WRITTEN BY: AURORA SIMIONESCU
We all wish we could experience time in a different way – something other than the inexorably regular tick-tock of your wrist watch or wall clock. Einstein’s genius consisted in daring to envision, and prove, that this wasn’t just a human dream, but a part of the fabric of reality. The consequences? Einstein’s special and general theories of relativity opened up a whole new, fascinating world that outdoes even the strangest ideas science fiction writers have ever come up with on their own.
Black holes are definitely among the VIPs of this brave new world – an endless generator of I-can’t-wrap-my-brain-around-this even for the most advanced mathematicians among us and, needless to add, a source of inspiration and speculation about anything from time travel to far-fetched end-of-the-world scenarios. Perhaps even more fascinating, black holes are also crucial in shaping the cosmic web we introduced in Episode 6 – despite their relatively minuscule size compared to galaxies and galaxy clusters. We will try to explain this intriguing connection for you in an upcoming blog post, together with its many aspects that are still a matter of scientific debate, and are the topic of some of our recent and future research.
But first things first – what actually is a black hole? In fact, the concept was around long before Einstein’s theory of general relativity. In November 1783, at a meeting of London’s Royal Society, reverend John Michell suggested that some stars might be dark, because their gravity could be too strong for light to escape. The argument goes something like this: if you throw a ball up in the air on the Earth, it will probably eventually fall back down to the surface. If you throw it with a higher and higher speed, it will go further and further and it will take longer and longer time until it falls back. Eventually, if you could throw it as fast as 11.2 km/s, it will never fall back anymore, but it will escape Earth’s gravity and be lost to space. If you were on the surface of the Sun (don’t try it, it’s very hot there), you’d need to throw the ball at least at 617.5 km/s so that it escapes the Sun’s gravity, which is stronger than Earth’s. Michell simply asked the question: does any star exist which is so dense and massive that something would need to move faster than light in order to escape its gravity? If it existed, Michell concluded that such a star would have to be dark, because no light from it could reach us.
It turns out, though, that there is a small flaw in the argument above: light doesn’t exactly behave like any other object, because it is made out of photons which have zero mass. The classical law of gravity discovered by Newton says that the gravitational force acting on an object depends on the mass of that object, so for a photon with zero mass, the gravitational force is also zero. Unlike balls and rockets, light should not feel any gravitational pull! In other words, Michell pretty much made the same mistake that would have made your high-school math teacher go ballistic: he inadvertently divided both sides of an equation by zero (not knowing, at that time, that photons are massless).
Einstein’s theory of general relativity later showed that light actually does feel the effect of gravity. It’s not because there is a gravitational force acting on the photons, but rather because gravity bends the space-time through which the light must pass, so the light rays get bent (it’s a bit like trying to draw a straight line onto the surface of a beach ball). And it is possible to have an object that is so extremely dense and warps the space-time around it so much that no light can escape. This was all worked out in very complicated calculations by Karl Schwarzschild, who determined how big such an object should be, depending on its mass. As it turns out, dividing by zero isn’t always bad, because in fact Michell did get the exact same answer as Schwarzschild (but for the wrong reasoning).
Doing the more complicated type of math allows us not only to predict how compact an object must be before it becomes a black hole, but also to understand how things behave and move close to and inside the so-called event horizon – which is, essentially, the edge of the black hole. And this leads us some of the most mind-bending (in addition to space-time bending) stuff in the Universe.
The best analogy I have found so far for sort-of helping our brain cope with how weird this is is the waterfall metaphor. It turns out that, mathematically speaking, it is possible to rewrite Schwarzschild’s equations to look like space itself were falling into the black hole. This was first discovered in 1921 by the Nobel prize-winner Alvar Gullstrand, and independently by Paul Painlevé, who was Prime Minister of France in 1917 and again in 1925. It seems at least some politicians back then were still remarkably good in math.
