Jim's Simple Physics Issue #3.14159: Black Holes this is your friendly neighbourhood Jim with another installment of Jim's Simple Physics. Today, I want to address the well-covered topic of black holes. While you can find many other posts on Physics.SE about why a black hole is black and whatnot, I want to take a slightly broader approach and make things easier to understand for any non-physicists out there.
In this post, I want to address the following questions:


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*How does a black hole form?

*What exactly is a black hole?

*Why do we say nothing can escape from it?

*Is it possible that all of this is just a dream and that, in reality, I'm actually just riding around on the back of a giant turtle?

 A: Okay, black holes. This is a bit of a tricky subject and I want to make sure everyone understands it properly. If you jump on Earth, gravity pulls you back down. This is good, because otherwise basketball would be a very dangerous sport. You can jump higher by pushing off the ground with more initial speed. So far so obvious. Naturally, there exists a speed you can give yourself when you jump off the ground that would allow you to escape Earth’s gravity entirely. But how do we know this speed? It’s all about energy.
We know that the force of gravity that a body exerts on you decreases the farther away from the center of mass you are. Using this knowledge, we can find the “Gravitational Potential Energy” of an object at any height above a body. It is also worth noting that the faster you move; the more energy you have. Knowing that, the potential energy is just the amount of kinetic energy (or speed energy) you would gain by falling a specified distance through a specified gravity field. I’m sure many of you have seen pictures of a gravity well similar to the one below. A gravity well is a representation of the gravitational potential energy at various distances from a body (a lot of places say it’s a picture of spacetime warping; sorry, that’s not the case). As you draw closer to an object, the amount of potential energy you have at your altitude decreases (makes sense; you can’t fall as far, so you can’t gain as much kinetic energy).

