movement. The law of physics are the same in the train or on the side (axiom 2), so for both of us, the ball goes up in the air and then goes down. For me on the side, in 3 seconds the ball has moved a higher distance than your hand, since your hand trajectory is a flat line and the ball is a triangle whose base is this line. As it all lasted the exact same time, it means the ball was faster than your hand. But your hand already goes at the speed of light. So the ball goes faster than the speed of light. But the ball can’t go faster than the speed of light (axiom 1). We have a contradiction there … Or ?
Time has stretched. Yes, time itself stretched such that t_for_me_on_side = gamma * t_for_you_in_train, where:
If you replace t by gamma*t in the equations of the train experience, you’ll see that everything match perfectly. The problem is solved, but then it means time depends on the referential. Time is not flat as we could think, the same everywhere. Time is just another component of space, and it can also be curved, as space.
Imagine the world in 2D. You put the (0, 0) coordinate somewhere, and add the two axis x and y. Now imagine me on the side is on coordinate (0, 0) and the train is on coordinate (10, c*t), so 10 meter away from me on the x axis, and moving in time on the y axis. Let’s add another axis, t, which is on the third dimension. Imagine that at every (x, y) point, you can draw a line parallel to the t axis. Special relativity is just saying that this line is curved. Like, seriously curved. The metric of this space, describing how the space is curved, is called the Minkowski metric:
The minus sign before dt² shows that time is not curved the same way as space. The thing is, to feel this curvature, you need to move fast. You need to change (x, y) coordinates at a pace similar to how you change time * c. Ie, you need to move at a speed close to c.
Space is now spacetime, and what we call movement is only a succession of points (x_i, y_i, t_i) in this 3 dimensional space. In reality, we have 3 coordinates of space (x, y and z) and 1 coordinate of time so spacetime is a 4 dimensional space. That’s why it’s so hard to understand how it works in our real world. It is far simpler in a 2D world.
General relativity is ‘just’ taking the 4 dimensional spacetime but is curving it in a different way that what we did with special relativity. For general relativity people, special relativity spacetime is even not really curved, they call it the ‘flat’ space. So, if you have nothing in your space (no mass object, no particles), the spacetime is naturally curved and we call it the ‘flat’ spacetime.
But where could other curvatures come from ? And that’s where the genius of Einstein came out: what if gravity could be translated into a curvature of spacetime. He wrote an equation giving you how the ‘flat’ spacetime would be further curved (in x, y, z but also t of course !) if massive objects are put in it. The same equation gives you how these same mass will move according to spacetime bent. This is Einstein’s equation.
how spacetime is curved by mass = mass move based on this curvature
Using the equation, we can derive how spacetime is curved if we have only one massive object at coordinate (0, 0, 0, t) in spacetime. The equation implies that this object won’t move based on its own induced spacetime curvature. Solving the equation gives what we call Schwarzschild metric, defining the curvature of space, which is:
This is saying that spacetime is curved on every coordinate (x, y, z and t), but it’s easier to write it using the radius r (square root of x²+y²+z²). In this formula, r_s = 2GD where G is the constant of gravitation and D is the density of the object. If you take D = 0 (no mass), then the Schwarzschild metric is exactly Minkowski metric, a flat space.
Now we can put an object in this spacetime (of small mass compared to M) and observe its trajectory !
Dynamics of black holes
Black holes density
Black holes are the densest objects of the universe, with around 2000000000000 tons/m3. This is enormous, therefore, given Schwarzschild metric, spacetime will be hugely curved around a black hole. That’s where general relativity is interesting.
Neutron stars are also very dense objects, where general relativity is implied. I suggest you learn more about how they formed, how we discovered them, and what is matter inside them here.
Schwarzschild radius and radius of lowest stable orbits
In the Schwarzschild metric, we call r_s ‘Schwarzschild radius’. This radius gives the frontier inside of which we don’t know what happens. The curvature of time and space become inversed, and many strange things can happen. This curvature unfortunately implies that nothing can go out from inside of this zone, not even light, and therefore we’ll never observe anything from there (leading to the impossibility to find contradictions in theories …).
The radius of lowest stable orbits is the radius under which no stable orbit can exist, ie everything has to fall into the blackhole, or rebound and get out. For most objects, this radius is far lower that Schwarzschild radius, so we don’t really use it since any object inside Schwarzschild radius is lost forever. However, for very dense object, it appears Schwarzschild radius can be lower (sometimes multiple times lower) than the radius of lowest stable orbits.
A lot of massive particles float in the universe, from protons to atoms to arranged matter to stars and galaxies. Highly massive black holes (so not only dense, but with a high mass as well), attract matter a lot. We can expect it to be swallowed by the black hole directly, but that’s not what happens.
To enter a blackhole (ie reach a lower radius than Schwarzschild radius), you need a sufficient amount of energy, which depends on Schwarzschild radius and therefore directly on the density of the black hole. As most particles don’t have this huge energy, they ‘rebound’ on Schwarzschild radius and gather on a stable orbit, on the radius of lowest stable orbit.
X-rays: finally, high energy photons
The thing is that, as time passes, these particles are very numerous and friction starts to happen. This energy is converted in heat, producing light (black-body radiation), and progressively reducing the radius of the orbit of some particles. But as, again, you need a sufficiently high energy to enter the black hole (and never get out again), the particles have to spin extremely fast to finally be swallowed, therefore losing a huge amount of energy in friction ! The X-ray radiation we observe from black holes does not come from the black hole itself, but from particles which can’t wait to enter it, fighting extremely hard between them (friction) and producing a lot of energy.
X-ray wavelength is around 0.1nm, corresponding to a temperature of around 10 million degrees celsius. For your information, at that temperature, matter is a plasma, soup of elementary particles (electrons, neutrons and protons).
The universe is extremely rich: very simple laws like gravity can lead to very interesting phenomenons. We are trying to model this world using theories, and the only criteria we have for a good theory is how many and how well it predicts phenomenons. Einstein and his general relativity is the successor of Newton, Galileo and even the greeks, explaining a large amounts of observations we have around gravity and its impact. The main idea behind it is that space and time are not separate concepts, they both live in one 4 dimensional space, called spacetime. This spacetime is naturally curved (special relativity) and can even be curved further by mass (general relativity). This curvature can move objects themselves (either massive particles or not), as water flows in a hole.
With this theory we can explain an interesting phenomenon: radiation of black holes, in X-ray frequencies. Only this powerful theory could do it, and it seems to be unbeaten yet, 100 years after its discovery.