So Andrew Hamilton (professor at the University of Colorado in Boulder) suggests to imagine that the space falling into the black hole is like the current of a waterfall, and the light rays, or photons, are like fishes swimming in the flowing water. Outside the event horizon, space is falling into the black hole at less than the speed of light, and photons moving upstream can make way against the flow. At the horizon, space is falling into the black hole at the speed of light, so a photon-fish swimming directly upstream will just stay there, swimming like crazy, but not going anywhere, the inward flow of space exactly cancelling the photon’s motion. Inside the horizon, the space falls faster than the speed of light, carrying everything with it. Even photons traveling directly upstream will be effectively dragged downwards. In this sense, Michell’s picture of the light traveling upward, like throwing a ball into the air, and then slowing down and falling back to the surface of a black hole is not correct; nothing can ever move upstream of the falling space, and nothing can stand still – just like, in the “normal” world outside a black hole, you cannot stop or turn back time. And if you took issue with space moving “faster than the speed of light”, remember that relativity only says that nothing can travel through space faster than light. Space itself can do whatever it likes.
With this analogy in mind, let’s look in more detail at what it means to be somewhere near or inside a black hole. To begin with, for all the fatalists out there, I must stress that it’s perfectly ok to get quite close to the black hole without being forced to fall inexorably to your painful death-through-being-ripped-apart-by-gravity. Just think of a real waterfall: the river is usually pretty calm, and the current is pretty mild, until you are very close to falling over the edge. You can safely orbit around the black hole at a distance as little as three times the size of its event horizon, and you would not be sucked in (though, if the radius of the black hole is small compared to your height, you’ll be wrapped around into a pretzel along this rather tight circular orbit, and that might not be that pleasant – but still, you would not be forced to fall in). If the Sun turned into a black hole (which it won’t), the Earth could still comfortably continue to go around it just as it’s doing right now – the gravitational pull of the Sun at distance of the Earth wouldn’t change a bit, though it would get pretty cold and dark over here.
What you would start to find strange, if you were safely in orbit fairly close to a black hole, would be the way things look. Because the light is starting to be swept away – diverted – by the current of space falling into the black hole, images of other things like stars and galaxies located in the same general direction as the black hole would start to appear very distorted, like looking at the world through a very strange lens – and a lot of things would appear piled up really close to the event horizon, even though they are in reality located somewhere else on the sky:
Things, of course, get a lot more exciting (and a lot more weird) once you’ve left the last stable orbit and are actually falling into the black hole. The inside of a black hole is a world completely unlike anything you’re used to, where the roles of space and time are reversed, where you can at once see the past and the future just as easily as you can now choose to move left or right – but this comes at the price of not being able to move upwards out of the black hole, just like we can’t move backwards in time in the “normal” world.
Our day-to-day intuition based on living outside of a black hole tells us the following rules: you can only know about and remember an event if (1) it happened in the past and (2) it happened at a distance from us which is smaller than the distance that light could have traveled since the event happened (else, you can’t have seen the event happening yet because its light hasn’t reached you yet). Inside a black hole, on the other hand, you can only see an event if (1) it happened further from the center of the black hole than where you are (remember, no light can flow upstream to you, so you have no way of seeing things that happen closer to the center of the black hole than you are) and (2) if T is the time that it would take light to travel to you from the location of the event, then it happened either no more than T hours ago according to your clock or (and this is the cool part) T hours into the future. So, in this sense, you can replace space with time and vice versa to change the rules that apply outside the black hole into the new rules that apply inside the black hole, which is why it is said that space becomes like time and time becomes like space (check out this article to read more on the topic; P.S. did I say “endless generator of I-can’t-wrap-my-brain-around-this”?).
You may have heard of Einstein’s conclusion that time is relative, meaning that clocks work very differently when you’re traveling close to the speed of light. And that gets a lot stranger when you are falling faster than the speed of light. This will allow you to quite literally see things before they happen; the further something is, the more into the future you can see it. But you can’t see infinitely into the future – just like, due to the finite speed of light, we can’t see infinitely far into space in the “normal” world.
Unfortunately, since you can only see things happening further from the black hole center than you are, and you can’t send any signal upstream, you would not be able to use that precious information about what will happen in the future. You can’t send a message to the people whose future you’re seeing and you couldn’t warn or advise them in any way – and they wouldn’t even know you are there.
