Do we really travel through time with the speed of light?

[Note: This transcript will not make much sense without the equations that I show in the video.]

Today I want to answer a question that was sent to me by Ed Catmull who writes:

“Twice, I have read books on relativity by PhDs who said that we travel through time at the speed of light, but I can’t find those books, and I haven’t seen it written anywhere else. Can you let me know if this is right or if this is utter nonsense.”

I really like this question because it’s one of those things that blow your mind when you learn about them first, but by the time you have your PhD you’ve all but forgotten about them. So, the brief answer is: It’s right, we do travel through time at the speed of light. But, as always, there is some fine-print to what exactly this means.

At first, it does not seem to make much sense to even talk about a speed in time. A speed is distance per time. So, if you travel in time, a speed would be time per time, and you would end up with the arguably correct but rather lame insight that we travel through time at one second per second.

This, however, is not where the statement that we travel through time at the speed of light comes from. It comes from good, old Albert Einstein. Yes, that guy again. Einstein based his theory of special relativity on an idea from Hermann Minkowski, which is that space and time belong together to a common entity called space-time. In space-time, you do not only have the usual three directions of space, you have a fourth direction, which is time. In the following, I want to show you a few equations, and for that I will, as usual, call the three directions of space, x, y, and z, and t stands for time.

Now, here’s the problem. You can add directions like North and West to get something like North-West. But you cannot add space and time because that’s like adding apples and oranges. Space and time have different units, so if you want to add them, you have to put a constant in front of one of them. It does not matter where you put that constant, but by convention we put it in front of the time-coordinate. The constant you have to put here so that you can add these directions must have units of space over time, so that’s a speed. Let’s call it “c”.

You all know that c is the speed of light, but, and this is really important, you do not need to know this if you formulate special relativity. You can put a dummy parameter there that could be any speed, and you will later find that it is the speed of massless particles. And since we experimentally know that the particles of light are to very good precision massless, that constant is then also the speed of light.

Now, of course there is a difference between time and space, so that can’t be all there is to space-time. You can move around in space either which way, but you cannot move around in time as you please. So what makes time different from space in Einstein’s space-time? What makes time different from space is the way you add them.

If you want to calculate a distance in space, you use Euclid’s formula. A distance, in three dimension, is the square-root of the of the sum of the squared distances in each direction of space. Here the Δx is a difference between two points in direction x, and Δy and Δz are likewise differences between two points in directions y and z.

But in space-time this works differently. A distance between two points in in space-time is usually called Δs, so that’s what we will call it too. A distance in space-time is now the square-root of minus the squares of the distances in each of the dimensions of space, plus c square times the squared distance in time.

Maybe let me mention that some old books on Special Relativity use a different notation, in which, instead of just putting a minus in the space-time distance, one uses a prefactor for the time-coordinate that is i times c. This has the exact same effect because the i square will give you a minus. The I turns out to be useless otherwise though, so this notation is not used today any more.

But why would you define a space-time distance like this, why not just all plusses? Well, for one, if you do it differently it doesn’t work. It would not correctly describe observation. That’s an answer, but not a very insightful one, so here is a better answer.

Einstein based special relativity on the idea that the speed of light is the same for all observers. You cannot do this in a Euclidean space where all the signs are plusses. But you can do it if one of the signs is different relative to the others. 

That’s because a space-time distance that is zero for one observer is zero for all observers. This is also the case in Euclidean space, but in Euclidean space, this just means zero in each of the directions of space. But what does a zero distance mean in space-time? Well, let’s find out. For simplicity, let us look at only one dimension of space. So if the distance in space-time is zero, this means that the distance in space divided by the distance in time equals plus or minus c. And that’s the same for all observers. So this speed, c, is an invariant speed.

But, well, we are not light, so we do not travel with the speed of light through space, and we do actually cover a distance in space-time. So let us look at this equation for the space-time distance again. Now let us divide this by the time difference. Now what you have on the left side is the space-time distance per time. And under the square root you have roughly something like the squares of the velocities in each of the directions of space. Plus c².

And there you have it. Relative to yourself, you do not move through space, so these velocities are zero. You then only move into the time-like direction, and in this direction, you move with the speed of light. So, we indeed all travel through time with the speed of light.

I always try to show you equations because physics is all about equations. But to really understand what these equations mean, you have to use them yourself. A great place to do this is Brilliant, who have been sponsoring this video. Brilliant offers a large variety of interactive courses on topics in science and mathematics. They do for example have a course on Special Relativity, that will teach you all you need to know about space-time diagrams, Lorentz-transformations, and 4-vectors.

To support this channel and learn more about Brilliant, go to brilliant.org/Sabine, and sign up for free. The first two-hundred people who go to that link will get twenty percent off the annual Premium subscription.

Your sudden enthusiasm for virtual meetings is beginning to worry me

Screenshot from Zoom meting. Image Source: Reshape.

I live about 100 kilometers away from my workplace. A round trip takes at least 2 hours, up to 4 by public transport. That’s why, for the past 5 years, I’ve had a home-office contract which allows me to do part of my job remotely.

