I’ve recently been asked to give a short lecture about the physics of black holes as depicted in a science fiction movie — I can’t say any more about it yet, you’ll just have to bear with my being enigmatic for the moment. A twenty minute talk isn’t enough to do justice to black holes, though, and since I talk about them a lot, I figured a series of posts looking at what they actually are — as well as the fascinating history of how we discovered them, first in theory and then observationally — would be a good thing to have.
Black holes are fundamentally a prediction from Einstein’s 1915 General Theory of Relativity, and various other aspects of early 20th Century physics. However, if we want to get an intuitive understanding for what a black hole is we can actually start before then, with a thought experiment proposed by John Michell in a letter where he speculated about what happened if a star became very heavy indeed.
Escape velocity
We’re all familiar with perhaps the fundamental quality of gravity: that what goes up must come down. No matter how hard you kick a football, it will always, eventually fall back to the ground, even if you kick it directly up. The force which you can apply to a football will never exceed the force which acts on the football to pull it back down to the earth. But that doesn’t mean that it’s impossible to apply such a force (I just suspect you’d break your foot if you tried it playing football).
The speed which our football needs to be kicked at to escape Earth’s gravity is what we call its escape velocity, which we can work out by working out the gravitational potential energy of the football, and working out how fast it would need to go to have more kinetic energy, and thus to break free of Earth’s gravity.
For the Earth this escape velocity is around 11.2 km/s. Which is faster than I think anyone can plausibly kick a football. Indeed, it’s substantially faster than the speed of a bullet, which might reach around a tenth of that. So while you might lose a football over the fence and into your neighbour’s garden, you’re probably safe from needing to appeal to NASA to get it back.
There’s a bit of calculus involved in working this speed out, but we reach quite a simple formula for it: \(v_{\rm esc} = \sqrt{\frac{2GM}{R}}\)
Where $G$ is the gravitational constant, $M$ is the mass of the object you’re trying to escape from, and $R$ is the distance from the centre of that object.
This tells us that the heavier the object we’re trying to escape, the faster we need to go to escape it (or the smaller, and tehrefore denser the object, the harder it is to get away).
Try it out for yourself with the escape velocity calculator, and see how the escape velocity changes for different planets, or even the Sun.
So the things which affect this velocity are the mass of the planet (or other body) we’re attempting to escape from, and how close we are to the centre of that body. This means that it’s harder to escape from a more massive planet than an asteroid, for example.
Choose a preset or enter your own values.
Now, you may be thinking, “but surely, we have sent a football into space”, and you’d be right (even if it was a gridiron football). When we think about escape velocities we’re thinking about the “ballistic” speed, which assumes that you give the object all its speed in one go. If you find some way of continuing to add energy to the object, for example by spewing hot gas in the opposite direction, then you can escape at a much lower speed (indeed, provided you can keep providing thrust of some kind, at any speed). But things like particles of gas, or other relatively simple objects don’t have engines, so need to move at the escape velocity to escape into space. (Particles of gas in the atmosphere can and do escape this way, so it’s not by any means impossible.)
Speed Limits
An interesting challenge arises if, for some reason, there is a limit on the speed you can travel at. By the time Michell was thinking about his very heavy stars, it was widely understood that light traveled at a specific, and finite, speed. So what if a star was so heavy its escape velocity was greater than the speed of light?
This would mean that light couldn’t escape from the surface of the star, and we would end up with a “dark star”, a serious impairment to the celestial navigator, since we would presumably only know about it when we hit it (or at least close enough to start to feel its gravitational attraction).
Too Dense for science?
Michell’s dark star idea falls apart because of one thing it effectively requires: an impossibly heavy star. Either the dark star needs to be hundreds of times larger than the Sun with its density (while we have observed stars which are over a thousand times larger than the Sun, they’re much less dense than it), or we need to make the star impossibly dense.
Radius fixed at R☉ = 695,700 km. Drag the slider or pick a preset to change the density.
Even if it was made of Osmium, the densest element present on Earth, which was discovered a few years after Michell’s musings, the Sun’s escape velocity would come nowhere near the speed of light. It would seem that we might be safe from these rather spooky objects.
You can try out different densities for a Sun-sized star with the escape velocity density calculator, and see how dense it would need to be to have an escape velocity greater than the speed of light.
Further, around this time the idea that light was made of particles which might be affected by gravity (which was Newton’s view) was being severely challenged by the idea that it might be a massless wave, unaffected by gravity.
It would take until the early 20th Century, and a number of quiet revolutions in our understanding of physics, to return to the notion of such a dark star, and only after we’d discovered something possibly even more bizarre, the white dwarf, which had a density absurdly higher than any known metal.
But that’s a story for the next post.