Not to mention that the really really strong gravity will stretch you into a quite painfully looking spaghetti while you’re falling in, and you’d lose consciousness (and likely get killed) before you could experience this mind-blowing world. And even if you survived crossing the black hole’s horizon, you would only have, by your clock, a tiny fraction of a second of being a psychic before you reach the center and get crushed for good. But at least you see why black holes are such an inspiration for science fiction movies and thinking about time and time travel in a completely new way! They give us hope to be able to see the future, grounded in the laws of physics. Black holes which have an electric charge, or are spinning, are even more complicated and yield even crazier possibilities.
But aren’t these objects too exotic? Did theorists just dream them up, or do they actually exist? And if they exist, but let no light signals escape, how can we know that they are there? In Episode 1, we already mentioned one way to create black holes, which is when a very very massive star runs out of fuel at the end of its life and collapses under the pull of gravity. In the next episode, we’ll talk about how to detect black holes, and how some of them grew to become extremely massive and shape today’s galaxies and the cosmic web!
Curious, if you are in the event horizon flowing with Space into the black hole, I see that you can’t see the things below you since they are going into the black hole – they are (the photons) are “downstream”. Why are the photons above me moving with space-time visible to me? Are they moving faster than me? And, why is this considered the future? Is it simply bring light quicker to me or am I totally wrong? Thanks. I always liked black holes.
David, I’ve lost at least one night of sleep over this question while trying to write this article, and to be quite honest I’m not sure I completely understand this either. So here’s what I think is happening, but you should probably ask Roger about it if you really want to be sure it’s right (wish I could ask him myself!).
The problem is with defining time and time intervals. Let’s start with special relativity where, as you probably know, if I am moving away from you close to the speed of light, my clock will appear to be ticking slow to you (AND your clock will appear to be ticking slow to me!). If I travel at the speed of light, then my clock will still seem to be ticking normal to myself, but your clock to me will appear to have stopped entirely (assuming you’re standing still). So if, by falling into the black hole, I manage to move effectively faster than the speed of light with respect to you, then it is possible for me to measure what to my clock will appear as a negative time interval (one of the two images of your clock which reaches me will appear to be going backwards). That’s not because your clock is actually going backwards, it just appears to do so (just like your clock hasn’t actually slowed down just because I’m traveling close to the speed of light in special relativity; it just appears to me to have slowed down).
This is not (directly) related to the fact that signals need time to propagate (otherwise whether your clock slows down would depend on whether I am moving towards or away from you in special relativity). Really all the problems our brains are having with understanding this is because space and time are so essentially different in our day-to-day life that the concept of mixing (leave alone switching) space with time is just impossible to imagine. Time inside a black hole simply isn’t what you think of when you say “time”!
Thanks. That is interesting. I’ll think about it for awhile. I see Roger once in awhile hanging around so if I am brave I’ll ask him. I remember what was neato to me back in school was that x,y,z,ct had to stay constant! Four dimensions with units of distance! Neato. Anyway, the trick is that somebody is going faster than the speed of light. I see. The stream analogy makes me think everybody is moving with the stream (water) so I was side tracked. Pretty dang wild stuff but who would not like to think about it? Luxury!
I got caught in the waterfall analogy a bit too much myself, until I realized that “space falling faster than light” is only half the story and doesn’t tell you anything about what happens to your clock while this whole crazy thing is going on. Anyway mathematically speaking it’s easy to see what happens, in the sense that signs get switched in the usual Minkowski metric element dx^2+dy^2+dz^2-c^2dt^2; the coefficient of dt^2 becomes positive and the coefficient of the radial coordinate becomes negative instead. So from the point of view of equations it’s all very simple, it’s just that our brains have a terrible time trading a time dimension for a spatial one…
By the way, I’ve also found this link to be quite interesting for understanding the time-space reversal further – although, in the beginning, it’s a bit slow getting to the point of why the stuff it describes is actually relevant to BH geometry. Give it a chance! http://www.einstein-online.info/spotlights/changing_places
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Even thou I was on the verge of failing math, physics or chemistry classes during my high school years (straight 5’s four years in a row – Aurora knows what this score means), I find these articles very interesting.
Moreover, the analogies used have been perfectly chosen, as they provide a comprehensive understanding even for “pupils” like me 🙂
Thank you for this great website. I am looking forward for the next episodes…
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