## Thursday, March 26, 2020

## Monday, March 23, 2020

### Are dark energy and dark matter scientific?

I have noticed that each time I talk or write about dark energy or dark matter, I get a lot of comments from people saying, oh that stuff doesn’t exist, you can’t just invent something invisible each time there’s an inconvenient measurement. Physicists have totally lost it. This is such a common misunderstanding that I thought I will dedicate a video to sorting this out. Dark energy and dark matter are entirely normal, and perfectly scientific hypotheses. They may turn out to be wrong, but that doesn’t mean it’s wrong to consider them in the first place.

Before I say anything else, here is a brief reminder what dark energy and dark matter are. Dark energy is what speeds up the expansion of the universe; it does not behave like normal energy. Dark matter has a gravitational pull like normal matter, but you can’t see it. Dark energy and dark matter are two different things. They may be related, but currently we have no good reason to think that they are related.

Why have physicists come up with this dark stuff? Well, we have two theories to describe the behavior of matter. The one is the standard model of particle physics, which describes the elementary particles and the forces between them, except gravity. The other is Einstein’s theory of general relativity, which describes the gravitational force that is generated by all types of matter and energy. The problem is, if you use Einstein’s theory for the matter that is in the standard model only, this does not describe what we see. The predictions you get from combining those two theories do not fit to the observations.

It’s not only one prediction that does not fit to observations, it’s many different ones. For dark matter it’s that galaxies rotate too fast, galaxies in clusters move too fast, gravitational lenses bend light too strongly, and neither the cosmic microwave background nor galactic filaments would look like we observe them without dark matter. I explained this in detail in an earlier video.

For dark energy the shooting gun signature is that the expansion of the universe is getting faster, which you can find out by observing how fast supernova in other galaxies speed away from us. The evidence for dark energy is not quite as solid as for dark matter. I explained this too in an earlier video.

So, what’s the scientist to do when they are faced with such a discrepancy between theory and observation? They look for new regularities in the observation and try to find a simple way to explain them. And that’s what dark energy and dark matter are. They are extremely simple terms to add to Einstein’s theory, that explain observed regularities, and make the theory agrees with the data again.

This is easy to see when it comes to dark energy because the presently accepted version of dark energy is just a constant, the so-called cosmological constant. This cosmological constant is just a constant of nature and it’s a free parameter in General Relativity. Indeed, it was introduced already by Einstein himself. And what explanation for an observation could possibly be simpler than a constant of nature?

For dark matter it’s not quite as simple as that. I frequently hear the criticism that dark matter explains nothing because it can be distributed in arbitrary amounts wherever needed, and therefore can fit any observation. But that’s just wrong. It’s wrong for two reasons.

First, the word “matter” in “dark matter” doesn’t just vaguely mean “stuff”. It’s a technical term that means “stuff with a very specific behavior”. Dark matter behaves like normal matter, except that, for all we currently know, it doesn’t have internal pressure. You cannot explain any arbitrary observation by attributing it to matter. It just happens to be the case that the observations we do have can be explained this way. That’s a non-trivial fact.

Let me emphasize that dark matter in cosmology is a kind of fluid. It does not have any substructure. Particle physicists, needless to say, like the idea that dark matter is made of a particle. This may or may not be case. We currently do not have any observation that tells us dark matter must have a microscopic substructure.

The other reason why it’s wrong to say that dark matter can fit anything is that you cannot distribute it as you wish. Dark matter starts with a random distribution in the early universe. As the universe expands, and matter in it cools, dark matter starts to clump and it forms structures. Normal matter then collects in the gravitational potentials generated by the dark matter. So, you do not get to distribute matter as you wish. It has to fit with the dynamical evolution of the universe.

This is why dark matter and dark energy are good scientific explanations. They are simple and yet explain a lot of data.

Now, to be clear, this is the standard story. If you look into the details it is, as usual, more complicated. That’s because the galactic structures that form with dark matter actually do not fit the data all that well, and they do not explain some regularities that astronomers have observed. So, there are good reasons for being skeptical that dark matter is ultimately the right story, but it isn’t as simple as just saying “it’s unscientific”.