My husband works for a company that has sections in several other countries, including India, the USA, and Great Britain. He, too, is used to teleconferences with participants from several time-zones.

This makes me think my family was probably better prepared for the covid lockdown than many others. For the same reason though, we also had more time to contemplate the pros and cons of remote collaboration.

The pros are clear: Less time wasted in transit. Less carbon dioxide emitted. Less germs circulated.

And with more people in the same situation, the pros have proliferated. I have, for example, been thrilled to see the spike in online seminars. Suddenly, even I am able to find seminars that are actually interesting for my research! Better still, if it turns out they’re not as interesting as anticipated, no one notices if I virtually sneak out. Also, asking for a virtual meeting has become routine. Everyone is now familiar with screen sharing and prepared to tolerate the hassle of lagging vids or chopped audios.

These have been positive developments, and many of them deserve to be carried forward. Traveling for seminars or colloquia has long been absurdly out-of-date. We all know that a lot of speakers will give the same seminar dozens of times to largely disinterested audiences, when those who actually wanted to hear it could as well have called into the same online meeting, or watched a recording.

Or consider this. I have frequently gotten invitations from overseas institutions that were prepared to fly me in and out for giving a one-hour talk. This isn’t only ecologically insane, it’s also a bad use of researchers’ time. A lot of my colleagues work while on planes and in airports, and of course I do, too, but let’s be honest: It’s not quality time. Traveling is disruptive, both mentally and metabolically. And that’s leaving aside that it screws up the work-life balance.

So, yes, scientists could certainly slim down those seminar series and cut back traveling quite a bit. But as researchers are becoming more familiar with virtual meetings and teleconferences, I fear some of them are getting carried away.

I’ve seen scientists on social media seriously discussing that seminar series should remain online-only even post-pandemic. Virtual conferences are supposedly better than the real thing. And if you listen to them, there’s nothing, it seems, you can’t get done on Zoom.

Let us therefore talk about the cons.

Virtual collaborations work well as long as you know the people in real life already. Even with both audio and video, a lot of information that humans draw on to efficiently communicate is missing. Through a screen, you neither get body language nor the context from chatter in the hallway or just from physically being in the same room. These cues are important for deliberation and argumentation to work properly.

I know this sounds somewhat Neanderthal, but fact is that evolution didn’t prepare us to communicate  through webcams.

This has long been known to sociologists who therefore recommend that teams which collaborate remotely meet in person at least a few times a year, a recommendation that my husband’s employer strictly follows. The occasional in-person meeting, so the idea, provides team members with the required information to understand where the others are coming from. It is especially important to introduce new members to a group.

A good starting point to get a sense of the troubles that remote collaboration can bring is the 2005 report by the (US-American) National Defense Research Institute on “Challenges in Virtual Collaboration”. Summarizing the published literature, they find that during video- and audio-conferences “local coalitions can form in which participants tend to agree more with those in the same room than with those on the other end of the line” and that computer-mediated communication has “shown to increase polarization, deindividuation, and disinhibition. That is, individuals may become more extreme in their thinking, less sensitive to interpersonal aspects of their messages, and more honest and candid.”

Online-only scientific collaboration and conferences would therefore most likely work well for some time, but eventually communication would suffer. Especially those who currently praise the zoomiverse for its supposed inclusivity, as this recent piece in SciAm, simply have not thought it through.

You see, regardless of how much effort we put into online conferencing and meeting, there will still be people who know each other in real life. These will be those who just happen to work or live near each other, or who have the funds to travel. Unless you actually want to forbid everyone to meet in real life, this will create a two-class community. Those who can meet. And those who can’t.

At present, most funding agencies acknowledge the need to occasionally see each other in person to collaborate effectively. If that would no longer be the case, then it would be especially the already disadvantaged people who would suffer because they would become remote-only participants. The Ivy League, I am sure, would find a way to continue having drinks together one way or the other.

None of this is to say that I am against virtual conferences or remote collaboration. But international collaboration has been a boon to science. And abstract ideas, like the ones we deal with in the foundations of physics, are hard to get across; having to pipe them through glass fiber cables doesn’t help. As we discuss how to reduce traveling, let us not forget that communication is absolutely essential to science. 

Flat Earth “Science”: Wrong, but not Stupid

I, as many people in science communication, am fascinated with flat earthers. Here you have a group of people steadfastly rejecting evidence that’s right in their face. Today, I want to tell you why I nevertheless think flat earthers are neither stupid nor anti-scientific. Most of them, anyway. More importantly, I also want to explain why you should not be embarrassed if you can’t remember how we know that the earth is round.

But first I have to tell you what flat earthers actually believe and how they got there. The most popular flat earth model is that of a disk where the North pole is in the middle and the south pole is an ice wall on the edge of the disk. But not all flat earthers sign up to this. An alternative is the so-called bipolar model where both poles are on the disk, surrounded by water that’s held by a rim of something, maybe ice or rocks. And a minority of flat earthers believe that earth is really an infinite plane.

