Transcription
Gravity from entropy is a new theory that quantifies the information content of the universe. It somehow challenges the reductionist approach.
This is Janestra Bianci, professor of applied mathematics at Queen Mary University of London, an architect of modern network science. Over the past few years, she went into the continuum despite coming from the discrete sector and published the radical paper "Gravity from Entropy."
On this channel, I, Kurt Chaungal, interview researchers regarding their theories of reality with rigor and technical depth. Today, Bianci's case that gravity can be derived from entropy. How dark energy emerges from her equations on its own, always positive, not anti-matter. And we close with the advice she gives her PhD students.
"What keeps me awake at night is the second quantization of this theory. Maybe there is no single static solution of the black hole in gravity from entropy. And maybe this singularity is avoided."
"What is gravity?"
"Well, uh, I think we know that gravity is a fundamental force, and we know this since, since Newton. But it's a particular fundamental force, uh, because it somehow, from my perspective, challenges the reductionist approach because it is about geometry. And this is what we learn from, from Einstein. And geometry is what allows all the other fundamental forces to occur in nature. So somehow, uh, I think that gravity is about geometry and how geometry interacts with matter fields. And this is a general question. I mean, for my perspective, this question, uh, goes also beyond gravity itself because it is a problem of the interplay between, generally speaking, structure and dynamics that is at the fundamental mathematical level common to many different other fields."
"How did a network topologist like yourself get interested in gravity?"
"Yeah. Yeah. So this is, this is a, a nice question. In, in my career, I was, I started doing research in, in this discrete structure that are networks, from nodes and links. And then I moved from networks to simplicial complexes, which, uh, allows to capture a discrete geometry and topology of a lot of systems and, and also real data that can be described with these systems. I cannot, maybe, overstate that maybe in most of my research since now, I have always focused on the interplay between structure and dynamics. And this is very central, for instance, in network theory as well. But, you know, there are important results in network science. So, for instance, how, you know, the presence, the topology of the network, so the presence of big hubs can affect epidemic spreading or things like that. But recently, it is becoming clear that, uh, topology and geometry also play a fundamental role in shaping the interplay between structure and dynamics. And, and I've been working a lot on how topology shapes dynamics in networks. And this is, um, an important mathematical problem that, you know, has implications for machine learning up to brain research. Um, but, you know, if you, uh, want to build this theory, that is a theory that includes topology, geometry, and dynamics, um, there are two aspects. On one aspect is that you want to write a theory that captures this interplay, um, using information theory, because ultimately you want to study, to describe the system in terms of their information content. And maybe we can go back to that. And, but on the other side, you don't have enough mathematics in the discrete. So also the notion of, uh, curvature is, um, not well-defined in the discrete setting. There are very important proposals. I was giving a seminar at ICTP in Trieste, in Italy, and somebody told me, 'But if you are doing this, why don't you do that in the continuum?'"
"Right?"
"And I answered, 'No, I will never go in the continuum.' And then I reflected on this and, you know, yes. So the continuum of this theory has to do a lot with gravity. And, and then, uh, why don't, pacing the, the real challenging problem that is quantum gravity and gravity, um, because, you know, somehow I think maybe a bit controversial, to, to understand the brain is more difficult to understand quantum gravity. I'd like to bring people up to your results to have them understand it. And one of the ways to do so is psychologically. You had a resistance to going into the continuum. Why? Why did you say no, I'm not going to go into the continuum?"
"Yeah. Because all my life I was doing in the, I was working in the discrete. And somehow, I, I was raised with a belief that maybe also quantum gravity should be discrete, or that, you know, nature should be discrete ultimately, um, which can be the case. But, um, yes, so then we have Gauss, we have Riemann, they're all what, in the continuum. So it's not that I will not think about turning gravity from entropy into a continuum in the discrete theory. But for the moment, I like the continuum because the mathematical quantities are well-defined. And, and conceptually, I can focus on the innovation of the, the idea."
"Okay. Can you tell me what you were thinking about, what led you to think about gravity? So your friend or colleague said you need to go into the continuum, but into the continuum for what? Like, what were you trying to do such that the advice was to go to the continuum?"
"Yes. So I, I was setting up a challenge that, that I think is, is urgent and emerging in, in the context of complexity and network theory. That is a comprehensive theory which includes, that addresses the interplay between structure and dynamics using topology and geometry, and viewing this under the lens of information theory. So I, I strongly believe that the advances in this context will be very important for, uh, the theory of complexity. But I, I think that also, you know, mathematically speaking, the problems are common, uh, with the interplay between geometry and matter fields in gravity."