Before I say anything else, here is a brief reminder what dark energy and dark matter are. Dark energy is what speeds up the expansion of the universe; it does not behave like normal energy. Dark matter has a gravitational pull like normal matter, but you can’t see it. Dark energy and dark matter are two different things. They may be related, but currently we have no good reason to think that they are related.

Why have physicists come up with this dark stuff? Well, we have two theories to describe the behavior of matter. The one is the standard model of particle physics, which describes the elementary particles and the forces between them, except gravity. The other is Einstein’s theory of general relativity, which describes the gravitational force that is generated by all types of matter and energy. The problem is, if you use Einstein’s theory for the matter that is in the standard model only, this does not describe what we see. The predictions you get from combining those two theories do not fit to the observations.

It’s not only one prediction that does not fit to observations, it’s many different ones. For dark matter it’s that galaxies rotate too fast, galaxies in clusters move too fast, gravitational lenses bend light too strongly, and neither the cosmic microwave background nor galactic filaments would look like we observe them without dark matter. I explained this in detail in an earlier video.

For dark energy the shooting gun signature is that the expansion of the universe is getting faster, which you can find out by observing how fast supernova in other galaxies speed away from us. The evidence for dark energy is not quite as solid as for dark matter. I explained this too in an earlier video.

So, what’s the scientist to do when they are faced with such a discrepancy between theory and observation? They look for new regularities in the observation and try to find a simple way to explain them. And that’s what dark energy and dark matter are. They are extremely simple terms to add to Einstein’s theory, that explain observed regularities, and make the theory agrees with the data again.

This is easy to see when it comes to dark energy because the presently accepted version of dark energy is just a constant, the so-called cosmological constant. This cosmological constant is just a constant of nature and it’s a free parameter in General Relativity. Indeed, it was introduced already by Einstein himself. And what explanation for an observation could possibly be simpler than a constant of nature?

For dark matter it’s not quite as simple as that. I frequently hear the criticism that dark matter explains nothing because it can be distributed in arbitrary amounts wherever needed, and therefore can fit any observation. But that’s just wrong. It’s wrong for two reasons.

First, the word “matter” in “dark matter” doesn’t just vaguely mean “stuff”. It’s a technical term that means “stuff with a very specific behavior”. Dark matter behaves like normal matter, except that, for all we currently know, it doesn’t have internal pressure. You cannot explain any arbitrary observation by attributing it to matter. It just happens to be the case that the observations we do have can be explained this way. That’s a non-trivial fact.

Let me emphasize that dark matter in cosmology is a kind of fluid. It does not have any substructure. Particle physicists, needless to say, like the idea that dark matter is made of a particle. This may or may not be case. We currently do not have any observation that tells us dark matter must have a microscopic substructure.

The other reason why it’s wrong to say that dark matter can fit anything is that you cannot distribute it as you wish. Dark matter starts with a random distribution in the early universe. As the universe expands, and matter in it cools, dark matter starts to clump and it forms structures. Normal matter then collects in the gravitational potentials generated by the dark matter. So, you do not get to distribute matter as you wish. It has to fit with the dynamical evolution of the universe.

This is why dark matter and dark energy are good scientific explanations. They are simple and yet explain a lot of data.

Now, to be clear, this is the standard story. If you look into the details it is, as usual, more complicated. That’s because the galactic structures that form with dark matter actually do not fit the data all that well, and they do not explain some regularities that astronomers have observed. So, there are good reasons for being skeptical that dark matter is ultimately the right story, but it isn’t as simple as just saying “it’s unscientific”.

## Monday, March 16, 2020

### Unpredictability, Undecidability, and Uncomputability

There are quite a number of mathematical theorems that prove the power of mathematics has its limits. But how relevant are these theorems for science? In this video I want to briefly summarize an essay that I just submitted to the essay contest of the Foundational Questions Institute. This year the topic is “Unpredictability, undecidability, and uncomputability”.

