The Woman Who Broke Gravity | Claudia de Rham

To be honest, I certainly had made a mistake. I really, I remembered going through it over and over again. It simply doesn't make sense. What if everything we thought we knew about gravity was wrong? What if the symmetries of Einstein's beautiful general relativity are broken at the fundamental level? We've been told that gravity is just the curvature of spacetime and not a force, but are we misguided? In this episode, we plunge into the world of massive gravity with Professor Claudia de Rham, a theoretical physicist at Imperial College London and author of The Beauty of Falling.

Professor de Rham has upended decades of scientific consensus with her recent radical theory suggesting that gravity itself may have mass, an idea that was long thought impossible. What does this mean for our understanding of the universe? Professor, please tell me, what's something about gravity that most physicists hold as an assumption that you think is deeply mistaken? Okay, that's a good one. I'm going to have most physicists against Me now. Okay, maybe there's two things which are probably related. The first one is the assumption that general relativity and our understanding of gravity as we have it

at the moment is based on Einstein's principles, which are the pillars of modern science, the pillars of general relativity related to the equivalence principle and related to symmetries. We don't need to have those pillars. I think you can derive the laws of gravity, you can derive general relativity, not based on additional assumptions and principles per se, but rather on the requirement that it is a self-consistent theory, that it is a stable theory, particularly when you embed it in a quantum field theory network. You can actually derive general relativity from the ground up, and therefore you

have Einstein's principles as consequences of stability and self-consistency as opposed to using them as original assumptions. That's probably one of them. Another thing which perhaps most of my direct colleagues would completely agree with, but as a whole, the general audience may think they disagree, although I think they would agree overall, Is the notion that gravity as described by general relativity is not a force. You may have heard that, and people say that. They emphasize a lot the difference between our representation and our understanding of gravity with other forces of nature, and I think actually that

is sometimes misleading. For many aspects, general relativity is actually a force like the other forces of nature, and we can describe general relativity as a force. Actually, the detection of gravitational waves is the proof that deep down gravity is a force. Gravitational waves is a representation through the stretching and squeezing of spacetime of the force that is hidden within gravity. Okay, well firstly, what are those symmetries you think that we can do away with when deriving general relativity? And then also, what is the definition of force? That's a good one. Definition of force is a

very hard one. Let me start with what I think is simpler, although maybe the words themselves are not as simple. What are the symmetries that general relativity, according to Einstein, is rooted within? It's a beautiful symmetry, which is the technical term is Perhaps coordinate transformation invariance. It is a non-linear diffeomorphism. That's a technical term, but actually that symmetry, in its sense, is very simple to understand. It is the realization that the laws of physics should be equivalent wherever we are, and however we describe a phenomenon, it should be equivalent independently of the observer. So whoever

you are, even though some elements are relative to one another, when it comes to measuring a physical quantity, we have to be able to describe it the way Einstein phrased it, in any frame of reference. So independently of how you decide to represent nature around you, however you decide to slice space and time, that shouldn't matter. That is your own prejudice. But then ultimately, when you come with something which is physically observable, we should all agree on that. It is a beautiful symmetry. It is a philosophical framework, if you want, in some sense. But to

my mind, this is actually so beautiful. It's something that is derived from the self-consistency of a theory, as opposed to setting it up as the basis, as the foundations upon which we're going to build General relativity. So that was the first question about symmetry. The second question is about what is a force. What do I mean by a force? I may get very deep to a point where terminology and vocabulary are not my strength. So what do we mean by a force? I think for most of us, including me, we think of force as a

contact force in itself. When I push against something, that's a notion of pressure. In itself, that's not really a force per se. It's more related to the electromagnetic bonds, etc. But that's not exactly what we mean by force, particularly not when we're dealing with the gravitational force. So if you feel you're sitting down, for instance, if you feel your chair, the bottom of your chair, this is not the gravitational force acting on you. This is the pressure and the contact interaction that is present between the cells in your body and the atoms on the chair.

And actually, that's related. That's very interesting. I'm going to attend to it. But that's related to Pauli's exclusion principle, which prevents two states, for instance, two fermions, two electrons, to occupy the same place at the same time. And so If we have already some of the states in the chair that are occupying a particular place at a particular time, I cannot just go through it. And so I'm feeling the effect from the pressure of this Pauli exclusion principle, which prevents me from having two states at the same place at the same time. That is not

what I mean by a force at the more fundamental level, which I more represent as something related to electromagnetism, for instance. I consider electromagnetism to be a force, and it is something that can act at a distance. So it's not something of a contact, it's something that can act at a distance. And we may be familiar with that with a magnetic force, for instance. If you take a magnet, and you go to take two magnets together, if you switch them side, switch them poles, then they will attract each other, or they will pulse each other.

And you can attract some iron with a magnet at a distance. That is happening through an electromagnetic force. And deep down, what happens is that there's a field there, there's an electromagnetic field that carries the force for us. And that can be represented Fundamentally in a particle, in a field theory level. For us, we understand the electromagnetic force as being carried by a messenger, which we call the electromagnetic waves, or which we call deep down the photon. The photon is a messenger for the electromagnetic force. And gravity can be represented as a force in exactly

the same way. So you may have heard of gravity, according to Einstein's theory of general relativity, is not a force, rather it is the representation of the curvature of space-time. And that is a beautiful, it's an extraordinarily beautiful and accurate description of gravity for all sorts of phenomena. But it doesn't mean that also deep down, it isn't represented as a fundamental force, just like electromagnetism. And so if we feel the gravitational attraction from the Earth, it is also mediated by a gravitational field, which is carrying a force. And fundamentally, the component of this gravitational field

is a particle, which we call the graviton. And there's a direct analogy there between electromagnetism being a fundamental force carried by photons, and gravity, which is also a fundamental force that's Mediated by a gravitational field and carried by a fundamental particle, which we call the graviton. Okay, so for the people listening, to go back, the general in general relativity comes from the theory being generally covariant, which is the same as being diffeomorphism invariant. So can you have general relativity if you remove that any coordinate system is an okay coordinate system? Isn't the whole point of

the language of bundles to describe constructs without reference to coordinates? So then do you remove the underpinnings of differential geometry to GR? So you don't want to do that directly, first of all, because general relativity is a beautiful framework that works extremely well in all sorts of settings. So as soon as you start removing some of the beauty of it directly, then you end up with elements which won't match observations. And as you said, it is a beautiful symmetry. It is also something that we like to have in a theory of a description of the

world where we would like to make abstraction as much as possible to anything which relies on us making a choice of any sort. And so in the world Of bundles or in different representations of general relativity, we can extract ourselves from expressing where we are in a frame of reference. However, there are some situations where we can think of a description of gravity, which is very well described by general relativity, up to a given level, but comes a point where this symmetry or this representation needs to have more to it. In reality, what happens is

that you never break everything down, but you start seeing a new structure emerging when you deep dive into it. To understand how to do that, you first need to embrace all of the beauty of general relativity. You need to understand how it works, and only in some special limit can you allow yourself to go beyond that and understand a generalized framework that has a symmetry and some limit, but then more generally, it behaves slightly differently. JS Now going back to a force, are you calling a force anything that has a force-carrying particle? And so that's

why you said gravity is a force because there's a graviton associated with it? HM So at the fundamental level, everything is quantized. It's Not because something has a particle associated with it that it will necessarily be a force, but fundamentally, all fundamental forces will, to my mind, necessarily have a particle as a messenger associated with it because everything is quantum. This is even more true for gravity because gravity connects to everything. And so if we know that gravity connects with a real, for instance, electromagnetism, for instance, electrons and other fundamental particles which are quantum, then

gravity has to be quantum as well. You can't just couple something which is fundamentally classical that satisfies fundamental classical probabilities with something which satisfies quantum probabilities. Actually, what you need to do is having a grander framework where everything is quantum, and in some limit, some sector may behave classical, but it's only in some limit. JS So what do you see as the difficulty in reconciling gravity or general relativity with quantum theory? HM We can do that, and we do that actually on an almost everyday basis in my work, but also in connecting with some people

doing observations when we're dealing with gravity in an Extreme environment. So when we're thinking about how gravity behaves on Earth, in the solar system, even in most of the galaxy, and in most of the universe actually, we can reconcile gravity with the quantum laws of nature and with the other quantum theories, the quantum theories of the forces, and there's no real issue associated with that. The distinction with gravity and where the problem arises is when we're trying to describe it when the curvature scale is very intense, so in a very extreme environment for gravity. And

there, what happens is if we took seriously the quantum laws of probabilities and we applied them to general relativity in those extreme environments, we would end up with some laws of probabilities that stop making sense. And so that doesn't mean that probability doesn't make sense. What it means is that we need a better framework to understanding how to reconcile gravity with those laws of quantum probabilities in those extreme environments. So we don't have access to all of the information, all of the description of how gravity behaves. So if you imagine that you are working with

some laws of probabilities, And at the end of the day, the outcome is not what you predicted, it must mean that something else must go on. Now for gravity, it's not like we've been in an extreme region in the universe, for instance, at the center of a black hole, or for instance, at the very beginning of the Big Bang, and tested gravity and the laws of probability there to be able to say that they don't work. That's not what we have done. Already, according to Einstein's theory of general relativity, when I use the standard laws

of quantum probabilities associated with it, I end up with outcomes which simply don't make sense. For instance, you can imagine that typically when you add probabilities, I have a probability for something to happen, I have a probability for something else to happen. When I add things up, I cannot end up with an outcome that has more than 100% probability to happen. But when we take general relativity and we're trying to apply the laws of quantum probability at the very center of the black hole, or for instance, at the very beginning of the universe, very close

