Yesterday I wrote a post that explored the flow of heat both forwards and backwards in time. I used this as a venue to introduce the notion of entropy and to describe one extreme example of the butterfly effect—where small changes in initial data can create big changes in the final result. That’s all fine and good and I stand by that.
But I said that the reverse heat equation, which runs the flow of heat backwards in time, was an example of chaos. And as this reddit user points out, this is very wrong. I have now fixed the original post so that it doesn’t say anything wrong. But I owe you all an explanation here.
The Heat Equation is Not Chaotic
You can never, ever actually solve the reverse heat equation. It is an example of a so-called ill-posed problem. And understanding which problems are well-posed or ill-posed is a very important topic in both physics and mathematics. (This is actually the reason I’m interested in the reverse heat equation. It’s the archetypical ill-posed problem.)
Truly chaotic systems, on the other hand, are well-posed. Although they depend strongly on their initial conditions, meaning that finding exact solutions is difficult, they can be solved. To illustrate the difference, let’s look again at the reverse heat equation, shown in figure 2.
Temperature differences just build on themselves exponentially until the whole thing becomes completely unmanageable. And this is the problem. Now let’s look at a genuinely chaotic system: the flow of water in a very shallow pond, as shown in figure 3. (You can find another good video here.)
Notice the vortices that form? The precise initial configuration of the water dramatically changes the positions of the vortices. However, although the vortices merge, they don’t grow so much that we can’t make predictions any more. And this is the important difference. This property, called topological mixing, is also what keeps the heat equation from being chaotic.
(There are other technical reasons that the heat equation is not chaotic. But this is the big one, and it’s the thing that I really failed to emphasize in my last post. So I’m emphasizing it here.)
As an aside, notice how small vortices become bigger? This is a property of fluids that are tightly confined in one direction like in a shallow pond or on the surface of the Earth. It’s actually why hurricanes form. Small vortices merge to become big vortices. In fluids without the confinement, the process goes the other way, big vortices become small.
As a physicist—and not a mathematician—I believed that I knew the definition of mathematical chaos when I did not. And instead of checking my facts, I just blithely went ahead and wrote about it.
Many physicists don’t know about mathematical chaos; I’m not ashamed of my ignorance. But I am ashamed of not doing my homework before writing about a topic with which I am unfamiliar. Many of you trust me as an authority on math and physics, and in yesterday’s post, I failed to live up to that trust.
Figure 1. The butterfly effect: a sinister insect plot?
The butterfly effect, shown comically in figure 1, is the idea that a very small change in one place on Earth can cause a very big change somewhere else. In this case, a butterfly flaps its wings and causes a tornado. This metaphor illustrates the mathematical concept of chaos, in which the Earth’s atmosphere is a chaotic system. While a single butterfly probably isn’t literally responsible for a tornado, mathematical chaos is very real and important. So this week, I’m going to try giving you some intuition for the butterfly effect using one extreme example from physics.
Suppose we take a flat, rectangular piece of metal and heat it up at four specific spots. Figure 2 shows what will happen to the metal: The four hot spots (shown in red at the start) will cool off as the heat spreads out, diffusing across the metal until the whole piece reaches the same temperature.
If we isolated the piece of metal beforehand, no heat can “escape” it, so it will never cool back down to its original temperature. The total amount of energy in the system will stay the same. The only thing that changes is how the heat is distributed over the metal’s surface. This “flow” of heat is described by the heat equation. Given any distribution of temperature across the metal, we can use the heat equation to know how hot each area of the metal will be at any point in the future.
But what if, instead of making a prediction about the future, we want to make a postdiction? What if we want to know the temperature of the metal at some point in the past?
Heat Flow Backwards?
Of course, we know the temperature change originated at the four spots we heated up, but let’s pretend we don’t. Suppose that we only saw our metal piece after its whole surface had reached the same temperature. Furthermore, suppose that we’re just a little uncertain about the temperature of the metal now. Maybe there are a few spots that are slightly hotter or colder than average—say, from us touching it, or from sunlight. Probably the best way to figure out what the metal looked like in the past is to take our best guess as to the temperature now, feed that number into the heat equation, and run it in reverse, right?
I did exactly that and figure 3 shows the result.
That doesn’t look anything like the four dots! What’s going on? The heat equation run in reverse, creatively called the reverse heat equation, suffers from the butterfly effect. Small uncertainties in the known temperature distribution cause huge variations in the “postdicted” temperature distribution. In the case of the reverse heat equation, this effect is so severe that we can’t make any useful statements.
Let’s try to understand what’s going on.
Understanding the Reverse Heat Equation
Why is the reverse heat equation so chaotic? What causes the butterfly effect here? Let’s think about how heat behaves. Heat spreads out, from hot regions into cooler regions. This makes hot regions cool down and cold regions warm up. Eventually everything becomes uniform.
If you reverse this behaviour, like rewinding a video, heat moves from cold regions to hot regions. Hot regions become even hotter and cold regions become even colder! This means that if you take a surface with a uniform temperature and randomly make some spots just a little hotter than others, those random warm spots will just keep getting warmer. Any difference from the average temperature, no matter how small, gets exaggerated exponentially. This means that if we want to work backwards from a near-uniform temperature distribution to find out how it originally looked, we need to be exactly certain of the temperature everywhere. And we can never be exactly certain. Measurement tools are flawed. And even if we did have perfect tools, quantum mechanics forbids infinitely precise measurements (at least, in finite time).
Worse, since heat diffuses, every original pattern—no matter how strange—leads to a uniform temperature across the metal. So even if the heat spread out perfectly, with every spot exactly the same temperature as every other spot, the reverse heat equation is still useless. Confronted with an infinite number of possible original patterns, it’s forced to just make an arbitrary decision. And while this process isn’t random, the solution that the equation picks will almost certainly be incorrect, since its odds are literally infinity to one.