Take Gödel’s theorem. Gödel’s theorem says that if a set of consistent axioms is sufficiently complex, you can formulate statements for which you can’t know whether they are right or wrong. So, mathematics is incomplete in this sense, and that is certainly a remarkable insight. But it’s irrelevant for scientific practice. That’s because one can always extend the original set of axioms with another axiom that simply says whether or not the previously undecidable statement is true.

How would we deal with mathematical incompleteness in physics? Well, in physics, theories are sets of mathematical axioms, like the ones that Gödel’s theorem deals with, but that’s not it. Physical theories also come with a prescription for how to identify mathematical structures with measurable quantities. Physics, after all, is science, not mathematics. So, if we had any statement that was undecidable, we’d decide it by making a measurement, and then add an axiom that agrees with the outcome. Or, if the undecidable statement has no observable consequences, then we can as well ignore it.

Mathematical theorems about uncomputability are likewise scientifically irrelevant but for a different reason. The problem with uncomputability is that it always comes from something being infinite. However, nothing real is infinite, so these theorems do not actually apply to anything we could find in nature.

The most famous example is Turing’s “Halting Problem”. Think of any algorithm that computes something. It will either halt at finite time and give you a result, or not. Turing says, now let us try to find a meta-algorithm that can tell us whether another algorithm halts or doesn’t halt. Then he proves that there is no such meta-algorithm which – and here is the important point – works for all possible algorithms. That’s an infinitely large class. In reality we will never need an algorithm that answers infinitely many questions.

Another not-quite as well-known example is Wang’s Domino problem. Wang said, take any set of squares with different colors for each side. Can you use them to fill up an infinite plane without gaps? It turns out that the question is undecidable for arbitrary sets of squares. But then, we never have to tile infinite planes.

We also know that most real numbers are uncomputable. They are uncomputable in the sense that there is no algorithm that will approximate them to a certain, finite precision, in finite time. But in science, we never deal with real numbers. We deal with numbers that have a finitely many digits and that have error bars. So this is another interesting mathematical curiosity, but has no scientific relevance.

What about unpredictability? Well, quantum mechanics has an unpredictable element, as I explained in an earlier video. But this unpredictability is rather uninteresting, since it’s there by postulate. More interesting is the unpredictability in chaotic systems.

Some chaotic systems suffer from a peculiar type of unpredictability. Even if you know the initial conditions arbitrarily precise, you can only make predictions for a finite amount of time. Whether this actually happens for any system in the real world is not presently clear.

The most famous candidate for such an unpredictable equation is the Navier-Stokes equation that is used, among other things, to make the weather forecast. Whether this equation sometimes leads to unpredictable situations is one of the Clay Institute’s Millennium Problems, one of the hardest open mathematical problems today.

But let us assume for a moment this problem was solved and it turned out that, with the Navier stokes equation, it is indeed impossible, in some circumstances, to make predictions beyond a finite time. What would this tell us about nature? Not a terrible lot, because we know already that the Navier-Stokes equation is only an approximation. Really gases and fluids are made of particles and should be described by quantum mechanics. And quantum mechanics does not have the chaotic type of unpredictability. Also, maybe quantum mechanics itself is not ultimately the right theory. So really, we can’t tell whether nature is predictable or not.

This is a general problem with applying impossibility-theorems to nature. We can never know whether the mathematical assumptions that we make in physics are really correct, or if not one day they will be replaced by something better. Physics is science, not math. We use math because it works, not because we think nature

All of this makes it sound like undecidability, unpredictability, and uncomputability are mathematical problems and irrelevant for science. But that would be jumping to conclusions. They are relevant for science. But they are relevant not because they tell us something deep about nature. They are relevant because in practice we use theories that may have these properties. So the theorems tell us what we can do with our theories.