They mostly agree though that gravity does not exist, and that the observations we normally attribute to gravity come instead from the upward acceleration of the flat earth. As a consequence, the apparent gravitational acceleration is the same everywhere on earth. I explained last week that this is in conflict with evidence – we know that the gravitational acceleration is most definitely not the same everywhere on earth.

The idea that gravity is due to upward acceleration also causes other problems. For example, you have to assume that the moon and the sun accelerate along with the flat earth so we don’t just run into them. That’s an ad-hoc assumption which disfavors the flat earth hypothesis against models where the orbits of the moon and the sun can be calculated from the gravitational law.

But that’s not the only problem. You also have to get the moon and the sun to somehow circle around over the disk to explain day and night and the phases of the moon. To get the day-night cycle to be noticeable, you have to shrink the sun and move it closer to the earth. 

You also have to somehow get the radiation of the sun to be directional. That’s many more ad hoc assumption. But even with those assumptions, the size of the sun will change during the day more than we observe. And no one has ever successfully predicted solar eclipses on a flat earth, or calculated the observed motions of the planets.

The bottom line is: it’s not easy to improve on today’s scientific standard. It was for good reasons that the hypothesis of a flat earth was abandoned more than two thousand years ago.

Some people suggested to me that flat earthers do not actually believe the earth is flat, they are just mocking people who take scientific evidence on trust. And that, let us to be honest, is something we all do to some extent every now and then. And it is probably the case that some flat earthers are indeed just pretending. But I find it exceedingly implausible they are all just faking it. 

To begin with, they would all have to be excellent actors. Just look at some of the videos on YouTube. Also, they’re putting quite some time and, in some cases, money behind their conviction. And that’s while most of them full well know coming out as flat earther will make others doubt their sanity. All that makes it unlikely they are just in for the fun.

Now, you may want to discard flat earthers as conspiracy theorists, which some fraction of them arguably are. But I think that would be somewhat unfair to most of them. To understand why, it helps to have a look at the history of the flat earth society.

The flat earth society goes back to an Englishman by name Samuel Rowbotham, who lived in the 19th century. He was a medical doctor who believed he had proved that the earth is flat and then complained for the rest of his life that the supposed scientific authorities ignored him. He referred to his methodology as “Zeteticism” after the Greek word zeteo, “to seek”. 

By “Zeteticism” he meant an extreme version of the philosophy of empiricism. Rowbotham’s philosophy, which is still the philosophy of flat earthers today, is that if you want to understand nature, you should only rely on information from your own senses. You can for example read on the website of the flat earth society:

“The world looks flat, the bottoms of clouds are flat, the movement of the Sun; these are all examples of your senses telling you that we do not live on a spherical heliocentric world. This is using what’s called an empirical approach, or an approach that relies on information from your senses. “

That flat earthers insist on evidence from your own senses only really is key to understanding their problem; I will come back to this. But first, let me tell you the rest of their history.

After Rowbotham’s death in 1884, the flat earth idea was carried forward by another British guy, Samuel Shenton, who once explained to a journalist:

“No man knows the ultimate shape of the earth, but that portion we life on is definitely flat. No one will ever know what the whole complexity is like, I suppose, because it goes beyond his sphere of observation, investigation and comprehension.”

Again, note the emphasis on personally collected evidence. In 1954, Shenton created the International Flat Earth Society. Few people cared. He died in 1971.

After his death, the Flat Earth Society was taken over by the US-American Charles Johnson. But even after the advent of the internet, flat earthers did not attract much attention. Johnson died in 2001, at which point the flat Earth society had 3500 or so members. The job then fell to another American, Daniel Shenton, who is not related to the earlier Shenton but whose logic falls right in line. He said in an interview with the Guardian in 2010:

“I don't think there is solid proof. I'm not intentionally being stubborn about it, but I feel our senses tell us these things, and it would take an extraordinarily level of evidence to counteract those. How many people have actually investigated it? Have you?”

Shenton had the idea to set up a wiki page for the flat earth community. Still no one cared. But in 2016, everything changed.

What happened in 2016 is that a few devout flat earthers put up videos, here, on YouTube. And that really got things going, by way of recruiting new believers. These videos have meanwhile been watched by millions of people. And that had consequences: In a 2018 poll in the United States, two percent of the respondents said they believe the earth is flat, while another 5 percent are not quite sure. 

Reliable numbers are hard to come by, but we are meanwhile probably talking about more than ten-thousand people in the developed world who reject science that was settled by the middle ages. Let that sink in for a moment.

How does someone end up rejecting something as scientifically well-established as the fact that the earth is round? 

There is not only one reason, of course. Some flat earthers find the idea is appealing for religious reasons, others are of the crowd who think NASA is evil, space a fake, and the moon landing didn’t happen. But mostly it’s because they think they are merely being rational skeptics. They have not themselves been able to prove the earth is round, so they believe they are only reasonable when they request evidence. CNN for example reports from a flat earth conference:

“Like most of the speakers at the event CNN spoke to, he was convinced after he decided he couldn’t prove the Earth’s roundness.”