Okay, now we understand gravity wells. Let’s go back to escaping from a body (like Earth). To escape any gravitational body, or rather to increase your altitude above a body, you need to have enough energy to move up to a new potential energy level. Total energy has to be conserved; to give yourself more potential energy (by going higher above a body), you need to take that energy from your speed (kinetic). So, to go from the bottom of the well in the image above to the top of it, you need to provide an equal amount of energy to cover the distance. There are a few ways to do this. On Earth, you could try jumping. Jumping gives you all the energy you need at once. You lift off the ground at the “escape velocity”, which is the speed at which you have just enough kinetic energy to escape to empty space. Alternatively, you could try using a rocket, which can provide the energy you need over time (the rockets continuously inputs more energy by burning fuel) and it allows you to escape while travelling at less than the escape velocity. This is a very good method; the escape velocity on Earth’s surface is 11 kilometers per second. That’s fast enough to burn up in the atmosphere. Using rockets lets you have a lower speed and avoid all that messy burning up nonsense (pro tip: don’t jump upwards at 11 kilometers per second). While both are good methods of escaping, we tend to only represent gravity wells in terms of the escape velocity at any given height. This allows us to easily convert it to the energy required to escape to empty space AND it gives us an indicator just how fast our spaceships have to go to travel other places. Now the fun thing about escape velocities is that it depends only on how far you are from the center of mass of the object AND on how much mass is closer to that center than you are. If you’re underground, the stuff above you isn’t pulling you down, so that’s not contributing to your escape speed. However, once you go above ground again, it will contribute. What’s this mean? It means that if we were to take Earth and make it bigger without changing its mass, the escape speed at its surface (the energy needed to escape to empty space) would be lower. If we made Earth smaller and denser, the escape speed would be higher (because you’re closer to the center of mass, and the amount of mass beneath you hasn’t changed). This may not seem fun or interesting now, but like my drawer of USB cables at home, all the loose ends tie together eventually (stupid ball of tangled USB hell).
Great! That was confusing. What does all this have to do with black holes? Well, stars are huge; really very massive. Naturally, they have way more gravity than Earth. Stars have so much gravity that they tend to want to collapse in on themselves. This creates so much pressure on the gases that make up the stars that they start to undergo nuclear fusion. To put that in more relate-able terms, gravity squeezes the hydrogen so much that it starts exploding like nuclear bombs. However, because the gravity is so strong, the star doesn’t fully explode. Instead, it reaches a balanced state where the outwards pressure from these nuclear explosions equals the inwards pressure from gravity (Whoa! That’s heavy, dude). Hydrogen explodes pretty easily and there’s a lot of it, but eventually a star will use up most of that fuel. The products of these reactions are elements that don’t “burn” as easily. This means, eventually, the outwards pressure decreases and can no longer resist gravity. The star collapses inwards again. For smaller stars, they can collapse to a point where the pressure is so great that even the remnant materials start exploding like nuclear bombs. This makes kind of a chain reaction and the entire star blows. This creates way more outwards pressure than gravity can counteract and the star explodes in a “supernova”. However, in larger stars, well before we reach the supernova point, the star collapses so much that the escape velocity at its surface exceeds the speed of light. Whoa, that is serious stuff.
What does it mean for the escape velocity to exceed the speed of light? It means that if you wanted to jump from the surface of this collapsing star (Kids, don’t try this at home!) and escape to empty space, you’d have to jump at faster than the speed of light. Too bad nothing can go faster than the speed of light. Us things with mass can’t even go AT the speed of light, let alone faster. So I’m pretty sure we’re not getting out, right? In fact, that also means light can’t escape from it now.
“But Jim, you over-zealous science-man,” you say, “you told us that rockets could allow you to escape while going slower than the escape velocity. Come to think of it, since light always travels at the speed of light and no slower, it’s not like gravity will slow it to a stop and then make it fall back. Shouldn’t rockets and light both be able to get out, even if it takes a long time?” That’s a great question. Good to see you were all paying attention. Remember that the escape velocity is a handy way of indicating how much energy you have to provide to travel from a given height all the way up to empty space. Additionally, while I didn’t cover this myself, remember that the speed of light is the maximum possible speed. Special relativity covered the math for this one (thank you Einstein); it turns out that as you approach the speed of light, the amount of kinetic energy you have approaches infinity. This means that an escape velocity greater than the speed of light means that the gravity well of this object would extend downwards infinitely; or rather, you’d need to provide an infinite (unlimited, unending, etc) amount of energy in order to raise your altitude from the surface of this collapsing star to some altitude where the gravity is weak enough that it effectively has no influence.
Infinity is big. What’s the largest number you can think of that isn’t infinity? Whatever it was, that is literally 0% the size of infinity. So let’s think about my earlier statement. You need to provide an infinite amount of energy (either all at once through jumping or over time with rockets, your choice) in order to go from the surface of our collapsed star out to empty space (note that empty space is not an infinite distance away). And let’s face it, we’re busy people, so it’d be nice if your escape didn’t take an infinite amount of time. Most of us can do the math on that and figure out that we can’t provide the energy required to escape from the surface. But there’s more to it than that. We are trying to raise our height by a finite (measureable) amount but can’t because it takes infinite energy to do so. So can we, at least, raise our height by some smaller amount? The answer is a mind-blowing “no”. If you need infinite energy to go up some distance, then you’d need infinite energy to go up half that distance (otherwise you could just go half the distance and go half again and you’d be there and then I’d be really confused how you performed such magic). This idea can be repeated until you realize that it would take infinite energy to raise your altitude by any amount. This is why rockets or light can’t escape; not only do you need to jump at greater than the speed of light to escape, you need to travel at greater than the speed of light to just move upwards a bit. This isn’t possible for anything. Light can constantly travel at the fastest possible speed, but it can’t travel fast enough to have the energy to move upwards even one centimeter.
This point, where the escape velocity at the surface of the star is equal to the speed of light (meaning the energy required to escape to empty space becomes infinite) is called the “Event Horizon”. Above this, the energy required to escape is still finite, which means it’s possible to eventually get free. Below this line, the energy you need to give yourself to escape to any higher altitude becomes infinity. Because of this, nothing that crosses the line can ever get back out, which means we can’t see anything beyond this line nor can anything that happens there affect us. Like how you can’t see beyond the horizon on Earth’s surface, neither can you see beyond the event horizon. Thankfully, the analogy stops there and things beyond the horizon on Earth can come back into view (no falling off the edge for us).
So what do we call this collapsed star? It’s something that things can only fall into, they can’t get out of it again. So it’s kind of like a hole in space. And light can’t even get out of it; without any light from it, it’ll probably look black. So we call it a Black Hole.
“But Jim, I’m having a bit of trouble understanding how light can be travelling at the speed of light upwards and still not move upwards. If it’s got a speed, doesn’t that mean it has to be moving? I get that it doesn’t have infinite energy and that apparently you’d need infinite energy to move upwards any amount, but I’d still like to understand why having speed doesn’t mean moving”. Yes, I thought you might ask that. Physics is a complex subject. General relativity describes this in an overly mathy way that, for anyone who’s studied for years and can grasp the math, makes perfect sense and explains that clearly. But that doesn’t help anyone that hasn’t studied for years to get the math of GR. So, while this isn’t really the physical reality of what’s going on, let me give you a way of picturing this in your head that will help you sleep better at night. Think of it like this, the gravity from a black hole is so strong at the event horizon that the space around it also falls into the black hole. At the event horizon, space is falling into the black hole at the speed of light, so if you travel through space at the speed of light, it’ll be like walking on a treadmill; the speeds will cancel out and you’ll just maintain one position above the black hole. To move upwards you’d have to go faster outwards than the speed of light, which is not possible. Again, I’d like to point out that this isn’t exactly what is happening, but it’s a good way of thinking about it and you can use this way of thinking about black holes to understand virtually everything about how they work (now that I think about it, if it works in all situations, that probably makes it a valid interpretation. Okay, go ahead and assume that space falling into the black hole is exactly what is physically happening). So there you have it; black holes are cosmic treadmills that have been turned up to the speed of light.
In conclusion, yes it's entirely possible this is a dream you're having while riding on the back of a giant turtle. But as part of your dream, I really can't prove to you which is the actual case. I'm willing to bet the turtle knows for sure. My advice? Ignore the question and just pretend this is reality; that'll cause less headaches.