to the Big Bang, I seem to be able to end up with outcomes which would have more Than 100% probability to happen, sometimes have a negative probability to happen, or a complex probability to happen. That simply tells me that I am missing something. I am not summing up all my probabilities given all the configuration that I'm allowing myself properly. Something is missing, and I need to understand gravity better. I need to understand how to go beyond the description of gravity using general relativity to being able to better appreciate how to reconcile my laws of quantum

probability with my description of gravity in those extreme environments. I see. Does it have to be that something's missing, or could it be that you overcounted in the case where it exceeds 100%? So even if it were that I overcounted, I need to understand why I overcounted, because the possible outcomes to my mind, not to my mind per se, but when we do the standard estimations, are possible outcomes, are possibilities which otherwise could have been realized. So if you want to think of an analogy, I can take two particles, which is something which is done,

for instance, in particle accelerometers. I can take two particles and I smash them together, And then I have a probability of a given outcome. The given outcome can be two other particles that are scattered with a different angle, or some of those particles may have transformed themselves into something different, and I can think of all the possible outcomes. And for general relativity, of course, I'm not going to smash gravitons together. I don't have access to gravitons, but I can perform those thought experiments, or I can think of doing them for other particles and understand what

would be the impact of having gravity in it as well. I can do all of this, and I can sum over all of the outcomes, the possibilities, and they seem to be realizable, and they seem to be making sense in themselves. So if now I'm overcounting in some of those extreme environments, I will need to understand what happens, what happens to those possible outcomes that are no longer a possibility at the center of a black hole. Irrespectively of what precisely happened, whether we are overcounting, undercounting, not counting it right, or giving too much weight in

some possible processes, when I say something is missing, it's not necessarily Something tangible is missing, but something in our understanding of what is happening is missing. So I want to get to this massive theory that you and your colleagues came up with in 2010, and literally massive. It's also outlined in your book, The Beauty of Falling, which will be on screen now, and people can click it in the description. The subtitle is A Life in Pursuit of Gravity. So firstly, why don't you tell us what is that book about and bring us through the journey

that led you to 2010 and that discovery? Okay, yes. So I can discuss about massive gravity afterwards. The journey itself is very much a journey towards the scientific exploration, the ups and downs of doing research, particularly doing research in theoretical physics, where the connection with the real world is still something that we need to develop and is ups and downs all the time. So it is about this journey as a researcher, and also associated with the journey of myself in going through different steps in my life, but also, ultimately, as a scientist. It is that,

and also in parallel, the journey that we have, us as humans, I would say, In our appreciation of the laws of nature, and particularly our appreciation of gravity, and how there's been ups and downs in our understanding of how gravity is being described. And actually, we are possibly hitting another possible down in the description of gravity, a failure or a falling down in gravity, related to what I was describing, that we know something is missing in our description of gravity. There's a point of failure, which in itself is part of how we do research. We

understand that we have a description of nature, we have a description of some phenomenon around us, which is quite fundamental, but is not fully basic. We need to go deeper in those laws of nature, in our description of nature around ourselves. And so, we can think of the points of failure, for instance, when general relativity breaks down, and we need to have a better description of gravity as one of the failures of general relativity, but actually is an opportunity to look for new underlying framework to better understand nature around us. So the book in itself

is framed within that premises of how we're doing research And how we're going along trying to understand things. But every day, almost, there's an up and down and embracing this level of knowing that at every stage, there will be a point where how we do research and how we connect with the world will have some elements of failure. And this is something to be proud of, actually, and this is something to very much embrace and keep exploring because it's through these points of failures that we're going to make discoveries and being able to get access

to new layers of understanding, new layers in understanding of how nature works around itself. So it's a description of this journey, but also associated with my own one, and through the understanding of a theory of gravity, which is very similar to Einstein's theory of general relativity, other than it is massive. And in that sense, being massive doesn't mean that it's huge. It's related to the fact that- It does sound like that. It does. It's a massive thing. Actually, it's quite the opposite. It makes gravity smaller in some sense, but I'll explain what I mean by

that. Yes, less far-reaching. Exactly, less far-reaching. So the idea behind massive Gravity is related to trying to tackle some of the other problems we have with gravity. And those are related not to the reconciliation of gravity with the quantum world, which is what happens when we're looking at very extreme regions of the universe where the curvature or the landscales, the curvature is very high, or the landscales are very small, like at the center of a black hole at the beginning of the universe. Rather, it's exploring what happens on the other side of the spectrum, where

we're looking at very, very small curvature scales. So not on the smallest possible distance scales, but actually quite the opposite, on the largest possible distance scales. What we mean by that is that we're exploring the behavior of nature around itself on cosmological scales, on the largest possible observable scales in the universe. So the size of the observable universe itself, which is spanning thousands of trillions of kilometers apart, as is the size of the observable universe. And so the idea is to try to understand how we can reconcile the behavior of the universe as we see

it cosmologically, with expectations from gravity and expectations from The other fundamental phenomenon in nature, in particular, the realm of particle physics, which postulates the existence of particles describing all the fundamental constituents of matter and the fundamental constituents of the forces. Now we know from particle physics that, for instance, the Higgs phenomenon is a phenomenon where the vacuum is not empty, it's filled with a Higgs vacuum, it's sort of a Higgs bath. It's a Higgs bath where you have interactions between the Higgs and all of the other massive particles, and it is these interactions between the

Higgs vacuum, this Higgs bath, that changes the dynamics of some of the massive particles, and in fact, give a mass to some of those particles. So we understand the Higgs phenomenon as the phenomenon that gives a mass to other particles of nature. So this is just an illustration to tell you that empty space is not at all this boring thing where nothing happens. Most of the universe, from our eyes, from our point of view, is filled with emptiness. We have huge cosmic voids, which are millions and millions of kilometers wide. They are huge. Most of

the universe is actually empty In that sense. It's filled with cosmic voids where galaxies and clusters of galaxies and filaments of dark matter are. That's just filaments that take in itself just a small fraction of the universe, and most of the universe is actually made out of these cosmic voids. And they seem empty from our perspective, but actually, at the very least, they should be filled with this Higgs bath because we know that it's thanks to this Higgs bath that other fundamental particles carry a mass. But now this Higgs bath or the vacuum should also

carry energy from possibly all the other particles. And that, according to Einstein's theory of general relativity itself, because of the equivalence principle, because gravity is so universal, if it has an effect on other particles, it should also connect with gravity. And so we should expect this Higgs bath and this energy in the vacuum to gravitate, to have an effect on gravity, to curve the structure of space-time. That is not something controversial. This is something that has been developed, has been understood already since the 30s, since the understanding of Einstein's theory of general relativity. And then

from the Beginning of quantum mechanics, the quantum realm of mechanics by Pauli and all the other fathers of quantum mechanics at the beginning of the last century, it was already understood that we should expect emptiness to be filled with something. That seems paradoxical, but emptiness should be filled by something and that something should gravitate. And if it gravitates, then we should expect it to have an effect on the evolution of the universe. And in fact, it was already understood already in the 1930s that the effect of, for instance, the energy of the electrons in the

vacuum should lead to an accelerated expansion of the universe, which would be going so fast that the space between us, the Earth, and the Moon should be stretching at a speed that exceeds the speed of light. And therefore, if we put those two things together, we shouldn't be able to see the Moon. Just a moment. So if in the vacuum there are these, are you referring to the virtual particles? Yes. Yeah. So if each of those virtual particles, and there are an infinite amount, if you go all the way down to zero, not the Planck

length and stop there, but there's an infinite, okay, whatever, There's a large amount, even if you put a cutoff, and each of those has some energy associated with it. That's right. And energy is associated with gravity. So you say they should gravitate, you mean that they should exert a gravitational force, but why would that force be repulsive and not contractive? Oh, okay. So that's very interesting of why the effect of this phenomenon is something which doesn't seem to be the same as an apple falling on the Earth, which I will classify, we typically classify as

something attractive, as opposed to the acceleration of the universe, which we seem to be looking at as a repulsion phenomenon. And indeed, if you look at this phenomenon, according to Newton gravity, you would think that this pushing away this accelerated expansion of the universe should be related to a repulsion or should be related to having an effective negative mass in there. That's how you would describe it according to Newton gravity. But we are in Einstein's theory of relativity, where things don't just happen in one dimension of time, or one dimension of space. What happens is

an intertwinement between space and time, Unified together. And so you have things happening in space, and you have things happening in time. Now, when you have an effect of energy, which is localized, you can think of this as something that happened in space to some extent. And it gives you what you would think a localized mass here, for instance, the Earth and us being attracted by the Earth. This is an attraction, but this is something which in the realm of general relativity is only in one special dimension, which is more related to time than to

space. There's a bit of technicalities there. Now, in general relativity, if something happens in space, it's also happening in time and vice versa. And for this vacuum energy, it's not just localized here at one point in space, it's everywhere all the time throughout the universe. And so it acts on all four of our dimensions. And the way it acts in the space and the time dimension are opposite. And since we have more dimensional space than we have of time, actually what happens along the space dimensions wins over. What happens along the space dimension looks like

it has the opposite sign as what happens in the time direction. And that's why It looks like you have a repulsion. But it's all attractive. It's just what attraction looks like in general relativity may have different ways to manifest itself. Attraction is something that according to space and time, it looks slightly different in how you represent itself. Maybe another way to see it is that rather than just being some energy localized in space, it is actually also some pressure which is localized in space and time. And this pressure has a negative sign. So it manifests

itself as something which looks repulsive in all direction, but actually that's just a manifestation of a negative pressure. So then in 2D gravity, would there be zero effect because there's one time and one space? So the way acceleration of the universe would work would look slightly different. That is correct, yeah. But there would still be a positive effect, like a repulsive effect? So you would still have a stretching of space. You would still have the stretching of the space direction in itself, yes. Okay, so now that we're on some opinions of yours that are controversial,