I want to leave aside here that, of course, you cannot strictly speaking prove any empirical fact; you can only prove mathematical identities, so more precisely we should speak of seeking evidence that disfavors the hypothesis that the earth is flat. Of which there is plenty, starting with the historical evidence about how stellar constellations shift if you travel, how the length of shadows changes, to Newton’s 1/R² force law that is the law for a sphere, not a disk, not to mention Einstein and gravitational redshift and the perihelion precession of mercury, and so on, and so forth.

The problem that flat earthers have is that they cannot do most of these observations themselves. So if you buy the idea that it’s only your personally collected evidence that you should accept, then it seems you cannot refute the idea that the earth is round, and so flat earthers philosophy forbids them to accept scientific fact. 

Needless to say, if you want to hold on to your convictions it helps if you refuse to do observations that could speak against them. There are actually many ways to convince yourself that the earth is round which are not that technically difficult. Buy a telescope and try to explain the motions of the moons of Jupiter, for example.

So what’s wrong with flat earther’s attitude? Isn’t it asking for evidence exactly what rational thinkers should do? Sure, evidence is key to scientific progress, but flat earthers’ philosophical approach by which they reject certain types of evidence is inconsistent and, ultimately, logically wrong. 

See, the only evidence anyone ever has of anything is evidence you collect with your own senses. Except, as Descartes pointed out, evidence of your own ability to think, but this is not relevant here. Relevant is that the distinction which flat earthers are trying to draw between different types of evidence does not exist.

All evidence you have is sensory input. If you hear an explanation of someone else’s experiment, if you read a paper laying out someone else’s argument, that’s your own sensory input. A distinction which does exist, however, is that some of our sensory input requires very little decoding, while some requires a lot. Flat earthers’ problem is that they refuse to decode difficult sensory input.

A good example for the need to decode sensory input by conscious thought are optical illusions. Your brain tries to interpret visual input in ways that sometimes gives a misleading result as in this example. You almost certainly think square A is darker than square B. It is not. 

To understand your sensory input correctly you need to draw on other information, in this case your knowledge about optical illusions. Your brain interprets this image as if it was a natural, 3-dimensional scene, and therefore calculates back to the original color of the squares taking into account what appears to be a shadow. This is the wrong interpretation if you want to know the actual color of pixels on the screen. The lesson is, if you do not think about your sensory input, if you do not properly decode it, you arrive at a wrong conclusion.

Flat earthers similarly arrive at the wrong conclusion by failing to decode evidence, indeed by simply ignoring a lot of evidence that their own senses deliver. This is evidence about how society and science works.

Whether we are scientists or not, we all constantly use this evidence to navigate life. And I am sure flat earthers are no exception. Just consider going to the supermarket and buying canned soup. Do you have evidence that what’s in the can is edible? Probably not. For one, the can’s closed. And if you are anything like me, you probably have no idea how or where or by whom it was produced. Why then are you not afraid of eating canned soup? Isn’t this entirely irrational?

No, because you do have evidence that canned soup is edible. You know how the legal system in your country works, you know that there are regulations on what can be sold as food in a supermarket, you know that if what’s in that can was harmful to you, then a lot of people along the food chain would be punished for their mistake, and they don’t want that. Your trust in canned food is an entirely reasonable inference from evidence, evidence that you collected with your own sense, because what else could you possibly have collected it with?

Now let’s come back to flat earthers. Most of you don’t have a physics degree and chances are that after learning in school how we know the earth is round you didn’t think much about it ever again. By and large you are probably confident it’s correct because what you learned in school was plausible, and you know it is widely taught to children, and you know that your government strives to give children in your country a scientifically accurate education. So you have good reason to think the knowledge you were taught is backed by solid scientific evidence.

There is no appealing to authority here. You have totally yourself collected all this evidence about how society works. You have also yourself collected lots of evidence that science works. Any airplane, any laptop, any pair of glasses is evidence that science works. It’s evidence that the system works. It’s evidence for how the whole world works.

So, if you cannot recall just what experiments demonstrate that the earth is not flat, or if you cannot immediately figure out what’s wrong with flat earther’s arguments, there’s no shame in rejecting their claims, because your rejection is based on evidence, evidence that science works.

What’s wrong is that flat earthers’ claim they are leading a scientific argument. But there is no scientific argument about whether the earth is flat. This argument was settled long ago. Instead, flat earthers’ argument is about whether you should trust evidence that other people have collected before you. And it’s an important argument because this trust is essential for society and science to progress. The only alternative we have is that each and every one of us has to start over from scratch with birth. You see, flat earthers would eventually figure out the earth is round. But it might take them a thousand years until they’ve reinvented modern science.

This is why I think scientists should take flat earthers’ philosophical problem seriously. It’s a problem that any scientifically advanced society must address. It is not possible for each and every one of us to redo all experiments in the history of science. It therefore becomes increasingly important that scientists provide evidence for how science works, so that people who cannot follow the research itself can instead rely on evidence that the system produces correct and useful descriptions of nature.

To me, therefore, flat earthers, are a warning sign that scientists should take seriously. The more difficult scientific experiments and arguments are to follow for non-experts, the more care we must take to explain how we lead those arguments.