"Okay. So now, under your theory, under your new result, what is gravity?"
"So, um, gravity from entropy is a new theory that stems from an action. And this action quantifies the information content of what are the microscopic degrees of freedom of the universe, essentially. And the idea is to describe this information content of this matter and geometrical degrees of freedom using the metric. So you have two metrics: the, the true metric of, uh, spacetime, and the metric induced by the matter field and curvature. And the action, that is the gravity from entropy action, describes this interplay between matter and geometry by using as Lagrangian what I call a geometric quantum relative entropy between these two metrics."
"Do you have a tension between these two metrics that are trying to be close to each other? So it's like the matter field telling the metric how they would like to see the true metric, and the true metric telling the matter field how to move? And, and this is essentially, you know, is building on our insight of Einstein's equations. But actually, Einstein's equations have this interplay as beautifully summarized by John Wheeler. But this interplay is not expressed at the level of the action. So instead, the gravity from entropy leverages this principle of the interplay between matter and geometry already at the level of the action with this information theory, um, Lagrangian."
"Let me see if I can summarize for myself and for the audience. So there's entropy, which has various incarnations like Shannon entropy, and von Neumann entropy, and Boltzmann entropy, and, and so on and so forth. Roughly speaking, they all earn the right to be named entropy by quantifying some distribution of what you don't know. Now, one of the various forms of entropy is called horizon/boundary entropy, which is defining some screen where you're saying, I don't know what the heck is going on beyond this screen. Yes."
"And I'm going to attach a number to how much I don't know what's going on, to how many microstates are consistent with whatever is the macroscopic view. That is what Erik Verlinde and Ted Jacobson use. But there's another kind of entropy called relative entropy. Which, for the people who have watched Carl Friston and heard about his free energy principle, it's roughly speaking that you have a model of the world, and then something happens, and then you want to know how surprised should I be? How much information should I glean? How much should I learn? And if something is drastically surprising, then you should learn plenty. But if your model of the world is pretty much correct, because you're, you're just a perfect person, well, then you don't have much entropy at all. You take this other approach, this relative entropy approach. You take this relative entropy, put it into an action, and then extremize it."
"Yes. So it's, uh, the concept of relative entropy is, is quite, uh, nice. Um, the gravity from entropy action is, is quite innovative also from the point of view of statistical mechanics. So, just to make things clear, right, the entropy is usually, you know, Boltzmann definition is the logarithm of the number of microstates compatible, uh, with the macrostate of a system. And, and this is, is huge, right? Because the entropy, uh, describes the second law of thermodynamics, which is, of course, one of the building blocks of, of physics. And because the entropy, why this is defined in this way, never decreases. So, as the tendency of increasing, and people are, are usually quite impressed by this, uh, second law of thermodynamics and things like that. Um, but for, for my perspective, it's just a, a measure of quantifying what you know about the system in terms of information theory. So the relative entropy is a different concept, right? It's, is when you want to compare two different states, for instance, two different quantum states, and express how much information in a quantum state is codified in the other quantum state. So if one quantum state describes the real geometry, and the other quantum state describes the geometry that the matter and the curvature would like to see, right, this gravity from entropy action describes this tension between these two. And actually, another, uh, property of this action is that it treats, in some sense, very symmetrically the geometry and the matter fields, because they are just, you know, described in terms of two metrics. And it's comparing these two metrics with this geometric quantum relative entropy. So you have this tendency of these two metrics to try to be close to each other, to approximate each other. But actually, the gravity from entropy action, it doesn't have only the Lagrangian, but has also the measure. And the measure actually, uh, plays also an important term, right? So conceptually, the gravity from entropy action wants to describe this tension between these two metrics, wanting to, to be close to each other. But then, at the same time, there is a nice mathematical aspect of it, because, um, this entropy is defined like the trace of a logarithm. And one of the properties, mathematical properties, is that the trace of the logarithm means the logarithm of the determinant. So when you express this, uh, relative entropy, you know, conceptually, the idea is that you have these two different, uh, entropies. But the idea is that this entropy also quantifies the number of microscopic degrees of freedom of, of, of this interplay between matter and geometry. So you have at the same time, you know, an understanding in terms of a relative entropy, and an understanding as a Boltzmann-like entropy. And I have, um, a paper that is now in, in press in PRD, which shows that actually, if you consider this action and you calculate this action on over Friedmann universe, which are an approximation of, of the solution of the modified gravity equations, you find that actually, um, this, uh, Lagrangian decreases in time. So that, you know, you have these two metrics kind of trying to be close to each other, but actually they are integrated. So the action, which integrates over the measure, and can be interpreted as an entropy increase in time. So the universe is consistent with an action that describes the, the total entropy that increases in time, while the relative entropy locally decreases in time. So that's an interesting aspect in terms of statistical, uh, physics."