what's your controversial take On the expansion of the universe? In other words, tell me the truth behind dark energy, professor. Okay, so dark energy is, I think that's not controversial to some extent. Dark energy is a placeholder for a lack of knowledge of what leads to the accelerated expansion of the universe. So we don't exactly, or we don't say necessarily we all agree on what leads to the accelerated expansion of the universe. So let's put us to our common ground, and let's call that the source for the acceleration of the universe, dark energy. Now there's

a different perspective on what dark energy could and could not be, but there's a very natural candidate for what dark energy can be, which is this vacuum energy, because the vacuum energy by itself leads to an accelerated expansion of the universe. So that is, I would say, a very natural candidate for the acceleration of the universe. However, it is controversial because the predicted rate of acceleration of the universe, if dark energy was indeed the vacuum energy, would be way too large, much faster than what we observe today. And that I already alluded to when I

was saying that if you just take the Vacuum energy from electrons that we know exist, it should lead to such a fast stretching of space, such a fast acceleration of the universe, that we wouldn't be able to see the Moon. That, of course, is not what is happening. So we do have an accelerated expansion of the universe, but by a rate which is way slower than what we would have expected if I accounted for all of this vacuum energy from particle physics. So faced with this dilemma of why doesn't the vacuum energy lead to a

much higher rate of accelerated expansion of the universe, instead, what one can do is say, well, maybe for a reason or another that I haven't yet found, this vacuum energy actually doesn't gravitate. Or maybe it's not there altogether, or maybe I don't understand it. So let me ignore it. Let me put it aside for a second, or for 100 years. And then instead, let me say there's another source for the accelerated expansion of the universe, which we call dark energy. The reality, something we should understand, is that if you want a natural source for the

accelerated expansion of the universe, which actually leads to the accelerated rate that we observe, this Issue that anything natural we would expect would still lead to a much higher rate of accelerated expansion is still there. So you have candidates for dark energy. There are various models. I can't come up with 100 different names of models of dark energy that explain the accelerated expansion of the universe. In every single one of them, you need to what we call fine-tune some parameters. So you need to really stretch some screws to a huge level of accuracy in a

way which is unstable. So if you look at the smallest quantum corrections to that, it will be much larger as compared to the level you tune it at. So it's unstable on the quantum corrections. All the models of dark energy that we have so far, or 99% of the models of dark energy that we have so far, are unstable against quantum correction in one way or another. So we will need to resolve what we call fine-tuning issues in those models as well. But in some cases, they're much better hidden, so it's harder to find where

the trucks of the matter lies. So that's where we're trying to come in. That's where we're trying to say, rather than postulating the existence of a new explanation for the accelerated Expansion of the universe, dark energy, which typically also comes with tuning in itself and is not stable on quantum corrections. Instead, let's just go back to this original idea that the vacuum energy is a natural candidate for the accelerated expansion of the universe. And for that, we need to understand why this huge level of vacuum energy doesn't lead to as high a level of acceleration

as what I would have expected. And there's a lot of models out there that try to address the vacuum energy itself, try to understand what it is in the quantum field framework. I haven't quite understood, so that the expected vacuum energy is not as large, it's actually much smaller. So there are some models that do try to do that. There's no successful model so far, but they try to do that still. Another alternative is to say, okay, well, let me accept, actually, that this is the way it is from the particle physics side. Because after

all, we have a very high level of control of what happens in the particle physics side. We understand particle physics very well, we actually have access to it, we understand how quantum corrections work extremely well. We look At quantum correction over quantum correction over quantum corrections, and this is very much under control in the particle physics side. So let me take that as it is, and instead, it is in making the connection between this and gravity, and understanding how the vacuum energy affects gravity, that I will try to tweak things. It's interesting, okay. And so

this is where I cannot have just general relativity, because general relativity, in all its beauty, it has this level of universality that everything and everyone is affected and affects gravity in a universal way. So I can't just decide that someone is going to be affected or is going to be affected by gravity in a different way. I'm not allowed to do that in general relativity. It is one of the pillars, to some extent, of general relativity. So if I want to have a phenomenon, if I want to have vacuum energy, which doesn't affect gravity in

quite the same way as other sources, as, for instance, a galaxy, as localized matter, then I need to tweak some of these pillars of general relativity every so slightly. Okay, you have to tweak it, but do you have to fine-tune it so that you're replacing One problem with an equally intractable problem? That's an excellent question. So how much is that tweaking fine-tuned in itself? That's an excellent question. And you have to tune it. You definitely have to tune it. The difference in the level of tuning that you need to make is that it is stable

under quantum corrections. So you need to have a very small number in the game. Whatever you do, however you try to resolve this paradox, you need to bring in a very small number in the game. In this case, the very small number is by how much you're modifying general relativity. And you do so with a very, very small, with a very, very thin brush. But when you have general relativity in itself, it isn't spontaneously affected by quantum corrections. So all of the symmetries of general relativity are protected by quantum corrections. If you have a symmetry

which is present and you do look at how quantum corrections affect them, they will preserve those symmetries typically, unless you have anomalies, but that's for a different question. So this coordinate invariance, for instance, this symmetry which we call covariance or diffeomorphism invariance, Or this notion that every observer should be equivalent irrespectively of their frame of reference, this is not a principle that gets modified by quantum corrections. So if you start with that, then you're not going to expect it to be modified by quantum corrections. But that means that if you depart from this symmetry every

so slightly, the amount by which you're going to be destabilized by quantum corrections is going to be proportional to how far away you are from it in the first place. So if you only displace it every so slightly, you're never going to move very far away from it. So in that sense, you have a very, very small number in the game, which is a small modification of gravity that you bring in, but this small modification doesn't start going completely ballistic. It doesn't start going completely crazy in the sense that it doesn't start growing over time,

if you want. So in our framework of quantum mechanics and naturalness, it is a tuning, but a technically natural tuning. So it's not a fine tuning in the sense that you don't need to keep tuning it at every level in your quantum corrections. But you understand That this is the case. I can say those things, and you may or may not believe me, but you understand that this is the case. You need to have actually a very rigorous framework in which you can explore those ideas and in which you can look at the quantum corrections.

And see how they affect this small modification of gravity. And this is precisely this idea behind massive gravity. It is an idea of a framework where gravity has a very, very small mass. The graviton, to be precise, the particle carrier of the gravitational force, the graviton, unlike being massless, as would be the case in general relativity, it acquires a very small mass. It's extremely small. It's the smallest possible mass that you can ever envision. It would be of the order of 10 to the minus 32, 33 electron volts. By comparison, the neutrino mass, which is

the lightest massive particle that we know of, for sure, that has a mass of 10 to the minus 3 electron volts, milli-electron volt, 10 to the minus 3 electron volt, roughly. So it's roughly 30 orders of magnitude below that. It's extremely, extremely small. So this very small departure of general relativity, You can now look at how the mass of the graviton could get corrected by quantum corrections. You know that in general relativity, the graviton is massless, so the mass is zero. And you know that you don't start asking yourself, is it true that the graviton

remains massless in general relativity when you include quantum corrections? Because the masslessness of the graviton is also related to the symmetries of general relativity. It's related to this equivalence principle. It's related to this universality. It's related to covariance. All of that is part of a big package in general relativity. And so, you can't just shake it a little bit around. It's a package that sticks together. Now we started to unfold it a little bit and to allow for the graviton to be every so slightly massive. And the amount by which all of the other implications

will start losing up is present, but it's the same amount by which you had started shaking it in the first place. And so, if the mass of the graviton is extremely small, you're going to have a small correction to the equivalence principle. You're going to have a small correction to all of the Phenomena that we discussed about, which is proportional to this graviton mass. And so, it also means that if you have a source, which is present here, like the Sun or the galaxy, or even a cluster of galaxy, these are distant scales, which are

still very small as compared to the quantum wavelengths of the graviton. So, the quantum wavelength is related to the inverse of the mass of the particle, if you want. So, if you have massive particles, one of the aspects of it, which we are after, is the fact that the force associated with it will have a finite range. So, it won't have an infinite reach, like in general relativity. It would actually just reach over a finite distance, and that's related to the quantum wavelength of the particle. And so, any structure that we're used to in the

universe, for instance, the solar system, a galaxy, even a local cluster of galaxy, will be within the quantum wavelength of the graviton, and therefore, a relatively small distance as compared to the scale at which gravity gets modified. And therefore, on smaller distances, gravity looks very similar as in general relativity, and we don't see a very Big departure from general relativity. It's only when you start looking at effects, which are much larger distances on the scale of the observable universe today, that you start seeing a departure of how those effects lead to a curvature of spacetime.