Theories of Everything [I've been singing again]

Understanding Quantum Mechanics #5: Decoherence

[Note: This transcript will not make much sense without the graphics in the video.]


I know I promised I would tell you what it takes to solve the measurement problem in quantum mechanics. But then I remembered that almost one of two physicists believes that the problem does not exist to begin with. So, I figured I should first make sure everyone – even the physicists – understand why the measurement problem has remained unsolved, despite a century of effort. This also means that if you watch this video to the end, you will understand what half of physicists do not understand.

That about half of physicists do not understand the measurement problem is not just anecdotal evidence, that’s poll results from 2016.This questionnaire was sent to a little more than one thousand two hundred physicists, from which about twelve percent responded. That’s a decent response rate for a survey, but note that the sample may not be representative for the global community. While the questionnaire was sent to physicists of all research areas, forty-four percent of them were Danish.

With those warnings ahead, a stunning seventeen percent of the survey-respondents said the measurement problem is a pseudoproblem. Even worse: twenty-nine percent erroneously think it has been solved by decoherence. So, this is what I want to explain today: What is decoherence and what does it have to do with quantum measurements? For this video, I will assume that you know the bra-ket notation for wave-functions. If you do not know it, please watch my earlier video.

In quantum mechanics, we describe a system by a wave-function that is a vector and can be expanded in a basis, which is a set of vectors of length one. The wave-function is usually denoted with the greek letter Psi. I will just label these basis vectors with numbers. A key feature of quantum mechanics is that the coefficients in the expansion of the wave-function, for which I used the letter a, can be complex numbers. Technically, there can be infinitely many basis-vectors, but that’s a complication we will not have to deal with here. We will just look at the simplest possible case, that of two basis vectors.

It is common to use basis vectors which describe possible measurement outcomes, and we will do the same. So, |1> and |2>, stand for two values of an observable that you could measure. The example that physicists typically have in mind for this are two different spin values of a particle, say +1 and -1. But the basis vectors could also describe something else that you measure, for example two different energy levels of an atom or two different sides of a detector, or what have you.

Once you have expanded the wave-function in a basis belonging to the measurement outcomes, then the square of the coefficient for a basis vector gives you the probability of getting the measurement outcome. This is Born’s rule. So if a coefficient was one over square root two, then the square is one half which means a fifty percent probability of finding this measurement outcome. Since the probabilities have to add up to 100%, this means the absolute squares of the coefficients have to add up to 1.

With these two basis vectors you can describe a superposition, which is a sum with factors in front of them. For more about superpositions, please watch my earlier video. The weird thing about quantum mechanics now is that if you have a state that is in a superposition of possible measurement outcomes, say, spin plus one and spin minus one, you never measure that superposition. You only measure either one or the other.

As example, let us use a superposition that is with equal probability in one of the possible measurement outcomes. Then the factor for each basis vector has to be the square root of one half. But this is quantum mechanics, so let us not forget that the coefficients are complex numbers. To take this into account, we will put in another factor here, which is a complex number with absolute value equal to one. We can write any such complex number as e to the I times theta, where theta is a real number.

The reason for doing this is that such a complex number does not change anything about the probabilities. See, if we ask what is the probability of finding this superposition in state |1>, then this would be (one over square root of two) times (e to the I theta) times the complex conjugate, which is (one over square root of two) times (e to the minus I theta). And that comes out to be one half, regardless of what theta is.

This theta also called the “phase” of the wave-function because you can decompose the complex number into a sine and cosine, and then it appears in the argument where a phase normally appears for an oscillation. There isn’t anything oscillating here, though, because there is no time-dependence. You could put another such complex number in front of the other coefficient, but this doesn’t change anything about the following.

Ok, so now we have this superposition that we never measure. The idea of decoherence is now to take into account that the superposition is not the only thing in our system. We prepare a state at some initial time, and then it travels to the detector. A detector is basically a device that amplifies a signal. A little quantum particle comes in one end and a number comes out on the other end. This necessarily means that the superposition which we want to measure interacts with many other particles, both along the way to the detector, and in the detector. This is what you want to describe with decoherence.

The easiest way to describe these constant bumps that the superposition has to endure is that each bump changes the phase of the state, so the theta, by a tiny little bit. To see what effect this has if you do a great many of these little bumps, we first have to calculate the density-matrix of the wave-function. It will become clear later, why.

As I explained in my previous video, the density matrix, usually denoted with the greek letter rho, is the ket-bra product of the wave-function with itself. For the simple case of our superposition, the density matrix looks like this. It has a one over two in each entry because of all the square roots of two, and the off-diagonal elements also have this complex factor with the phase. The idea of decoherence is then to say that each time the particle bumps into some other particle, this phase randomly changes and what you actually measure, is the average over all those random changes.

So, understanding decoherence comes down to averaging this complex number. To see what goes on, it helps to draw the complex plane. Here is the complex plane. Now, every number with an absolute value of 1 lies on a circle of radius one around zero. On this circle, you therefore find all the numbers of the form e to the I times theta, with theta a real number. If you turn theta from 0 to 2 \Pi, you go once around the circle. That’s Euler’s formula, basically.