What Makes Heat Special?
The inability to make postdictions about temperature is surprising. Most of the laws of physics work perfectly well in reverse. If I know the height of waves in a pond—like the one shown in figure 4, for example—at the present moment, then I can say what the pond will be doing at any moment in time, whether past or future. (At least in principle. In reality, friction will convert much of the wave motion into heat. The waves also need to be sufficiently low-energy; otherwise, water can become chaotic. I’ll get to that in a bit.)
So why is heat special? Roughly speaking, the temperature of a metal is actually an average of the energy of the atoms that make it up. In principle, we could track the motion of every individual atom and make a prediction of their motion after heating the metal up with a laser. Then we could make a good postdiction by tracking the atomic motion back in time.
Of course, this is impossible in practice. There’s way too many particles and way too much information to keep track of, so we’d need a practically infinite amount of computing power. So instead, we use the abstraction of temperature, which averages over the particles.
This abstraction has a price, however. We are intentionally hiding information from ourselves: the precise configuration of the metal. And so it should come as no surprise that we can’t use the heat equation in reverse. We lack the necessary information to do so! We can even quantify how much information we’ve hidden from ourselves. The quantity that tells us this is the entropy of the system. And one way to understand the Second Law of Thermodynamics (“entropy never decreases”) is that, as we step forward in time using the heat equation, we forget more and more about the initial configuration of our metal.
(I want to note that, although I’ve been talking about tracking particles, which are classical, quantum mechanics has analogous ideas. Instead of tracking particles, you track—or average over—a wavefunction whose amplitude represents the probability of measuring all the of the positions of a huge number of particles.)
The reverse heat equation is totally unusable. There is no saving it. But it is an extreme example of the butterfly effect. And it’s not actually chaotic. True chaos is more manageable because it is well-posed, meaning that predictions are, in principle, possible.
Manageable chaos emerges naturally in many areas of science. If the pressure is strong enough, or the temperature or speeds high enough, fluids like air and water are actually chaotic, but in a way that we can handle. Because it takes a lot of computing power to handle the chaos in the atmosphere, it’s very difficult to make concrete predictions about the weather…but it’s not impossible.
Large-scale phenomena, like planetary motion, can also be chaotic. Two objects gravitationally attracted to each other will behave pretty predictably, but adding even one more mass to the system can cause their motion to become chaotic. Satellites under the gravitational influence of both the Earth and the moon, or both the sun and Jupiter, are important examples of such three-body systems.
Understanding chaotic systems is very difficult, but it’s also essential if we are to understand much of the universe. And in many cases, we can manage the chaos.
If you enjoyed this post, you may enjoy some of my other posts on mathematics.
In this post, I describe the many sizes of infinity.
In this post, I describe the history of imaginary numbers.
If you’re curious how I produced those images, I put my code in the IPython notebooks in this bitbucket repository. Feel free to play around with them. I’m afraid there’s no documentation at the moment.
You can find a more technical discussion of the heat equation and reverse heat equation in this blog post by an engineering Ph.D. student.
And here‘s an in-depth discussion of entropy as “lost information.”
My previous post was a description of the shape of spacetime around the Earth. I framed the discussion by asking what happens when I drop a ball from rest above the surface of the Earth. Spacetime is curved. And the ball takes the straightest possible path through spacetime. So what does that look like? Last time I generated a representation of the spacetime to illustrate.
However, I generated some confusion by claiming that it “should be obvious” that the straightest possible path is curved towards or away from the Earth. When a textbook author says “the proof is trivial” usually what they mean is that they don’t want to go through the work of writing a proof. The same is true here, I didn’t want to generate a picture with the path of the ball in it. Since this was confusing however, I apologize. And to make it up to you, I’ve plotted the path of the ball, shown in figure 1.
Note that it approaches a straight line. That’s because as it accelerates it’s approaching the speed of light (we are neglecting air resistance and exaggerating the distance from the surface of the Earth to make that happen). The path of the ball is curved—it curves with the surface, after all. But it’s as straight as it possibly can be. And that’s what makes it a geodesic.
Note also that the speed of light is a straight line that’s wider than 45 degrees. I told you last time that in Minkowski space light travels at 45 degree angles. However, to make the curvature of the spacetime visible, I stretched out lengths radially (the direction of the red arrow) a bit. So actually light cones in this plot are wider. I didn’t think this would be visible when I made the plot before, but it’s quite clear if you include the geodesics. So I apologize for that slight misrepresentation last time.
I’ve updated the previous post to include this plot. So this week’s post is only for those of you who read the last post.
General relativity tells us that mass (and energy) bend spacetime. And when people visualize the effect of a planet on spacetime, they usually imagine something like in figure 1, where the planet creates a “dip” in spacetime much like a “gravitational well.” But today I’m going to show you what spacetime actually looks like near a planet… and it doesn’t look anything like the common picture.
As we learned, general relativity tells us that gravity is really a distortion in how we measure distance and duration. In the presence of mass, spacetime distorts so that distances are longer or shorter and time flows more or less quickly. Then objects (under no forces) travel along the straightest possible path through this distorted spacetime. And this motion, which doesn’t look straight, is what we perceive as gravity.
But what does this curvature look like? It’s hard to visualize. And as a result, I often get the following question: how does all this work on Earth? If I stand at the top of a cliff and drop a bowling ball, as shown in figure 2, what causes it to accelerate towards the Earth? How does the structure of spacetime make that happen? Why doesn’t it, for example, simply fall at a constant speed? Or simply hold still in the air?
To understand this, we’re going to try and visualize our local spacetime.
Before we talk about curved spacetime, though, I want to remind you what spacetime looks like in the absence of gravity… i.e., when it’s flat. That’s the domain of special relativity. Flat spacetime is called Minkowski space.
In Minkowski space, we give each point (or event) a position in space and a position in time, as shown in figure 3.