And this is precisely where the vacuum energy is coming in. So, we have vacuum energy on the whole of the universe, since the beginning of time, over billions of years, over billions of light-years across in distance. And over those huge distances, this is where we start seeing a weakening of gravity. And therefore, the effect of this vacuum energy on gravity, on cosmology, and on the evolution of the universe is much weaker as compared to what we would have expected in general relativity. Interesting. So, does this then give an alternate explanation for dark matter? Ah, that's

a good question. So, if you want to understand dark matter, you can, and people have indeed tried to understand whether you can use a similar framework to understand dark matter. Naturally, it is difficult to do both at the same time, because the scale involved is actually quite different. We have a very good understanding Of the presence of dark matter or something that looks like dark matter already on galactic scales that is very much present there. And so, the scale at which you need to see this effect to consider it as an alternative to dark matter

would be on much smaller distances as compared to what you would need to have it for dark energy. So, in principle, you can do this in different layers, and you can have a modification at one scale and then another modification at another scale. You could do that in principle, but the reality is these are relatively separate phenomenon. So, you might as well just consider first what happens for dark matter, and then as a separate phenomenon, what happens for dark energy. Now, people have tried to do that for dark matter indeed, and their models were similarly

to considering what would happen for massive gravity. What if gravity had a mass? What they're considering is a model of what we call multi-gravity. So, rather than having just gravity as we know it, there's many different layers of gravity. There's many different notions of gravity, and you can have some of these alternative gravities which act As dark matter for observations. So, this is not something I have worked on. These are the models that are taking on some of the aspects of massive gravity, bringing them on top of gravity itself, on top of general relativity, as

a new source for dark matter and for other phenomenon in cosmology or in particle physics even. I see. So, for people who are listening and thinking, okay, a massive graviton, the graviton was thought to be massless. Why didn't physicists think about a massive graviton earlier? They did, and there were two problems. One was the VDZ. V-D-V-Z discontinuity, if I'm not mistaken, which we can talk about. And then another was that there are some ghosts. There are two types of ghosts, generally speaking. Yeah, exactly. Exactly. There's the Faddeev-Popov, which are the wanted ghosts or the benign

ghosts, and then there's Pauli-Villars, if I'm pronouncing that correctly. Yes, that's right. Exactly. Exactly. Okay, so you got everything right. So people have indeed considered the idea that gravity could have a finite reach, which is the essence behind massive gravity. In fact, I should say, Newton himself, he thought About this idea that according to his law of Newton's law, square law of gravity, as he added, this is a phenomenon that has an infinite range. So gravity gets diluted like the square law, so it gets diluted like the distance, and this is in itself very geometrical.

But himself, he was thinking about whether gravity could have a finite reach at the end of the day, and trying to understand how to make sense of that from a Newton perspective. Other scientists since then, like Laplace, also considered that. Now, if you wanted to do it just at a level of Newton law, that wouldn't be too challenging. The challenge is to do it at the level of a fully-fledged nonlinear theory of gravity as in Einstein's theory of general relativity, with everything that we know about general relativity, and then further embed it into a quantum

field theory framework, as we know has to be the case. And so, since we know much more since then on how gravity works, we need to make sure, in thinking about our theory of massive gravity, that it still satisfies all of the other qualities of gravity as we know them. And in fact, Pauli himself, in the 1930s, Fierz and Pauli, they first started looking at a theory of gravity where the graviton could have a mass. One of the issues was pointed out by what you mentioned, these VDVZ discontinuities, and that was in the 70s. So

VDVZ stands for Van Damme, Veltman, and Zakharov. In the same year, in 1970, they realized that if you take just a theory of massive gravity, and then you compare it with a theory of general relativity, you seem to be getting some effects which are different for both cases, even in the limit where the mass is extremely small. So you may have the impression that you can look at, let me call it, the force of gravity between the Earth and the Moon, and what you would obtain in general relativity is the result that we know of,

and what you would obtain in a theory of massive gravity is a different result, which is different even when the mass is as small as you want. It's different by order one, no matter what. And the reason for that is quite simple to understand, actually. And it's going back to this idea of what carries the force, what happens in there. Okay. And the idea that you have a messenger for gravity, which is Related to gravitational waves. The real force of gravity is carried by gravitational waves, by gravitons through gravitational waves. And that part is uncontroversial?

That's not just you saying it? No, that part is uncontroversial. Got it. Uncontroversial. So this is fine. I'll tell you when I start. I'll stop being controversial. The controversy is whether you can have a theory of massive gravity. That's where the controversy is. But in terms of how gravity behaves and what the issues were at the time, all the way up to a few years ago with massive gravity, that's also uncontroversial. So I can go through them and what those issues were. So if you think of gravitational waves that we have detected, we have detected

gravitational waves coming from very far away events. And the way they work is as the gravitational waves propagate through space and time, they actually affect the notion of distance. They affect space along the line orthogonal to the line of propagation. So they are what we call transverse polarizations. So I should do it like this. Sure. It goes like this. You have a squeeze in the stretching like This in the opposite direction. It's what we call a quadrupole. And they go along the line transverse to the line of propagation. That's uncontroversial. Now, if you think of

a theory of massive gravity, what happens there is that rather than being a massless particle, you have a massive particle. And so if you think of the idea that light travels at the speed of light in the vacuum, light travels at the speed of light in the vacuum because it is a massless particle, because it's carried by a massless particle. You and me, no offense, but we are massive objects in the sense that we're very important. We are massive objects, and I don't typically travel anywhere close to the speed of light because I'm quite massive.

And so if you have a massive object, you no longer travel at the speed of light anymore. You actually can't. You can try to go very close to it, but you can never actually go quite at the speed of light in the first place. But that means also what we can do, though, is control our speed. I can decide to be at rest, and I can decide to speed up or slow down. I can do those things. Being a massive particle is actually quite a positive thing. And So the same thing would be true for

massive gravity. And gravitational waves could also speed up every so slightly or slow down every so slightly. And so that means that in addition to just having polarization which are transverse to the line of propagation, you could also play with the longitude in our direction, a little bit like sound waves. If you're thinking of how, not for you and me right now, but how you hear each other, it is a sound wave and it's a compression of the air pressure and more pressure and less pressure in the air along the line of propagation of the

wave. Or just if you drop a stone on a pond, you'll see some waves traveling along the surface of the pond. And those are what we call longitudinal waves, because they go along the line of propagation of the wave. And so in massive gravity, you have this additional freedom as well in how gravitational waves can evolve. And this additional freedom may seem like, okay, it's great, you can do that as well. It also means that in terms of the force of gravity, because it carries additional channels in which gravity can be mediated, you would expect

gravity in Massive gravity on short distances to actually be stronger. And that's counterintuitive. We came up with a theory, or not we, but overall when people think of massive gravity in the sense that the particle has a mass, one wants to do that because it weakens the behavior of gravity at large distances. But what seemed at the time a price to pay for that would be to have additional channels of communications for gravity. So an additional way to transmit the force of gravity through longitudinal polarizations, which would also mean that gravity on shorter distances would

seem at the time to be stronger. And so this is what V, D, V, and C discovered in 1970, that because of this additional channel, which seems to be present even when the gravitational mass is as small as you want, that means that there's a discontinuity between what happens in term relativity, where the mass is exactly zero, as compared to what happens in the massless limit of massive gravity. So that was the original issue, controversies set up by V, D, V, and C in 1970. But then two years later in 1972, Weinstein understood what happened.

Arkady Weinstein came along and realized that Actually the remit within which this understanding was done, these calculations were done, and how you think of a notion of force in terms of being mediated by these different channels, these different polarizations, does make sense in some limit. But when you really want to understand what happens in very small masses, actually you can't just neglect all sorts of other things that should otherwise be present. So actually while it is true that those additional polarizations are present when the mass is finite, actually when the mass is becoming very, very

small, it becomes extremely hard to excite them. They actually themselves freeze in some sense. But you understand this freezing mechanism, what we now call actually the Weinstein mechanism, it's a screening mechanism. You need to understand how the self-interactions of gravity allow for specific polarizations, which otherwise would not be there in general relativity, to freeze themselves. So it's almost if I'm taking too much… What is meant by this freezing? So it's almost as if I'm putting too much luggage on myself that I'm no longer able, it completely inhibits my motion And my ability to communicate. So

when we're thinking of the force being mediated by gravity at the level of this different polarization, we have a very simple picture in mind. But when the mass of the graviton becomes very, very small, this additional polarization interacts with itself. So it's almost playing the role of a honey, in which it prevents its own dynamics. It's no longer free to move at wish, and it does that by itself through its own interaction. So it is what we call a phenomenon of strong coupling. The self-interactions of the graviton in that particular sector become so important that

they resemble something which is very different as compared to what one would have expected in the first place. So Weinstein understood that a phenomenon like that had to be the case. But to understand how this is implemented in practice, how it happens in practice, you need to have a fully nonlinear theory of gravity, where all of the nonlinearities, all of the interactions of the graviton come along, just like would happen in general relativity. Now you can think of what happens for general Relativity when the interactions become very strong. When the curvature becomes too strong, you

can imagine having a black hole. You can imagine that if you have general relativity and you have a regime where actually not necessarily the curvature, but the nonlinearities of gravity become important, that's where you're actually quite far away from Newton and gravity. And this is precisely what happened at the onset of a black hole horizon. A black hole is precisely where things will be very different as compared to what you would have expected in Newton gravity, because you're no longer in the weak gravity regime. You're starting having important interactions for gravity. This is the whole

realm of black holes testing some new aspects of general relativity in a regime that wouldn't otherwise be the case in the solar system, for instance, where even though we do understand the subtle difference between Newton and gravity and general relativity in the solar system, there are still very subtle differences. They're not all the one difference. But when you get close to a black hole, actually the difference between Newton and Gravity and general relativity, they are very big. They're very noticeable. Now for massive gravity, in addition to what would seem to be this distance associated with

the size of the horizon of a black hole, you have an additional distance scale related to where the nonlinearities now for the additional polarizations become important. So you have two distance scales. You have the nonlinearities that are important for the additional modes of the graviton, and then you have a much smaller distance, which is your standard Schwarzschild radius if you want in Einstein's theory of general relativity. I don't know how familiar people are with the Schwarzschild radius and the idea of horizon. So what we need to understand is how to make this transition between what

happens at very, very far, very big distances, where there we understand gravity should be weaker, and it is in a linear regime. But as you go to look in a theory of massive gravity, and you start looking at smaller distances, you need to start kicking in the nonlinearities for some sector of gravity, which will then suppress the effect of The additional polarization and the departure of massive gravity as compared to general relativity within that radius. We call that the Wernstein radius. And when the graviton mass becomes very, very small, this Wernstein radius becomes larger and