An example. Maybe the Navier-Stokes equation is not fundamentally the right equation for weather predictions. But it

The plasma sometimes develops instabilities that can damage the containment vessel. Therefore, if an instability is coming on, the fusion process must be shut down quickly. This makes fusion very inefficient. If we’d better understand when unpredictable situations are about to occur, we may be able to prevent them from happening in the first place. This would also be useful, for example, for the financial system.

In summary, mathematical impossibility-theorems are relevant in science, not because they tell us something about nature itself, but because we use mathematics in practice to understand observations, and the theorems tell us what can expect of our theories.

You can read all the essays in the contest over at the FQXi website. The deadline was moved to April 24, so you still have time to submit your essay!

Take Gödel’s theorem. Gödel’s theorem says that if a set of consistent axioms is sufficiently complex, you can formulate statements for which you can’t know whether they are right or wrong. So, mathematics is incomplete in this sense, and that is certainly a remarkable insight. But it’s irrelevant for scientific practice. That’s because one can always extend the original set of axioms with another axiom that simply says whether or not the previously undecidable statement is true.

How would we deal with mathematical incompleteness in physics? Well, in physics, theories are sets of mathematical axioms, like the ones that Gödel’s theorem deals with, but that’s not it. Physical theories also come with a prescription for how to identify mathematical structures with measurable quantities. Physics, after all, is science, not mathematics. So, if we had any statement that was undecidable, we’d decide it by making a measurement, and then add an axiom that agrees with the outcome. Or, if the undecidable statement has no observable consequences, then we can as well ignore it.

Mathematical theorems about uncomputability are likewise scientifically irrelevant but for a different reason. The problem with uncomputability is that it always comes from something being infinite. However, nothing real is infinite, so these theorems do not actually apply to anything we could find in nature.

The most famous example is Turing’s “Halting Problem”. Think of any algorithm that computes something. It will either halt at finite time and give you a result, or not. Turing says, now let us try to find a meta-algorithm that can tell us whether another algorithm halts or doesn’t halt. Then he proves that there is no such meta-algorithm which – and here is the important point – works for all possible algorithms. That’s an infinitely large class. In reality we will never need an algorithm that answers infinitely many questions.

Another not-quite as well-known example is Wang’s Domino problem. Wang said, take any set of squares with different colors for each side. Can you use them to fill up an infinite plane without gaps? It turns out that the question is undecidable for arbitrary sets of squares. But then, we never have to tile infinite planes.

We also know that most real numbers are uncomputable. They are uncomputable in the sense that there is no algorithm that will approximate them to a certain, finite precision, in finite time. But in science, we never deal with real numbers. We deal with numbers that have a finitely many digits and that have error bars. So this is another interesting mathematical curiosity, but has no scientific relevance.

What about unpredictability? Well, quantum mechanics has an unpredictable element, as I explained in an earlier video. But this unpredictability is rather uninteresting, since it’s there by postulate. More interesting is the unpredictability in chaotic systems.

Some chaotic systems suffer from a peculiar type of unpredictability. Even if you know the initial conditions arbitrarily precise, you can only make predictions for a finite amount of time. Whether this actually happens for any system in the real world is not presently clear.

The most famous candidate for such an unpredictable equation is the Navier-Stokes equation that is used, among other things, to make the weather forecast. Whether this equation sometimes leads to unpredictable situations is one of the Clay Institute’s Millennium Problems, one of the hardest open mathematical problems today.

But let us assume for a moment this problem was solved and it turned out that, with the Navier stokes equation, it is indeed impossible, in some circumstances, to make predictions beyond a finite time. What would this tell us about nature? Not a terrible lot, because we know already that the Navier-Stokes equation is only an approximation. Really gases and fluids are made of particles and should be described by quantum mechanics. And quantum mechanics does not have the chaotic type of unpredictability. Also, maybe quantum mechanics itself is not ultimately the right theory. So really, we can’t tell whether nature is predictable or not.