The whole magic of decoherence is in the following insight. If you randomly select points on this circle and average over them, then the average will not lie on the circle. Instead, it will converge to the middle of the circle, which is at zero. So, if you average over all the random kicks, you get zero. The easiest way to see this is to think of the random points as little masses and the average as the center of mass.

Now let us look at the density matrix again. We just learned that if we average over the random kicks, then these off-diagonal entries go to zero. Nothing happens with the diagonal entries. That’s decoherence.

The reason this is called “decoherence” is that the random changes to the phase destroy the ability of the state to make an interference pattern with itself. If you randomly shift around the phase of a wave, you don’t get any pattern. A state that has a well-defined phase and can interfere with itself, is called “coherent”. But the terminology isn’t the interesting bit. The interesting bit is what has happened with the density matrix.

This looks utterly unremarkable. It’s just a matrix with one over two’s on the diagonal. But what’s interesting about it is that there is no wave-function that will give you this density matrix. To see this, look again at the density matrix for an arbitrary wave-function in two dimensions. Now take for example this off-diagonal entry. If this entry is zero, then one of these coefficients has to be zero, but then one of the diagonal elements is also zero, which is not what the decohered density matrix looks like. So, the matrix that we got after decoherence no longer corresponds to a wave-function.

That’s why we use density matrices in the first place. Every wave-function gives you a density matrix. But not every density matrix gives you a wave-function. If you want to describe how a system loses coherence, you therefore need to use density matrices.
br> What does this density matrix after decoherence describe? It describes classical probabilities. The diagonal entries tell you the probability for each of the possible measurement outcomes, like in quantum mechanics. But all the quantum-ness of the system, that was in the ability of the wave-function to interfere with itself, have gone away with the off-diagonal entries.

So, decoherence converts quantum probabilities to classical probabilities. It therefore explains why we never observe any strange quantum behavior in every-day life. It’s because this quantum behavior goes away very quickly with all the many interactions that every particle constantly has, whether or not you measure them. Decoherence gives you the right classical probabilities.

But it does not tell you what happens with the system itself. To see this, keep in mind that the density matrix in general does not describe a collection of particles or a sequence of measurements. It might well just describe one single particle. And after you have measured the particle, it is with probability 1 either in one state, or in the other. But this would correspond to a density matrix which has one diagonal entry that is 1 and all other entries zero. The state after measurement is not in a fifty-fifty probability-state, that just isn’t a thing. So, decoherence does not actually tell you what happens with the system itself when you measure it. It merely gives you probabilities for what you observe.

This is why decoherence only partially solves the measurement problem. It tells you why we do not normally observe quantum effects for large objects. It does not tell you, however, how it happens that a particle ends up in one, and only one, possible measurement outcome.

The best way to understand a new subject is to actively engage with it, and as much as I love doing these videos, this is something you have to do yourself. A great place to start engaging with quantum mechanics on your own is Brilliant, who have been sponsoring this video. Brilliant offers interactive courses on a large variety of topics in science and mathematics. To make sense of what I just told you about density matrices, for example, have a look at their courses on linear algebra, probabilities, and on quantum objects.

To support this channel and learn more about Brilliant, go to brilliant.org/Sabine, and sign up for free. The first two-hundred people who go to that link will get twenty percent off the annual Premium subscription.

Really Big Experiments That Physicists Dream Of

This week, I have something for your intellectual entertainment; I want to tell you about some really big experiments that physicists dream of.


Before I get to the futuristic ideas that physicists have, let me for reference first tell you about the currently biggest experiment in operation, that is the Large Hadron Collider, or LHC for short. Well, actually the LHC is currently on pause for an upgrade, but it is scheduled to be running again in May 2021. The LHC accelerates protons in a circular tunnel that is 27 kilometer long. Accelerating the protons requires powerful magnets that, to function properly, have to be cooled to only a few degrees above absolute zero. With this, the LHC reaches collision energies of about 14 Tera Electron Volt, or TeV.

Unless you are a particle physicist, this unit of energy probably does not tell you much. It helps to know that the collision energy is roughly speaking inversely proportional to the distances you can test. So, with higher collision energies, you can test smaller structures. That’s why particle physicists build bigger colliders. The fourteen TeV that the LHC produces correspond to about ten to the minus nineteen meters. For comparison, the typical size of an atom is ten to the minus ten meters, and a proton roughly has a size of ten to the minus fifteen meters. So, the LHC tests structures a thousand times smaller than the diameter of a proton.

As you may have read in the news recently, CERN announced that particle physicists want a bigger collider. The new machine, called the “Future Circular Collider” is supposed to have a tunnel that’s one-hundred kilometers long and it should ultimately reach one-hundred TeV collision energy, so that’s about six times as much as what the LHC can do. What do they want to do with the bigger collider? That’s a very good question, thanks for asking. They want to measure more precisely some properties of some particles. What is the use given that these particles live some microseconds on the outside? Nothing, really, but it keeps particle physicists employed.