larger. And as the graviton mass becomes zero, this Wernstein radius becomes infinite. And so the whole universe is within its own Wernstein radius, which means that it looks identical to GR. We need to understand the nonlinearities to make that happen. So it exactly smoothed it out. It didn't just temper the discontinuity. That's right. It smoothed it out. And so now we have exact realizations of massive gravity where we can see precisely this transition, where we can think of, for instance, the force between the Earth and the Moon in a theory of massive gravity. And we

know precisely what happens when the mass of the graviton is smaller and smaller and smaller, and we recover precisely the same result as in general relativity when the mass is exactly zero. So we understand that. And therefore, in a theory of massive gravity, if the mass is sufficiently small, that's what we would Want anyways, what we know is that the prediction for gravity in massive gravity would be extremely similar to what they are in general relativity. The departure would be extremely small. It doesn't mean that we may never see them in the solar system, because

actually we have very, very precise tests of gravity in the solar system, but they are very, very suppressed. Okay, so let's see if I can do a summary so far. So the VDVZ says that if you have a massive gravity theory, sorry, not VDVZ, but generally, you should have your theory as you take some parameter and you deform it down to zero, agree with a theory where it has zero if you're trying to recover that theory. So for people who are familiar with quantum mechanics, there's H-bar and the correspondence principle. So what that means is,

as you set H-bar to zero, if H-bar is supposedly supposed to measure the quantumness and you set H-bar to zero, then you should recover classical mechanics. There's a way you can do that with a theorem, although there's some hand waviness, I believe, but it doesn't matter. The point is, there's no discontinuity there. And then you would think, Okay, well, if I have a massive theory of gravity, if I have a massive graviton, it should be straightforward to just put that mass down to zero and recover GR. But it turns out you don't, and that's quite

odd. But then the reason why it's odd is because this Van Stein guy realized that we weren't taking into account the nonlinearities and the other mode of polarization. That's right. That's right. Exactly. So you got that exactly right. There's a lot of subtleties going on, but this is exactly the gist of the story so far. So we're still in 1972, and then the story is not over because still in 1972, the same year as what Van Stein came along and resolved this vDVZ discontinuity and said, hang on a second, it can't just be the calculations

that were done so far, where they were much more similar to what happens in Newton and gravity as what happens in a fully nonlinear theory of gravity like general relativity. And he said, you need to account for all of those subtleties. In reality, already in general relativity, we need to account for many subtleties that arise for nonlinear interactions. We need to account for them because that's Precisely how we understand black holes. This nonlinear effect of gravity is important there. So it is the case that general relativity is also something that has very important interactions that

manifest itself in specific frameworks. And so for massive gravity, it also has to be the case that its interactions have to be accounted for. And we understand how accounting for these non-trivial interactions help us understanding how to take the small mass limit of gravity smoothly to zero. I should say, it's not an ad hoc, it's not something we'll put in by hand at the end of the story to fudge things so that they work with observation or so that they satisfy a principle that we had imposed on ourselves in advance. It is something that comes

in naturally. You're thinking of a theory of gravity, naturally it has to be something which carries non-trivial interactions. And when you account for these non-trivial interactions, naturally you see this Wernstein mechanism emerging and a smooth limit to general relativity when the mass is small. So it's not something ad hoc, it's something fully fledged in the model itself. But then in the same year, What happened in 1972, Stanley Deser and Boulware, so two physicists, realized that when you account for these non-trivial interactions in massive gravity, they seem to always come in hand-in-hand with what we call

a ghost. And it is a ghost which seems to be present in the physical sector. All right, please explain what ghost particles are. So a ghost should not exist. People who are not physicists, they just think that this is a different channel now. I think the new Ghostbusters movie came out. Yes, that's basically what you did in 2010. And I have a Ghostbusters t-shirt. I'm not going to go too crazy, but the word sounds slightly science fiction, but it's all really based in the realm of very scientific. I didn't come up with the term ghost

myself. It's something that was known since 100 years as well, or since the 1930s, so yeah, almost 100 years already. So a ghost is, at the most basic level, a particle with negative energy. What we mean by negative energy there is negative kinetic energy, which means that if you have a ghost and you make it run around at a given speed, its energy will decrease. So it will release energy to the system. And the faster it goes, the more energy it releases to the system. And the more it does that, the more other particles, which

are normal particles, can absorb that energy. And that means that the universe as we see it would be completely unstable, because you would have this possibility to trade up positive and negative energy. So we, all of us, would be allowed to be as excited as we want and take over the energy that the ghost is releasing, and by making the ghost go as fast as they want. So this seems like a very unstable process. It is different. It is different from having a particle whose, you may have heard of having a particle which has a

negative potential or a negative mass square. Some of the phases of the Higgs boson may have had that in their history. That is just a transition phase. When that happens, it's a tachyon. It's just a transition phase where, for a while, for instance, you go up the hill with your bike, and when you're at the top of the hill, you can very quickly speed down the hill. That's just a transition phase up to the point where you find, again, yourself at the bottom of the hill, and Then there'll be no way to go from there

on. Just when you're at the top of the hill, you're not on the most stable position, but then you always can go and decay to something which is stable, and that's around that stable point. The point is that it's bounded from below? It's exactly that. Yeah. It's bounded from below. Because some people will say the problem is negative energy, but would it be more accurate to say the problem is that it's not bounded from below, you could have negative energy that's bounded? Exactly. That's exactly right. You're absolutely right. So the problem is that it's not

bounded from below. So you're absolutely right. So a tachyon is a particle with negative energy, but it will still be bounded from below. In that case, for the tachyon, it has negative mass squared, and it will still be bounded from below once you find the right vacuum, whereas for a ghost, it's unbounded from below, and you know it's unbounded from below because you can decrease its energy by going faster and faster, and there's no limit to how close to the speed of light you want it to go. So it can go as close to the

speed of light as you want, And that would lead it to as a negative energy as you want it to have. So that's the real issue with a ghost. You're exactly right. The real issue is that it's unbounded from below, and so there's no sense in which we can start a life around a stable vacuum where the particles don't go all crazy, and where we can build a model from the ground up from that stable vacuum, because there's nothing to start from. There's no ground basis. There's no ground zero. The energy is unbounded from below

in the case of the existence of a ghost, of a ghost particle. And that's for any type of ghost? Yeah, so this is in principle for any type of ghost. So if the ghost is really there in your theory, as seemed to be the case as observed by Boulware and Deser in 1972 for massive gravity, then it is an issue, and then that's it. This is a bit different from the Fadiyev-Popov ghosts, which are not really ghosts which are present in your theory. For the Fadiyev-Popov ghosts, they are a mathematical trick. They are a trick

in when you try to quantize some fields which have some nice symmetry embedded into them, sometimes it's easier to use an extended framework where You have the impression that they have additional ways to excite themselves, which is not the case, and then to cancel them directly one by one with Fadiyev-Popov ghosts. So the Fadiyev-Popov ghosts don't really exist, and the additional modes of the fields that you're trying to quantize are not really there, but you promote them to being there because it makes it easier to look at all this formulation. So it is a mathematical

trick in some sense, but you do so that there's an exact cancellation between two pairs, two pairs, etc., of Fadiyev-Popov ghosts and what would otherwise have been a mode that you artificially introduced in your theory. So it's not something physical, it's a mathematical way to shortcut some of the framework that you're trying to establish when you're trying to quantize it. So the Fadiyev-Popov ghosts, even though the word ghost is present in this case, it's a good ghost in the sense that it's been specifically engineered to patch another mode that you artificially introduced. So everything is

under control there. The terminology seems to be the same, but in this case, it's doing surgery in a way that you know precisely What you include, like with like, in such a way that everything is fine. But for massive gravity, no one ever came along and said, oh, I'm going to introduce a ghost so that it can patch something up. No, it just comes up by itself. And that is really the issue because it is there, it is physically there, and as soon as it is present, it leads to energy being unbounded from below. Its

very existence means that it can reduce the whole of the energy of the universe to as much as it wants by simply going as fast as it wants. And that's the real issue with that. Now, I'd like you to take us through to 2010 when you and your colleagues circumvented that ghost and what that experience was like when you busted that ghost, I should say. It was exactly like in the movies, exactly like that. So yeah, so that was in 1972, and Boulware showed that in a theory of massive gravity or in a theory of

gravity that has a finite range, it seemed to be always the case that they come hand in hand with the existence of a ghost. Whether you want it or not, it is present. And many different people re-explored that analysis, Boulware, re-explored this in Different languages. Actually, you can think of it at the level of particles, at the level of energy, various different types of levels. And then in the middle of the 80s, et cetera, people had given up on the idea of massive gravity because it seemed to be always the case that you can't make

it work. To have a smooth limit to general relativity where the graviton is massless, you need to implement these nonlinearities, but these nonlinearities always seem to lead to an excitation of a ghost, which seemed to be impossible. So that was up to the mid 80s. But in 1998, there was the confirmation from different groups, from supernova observations, that the expansion rate of the universe was not slowing down as was expected. Rather, it was going faster and faster. So 1998 was when it was confirmation that the universe is accelerating. And so from there on, there was

this whole emergence of new ideas in understanding whether it is the vacuum energy, whether it is something like the cosmological constant that also plays the role of the vacuum energy, whether it's dark energy and all these tuning and self-tuning Issues. And so along with all of this, we came to the idea that gravity could possibly be modified. It could have a finite range so as to tackle what we call this cosmological constant problem or the vacuum energy problem related to the accelerated rate of expansion of the universe. And this is not where I came in.