In Minkowski space, people and objects exist at all times (between birth and death at least), but move between places. The line representing someone or something’s path through space and time is called a worldline. If an object is stationary, the worldline is vertical. If an object is moving, the worldline is at an angle, and the slope of the line is based on the speed at which the object is moving, as shown in figure 4.
When working in Minkowski space, it is customary to work in units where the speed of light is one. We do this so that we can convert between position and time, which we treat as two different types of distance. (For instance, a second is the amount of time it takes light to travel meters.)
Using such units, the worldline of a photon is a line forty-five degrees off of each axis–i.e., a line whose slope is one. The worldlines of light traveling away from a point in every direction thus form the light cone for that point.The light cones traveling into the future are future-directed and the light cones traveling into the past are past-directed, as shown in figure 5.
Because nothing can travel faster than light, the light cones determine what events in the past can affect current events and what events in the future can be affected by the present. As shown in figure 6, if event B is in the past-directed light cone of event A, it would be possible for event B to affect event A. However, since event C is outside of the light cone, it can’t possibly affect event A.
Visualizing Far From the Earth
Since we can’t visualize a four-dimensional spacetime, we’re going to make some simplifying assumptions. We’re going to imagine that spacetime only depends on how far we are from the Earth, and we’re going to ignore things like lattitude and longitude. This brings us from a four-dimensional spacetime to a two-dimensional one, which we can visualize by putting it into a three-dimensional volume.
However, things are still tricky because we want distances one travels on our two-dimensional spacetime to match up with the distances one travels in the real four-dimensional spacetime. And this is going to distort the image slightly from what we would intuitively expect. Because our visualization preserves distances in this way, it’s called an isometric embedding.
Far from the Earth, we can get the shape of spacetime in our visualization by taking piece of paper with the graph of Minkowski space in figure 3, putting one hand each on the top and bottom of the paper, and lifting it so that the centre sags, as shown in figure 7. Because paper isn’t stretchy and the graph paper didn’t rip, we know distances were preserved.
But wait! I said that spacetime far from the Earth was flat! So in that case, shouldn’t it just look like figure 3 and not be bent like it is in figure 7 at all? It turns out that, in the sense that we care about, bothfigure 3andfigure 7 are flat. The kind of curvature we’re interested in is exactly equivalent to a distortion of how we measure distance. If the graph paper doesn’t rip, it’s flat. In this sense, any shape you can make from a sheet of paper is flat.
This type of curvature is called intrinsic curvature. A two-dimensional shape is intrinsically curved if one would need to stretch or distort or cut a piece of paper to make it. In other words, if distance changes on the surface of the shape. (There are higher-dimensional generalizations of this too.) There’s another type of curvature called extrinsic curvature, which describes how a surface looks when you put it in a volume. Figure 7 is extrinsically curved while figure 3 is not.
But why do we insist on figure 7 if both figures are flat? Well, flat spacetime certainly could look like figure 3, but if it did, we would run into trouble when we got closer to the Earth. Not all two-dimensional shapes fit in three dimensions and if we want the shape of spacetime near the Earth to fit, while at the same time preserving distances, then the bit of spacetime far from the Earth has to look like figure 7.
Our Local Spacetime
Now that we know what spacetime looks like far from the Earth, we’re ready to explore what it looks like near Earth. Our local spacetime is shown in figure 8.
The lines parallel to the red arrow are lines of constant time, and the lines parallel to the blue arrow are lines of constant distance from the Earth. Notice that the surface of the Earth, the big solid black line, is not a point but a line. This is the worldline of the surface of the Earth. Notice also that the lines scrunch together as you approach the surface of the Earth. This is because lengths and durations are actually shrinking near the Earth. We age slightly slower at sea level than we do on an airplane. (This is related to the gravitational redshift I discussed in an older post.)
If it looks like that scrunching together would eventually lead to the lines of constant distance lying on top of each other, you’re right! If I made the surface of the Earth a smaller and smaller radius, then the lines would eventually lie on top of each other. And that would be the event horizon of a black hole. The spacetime wouldn’t stop at the event horizon, of course. It would happily continue. But that’s a story for another time.
I should note that to make the curvature more visible, I’ve stretched out the axis along the red arrow. This means light travels at about 30 degrees off of horizontal, not 45 degrees.
Dropping the Ball Again
So what happens when I stand on a cliff and drop a ball from the top of the cliff? The ball wants to take the straightest possible path through spacetime. Since I don’t throw the ball, I just drop it, it starts in a path roughly like that of the blue arrow. This is a path of constant radius where the only motion is forward in time. It should be roughly visible in the picture that such a path is extremely bendy. The more the ball moves either towards or away from the Earth, the straighter the path.
Of course, because the ball can’t travel faster than light. So a path like that of the red arrow, which is almost a straight line, isn’t valid. The ball has to be within my light cone. Therefore, the worldline of the ball will be some path that travels both forward in time and towards the Earth. And because of the way space and time curve, this will appear as an “accelerating” path.
I plot the geodesic for the ball in figure 9. Note that it approaches a straight line. That’s because as it accelerates it’s approaching the speed of light (we are neglecting air resistance and exaggerating the distance from the surface of the Earth to make that happen). Note also that the speed of light is a straight line that’s wider than 45 degrees. That’s because of the stretched axis. The path of the ball is curved—it curves with the surface, after all. But it’s as straight as it possibly can be. And that’s what makes it a geodesic.
It’s worth noting that a path away from the Earth would also be a valid worldline. And indeed, it would be just as straight as the path towards the Earth. If, instead of dropping my ball, I threw it upwards at escape velocity, this is indeed the worldline it would choose.
If we’d somehow included lattitude and longitude in our visualization, we could have seen worldlines where the ball orbited the Earth too.
Cool, huh? I think that’s enough for now.