This is a general problem with applying impossibility-theorems to nature. We can never know whether the mathematical assumptions that we make in physics are really correct, or if not one day they will be replaced by something better. Physics is science, not math. We use math because it works, not because we think nature

*is*math.All of this makes it sound like undecidability, unpredictability, and uncomputability are mathematical problems and irrelevant for science. But that would be jumping to conclusions. They are relevant for science. But they are relevant not because they tell us something deep about nature. They are relevant because in practice we use theories that may have these properties. So the theorems tell us what we can do with our theories.

An example. Maybe the Navier-Stokes equation is not fundamentally the right equation for weather predictions. But it

*is*the one that we currently use. And therefore, knowing when an unpredictable situation is coming up matters. Indeed, we might want to know when an unpredictability is coming up to avoid it. This is not really feasible for the weather, but it may be feasible for another partly chaotic system, that is plasma in nuclear fusion plants.The plasma sometimes develops instabilities that can damage the containment vessel. Therefore, if an instability is coming on, the fusion process must be shut down quickly. This makes fusion very inefficient. If we’d better understand when unpredictable situations are about to occur, we may be able to prevent them from happening in the first place. This would also be useful, for example, for the financial system.

In summary, mathematical impossibility-theorems are relevant in science, not because they tell us something about nature itself, but because we use mathematics in practice to understand observations, and the theorems tell us what can expect of our theories.

You can read all the essays in the contest over at the FQXi website. The deadline was moved to April 24, so you still have time to submit your essay!

## Saturday, March 14, 2020

### Coronavirus? I have nothing to add.

I keep getting requests from people that I comment on the coronavirus pandemic, disease models, or measures taken to contain and mitigate the outbreak. While I appreciate the faith you put into me, it also leaves me somewhat perplexed. I am not an epidemiologist; I’m a physicist. I have nothing original to say about coronavirus. Sure, I could tell you what I have taken away from other people’s writings – a social media strain of Chinese Whispers, if you wish – but I don’t think this aids information flow, it merely introduces mistakes.

I will therefore keep my mouth shut and just encourage you to get your information from more reliable sources. When it comes to public health, I personally prefer institutional and governmental websites over the mass media, largely because the media has an incentive to make the situation sound more dramatic than it really is. In Germany, I would suggest the Federal Ministry of Health (in English) and the Robert Koch Institute (in German). And regardless of where you live, the websites of the WHO are worth checking out.

I have not come across a prediction for the spread of the disease that looked remotely reliable, but

Having said that, what I can do is offer you a forum to commiserate. I got caught in the midst of organizing a workshop that was supposed to take place in May in the UK. We monitored the situation in Europe for the past weeks, but eventually had to conclude there’s no way around postponing the workshop.

Almost everyone from overseas had to cancel their participation because they weren’t allowed to travel, or, if they had, their health insurance wouldn’t have covered had they contracted the virus. At present only Italy is considered a high risk country in Europe. But it’s likely that in the coming weeks several other European countries will be in a similar situation, which will probably bring more travel restrictions. Finally, most universities here in Germany and in the UK have for now issued a policy to cancel all kinds of meetings on their premises so that we might have ended up without a room for the event.

We presently don’t know when the workshop will take place, but hopefully some time in the fall.

I was supposed to be on a panel discussion in Zurich next week, but that was also cancelled. I am scheduled to give a public lecture in two weeks which has not been cancelled. This comes to me as some surprise because it’s in the German state that, so far, has been hit the worst by coronavirus. I kind of expect this to also be cancelled.

Where we live, most employers have asked employees to work from home if anyhow possible. Schools will be closed next week until after the Easter break – for now. All large events have been cancelled. This puts us in a situation that many people are facing right now: We’ll be stuck at home with bored children. I am actually on vacation for the next two weeks, but looks like it won’t be much of a vacation.

I’m not keen on contracting an infectious disease but believe sooner or later we’ll get it anyway. Even if there’s a vaccine, this may not work for variants of the original strain. We are lucky in that no one in our close family has a pre-existing condition that would put them at an elevated risk, though we worry of course about the grandparents. Shopping panic here has been moderate; the demand on disinfectants, soap and, yes, toilet paper, seems to be abnormally high, but that’s about it. By and large I think the German government has been handling the situation well and Trump’s travel ban is doing Europe a great favor because shit’s about to hit the fan over there.