"I'll be placing links as well as visuals to your work on screen so that people can dive even deeper and then even earlier to some of what you're saying, like simplicial complexes and so forth. So, if you're listening to this and you're driving and you, you're wondering what the heck does this correspond to in terms of what does it look like? You can feel free to watch the podcast safely as you drive. Don't do it while you drive. I mean, pull over. Okay. What are these two metrics then? Like, what is one? Is one just flat, the flat case? And then the other is whatever the Einstein curvature would be? Like, it seems like it sounds, which is why I'm putting your work on screen, but it sounds like you're putting Einstein's equations in already to get out Einstein's equations. Like, if we're already motivated by, let's see how matter would tell spacetime how to curve. Well, then we have some idea of Einstein's equations going into it. That's what it sounds like. So dispel, tell me, what is going on with these two metrics? What are they?"
"So now, there is no assumption of any Minkowski background. Absolutely not. So the true metric is the true metric, the one that defines the Ricci scalar, the Riemann curvature. So it is the true metric as it is. And the metric induced by the matter field and curvature, it's a geometricization of the matter field. And this builds on a different insight. The first insight is, uh, Gauss. The first fundamental form of Gauss, which expresses practically, you know, in, in the simplest setting, you can say, you know, you have your manifold, which might be curved or whatever, your general Lorentzian manifold. And then you have, let's say, a scalar field, which defines a dimension, an additional dimension. And then this scalar field, you can imagine as a surface defined on your original manifold, because it's, you know, your scalar field defined on the manifold. Now, for this surface, there is a notion of metric induced by, by this function. And, and this is described by the first fundamental form of Gauss. And this is the metric induced by, by this field. So the idea is, in, in gravity from entropy, is that there is not only a scalar, but there is a higher-order description of the matter field. So you have at each point, you have a scalar, a one-form, and a two-form. And here the whole setting of differential geometry comes about, that is very fundamental for this theory. And then you define the metric induced by the matter field and curvature, similarly as, you know, extending somehow Gauss's first fundamental form of Gauss. And this is an, an expression of this metric induced by the matter field and curvature is inspired by the literature in, in Bounded algebra. And, you know, there is a beautiful paper by, by Witten in Reviews of Modern Physics that discusses a, a closely related definition of entropy, which is again, a relative entropy, which is called the Iraqi entropy for Bounded algebra, as a, a very important measure for entanglement that can overcome the problems of, uh, you know, entanglement entropy for, for quantum field theory that is ultraviolet divergent. So the connection is not established fully, but it's been, it's been very important for the formulation of, of this theory for me. And, uh, there are things to explore whether, again, what is the connection with the, between the gravity from entropy action and entanglement and Iraqi entropy. But the idea is that, yes, this action treats matter field and geometry on the same footing by geometrizing the matter field and the curvature. And from this aspect, it's much more symmetric than the Einstein action plus the matter field, right? Because the matter field has the matter action has only minimal coupling. And there is no symmetric way of treating the two. While in gravity from entropy, it's fully symmetric trick, and you treat them on the same footing and interpret them in the light of their mutual information content, uh, relation. And the beauty is that this action leads to modified gravity equations. So it might lead to to testable predictions that go beyond Einstein, uh, equations. But it reduces to Einstein equations in the low energy limit. So everything we know in the low energy limit remains valid. Um, but of course, uh, the, the gravity from entropy equations of motion can be used to, to probe the high energy limit."
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"Okay, one of the questions occurring in in the audience's mind may be, okay, there are many different reservations of Einstein's field equations or or GR. And so then the question is, okay, so what? So what if we found some new formulation? You're saying the, so what is at least that well, matter and geometry are now on the same footing, whereas before they were seen as separate. You treat them both with mutual information. Then another is that you get modified gravity, which will lead to testable interpretations. And furthermore, that this has implications for quantum gravity."
"Yes. So, I mean, this aspect is, is very crucial because, uh, you know, gravity from entropy action stems from this idea that you can treat the, the metric as quantum operators and practically write a quantum relative entropy among them. But it's still not in second quantization. So the program is there, and I'm still working on, you know, even more high energy limits, right? Because it could be that this is a theory being classical. Maybe it could be that, you know, if you really address the full second quantization aspect, you get still more, more effects. Yes, this is extremely new, less than two years old. The, your, your paper, I think, is from 2025, if I'm not mistaken."