I started thinking about those ideas a few years later. I was doing my undergrads at the time. I started my PhD in 2002. And there, it was already well-established. People had tried again to understand whether you can have a theory of massive gravity which would have no ghost. And it seemed to be always coming up with the same impossibility to understand how to make the algebra of the interactions work out in a way that there was no ghost. And actually, throughout the noughties, I think you call this the noughties from 2000? Oh, yeah. Okay. Got

it. Is that the noughties? The zeroes, yeah. The zeroes. There's been a... Whatever that period was called. The Lady Gaga period. That's how I remember it. Okay. That's a good name. That's a good name for it. So during that period, there's been quite a few very systematic papers being published going Precisely through the different proofs in different languages, with different logics, ensuring how for every single one of these ways of thinking about it, we always end up with the same issue that you can't have a theory of massive gravity with these nonlinearities without also coming

up with a ghost. So it concluded in a set of what we call no-go theorems. A no-go theorem is exactly what the name indicates. It tells you that it is a rigorous mathematical theorem, which the answer is no. There's a no-go. You can't do that. And what was this theorem's name? The no-go theorem, or the no-go theorems in this case. Yes, there was no-go theorems for massive gravity. You can look it up, no-go theorem for massive gravity or for finite range gravity, where you go through an analysis. They wouldn't have had a better name than

that in themselves. Okay, so it's not like the Coleman-Mandula theorem that doesn't have a name. No, it wouldn't be like that, yeah. So this was the situation where I was a researcher, in fact, at the Perimeter Institute and at McMaster, and then in Geneva in late 2009-2010. And instead of thinking of theory of massive gravity, because At the time I was convinced that all of this made sense, we were trying instead to look for a model of gravity which is based on having extra dimensions. Extra dimensions were very big at the time. Still are. Still

are, yes, yeah. That could have some of the properties of massive gravity without this issue with the ghost. And we did come up with a model, a model based on extra dimension. In retrospect, it wasn't fully finite, so it still had some issues at some level. But when we looked at how it was consistent with the current framework in simply four dimensions, it seemed to be leading to a theory from a four-dimensional perspective that would have looked like a theory of massive gravity with a finite range. And yet, by going through this extra dimension, we

seemed to be able to implement a framework which was free of the ghost. So when we were going through the proof for why there should be a ghost, we were going through every stage, and somehow everything would agree up to a given level. But then at some point, we saw that the outcome for our theory was to be absent of the ghost, whereas in the no-ghost theorem, the ghost should have been There. And even though the model we were working on was a model that was based in a structure in the way that we engineered

it was based on extra dimensions, and really what that did is that the type of bricks or the type of Legos that we used to build it were within the logic of the extra dimension as opposed to the logic of four dimensions. So we were using a language which was more appropriate for the symmetries in five dimensions as compared to what we would have used had we been in four dimensions from the outset. However, even though we did that, it doesn't matter. We can still think of a theory based on extra dimension and ask ourselves

the question of what would a four-dimensional observer see in that model? What would be the four-dimensional characteristics of the theory of gravity from a purely four-dimensional perspective? And on the one hand, it was leading to a theory that would look like a theory of massive gravity that would have a finite-range gravity where the would be four-dimensional graviton would be massive. And on the other hand, we couldn't see any sign of that ghost, at least not to the level where it was Indicated in all of these no-ghost theorems. And this is really where it pushed us

to understand where was the discrepancy between the model that we were having and all of those no-ghost theorems. And to be honest, I was certain I had made a mistake. I remember going through it over and over again. I said, it simply doesn't make sense. The ghost must be there. Where is it? I must have hidden it somewhere. It's very easy to do that. It's very easy to convince yourself that things are fine, but then the problem is hidden much deeper. So I spent a year trying to understand where these pathologies in my model should

be hidden, because surely it should be there somehow, but I can't quite see it at the first sight. And understanding whether I made a mistake or whether things were actually more subtle, but the pathology would manifest itself in a given way, until I realized going back through all of the proofs that actually there had been some shortcut being used. In some cases, there had been some implicit assumption being used. In some cases, there had been too many shortcuts being used in such a way that the answer, the result wasn't as General, and it wasn't applicable

to all possible situations as compared to what people thought. And the example we had found was precisely, almost by miracle, one that fit precisely in the box of things that could work out, that could make the ghost disappear. You evaded those assumptions. Yeah. Sorry, what would be an example of one of those implicit assumptions? And by the way, when you say that, you mean to say that the paper itself didn't make clear or explicit the assumption? It was just embedded in the ethos of their argument, but it was subtle? That's right, that's right. So, it's

a little bit how, what do you mean by a ghost in some sense? When you try to understand what a ghost is, it's very clear what you mean by that when you can identify different particles, and when you can identify the energy, the kinetic energy of every single particle. And that is something we all know how to do very well in the simplest scenario, when we're in flat spacetime and when things are relatively simple. But this is not what we're interested in. We were interested in having a theory of gravity where you want to think

of it in potentially quite different Geometries, where you're not in Minkowski flat spacetime, where you understand precisely what is your notion of energy, what is your notion of pressure. We want to understand it in curved spacetime, where things become much more murky on precisely how to separate out your notion of energy with the other notions. And that's where you can have some mixing between what you think is energy, what you think is pressure. These are just examples. And the different modes, actually, the different modes of the gravitational waves, start interacting with one another in a

very subtle way, so that the way you identify what you would have traditionally called the standard gravitational polarizations along the transverse mode, that in a very curved spacetime starts getting mixed up with what you would have called being the longitudinal directions, and what you would have called the ghost. So we needed to formulate a new framework which would allow you to separate out these different characteristics. Imagine you're in a pond, it's a clear, beautiful day, and you drop a stone, and you see these beautiful waves going on the surface. And we can all agree, those

Are the waves on the flat surface of the pond, and it's beautiful. Now imagine you're not on a pond, you're in the middle of the ocean, it's the hugest storm of the century, with waves which are bigger than your boat. Who is to say where starts the small fluctuation from the little stone that you dropped, and where are the underlying huge fluctuations, huge waves, which are bigger than anything else, than you want? It's very hard on the first sight to distinguish one from another. And yet, you need to do that to separate out what you

mean by the different polarizations of the graviton in that situation, and how they interact with one another, and what type of energy they carry out. You need to be able to separate all of those things. So there was some implicit level of assumption of how you separate those things out. They're a little bit technical in how to do that, but it's a little bit as if you imagine for your ocean, you say, okay, I'm going to say this is the zero depth of the ocean, and I'm going to just calibrate them in the way I

would have thought of doing it if I were on a pond. But sometimes that's not the right way to do it. You really need To change your perspective completely, the way you're going to characterize all of those things, so that you end up with something where you can separate out the different modes of the graviton. So when people were identifying the existence of a ghost for the graviton, actually what they were identifying was one of the normal modes of the graviton. But because it's difficult to really understand what type of energy it carries, they were

just thinking that this was corresponding to the ghost, when in reality, it was just one of the healthy modes of the graviton. Okay, so you spent a year checking this over and over? Yeah, it's like that in research. Yes. I mean, I spent a lot of time, a lot of nights going through all of this. And even when we understood that this could be a possibility, it's not enough that you understand it's a possibility that you need to understand better how these things work. You still need to understand what is the best way to frame

all of these things and to have an explicit realization so that you see whether it can be fully fledged in a full theory. So the model we had in the model of extra-dimension, That was a model where we could have a first insight of how it could work out in principle in a limited context. But it wasn't the full story because it was still breaking down at some point. But that was enough because we had the seed of the idea of how things could work out in principle. And then what you have to do is

engineer a model which fits precisely in that box, satisfies precisely what was falling between the cracks, feel those cracks in precisely the right way, so that you can end up with a theory of massive gravity that evades all of these no-go theorems. So to the young theorist who's watching, this sounds inspirational because they may have some theory and their advisor may push against it because it violates some no-go theorem or it produces an anomaly. What would be your advice then to them? So I mean one of the things is that of course, that's the beauty

of falling, right? Sometimes things don't work and then it is in the beauty of understanding why they don't work, why they don't work precisely that you may discover something new, something beautiful. Even if that thing is not useful for precisely what you wanted, Even in a theory of massive gravity may not be for the description of the universe as we wanted, it still has some structure in itself. It still has some level of inner beauty, which I think is worth in and of itself. So absolutely, sometimes, and that has happened multiple times where there's a

consensus of what is accepted, what are the theorems, and you should in reality take nothing for granted. However, in most of the case, it is true that there's deep reasons for why things have been fledged in a particular way. So it's not like I just woke up in the morning and thought, okay, I'm just going to go against the norm and come up with all sorts of different ideas that don't fit the box. I think you always need to go back to what has been proven, what has been understood so far and really understand this

to that depth. It's by understanding the work that has been done so far to its depth that then you can also understand how what you want to do can come along and how it can complement or how it can even contradict what has been presented. But throughout trying to understand whether you can have a theory of gravity which goes beyond general relativity, I think I'm in a very good position to tell you how incredible general relativity is, how you may think that it's set up like that because of some assumptions of the pillars of Einstein

and some particular assumptions that were pre-set in advance. But actually, for me, it's much more self-consistent in itself. And I can see that all of the beauty that is present in general relativity is completely self-consistent from the outset. It's extremely hard to just challenge a tiny little thing in itself. You need to understand all of this beautiful structure first before you understand how you can start dismantling maybe a small little piece without everything falling apart. Yes, okay. So it's not throwing away the history of physics, it's deeply understanding it in order to snake your way

through a narrow path. That's right, that's right. So this is an extremely bold new theory of gravity that in some ways shouldn't exist. Does it solve any other problems like the information paradox or tell you what happens in the inside of a black hole near the singularity? So by design, the theory of massive gravity is A theory of gravity that generalizes or goes beyond Einstein's theory of general relativity on very large distances or very small curvature regions of the universe, which is precisely the opposite end of the spectrum as compared to what we're interested in