Spacetime Isn’t Curved Into Anything
Our visualization exercise today may have lead you to believe that spacetime must be curved inside some higher-dimensional space. After all, to show you the curvature of spacetime near the Earth, I took a two-dimensional spacetime and put it in a three-dimensional volume. But I did this out of convenience, to help us understand what goes on near a planet. In truth, all you need for spacetime to be curved is for distances and durations to distort. And they can distort all by themselves, without depending on a higher-dimensional space.
Play With it Yourself
If you’re interested in exploring our local spacetime, good news! I wrote a Python script that generates the surface I showed you in figure 8. You can find it in the following github repository:
Gravitational waves are “ripples in space time” that propagate through it like waves on water. That’s the common story and, for the most part, it’s right. But what does that mean? This is part four in my many-part series on general relativity. The first three parts introduce general relativity from the ground up. You can find them here:
First, I want to help you get an intuition for why gravitational waves should exist. So before we dive into the relativity, let’s step back for a moment and imagine boring old Newtonian gravity. Suppose we have a bowling ball (blue) and a marble (red), as shown in figure 2. We take the bowling ball and we move it periodically towards and away from the marble. As we do, we measure the strength of the gravitational pull the bowling ball exerts on the marble. It gets stronger as the bowling ball gets near and weaker as it moves further away. This is plotted in the bottom of figure 2.
Notice how wavy the gravitational strength looks? At this point, you might be tempted to call it a gravitational wave. But that temptation is leading you astray. See, an important property of waves is that they travel at finite speed. Information can’t travel instantly. But in Newtonian physics, the marble feels the change in the gravitational pull of the bowling ball instantly.
Since special relativity is basically true, and we feel gravitational forces, this should convince you that gravitational waves should exist. And it should also give you a sense on what a gravitational wave should be like. We should feel a temporary change in the “pull” from the gravity a distant object, which is an echo of its motion.
Gravitational Waves in General Relativity
But of course, gravitational waves don’t actually work the way I just described. Gravity is not a force, it’s a distortion in the way we measure distance. So how do gravitational waves work in this context? Well, in some sense, I already told you. Gravity is a distortion of how we measure distance. So a gravitational wave is a distortion in how we measure distance that travels.
Of course, there are some caveats, most of which I won’t get into. The most important caveat is how the distance distorts. Distances don’t just grow and shrink evenly in every direction. They grow in one direction and shrink in another. For example, if you took a circular ring of particles, I’m a fan of marbles, floating in outer space, and a gravitational wave passed by, you’d observe them distort into one ellipse and then another, as shown in figure 3. And this happens because the distances between the particles are changing. (For experts: I’m showing the + polarization. If you rotate by 45 degrees, you get the x polarization.)
Fortunately, we have one. An artist’s impression is shown in figure 4. Except our marbles are all neutron stars and our ring is millions of lightyears wide. Basically, each marble is a type of star called a millisecond pulsar, which is a neutron star that’s rotating very fast. For reasons I won’t get into, this makes it emit light (though usually not visible light) in a beam. And as it rotates, we see a pulse as the beam points towards us, like a lighthouse. To measure a distortion in spacetime due to a gravitational wave, we measure how long a pulse takes to reach us over many many pulses. If a pulse comes before or later than it should, that might be a gravitational wave! To see if it is, we need to check with all the pulsars in the “ring” to see if they distorted in the right way and do some fancy math.
This whole scheme is called pulsar timing, which is done with pulsar timing arrays. A pulsar timing array is a collaboration of people who use telescopes, like the one at Arecibo shown in figure 5, that keep track of millisecond pulsars and do statistics to see if they’ve detected a gravitational wave.
Detecting Gravitational Waves, Part 2
Pulsar timing is great and all… but is there a more… direct way we can find gravitational waves? Maybe something we can build on Earth? I’m glad you asked! We don’t really need a ring of particles, right? All we actually need are two very very precise rulers… set up so that we can measure distance growing in one direction and shrinking in another.
Fortunately, light makes an incredibly good ruler. So we can make our rulers out of laser light and compare them to detect a gravitational wave. That’s how the two LIGO detectors and detectors like them work. One of the detectors is shown in figure 6.
Each LIGO detector has two 4km long, vacuum-sealed, seismically isolated, supercooled laser arms that measure distance incredibly accurately. If you compare the distances measured in the two arms (which is actually all you can do because LIGO is a laser interferometer), the measurement in the difference is accurate to better than one part in . This means they can measure a change in distance one one-thousandth of the width of a proton.
The LIGO systems were recently upgraded and they’re coming online this year. So stay tuned in the following years for news of a gravitational wave detection!
I should mention that moving a mass in a straight line back and forth, as in figure 2, is not enough to excite a gravitational wave in general relativity. The motion of the mass needs to have a so-called quadrupole moment. Most motions in the real world, such as orbiting a star, do have a quadrupole moment. But I wanted to mention this so that you’re not under the impression that all motion produces gravitational waves. Just most motion.
I have a lot more to say about gravitational waves. But I think this is enough for now. In future posts, look forward to learning about the astrophysical systems that produce gravitational waves and listening to the sound of two black holes colliding.
I didn’t pull my description from a single source, this time. I used a bunch of textbooks, such as Spacetime and Geometry by Sean Carroll and Introduction to 3+1 Numerical Relativity by Miguel Alcubierre. But here’s some more accessible resources:
Figure 1 shows light from a distant blue galaxy that is distorted into a so-called Einstein ring by the curvature of spacetime around a red galaxy. This is called gravitational lensing and today we’ll learn how it works.
(If you haven’t read parts one and two, I recommend you do so now. You can find them here and here.)
When Distance Warps, Space Curves
First, let’s try to understand what a warping of distance means. We’re going to find that it’s the same as curvature. To understand the connection, let’s go closer to home and imagine a curved space we’re all familiar with: the surface of the Earth.