In any case, I feel like there isn’t much we can do right now other than washing our hands and not coughing other people in the face. I have two papers to finish which will keep me busy for the next weeks. Wherever you are, I hope you stay safe and healthy.

Update: As anticipated, I just got an email saying that the public lecture in April has also been cancelled.

I will therefore keep my mouth shut and just encourage you to get your information from more reliable sources. When it comes to public health, I personally prefer institutional and governmental websites over the mass media, largely because the media has an incentive to make the situation sound more dramatic than it really is. In Germany, I would suggest the Federal Ministry of Health (in English) and the Robert Koch Institute (in German). And regardless of where you live, the websites of the WHO are worth checking out.

I have not come across a prediction for the spread of the disease that looked remotely reliable, but

*Our World in Data*has some neat visualization tools for the case numbers from the WHO (example below).Having said that, what I can do is offer you a forum to commiserate. I got caught in the midst of organizing a workshop that was supposed to take place in May in the UK. We monitored the situation in Europe for the past weeks, but eventually had to conclude there’s no way around postponing the workshop.

Almost everyone from overseas had to cancel their participation because they weren’t allowed to travel, or, if they had, their health insurance wouldn’t have covered had they contracted the virus. At present only Italy is considered a high risk country in Europe. But it’s likely that in the coming weeks several other European countries will be in a similar situation, which will probably bring more travel restrictions. Finally, most universities here in Germany and in the UK have for now issued a policy to cancel all kinds of meetings on their premises so that we might have ended up without a room for the event.

We presently don’t know when the workshop will take place, but hopefully some time in the fall.

I was supposed to be on a panel discussion in Zurich next week, but that was also cancelled. I am scheduled to give a public lecture in two weeks which has not been cancelled. This comes to me as some surprise because it’s in the German state that, so far, has been hit the worst by coronavirus. I kind of expect this to also be cancelled.

Where we live, most employers have asked employees to work from home if anyhow possible. Schools will be closed next week until after the Easter break – for now. All large events have been cancelled. This puts us in a situation that many people are facing right now: We’ll be stuck at home with bored children. I am actually on vacation for the next two weeks, but looks like it won’t be much of a vacation.

I’m not keen on contracting an infectious disease but believe sooner or later we’ll get it anyway. Even if there’s a vaccine, this may not work for variants of the original strain. We are lucky in that no one in our close family has a pre-existing condition that would put them at an elevated risk, though we worry of course about the grandparents. Shopping panic here has been moderate; the demand on disinfectants, soap and, yes, toilet paper, seems to be abnormally high, but that’s about it. By and large I think the German government has been handling the situation well and Trump’s travel ban is doing Europe a great favor because shit’s about to hit the fan over there.

In any case, I feel like there isn’t much we can do right now other than washing our hands and not coughing other people in the face. I have two papers to finish which will keep me busy for the next weeks. Wherever you are, I hope you stay safe and healthy.

Update: As anticipated, I just got an email saying that the public lecture in April has also been cancelled.

## Thursday, March 12, 2020

### Essays, Elsewhere

Just a brief note that Tim and I have an essay up at Nautilus

Head over to Nautilus to read the whole thing. It’s a great magazine, btw, and I warmly recommend you follow it.

If you found that interesting, you may also be interested in my contribution to this year’s essay contest from the Foundational Questions Institute on Undecidability, Uncomputability, and Unpredictability:

How to Make Sense of Quantum Physics

Superdeterminism, a long-abandoned idea, may help us overcome the current crisis in physics.

BY SABINE HOSSENFELDER & TIM PALMER

Quantum mechanics isn’t rocket science. But it’s well on the way to take the place of rocket science as the go-to metaphor for unintelligible math. Quantum mechanics, you have certainly heard, is infamously difficult to understand. It defies intuition. It makes no sense. Popular science accounts inevitably refer to it as “strange,” “weird,” “mind-boggling,” or all of the above.