The former Chief Scientific Advisor of the British government, Prof Sir David King, commented on the new collider plans in a BBC interview: “We have to draw a line somewhere, otherwise we end up with a collider that is so large that it goes around the equator. And if it doesn't end there perhaps there will be a request for one that goes to the Moon and back.”

Particle physicists don’t currently have plans for an accelerator around the equator, but some of them proposed we could place a collider with one-thousand-nine-hundred kilometer circumference in the gulf of Mexico. What for? Well, you could reach higher collision energies.

However, even particle physicists agree that a collider the size of the Milky Ways is not realistic. That’s because, as the particle physicist James Beachman explained in an interview with Gizmodo, unfortunately even interstellar space, with a temperature of about 3 degrees above absolute zero, is still too warm for the magnets. This means you’d need a huge amount of Helium to cool the magnets. And where would you get this?

But even a collider around the equator would be a technological challenge. Not only because of the differences in altitude, also because the diameter of Earth pulses with a period of about 21 minutes. That’s one of the fundamental vibrational modes of the Earth and, by the way, more evidence that the earth is not flat. The fundamental vibrational modes get constantly excited through earthquakes. But, as a lot of physicists have noticed, this is a problem which you would not have --- on the moon. The moon has very little seismic activity, and there’s also no life crawling around on it, so, except for the occasional asteroid impact, it’s very quiet there.

Better still, the moon has no atmosphere which can cloud up the view of the night sky. Which is why physicists have long dreamed of putting a radio telescope on the far side of the moon. Such a telescope would be exciting because it could measure signals from the “dark ages” of the early universe. This period has so-far been studied very little due to lack of data.

The dark ages begin after the emission of the cosmic microwave background but before the formation of the first stars, and they could tell us much about both, the behavior of normal matter and that of dark matter.

The dark ages, luckily, were not entirely dark, just very, very dim. That’s because back then the universe was filled mostly by lots of hydrogen atoms. If these bump into each other, they can emit light at a very specific wavelength, 21 cm. This wavelength then stretches with the expansion of the universe and should be measureable to day with radio telescopes. Physicists call this “21 centimeter astronomy” and a few telescopes are already looking out for this signal from the dark ages. But the signal is very weak and hard to measure. Putting a telescope on the moon would certainly help.

This is not the only experiment that physicist would like to put on the moon, if we’d just let them. Just in February this year, for example, a proposal appeared to put a neutrino source on the moon and send a beam of neutrinos from there to earth. This would allow physicists to better study what happens to neutrinos as they travel. This information could be interesting because we know that neutrinos can “oscillate” between different types as they travel – for example an electron-neutrino can oscillate into a muon-neutrino – but there are some oddities in the existing measurements that could mean we are missing something.

And only a few weeks ago, some physicists proposed to put a gravitational wave interferometer on the moon, though this idea was originally proposed already in the 1990s. Again the reason is that the moon is far less noisy than our densely populated and seismically active planet. The downside is, well, there are no people on the moon to actually build the machine.

That’s why I am more excited about another proposal that was put forward some years ago by two physicists from Harvard University. These guys suggested that to better measure gravitational waves, we could leave a trail of atomic clocks behind us on our annual path around the sun. When a gravitational wave passes through the solar system, the time that it takes signals to travel between the atomic clocks and earth slightly changes. The cool thing about it is that this would allow physicists to detect gravitational waves with much longer wavelengths than what is possible with interferometers on earth or on the moon. Gravitational waves with such long wavelengths should be created in the collisions of supermassive black holes and therefore could tell us something about what goes on in galactic cores.

These experiments have in common that they would be great to have, if you are a physicist. They also have in common that they are big. And since they are big, they are expensive, which means chances are slim any of those will ever become reality. Unfortunately, ignoring economic reality is common for physicists. Instead of thinking about ways to make experiments smaller, easier to handle, and cheaper to produce, their vision is to do the same thing again, just bigger. But, well, bigger isn’t always better.

What is the equivalence principle?

Folks, I recently read the website of the Flat Earth Society. I’m serious! It’s a most remarkable collection of… nonsense. Maybe most remarkable is how it throws together physical facts that are correct – but then gets their consequences completely wrong! This is most evident when it comes to flat earthers’ elaborations on Einstein’s equivalence principle.


The equivalence principle is experimentally extremely well-confirmed, yes. But flat earthers misconstrue evidence for the equivalence principle as “evidence for universal acceleration” or what they call the “universal accelerator”. By this they mean that the gravitational acceleration is the same everywhere on earth. It is not. But, you see, they believe that on their flat earth, there is no gravity. Instead, the flat earth is accelerating upwards. So, if you drop an apple, it’s not that gravity is pulling it down, it’s that the earth comes up and hits the apple.

The interesting thing is now that flat earthers’ claim Einstein said you cannot distinguish upward acceleration from downward gravity. That’s the equivalence principle, supposedly. So, you see, Einstein said it and therefore the earth is flat.

You can read on their website:

“Why does the physics of gravity behave exactly as if the earth were accelerating upwards? The Universal Accelerator answers this long-standing mystery, which has baffled generations of scientists, by positing that the earth is accelerating upwards.”