Imagine that you’re driving from your home town of City to the capital, Metropolis, and that there’s a mountain in the way, as shown on the left in figure 2. Travelling over the mountain takes more time than travelling around, both because the mountain is tall and because the vertical climb is more difficult.
A three-dimensional picture of what’s going on would show that the ground is curved upward into the shape of a mountain, forcing you to go around. However, it’s possible to encode the same information in two dimensions. If we draw the two paths on a map, as shown on the right in figure 2, the path over the mountain looks straight and the path around it looks curved. However, we define the straight path to be longer than the curved one, even though our Euclidean eyes tell us otherwise.
This tells us that a curved surface (in this case, the region around City and Metropolis, which bulges out with a mountain) is the same as a surface where distance is distorted. And we can go the other way. A distortion in the way we measure distance implies curvature.
In the context of general relativity, this is what we mean when we say spacetime is curved. Distance has warped such that the straightest possible path is not what you expect.
In Curved Spacetime, Straight Paths Look Curved
Let’s get some better intuition for how curved spaces work. The curved surface we’re most familiar with is the Earth, so let’s see if we can’t get some feel for curvature by exploring how we move around on Earth.
Say you want to go from Narita, Japan to San Diego, U.S.A. What’s the shortest route? Naively, you’d look at a map and draw a straight line between the two cities. However, if you look at Japan Airline’s route map, shown in figure 3, you’ll see something quite different.
What’s going on (other than the effects of prevailing winds)? It’ll help if we look at the Earth as a sphere instead of as a plane, as shown in figure 4. The straight line between the two cities goes through the Earth, so that’s a no-go. The naive path is just a straight line on a flat map, which in this case keeps our latitude more or less constant; this is doable, but not the best we can do. The best path is a path that goes a bit north.
What’s so special about this last path? Every path on the Earth must curve, because the sphere curves. However, there’s a portion of the curvature of the path that comes from the curvature of the Earth and there’s a portion that comes from the curvature of the path itself. The latter is called the geodesic curvature. A path that’s as straight as possible—i.e., whose only curvature is the curvature of the surface it’s on—is called a geodesic. This straightest possible path, which has no curvature of its own, will always also be the shortest possible path between two points.
The geodesics for planet Earth are the great circles. These are the circles with the same radius as the Earth; in other words, take the circle formed by the equator and rotate it to make it pass through any two points, as shown in figure 5. A great circle will always cut the Earth into two hemispheres of equal size. However, they will no longer be the Northern and Southern Hemispheres we’re used to.
The lesson that I want you to take away from this is that, in a curved space (or spacetime!), straight doesn’t mean what you think it means. In flat space, a geodesic is a straight line. But in curved space, a geodesic is not a straight line. But it’s the closest thing to a straight line you can get. Indeed, it’s the appropriate definition of straightness.
In curved spacetime, straight lines look curved.
Gravity, Curvature, and Lensing
I told you gravity isn’t a force, but looks like one. We’re now almost ready to understand that. Let’s walk through the argument. The presence of mass, which we typically think of as gravity, distorts distance and time nearby. This, as we just learned, curves spacetime. And in a curved spacetime, straight lines don’t look straight.
Now here’s the clincher.
In the absence of an external force, objects travel along the straightest possible paths, geodesics, through spacetime.
In the absence of gravity, those paths look like the straight lines we’re all used to. But in the presence of mass, they can look very curved.
That, my friends is the gravitational “force.” And let’s be clear. It’s not a force! Particles under the influence of gravity aren’t moving, at least not in the traditional sense. (They’re moving forward in time only.) It’s just that, to us, they appear to be moving because spacetime is curved. This is why, in Galileo’s famous experiment at the leaning tower of Pisa, the feather and the bowling ball fall at the same rate: they’re not falling at all.
We’re now ready to discuss the gravitational lensing shown in figure 1. The red galaxy distorts the spacetime around it, very much like the “mountain” in figure 2 so that the straightest possible path light coming from the distant blue galaxy behind it is curved. The result is that light gets spread out and “lensed” to form the Einstein ring you see in the image.
Now, before I conclude, there’s a common misconception that I want to nip in the bud. People think that, because the universe is curved, it has to be curved into something. In other words, in the same way that the surface of the Earth is a curved two-dimensional sphere embedded in three-dimensional space, the curved four-dimensional universe must be embedded in some higher-dimensional space.
This is wrong.
The universe doesn’t need to be embedded in a bigger space to be curved. All it needs is for the way we measure distance in our own four dimensions to be distorted. We can understand and encode all the information we need about the curvature of spacetime in how distances shrink and durations stretch out. In other words, if you look at figure 2, the left picture isn’t important. Only the right picture matters.
Okay, that’s all for now, folks. Starting next time, I’m going to discuss various different cool properties of general relativity… black holes, gravitational waves, that sort of thing. Exciting!
If you liked this post, you may be interested in some of my older posts on gravity, curvature, and all that.
I took my treatment of general relativity from Sean Carroll‘s excellent text, Spacetime and Geometry. However, there are some great, less technical resources online. Currently, my favorite is this five-part series by PBS Space Time on youtube:
If you’d like some information on the history and context of general relativity and the measurements we’ve made that tell us it’s true, check out these great articles by Ethan Siegal and Brian Koberlein:
This is part two in a many-part series on general relativity. Last time, I described how Galileo almost discovered general relativity. In particular, I told you that gravity isn’t a force. In fact, gravity is the same as acceleration. Now, this is a completely crazy idea. After all, we’re all sitting in the gravitational field of the Earth right now, but we don’t feel like we’re moving, let alone accelerating. But let’s take this crazy idea at face value and see where it leads us.
(Of course, the Earth is spinning, which is an acceleration. And it’s orbiting the sun, which is an acceleration. And the sun is moving in the galaxy. But let’s ignore all that. It’s not important for the argument I want to make.)
But first, we need to make a brief detour and discuss the Doppler effect.