We beg to differ. Quantum mechanics is perfectly comprehensible. It’s just that physicists abandoned the only way to make sense of it half a century ago. Fast forward to today and progress in the foundations of physics has all but stalled. The big questions that were open then are still open today. We still don’t know what dark matter is, we still have not resolved the disagreement between Einstein’s theory of gravity and the standard model of particle physics, and we still do not understand how measurements work in quantum mechanics.

How can we overcome this crisis? We think it’s about time to revisit a long-forgotten solution, Superdeterminism, the idea that no two places in the universe are truly independent of each other. This solution gives us a physical understanding of quantum measurements, and promises to improve quantum theory. Revising quantum theory would be a game changer for physicists’ efforts to solve the other problems in their discipline and to find novel applications of quantum technology.

Head over to Nautilus to read the whole thing. It’s a great magazine, btw, and I warmly recommend you follow it.

If you found that interesting, you may also be interested in my contribution to this year’s essay contest from the Foundational Questions Institute on Undecidability, Uncomputability, and Unpredictability:

Math Matters

By Sabine Hossenfelder

Gödel taught us that mathematics is incomplete. Turing taught us some problems are undecidable. Lorenz taught us that, try as we might, some things will remain unpredictable. Are such theorems relevant for the real world or are they merely academic curiosities? In this essay, I first explain why one can rightfully be skeptical of the scientific relevance of mathematically proved impossibilities, but that, upon closer inspection, they are both interesting and important.

## Saturday, March 07, 2020

### Is Gravity a Force?

I was sick last week and lost like 10 pounds in 3 days, which brings up the question, what is weight?

Weight is actually the force that acts on your body due to the pull of gravity. Now, the gravitational force depends on the mass of the object that is generating the force, in this case, planet Earth. So you can lose weight by simply moving to the moon. Technically, therefore, I should have said I lost mass, not weight.

Why do we normally not make this distinction? That’s because in practice it doesn’t matter. Mass just a number – a “scalar” – as physicists say, but weight, since it is a force, has a direction. So if you wanted to be very annoying, I mean very accurate, then whenever you’d refer to weight you’d have to say which direction you are talking about. The weight in East direction? The weight in North direction? Why doesn’t anyone ever mention this?

We don’t usually mention this because we all agree that we mean the force pulling down, and since we all know what we are talking about, we treat weight as if it was a scalar, omitting the direction. Moreover, the gravitational attraction downwards is pretty much the same everywhere on our planet, which means it is unnecessary to distinguishing between weight and mass in everyday life. Technically, it’s correct: mass and weight are not the same thing. Practically, the difference doesn’t matter.

But wait. Didn’t Einstein say that gravity is not a force to begin with? Ah, yes, there’s that.

Einstein’s theory of general relativity tells us that the effect we call gravity is different from normal forces. In General Relativity, space and time are not flat, like a sheet of paper, but curved, like the often-named rubber sheet. This curvature is caused by all types of mass and energy, and the motion of mass and energy is in return affected by the curvature. This gives you a self-consistent, closed, set of equations know as Einstein’s Field Equations. In Einstein’s theory, then, there is no force acting on masses. The masses are just navigating the curved space-time. We cannot see the curvature directly. We only see its effects. And those effects are what we call gravity.

Now, Einstein’s theory of General Relativity rests on the equivalence principle. The equivalence principle says that locally the effects of gravity are the same as the effects of acceleration in flat space. “Locally” here roughly means “nearby”. And acceleration in flat space is described by Einstein’s theory of Special Relativity. So, with the equivalence principle, you can generalize Special Relativity to General Relativity. Special Relativity is the special case in which space-time is flat, and there is no gravity.

The equivalence principle was well illustrated by Einstein himself. He said, let us consider you are in an elevator that is being pulled up at constant acceleration. There is one force acting on you, which is the floor pushing up. Now, Einstein says, gravity has the very same effect without something pulling up the elevator. And again, there is only one force acting on you, which is the floor pushing up.