Ingenious! Why didn’t Einstein think of this? Well, because it’s wrong. And in this video, I will explain why it’s wrong. So, what is the equivalence principle? The equivalence principle says that:

“Acceleration in a flat space-time is locally indistinguishable from gravity.”

Okay, that sounds somewhat technical, so let us go through this step by step. I assume you know what acceleration is because otherwise you would not be watching a physics channel. Flat space-time means you are dealing with special relativity. So, you have combined space and time, as Einstein told us to do, but they are not curved; they’re flat, like a sheet of paper. “Locally” means in a small region. So, the equivalence principle says: If you can only make measurements in a small region around you, then you cannot tell acceleration apart from gravity. You can only tell them apart if you can make measurements over a large enough distances.

This is what Einstein’s thought experiment with the elevator was all about. I talked about this in an earlier video. If you’re in the elevator, you don’t know whether the elevator is sitting on the surface of a planet and gravity is pulling down, or if the elevator is accelerating upward.

The historical relevance of the equivalence principle is that it allowed Einstein to make the step from special relativity to general relativity. This worked because he already knew how to describe acceleration in flat space – you can do that with special relativity. In general relativity then, space-time is curved, but locally it is flat. So you can use special relativity locally and get general relativity. The equivalence principle connects both – that was Einstein’s great insight.

So, the equivalence principle says that you cannot tell gravity from acceleration in a small region. That sounds indeed very much like what flat earthers say. But here’s the important point: How large the region needs to be to tell apart gravity from acceleration depends on how precisely you can measure and how far you are willing to walk. If you cannot measure very precisely, you may have to climb on a mountain top. You then find that the acceleration up there is smaller than at sea level. Why? Because the gravitational force decreases with the distance to the center of the earth. That’s Newton’s 1/R² force. Indeed, since the earth is not exactly a sphere, the acceleration also differs somewhat between the equator and the poles. This can and has been measured to great precision.

Yeah, we’ve know all this for some while. If the acceleration we normally assign to gravity was the same everywhere on earth, that would contradict a huge number of measurements. Evidence strongly speaks against it. If you measure very precisely, you can even find evidence for the non-universality of the gravitational pull in the laboratory. Mountains themselves, for example, have a non-negligible gravitational pull. This can, and has been measured, already in the 18th century. The gravitational acceleration caused by the ground underneath your feet has also local variations at constant altitude just because in some places the density of the ground is higher than in others.

So, explaining gravity as a universal acceleration is in conflict with a lot of evidence. But can you instead just give the flat earth a gravitational pull? No, that does not fit with evidence either. Because for a disk the gravitational acceleration does not drop with 1/R². It falls more slowly with the distance from the disk. Exactly how depends on how far you are from the edge of the disk. In any case, it’s clearly wrong.

The equivalence principle is sometimes stated differently than I put it, namely as the equality of inertial and gravitational mass. Physicists don’t particularly like this way of formulating the equivalence principle because it’s not only mass that gravitates. All kinds of energy densities and momentum flow and pressure and so on also gravitate. So, strictly speaking it’s not correct to merely say inertial mass equals gravitational mass.

But in the special case when you are looking at a slowly moving point particle with a mass that is very small compared to earth, then the equality of inertial and gravitational mass is a good way to think of the equivalence principle. If you use the approximation of Newtonian gravity, then you would describe this by saying that F equals m_i times a, with m_i the inertial mass and a the acceleration, and that must be balanced with the gravitational force that is m_g, the gravitational mass of the particle, times the mass of earth divided by R^2, where R is the distance from the center of earth which is, excuse me, a sphere. So, if the inertial mass is equal to the gravitational mass of the particle, then these masses cancel out. If you calculate the path on which the particle moves, it will therefore not depend on the mass.

In general relativity, the equivalence of inertial and gravitational mass for a point particle has a very simple interpretation. Remember that, in general relativity, gravity is not a force. Gravity is really caused by the curvature of space-time. In this curved space-time a point particle just takes the path of the longest possible proper time between two places. This is an entirely geometrical requirement and does not depend on the mass of the particle.

Let me add that physicists use a few subtle distinctions of equivalence principles, in particular for quantum objects. If you want to know the technical details, please check the information below the video for a reference.

In summary, if you encounter a flat earther who wants to impress you with going on about the equivalence principle, all you need to know is that the equivalence principle is not evidence for universal acceleration. This is most definitely not what Einstein said.

If this video left you wishing you understood Einstein’s work better, I suggest you have a look at Brilliant dot org, who have been sponsoring this video. Brilliant offers online courses on a large variety of topics in mathematics and science, including physics. They have interactive courses on special relativity, general relativity, and even gravitational physics, where they explore the equivalence principle specifically. Brilliant is a great starting point to really understand how Einstein’s theories work and also test your understanding along the way.

To support this channel and learn more about Brilliant, go to brilliant.org/Sabine, and sign up for free. The first two-hundred people who go to that link will get twenty percent off the annual Premium subscription.

Thanks for watching, see you next week.