The Doppler effect is a bit complicated (especially for light), so I won’t go into too much depth. Instead, I’ll describe it by analogy. (I’ve given the same analogy before, in my article on the expanding universe. So if you remember, you can skip all this.)
Imagine that Paul Dirac and Leopold Kronecker are playing catch, as in figure 2. Each second, Kronecker throws a ball to Dirac, who catches it. Thus, the frequency of balls that Dirac catches is 1 Hertz (Hz)—one per second, or one inverse second.
But now imagine that Dirac starts backing away from Kronecker, as shown in figure 3. Kronecker continues to throw at a rate of one ball per second. However, since Dirac is moving away from the balls, each one takes longer to get to him. Thus, he catches the balls at a rate slower than one per second…say, one every 1.5 seconds.
A similar thing happens with both light and sound. (In the case of sound, we call it the acoustic Doppler effect.) Light is a wave. It has peaks and troughs which wiggle up and down in time, as shown in figure 4. The number of peaks (or troughs) per meter is called the wave number. The speed at which it wiggles up and down in time is called the frequency. The two are related by the speed of the light wave, which is always constant, so they’re basically interchange-able.
The frequency of a light wave is analogous to the frequency at which Kronecker throws balls at Dirac. Instead of counting the number of times Dirac throws the ball, we count the peaks of the wave. The frequency of a light wave also determines its color; high frequencies are blue, while low frequencies are red.
This means that if Kronecker fires a green laser at Dirac, and Dirac moves away from him, the laser light will appear more reddish to Dirac than it does to Kronecker. This is called a redshift. If Dirac were moving away from from Kronecker at an increasing rate, in other words if Dirac were accelerating, the redshift would be even more pronounced.
So what does all this have to do with gravity? Well remember, gravity is acceleration. So we should be able to see a Doppler-like effect just by moving from a region with strong gravity into a region with weak gravity, or vice-versa. To see what I mean, imagine that Kronecker and Dirac are up to their old tricks. But this time, imagine that Kronecker is on Earth, and Dirac is in space, as shown in figure 5.
Kronecker fires a green laser up at Dirac. Now, remember: gravity is acceleration. Both Kronecker and Dirac are in a gravitational field, so they’re both accelerating. But Kronecker is in a stronger field, so he’s accelerating more. This means that, from Dirac’s perspective, Kronecker is accelerating away from him. Therefore, by the time the light reaches Dirac, he sees it redshifted because of the Doppler effect.
In the context of general relativity, we call this gravitational redshift, and it’s a real effect. We need to take it into account when we read signals sent to us from gps satellites, for example.
Redshift, Distance, Time
The peaks and troughs of light make it an extraordinarily good ruler. If you know the wave number of a wave of light, you can count the number of peaks and in the wave between two places and calculate how far away those two places are from each other. In a very real sense, distance is defined by this procedure.
How, then, do we interpret the redshifted light that Dirac sees? If light on Earth is redshifted when it goes into space, that light stretches out. The distance between adjacent peaks in the light wave grows. Does this mean that distance itself grows?
Yes. It means exactly that.
In a strong gravitational field, distances are shorter than in a weak gravitational field. Indeed, because the wave number of a wave and the frequency of a wave are interchange-able, this also means that times are longer in duration in a strong gravitational field than in a weak gravitational field.
We started with the crazy (but true!) idea that gravity is the same as acceleration. But this has lead us to an even crazier (but still true!) idea: gravity shrinks distance and stretches duration.
This is what people mean when they say that gravity is a warping of space and time (or suggestively, spacetime). The very way that we measure distance is distorted by a gravitational field.
And general relativity is the dynamics of distance.
Next time we’ll talk about how a warped spacetime creates the illusion of a gravitational force.
I took the gravitational redshift argument directly out of the excellent textbookSpacetime and Geometry by Sean Carrol. If you have a good background in math and you want to learn general relativity, I highly recommend it. Here are some other resources:
We all know the (probably apocryphal) story. Galileo Galilei, all around physics bad-ass, went up to the top of Leaning Tower of Pisa and dropped stuff off the top. He found that objects of vastly different weights, like bowling balls and feathers for example, would fall at exactly the same rate and hit the ground at exactly the same time. Air resistance gets in the way, of course. But if you perform the experiment in vacuum, as these guys did, then you do find the bowling ball and the feather land at exactly the same time:
This leads to a fundamental truth we’ve all memorized in school: The acceleration due to gravity is constant. But there’s a more fundamental truth underneath that one, a truth that sat unrecognised until the time of Einstein: Gravity is not a force. To get the full story, you’ll need to wait until next time, when I start to describe general relativity. But for now, let’s explore how Galileo’s experiment shows that gravity is incredibly special.
Electric Bowling Ball, Electric Feather
To understand why gravity is weird, we have to understand how the other forces work. So let’s set up an experiment analogous to Galileo’s, but with electricity, and see what happens. So here’s the experiment (shown in figure 2).
We take two metal plates out into space, far enough away that there’s no gravity. Then we connect the plates to a battery so that one plate gets a positive charge (red) and one gets a negative charge (blue). This creates a constant electric field, much like the constant gravitational field near the Earth. Finally, we place two particles of equal mass at rest at the same position between the plates. We give one particle a very large positive charge (right), and one particle a smaller positive charge (left). Opposite charges attract and like charges repel, so both particles will move towards the blue plate.
The particle on the right will absolutely reach the plate before the particle on the left.
Okay, that’s strange. In this experiment, electric charge played the role of “mass” in the sense that it controls how strong the electric force that acts on the particles is. In electromagnetism, unlike in gravity, the stronger force resulted in a bigger acceleration. But is that the whole story? To find out, let’s try the same experiment, but with a twist.