If there was nothing pulling the elevator (so, if there was no acceleration) you would feel no force at all. In General Relativity, this corresponds to freely falling in a gravitational field. That’s the key point of Einstein’s insight: If you freely fall, there is no force acting on you. And in that, Einstein and Newton differ. Newton would say, if you jump off a roof, the force of gravity is pulling you down. Einstein says, nope, if you jump off a roof, you take away the force that was pushing you up.

Again, however, the distinction between the two cases is rather technical and one we do not have to bother with in daily life. That is because in daily life we do not need to use the full blown apparatus of General Relativity. Newton’s theory works just fine, for all practical purposes, unless possibly, you plan to visit a black hole.

Weight is actually the force that acts on your body due to the pull of gravity. Now, the gravitational force depends on the mass of the object that is generating the force, in this case, planet Earth. So you can lose weight by simply moving to the moon. Technically, therefore, I should have said I lost mass, not weight.

Why do we normally not make this distinction? That’s because in practice it doesn’t matter. Mass just a number – a “scalar” – as physicists say, but weight, since it is a force, has a direction. So if you wanted to be very annoying, I mean very accurate, then whenever you’d refer to weight you’d have to say which direction you are talking about. The weight in East direction? The weight in North direction? Why doesn’t anyone ever mention this?

We don’t usually mention this because we all agree that we mean the force pulling down, and since we all know what we are talking about, we treat weight as if it was a scalar, omitting the direction. Moreover, the gravitational attraction downwards is pretty much the same everywhere on our planet, which means it is unnecessary to distinguishing between weight and mass in everyday life. Technically, it’s correct: mass and weight are not the same thing. Practically, the difference doesn’t matter.

But wait. Didn’t Einstein say that gravity is not a force to begin with? Ah, yes, there’s that.

Einstein’s theory of general relativity tells us that the effect we call gravity is different from normal forces. In General Relativity, space and time are not flat, like a sheet of paper, but curved, like the often-named rubber sheet. This curvature is caused by all types of mass and energy, and the motion of mass and energy is in return affected by the curvature. This gives you a self-consistent, closed, set of equations know as Einstein’s Field Equations. In Einstein’s theory, then, there is no force acting on masses. The masses are just navigating the curved space-time. We cannot see the curvature directly. We only see its effects. And those effects are what we call gravity.

Now, Einstein’s theory of General Relativity rests on the equivalence principle. The equivalence principle says that locally the effects of gravity are the same as the effects of acceleration in flat space. “Locally” here roughly means “nearby”. And acceleration in flat space is described by Einstein’s theory of Special Relativity. So, with the equivalence principle, you can generalize Special Relativity to General Relativity. Special Relativity is the special case in which space-time is flat, and there is no gravity.

The equivalence principle was well illustrated by Einstein himself. He said, let us consider you are in an elevator that is being pulled up at constant acceleration. There is one force acting on you, which is the floor pushing up. Now, Einstein says, gravity has the very same effect without something pulling up the elevator. And again, there is only one force acting on you, which is the floor pushing up.

If there was nothing pulling the elevator (so, if there was no acceleration) you would feel no force at all. In General Relativity, this corresponds to freely falling in a gravitational field. That’s the key point of Einstein’s insight: If you freely fall, there is no force acting on you. And in that, Einstein and Newton differ. Newton would say, if you jump off a roof, the force of gravity is pulling you down. Einstein says, nope, if you jump off a roof, you take away the force that was pushing you up.

Again, however, the distinction between the two cases is rather technical and one we do not have to bother with in daily life. That is because in daily life we do not need to use the full blown apparatus of General Relativity. Newton’s theory works just fine, for all practical purposes, unless possibly, you plan to visit a black hole.

## Sunday, March 01, 2020

### How good is the evidence for Dark Energy?

I spoke with Professor Subir Sarkar from the University of Oxford about his claim that dark energy might not exist and the so-called Hubble-tension isn't.

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