in black holes, for instance, the horizon of black holes or in black holes themselves. So in its very construction, the theory of massive gravity, I would say from the outset, shouldn't have anything to say about those aspects of quantum gravity at a very large scale. So you're agnostic. You're agnostic when it comes to string theory or loop quantum gravity or causal dynamical triangulations. Yes, I am completely agnostic, but let me tell you something. So I'm completely agnostic and I think it's good to have a spectrum of different point of views because I think it's good

to understand how to connect between those. So far, I have no particular preference necessarily on which one is right or wrong. I think it is useful to understand them all. It is useful actually to very much understand them all and very much understand what are the common points and what are the differences so That then we can try much more to distinguish them at different levels. However, let me just say that for the theory of massive gravity, even though I don't have a preference per se on what is the ultimate high-energy completion of it, because

it has some of these additional modes, some of those additional polarizations and a radius associated with it, which we call the Einstein radius, which is much bigger as compared to the horizon of our would-be black hole. And so, in this theory of massive gravity, even though from the outset that's not the reason it was engineered, it comes in as well with some features where you start needing to understand some phenomenon about, let me call them, strong coupling or quantum nature of some of this polarization of gravity already at the scale of the Einstein radius and

not at the scale of the horizon. So, some of the mysteries of gravity that are present in general relativity are very much present in massive gravity, and actually, they manifest themselves in some of the modes of the graviton before they would have manifested themselves for general relativity. On one side, you may say, okay, It's going to be very, very complicated. It is very, very complicated, but on the other hand, you can also see it as a playfield to explore some of these ideas that you would want to explore for quantum general relativity, for quantum gravity,

in a framework of massive gravity where this comes in for just one of the modes. So, you can just explore the effect for one of these modes and maybe understand some of the phenomenology of quantum gravity in a simplified model. Now, there's something called the Higuchi bound, which imposes a limit on the mass of a spin-2 particle in de Sitter space. Yes. So, are you within that bound? Are you above it? Are you below it? Yes, yes. So, the Higuchi bound is absolutely right if you are in de Sitter space. So, if you're in an

accelerated expanding universe, let's say we're close to de Sitter space now in the universe, it tells you that the mass of the graviton has to be either zero, or it has to be larger than twice the Hubble parameter, or actually the square root of the Hubble parameter. It's interesting. It's like a Yang-Mills mass gap. There must be a mass gap. Yeah, exactly. The square root of the Hubble parameter. Otherwise, you end up with a ghost, which is not the Boulware's ghost. This time, it's called the Higuchi ghost. It's not an additional mode, which is a

new ghost mode. It's actually one of the modes of the graviton, which we discussed before, which itself has a negative mass. So, this is okay. If you want your theory to take off even further, you should have called it ghost gravity. That's such a catchy name. Many books can be written about that. I'll do that. I know it's not a ghost gravity. That's the whole point. But it has potential for a lot of ghosts. Yes, yes. I give you the trademark. Thank you. Okay. So, going back to your question on whether this is okay, it's

fine. It's such today, because the Hubble parameter today is actually of the same order of the mass of the graviton we would like it to have today. We want the mass of the graviton to be slightly larger than the Hubble parameter today. So, we would have expected the graviton to be within the realm of mass, which satisfies the Higuchi bound. However, if you want to understand how this occurred throughout the History of the universe, so if you are at the very beginning of the universe, you were close to another De Sitter region in the universe,

where now the Hubble parameter was way higher. And so, you would have expected to have the mass of the graviton at the time to need to satisfy an Higuchi bound, which is much stronger, or the graviton mass would have needed to be much higher to satisfy the Higuchi bound at the time. What we think at the moment is actually there is some sort of redressing mechanism, and that comes in naturally. We still have examples where we see that naturally occurring from the environment. So, when we are in the very early regions of the universe, the

effective mass, or what you call the effective mass of the graviton, you identify as the mass of the graviton, is actually redressed by its environment. So, it leads to effects on the graviton mass, which means that the graviton mass is actually carried by the environment and carried by the Hubble parameter at the time. So, it satisfies the Higuchi bound at the time. And as you have an evolution of the universe, the mass of the graviton also effectively evolves With the curvature and with the evolution of the universe, so that it becomes very small today. But

it could have been much larger in the earlier parts of the universe, and that in itself is also consistent with observations. JS Interesting. So, are you suggesting that, just like the Higgs so-called gives mass to the fermions, that there's something that couples, like a Yukawa coupling for the graviton that gives mass to the graviton? CM So, I'm certainly not suggesting a Higgs mechanism for graviton. That mechanism is not so much a mechanism where you're in the same way as the Higgs mechanism, where it's in the interplay between the Higgs bath and the fermions, for instance,

that you see an affecting of the dynamics and affecting of the inertia of the particles, which lead to an effective mass of the particles through a Higgs mechanism. It's something slightly different where, because of the existence of an environment being present, it does carry along with the graviton. So, the graviton itself carries along the environment in which it is living, but you don't need an extra particle like the Higgs to lead to This environment in some sense. It is something that you can have at the completely classical level. JS Now there's an article by The

Economist I'll put on screen called, The Dominant Model of the Universe is Creaking, and it's about the DESI results from the past couple of months or so. Can you go over either the DESI results or any new data that validates that you're on the right track? CM So, there's been, in the past almost 10 years or so, while all observations completely agree that the universe is accelerated within a given rate, there's been a deepening in the level at which different types of observations lead to a slightly different rate for the Hubble parameter, which we call

the Hubble tension, or if you want, the rate of accelerated expansion of the universe. There seems to be some slight discrepancy between the rate you seem to be observing, depending on the type of observation, whether you're dealing with observations which are later on in the age of the universe as compared to early on in the age of the universe, and depending on the scale of those observations. So, for instance, between supernovae or what you would have from other observations Or from observation of the cosmic microwave background. So, the DESI results, which are more recent, and

we're still expecting much more from the DESI results, they're extremely interesting. I personally think it's a little early to read too much into the result. I think they will have a full many years of observations where they can consolidate some of the results. But taken at face value today, they seem to suggest that while in principle, having accelerated expansion of the universe and evolution of the universe, which is consistent with it being driven simply by a pure cosmological constant, it seems to be slightly favored to have dynamical dark energy, so the equation of state parameter

that changes every so slightly over time. And what seems to be also very interesting, if correct, is that the equation of state parameter for the would-be dark energy is not within a regime we would have anticipated it should be based on typical scalar field models a priori. And so, all of this may seem to suggest that there's much more to the picture of dark energy and to the accelerated expansion of the universe as compared to what The most vanilla, pure, constant cosmological constant model would seem to suggest. I mean, it is possible that there's some

systematic effects between different types of observations which are playing a role into that. I'm not at all within this data, I can't say anything about that. But if it is all entirely correct, it seems to suggest that there is some dynamics within the evolution of the accelerated expansion of the universe. And so, something else than simply a pure cosmological constant or a pure vacuum energy, with the right order of magnitude, should be at play to explain those observations. To me, that really is fascinating because it's a signpost for potential science of new physics, and whether

it is dynamical dark energy or a modification of gravity or anything in between is very much something we have to better understand. But it tells us that there's something which is beyond the simplest possible model. And so, massive gravity or other models of modified gravity, where you can have a behavior of gravity which is every so slightly different throughout the ages of the universe, is something which could, in principle, Help with understanding some of those questions. Does massive gravity have any implications for anti-gravity? For anti-gravity, as you think of it, in the sense of having

two masses repulsing each other, is that what you mean? So, the reality is massive gravity is so anchored within the framework of general relativity with very small departure that you're not going to end up with a result ever which is so radically different as compared to general relativity. It's not going to make something flipside in itself. This very notion that you could have anti-gravity or that things could start really fundamentally becoming repulsive as opposed to attractive, as in two masses being repulsive, as much in massive gravity than in general relativity, leads to some instabilities also

related to negative mass, if you want, or negative energy, which is unstable. So, a lot of what we do in massive gravity is still following very much the same rules as in general relativity. So, those type of things would not be directly applicable for massive gravity. And does the Witten-Weinberg no-go theorem about the massless spin-2 particle, Does it apply to yours, or do you see it as evading it? Okay, so what the Weinberg theorem tells you is that you can't have... Let me remind me what he tells me again. Sorry. Yeah, it has to do

with if you're in four dimensions, technically three plus one, that if you have a conserved current, then you have some limitations on its spin when it's greater than one or greater than one half. Then you also have limitations on its mass and its charge. Sorry, let me go through that. I knew it. Massless particle with spin greater than half cannot carry a Lorentz current. So, some people saw that as suggesting that the graviton shouldn't exist in four dimensions. One of the ways around it is to exceed or shorten your dimensions, or to say that the

graviton is a composite particle. But it sounds like massive gravity is an evasion of this theorem. Yeah, so the Weinberg-Witten theorem tells you that a massless graviton in four dimensions cannot be composite particles. So that's okay for the graviton in general relativity, which is a massless particle, but it would be a fundamental particle. Now, if it is a massive particle, that Evades the theorem altogether. So in principle, it could be a composite particle. So Professor, what are you working on now? What are you most looking forward to? Yes, so what I do at the moment

is some of these aspects are quite different as compared to massive gravity. They started with massive gravity to some extent, and trying very much to understand how to make connection with the theories that we use on a daily basis to describe the world around us, still as a theorist, as a quantum field theorist. So we have the framework of effective field theories. For instance, the Standard Model of particle physics is an effective description for all of the Standard Model, all of the constituents of matter and the other forces of nature, aside from gravity. Or general

relativity is an effective description of gravity, which works extremely well, we believe, on low-energy scales. But we know, because of the issues related to embedded general relativity in a quantum world at high energy, that at some point, we need to have a better description of gravity. And so that can be string theory, it can be loop gravity, it can be causal sets, It can be all sorts of different alternatives. It is possible that it is another UV, high-energy, ultraviolet high-energy completion of gravity, which we haven't yet come across, which we haven't yet envisioned. So there's

all sorts of different possibilities at high energy. And myself, I don't want to be too specific on which particular completion I want to commit into. So as you mentioned before, I like to remain agnostic on the type of completion that I will allow for myself, being in string theory, etc. But I still want physics to make sense, ultimately. So for instance, I don't need to know precisely what the laws of physics are at infinitely high energy. But it still is meaningful for me to ask that whatever they are, they satisfy what we call unitarity. So

they satisfy some laws of quantum probability, so that a thing sums up to one. That's what I mean by unitarity to some extent. It's a bit more than that, but to some extent, I can think of it like that. I can also ask, for instance, for causality. We understand which that is probably not controversial, although how you formulate this is probably more controversial. But at the bare Level, the notion of causality is that I would like the consequence of an effect to happen after the effect, not before. So if I were to kick this table,