Let’s take the same battery-powered metal plates into space. But this time, we put two particles of equal electric charge between the plates, as shown in figure 3. But we give one particle (on the left) much more mass by attaching some extra weights to it. Now the electric forces on the two particles are the same. Should they fall at the same rate?
Not so fast. Now the less massive particle (on the right) falls faster.
There Are Two Types of Mass
What we’ve discovered with these two experiments is that there are two types of mass. There’s the gravitational mass, which controls how strong the force of gravity is for a particular object. This is directly analogous to electric charge. The bigger an electric charge, the bigger the electric force. The bigger a gravitational mass, the stronger gravity is. This is what we saw in our first experiment.
But there’s also the inertial mass. This mass controls how difficult it is to change an object’s motion. This is the mass in Newton’s second law of motion, . The bigger the mass is, the more force is required to move it. We all know this intuitively: the heavier an object, the harder it is to push. This is what we observed in our second experiment. More mass means less motion.
But when Galileo performed his experiment at Pisa, he discovered something incredible: for gravity and only gravity, these two types of mass are the same. This is unlike every other force. The electromagnetic force, the strong force, and the weak force, all have a charge, which controls how strong the force is, that is separate from their inertial mass. But the gravitational “charge” is the same as the inertial mass.
And if you allow inertial and gravitational mass to be the same, something changes. Instead of “the gravitational field due to the Earth is the same everywhere,” we get “the acceleration due to the gravity of the Earth is always the same everywhere.”
Gravity becomes the same as acceleration.
That’s very weird. And very special. And it is this fact that lead Einstein to develop general relativity. But through his experiment, Galileo almost found it first.
To learn how this leads to general relativity. Tune in next week.
The thought experiments I described here are modifications of Einstein’s famous elevator thought experiments. You can find many descriptions of those thought experiments online. Here’s a few:
However, there’s another type of supernova, one in which a star whose nuclear fires long ago petered out is reignited, causing a catastrophic explosion. This is the type Ia supernova. We start our story with the type of star that explodes: the white dwarf.
A star is a balancing act. On the one hand, these massive objects exert an enormous gravitational pull on themselves, driving all the gas to collapse towards the centre of the star. On the other hand, the nuclear fusion reaction at the core of the star heats it up, and hot gas likes to expand, holding the star apart. Paradoxically, the driver of this nuclear reaction is the gravitational pull of the star itself. The weight of the star pushes the stuff in the core together so much that the atoms fuse together, releasing huge amounts of energy.
(Surprisingly, stars need quantum mechanics to burn. When atoms fuse together in a star, the fusion only occurs because the atoms quantum tunnel together. Astrophysicist Brian Koberlein has a nice article on this.)
The eventual fate of a main sequence star like our sun depends on its mass. If the star is more than about 1.4 times the mass of our sun (this is called the Chandrasekhar limit) then, once the nuclear reaction stops, the star collapses under its own weight, triggering a core-collapse supernova explosion. However, if the star is less massive, something amazing happens: the star collapses down to a tiny fraction of its original size–a white dwarf star might have a radius only 4 times or so larger than that of the Earth–but it doesn’t explode. Now the star isn’t held up by heat or nuclear fusion. It’s held up by a quantum-mechanical effect called Pauli exclusion principle.
Basically, a white dwarf is a hot, ultradense fluid made of electrons and atomic nuclei, packed together so tightly that the only thing holding them apart is their inability to occupy the same physical space. This means white dwarfs are incredibly dense. A tablespoon white dwarf starstuff would weigh about 100 tonnes. Figure 2 shows a white dwarf star next to a larger type A main sequence star on the left and our sun on the right. Keep in mind: that tiny little white dwarf star has the same amount of mass as our sun.
(Neutron stars are very much like white dwarfs, and they are held apart by similar principles. However neutron stars are, unsurprisingly, made mostly of neutrons, and can be about ten times denser and smaller than white dwarfs.)
But sometimes, a white dwarf can reignite. And the results are explosive.
The nuclear fires of a white dwarf have died down. But these fires were first produced by intense pressure. So if the pressure in the core of the white dwarf is ever high enough, then the carbon atoms in the core of the star will start fusing and, temporarily, the nuclear furnace will reignite. Figure 3 shows a computer simulation of the beginning of this process. The core of the star becomes hot due to nuclear fusion and this spreads across the star.
The end results of stellar nuclear fusion are carbon and oxygen. So a white dwarf is made up of carbon and oxygen nuclei… and as we know, oxygen reactions are what make fire. So once the nuclear fires reignite, the star doesn’t just become hotter or expand. The entire star literally burns. That’s what figure 3 is showing. The bright orange stuff in the images is actually ash.
Although the fusion reaction ignites the star, it doesn’t produce enough energy to make the star explode completely. Instead, all of the fire that spread across the star eventually concentrates on one side of the star in a concentrated burst, which can accelerate the star up to thousands of kilometres per second like a rocket. Stars moving this fast are, awesomely, called hypervelocity stars. Figure 4 shows the next part of the simulation in figure 3, where now one side of the star explodes in a pulse.
After the burning in figure 3 and the explosion in figure 4, things calm down. The nuclear fusion in the star stops, and it returns to normal… albeit with a very different velocity.
Before the Explosion
So now I’ve described how the star explodes… but I still haven’t told you why it explodes. I said that if the pressure in the core of the star becomes high enough, it can re-ignite. But how does that happen? Quite simply, the star has to put on weight. Usually, this means that the white dwarf in question has a companion star–another star nearby such that the two stars orbit each other. And over time, the white dwarf steals material from the companion until it gains enough mass that the weight of the star on the core causes it to reignite.
It’s not known what type of star the companion must be. One possibility is that it must be a massive star near the end of its life. Figure 5 shows the stellar evolution process that might result in a white dwarf stealing from a massive companion. Another possibility is that two white dwarfs might collide. Distinguishing between these models, or perhaps some combination of the two, and identifying which stars will become supernovae is a long-standing problem in astrophysics.
Different Models and The Ignition Problem
It is worth noting that the precise mechanism by which the nuclear fusion restarts in the star is not completely known. There are also a number of models that describe the details of the supernova explosion. The simulation I showed is one such model, but there are others. However, all models are qualitatively the same and they all produce predictions that match the supernovae we observe in the sky.
As we’ve learned, core-collapse supernovae are not the only kind of supernovae. Indeed, the study of white dwarfs and type 1a supernovae is an active field of research with a rich history. Here I include some resources for your reading enjoyment.
That is, of course…if there actually is an event horizon, not just something that looks like one. Carlo Rovelli , one of the founders of loop quantum gravity, recently proposed something crazy: Not only do black holes not really have event horizons, they eventually explode.
The conclusion is crazy, but the reasoning is surprisingly elegant. Let me walk you through it.
(DISCLAIMER: I want to emphasize that, although the science in this post is peer-reviewed, it’s extremely speculative. The quantum gravity predictions I describe in this post are not guaranteed or even likely to be true.)
The typical story of black hole formation (at least for stellar-mass black holes) goes something like this: A massive star runs out of nuclear fuel, and the fusion reaction keeping the star alive peters out. Without the energy from the fusion, the star can no longer resist its own gravitational pull and collapses in on itself. The resulting compression of its gases triggers a catastrophic explosion, ejecting a fair amount of the gas to leave behind the stellar core, which becomes increasingly dense. If the star is massive enough, the collapsing core squeezes into such a dense ball that it forms an event horizon and becomes a black hole. (If the star isn’t quite massive enough, the core remnant is pushed outward by the Pauli exclusion principle and becomes a neutron star.) This is called a core-collapse supernova. Here’s a video of a simulation of a supernova that results in a neutron star:
(I am, of course, glossing over a huge number of details. Core-collapse supernovae are not fully understood and there is a rich body of work devoted to understanding them…which many of my friends and collaborators are contributing to. See the bottom of the article for a small, hopefully accessible sampling of current research in core-collapse supernovae.)
Once the event horizon forms around the collapsing matter, no information can emerge from the black hole, so we don’t know what’s going on inside. General relativity predicts that the matter will keep collapsing until it forms an infinitely dense singularity. But the modern view among physicists is that this isn’t what actually happens. Rather, the singularity is a sign that the theory of general relativity is incomplete. What happens inside the black hole can only be described by quantum gravity. We don’t have a theory of quantum gravity, but we are actively searching for one and making (slow) progress.
The Quantum Bounce
Rovelli and his collaborators speculate that these quantum gravity effects not only prevent the singularity from forming, but may in fact cause the black hole to explode.
If black holes do experience a quantum bounce and form neither singularities nor event horizons, and if the bounce happens at the end of collapse, where are all the explosions? Surely we would have seen them!
If Rovelli and collaborators are right, the first black holes that formed in the universe, which formed many billions of years ago, should be exploding about now. And when they explode, they should release a huge amount of energy. Some of this energy will be emitted as light, which we can detect.
The earlier the exploding black hole formed in the history of the universe, the less massive it will be. And this corresponds to a shorter wavelength of the emitted light. But, because the speed of light is constant, looking further away from Earth means looking back in time. So the wavelength of light emitted by exploding black holes should change depending on how far away the black hole is. After correcting for cosmological redshift, this results in a very peculiar and distinct wavelength of light as a function of distance, shown in figure 4.
So all we have to do is look for some light coming from outside the galaxy and see if we can compare the wavelength of the light to its distance from us. If it matches the curve in figure 4, then Rovelli and collaborators are right. Otherwise, they’re not.
Rovelli and collaborators suggest using fast radio bursts, which have approximately the right wavelength and may be of extragalactic origin, to test the model. So far, we don’t know very much about fast radio bursts. If they turn out to come from exploding black holes, this would be very exciting, because it would be a real probe of quantum gravity.
These proposals are all motivated by the so-called black hole information paradox. Basically, we believe that information in the universe is conserved. It cannot be created or destroyed. When information falls into a black hole, it is irretrievable. This wouldn’t be so bad, except that the black hole eventually disappears because it gives up its energy to Hawking radiation, which doesn’t transmit all the information in the black hole. Therefore, once the black hole evaporates, all the information that fell into it is lost forever…simply gone from the universe. But that seems to break the law of conservation of information.
Rovelli’s proposal gets around the paradox by proposing that black holes explode and eject all information they contain. And this is certainly one motivation for him considering it.
I want to emphasize that Rovelli’s proposal is ridiculously speculative. He is relying on arguments from quantum gravity, which we don’t even remotely understand. And even the arguments that don’t use quantum gravity are rather contrived.
Rovelli writes down a quantitative model of the collapsing and bouncing star, but it’s very simplistic…in fact, I’d call it the general relativity version of a “spherical cow.” The spacetime has a region in which quantum gravity is non-negligible, which means a region in which physics we don’t understand take place. And the collapsing star is modelled as a thin spherical shell of matter, which is way too simple. (Furthermore, spherical shells of matter are known to have pathologies.) Worse yet, the expansion of the matter post-bounce is modelled as a white hole, which is known to be intrinsically unstable.
Yet, despite all that, Rovelli’s proposal is a cool idea. And I like it.
You can find Rovelli and collaborators’ first paper on the bouncing black holes here. The paper where they predict that fast radio bursts come from exploding black holes is here.
For a review of the physics of core-collapse supernovae, first published in Nature, check out this article.
The physics of core-collapse supernovae are very complicated, and accurately modelling this phenomenon is an open problem in the numerical relativity community. Professor Christian Ott wrote an awesome article about some of the challenges the community faces (revealed by his and his collaborators’ research), which you can find here.
This is a nice article by PBS on Hawking’s recent claim that black holes don’t exist and how it relates to the black hole information paradox.