I want to be hurting after I kick it and not feel it before I kick it. This is my notion of causality. We can state that in more formal, more rigorous terms in saying that I want to have no support of my propagator outside my light cone. I can state it like that because if I also think of the Lorentz invariance, then different boosted observers should be equivalent with respect to one another. So if I have something which seems to propagate outside my light cone, then for an observer which is boosted with respect to me,

that may seem to be perceived as something that goes backwards in time. So there are some relatively easy-to-formulate or relatively general statements about physics which are not too controversial in themselves, and I still want them to be satisfied in physics in general. So even within a realm of physics for which I don't have direct access, neither theoretically nor observationally, I have no access to it, and yet I want to make sure that Physics satisfies those notions because they make sense. Conceptually, they make sense. If things were starting to become a causal, I would need to

rethink about everything from the ground up. If things were not satisfying unitarity, then I would need to rethink completely about the laws of quantum mechanics. So it's not many of them. There's a few set of properties of physics. I want to make sure they'd be satisfied at very high energy, but those in themselves, whether they are realized in the way that string theory realizes them, or loop quantum gravity realizes them, or other type of UV completion realizes them, it doesn't matter in which one. They still have consequences for the laws of physics in the way

that I observe them at the moment, and in the way that I can actually probe with my observations, not mine, but with observation at our disposal, or experiments in particle colliders, for instance. So there are some features, some imprints of high-energy physics based on those assumptions, which should be present on the low-energy framework that I'm using to describe the world around me. And we're used to those, Actually. We're used to knowing that the notion of causality at very high energy has for effect that no one can travel faster than the speed of light at low

energy, the speed of light in the vacuum. The notion of causality is something which is actually embedded at very high energy, at infinite energy, because it's related to what we call the front velocity, the infinite frequency limit of the phase velocity. The front velocity is something I think we all hear, but maybe we don't all remember. We all hear, for instance, that the phase velocity can be superluminal, so long as the group velocity can be subluminal, because we are thinking of the group velocity as generally carrying the information, whereas the phase velocity is something more

artificial. That is actually incorrect. There are experiments for which the group velocity is superluminal, and it doesn't actually violate causality, because the very notion of causality is actually set up in the infinite frequency limit of the phase velocity. So if you're thinking of the notion of causality, I want almost to have a discontinuity. I want to send you a signal. So it cannot be the case that I was sending you a signal since the beginning of time. There needs to be a time where I'm not sending you a signal, and then it starts kicking in,

I'm sending you a signal. This is what has to happen. And so there's a discontinuity there. This is me with no signal sending to you, and now I'm sending you a signal. And so it is in this discontinuity that most of the notion of causality starts kicking in. But because it's discontinuous, it's something, if I were to do a Fourier transform, I don't know, or if you want to think of the frequency associated to that, that's something that lives at infinite frequency, at infinite energy. So I don't know how familiar your audience is with those

terminologies, but it is actually something that doesn't... I can't realize this exactly within the realm in which I have contact, right? You can imagine a real discontinuity is something that would require so much precision that I can't do it exactly. It's the same reason that in Heisenberg's uncertainty, you can't reduce the position down to a direct delta function, down to just one point. Exactly, it's exactly the same thing. So it's Something which is not within my realm to achieve. It is something that would require me an infinite energy to be able to achieve. So really,

to probe the notion of causality, to very probe the notion of what happens if I make the transition between not sending you a signal and starting t zero or start sending you a signal, then I need to be living at infinite frequency and infinite energy. So this is something within the realm of the UV completion of everything, in the grand theory of everything. That's where the notion of causality really resides. But it doesn't mean that it's completely disconnected to how we experience it in our everyday life. And we still know that this very notion of

causality, as embedded at very high energy, has consequences for our everyday life. And we know that as a consequence, it means we cannot be traveling faster than the speed of light in the vacuum. That has a consequence, unless there's some different features that emerge, some small violation of unitarity in some particular fluids that we can engineer locally, or other things like that. This is how this is being played out in all of this engineering System, where they manage to achieve having a good velocity which is faster than light in the vacuum. So that's just one

example, but there's actually an infinite number of consequences that can be explored, that we can use to better understand how physics gets implemented at high energy. So it's almost a two-way street, where in putting some assumption at high energy, it can guide us on how to think about the physics in the way we describe it at low energy. And it can guide us where to look for signals, for instance, at the LHC, or for instance, for gravitational waves observation or cosmological observations, where we have a huge level of data, so much data, we need to

make some prior decisions. We need to make some biases in how we gonna sort out our data ahead of time to better analyze it. And so if we can make some of those prior based on information on how meaningful physics is at high energy, that can guide us searches for new physics at low energy. That's one way to think about it. Another way to think about it is in exploring how high energy physics imprints itself at low energy. In exploring how energy looks like, we may Also be able to get a better understanding of how

high energy physics looks like. And whether some of the assumptions that we think we should impose on ourselves at high energy, whether they are justified or not, maybe we're observing low energy physics in such a way that some of those assumptions at high energy should be violated. And so maybe that could guide us to understanding where we should be in our realm of high energy completion. Is it more towards string theory? Is it more towards something else? One of the beauty of this way of trying to make connection with the high energy world, which is

not specific to string theory, is very much in addressing this notion that you may think, you may have heard that string theory is not a theory because it doesn't have a specific observable. And actually, that may not be true. There may be some ways that you can falsify string theory, because it comes in with some assumptions which have consequences for physics that we can observe, that we can test within our realm today. And so we could come up with a result of an experiment or observation that would falsify string theory, the Very underlying assumptions of

string theory. Can you talk briefly once more about how is it that if you were to exceed the speed of light, it wouldn't break causality? Because in the traditional model that you learn in university, you have the light cone, and as soon as you tilt past that, then you can transform your vector in any which way and you would violate causality. It's complicated, so let me see if I can. So when we go through the standard explanation of if you have a wave, let's imagine you have a wave, which is your signal, and depending on

the velocity of the wave, if the group velocity of the wave exceeds the speed of light, the traditional picture is then to say that this is signaling that something is outside of your light cone, and then you can go into a frame of reference where an observer is boosted with respect to you, and something which is outside your light cone for them looks like it's traveling backwards in time, according to their frame of reference. So this is how it seems typically to suggest that traveling faster than the speed of light, in the sense that if

you have a group velocity for a signal, a wave With a particular frequency going faster than the speed of light, this seems to suggest a violation of causality. But this argument is mainly true. The reality is this is an idealized scenario where you imagine a signal being emitted by a wave, and a wave which has been there since the beginning of time. There's no beginning or end to the wave, because if there were a beginning or end to the wave, then it wouldn't correspond to a wave with just a particular frequency. You would need to

include the frequency associated with the physics of this dying in and dying out. So when we really think much more about the notion of causality, I want to think of what it means to send you a signal and go from the transition between me not telling you anything—it's embargoed—to me starting to tell you something. And so I need to switch on my signal, which means I can't just be communicating this information with a single frequency wave which has been there since the beginning of time. It won't just be one frequency wave. And just switching on

my signal would lead to a spectrum which also includes infinite frequencies, so abruptly sharp wavelengths in the Signal that I'm trying to send you. And so it is very much in this very, very sharp, very, very small wavelength part of the signal that information about the high-energy physics is encoded. And causality is therefore very much encoded at high energy. It's not something that I can simply diagnose at low energy. But that's an example where high-energy physics actually has an imprint in how we think about physics at low energy and how it still tells us that

we shouldn't be able to travel faster than the speed of light in the vacuum. Professor, speaking of sharp, your book, The Beauty of Falling, is out now. People can read it. They can get it in the link in the description, if they like. And I recommend it. It's an honor to be able to speak with you. Thank you. You're great at explaining concepts extremely simply. And you have an effervescence about you that I appreciate and the audience can relate to, I'm sure. Thank you, Curt. Thank you. That was great. Thanks. Really, really fun. Yeah, it

feels a lot more very technical. I hope your audience likes it. Yeah. No, thank you. Thanks for the questions. They were really great. So you Did a PhD in Toronto? No, I did a bachelor's in University of Toronto. Wow, but you know everything. You're incredible. Oh, no, no. It's incredible. Wow. No, I do my homework. Yeah, you do. So actually, I spoke with Faye, Fay Dowker, who is also at Imperial. And she was telling me how great he was to talking with you. She really enjoyed it. She showed me your video as well. And Oh,

wonderful. Yeah. She said, yeah, how much, how knowledgeable you are about everything. Also, thank you to our partner, The Economist. Firstly, thank you for watching. Thank you for listening. There's now a website, Curt Jaimungal.org. And that has a mailing list. The reason being that large platforms like YouTube, like Patreon, they can disable you for whatever reason, whenever they like. That's just part of the term. In terms of service. Now, a direct mailing list ensures that I have an untrammeled communication with you. Plus, soon I'll be releasing a one-page PDF of my top 10 TOEs. It's

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