"Yes."
"Someone also in December just took your, your work even further. Inflation without the inflaton, something like that. There was a paper that also acknowledged you."
"Yes. Yes. This is a group, uh, in, in based in China. They have started the challenging task to integrate these equations and they looked at at this model and they showed that it might lead to inflationary behavior also in the absence of a scalar field. Yeah. But I mean, it's also, it's also interesting. I mean, one of the predictions of, of the model, and this comes from my original paper, is actually that, you know, this modified gravity equations are consistent with a dark energy term, which is a dynamical cosmological constant, which is always positive and vanishing in the low energy limit. But it is, is dynamical and is expressed in terms of a new quantity, which I introduce, which is called the G field, which is an emergent field of the theory, which encodes this interplay between structure and dynamics. And is responsible for this dark energy term. It's a bit like, you know, it's emerging mathematically as a Lagrange multiplier of the theory. But we know that Lagrange multipliers in statistical mechanics can have a physical meaning. Like, for instance, the temperature is a Lagrange multiplier of the energy and has a physical meaning. In the same way, you know, this G field that is emerging has a Lagrange multiplier. We want to give a physical meaning to that. And, and when we do that, you know, this action that it looks like the, the relative entropy, when you express it in terms of this G field, becomes much closer to the Einstein action. And the differences are, are mainly two. One is that the metric is dressed by this G field. So it's like the true metric is not exactly what interacts with matter, but what interacts with matter is a dressed metric with this G field. And you have a cosmological, a dark energy term, which is a dynamical cosmological constant that depends on this G field. So an important testable prediction is to see if this can can shed some light into, you know, the Hubble tension or other properties."
"What do you mean when you say that the metric is dressed?"
"Yeah. So practically, you know, this, uh, geometric, uh, relative entropy can be written in terms of this G field. And then there will be, in terms of the G field, it appears to have three terms. One that you can identify with something closer to Einstein-Hilbert action. It's a sort of dressed Ricci scalar, because it's formed by the Ricci tensor contracted with this dressed metric, and the Riemann tensor contracted with the dressed metric. And, uh, another term that is the matter action, that, you know, is this thing that comes from, from the geometry induced by the matter field and curvature. It is contracted with this dressed metric. So it's just dressed metric instead of the metric. You have the metric contracted with this G field. So it's like the matter feels this dressed metric."
"Earlier, when talking about that, there are disadvantages of discrete approaches, for instance, one was that curvature isn't well-defined. However, there are some discrete approaches to quantum gravity like causal set theory and causal dynamical triangulations. Even Wolfram has an approach that's discreet. So what are the errors in those approaches, in your opinion? There must be something that's not convincing about them."
"Well, I, I think one advantage of the gravity from entropy approach is that, uh, it, it leads to, um, Einstein equations in the low energy limit. This, to my understanding, is not always the case for many quantum gravity approaches, or at least might be quite difficult to obtain and respect. With respect to other modified gravity, that, you know, there is a huge literature on modified gravity, but the gravity from entropy, it has the advantage that there is a motivation. There is a physical motivation for choosing this action, right? It's not just, you know, the next term in the power series or things like that."
"There is this information theory and statistical mechanics understanding of the interplay between matter and geometry. Respect to other, you know, approaches using entropy, of course, the approach is fully statistical mechanics. So it embraces the microscopic degrees of freedom, while, you know, approaches that are based on horizon entropy, they are typically thermodynamics in spirit. So they start with the area law."
"Right."
"Which is, of course, fundamental. But it's the area law, to my understanding, is mostly coming from a thermodynamics description of gravity. While I think it is important to have a microscopic description. And also in this respect, you know, people that try to do kind of entropic approaches to gravity, sometimes they are coming from the perspective of theoretical physics and they try to be sympathetic with, you know, people that do research in statistical mechanics or, you know, condensed matter, which is so, was seen as, you know, very applied. But actually, I'm coming from, you know, complexity theory, network theory, but I have a very solid statistical mechanics understanding. And from my perspective, actually, statistical mechanics and field theory are the same thing, right? It's just changing an eye with, with a, you know, in the big rotation, right? So statistical mechanics is a fundamental theory, actually. And it is possible to write an action that is a field theory action and at the same part time, a statistical physics action. It's not that if you want to go towards statistical physics, you need to go toward, you know, soft matter or things like that."
"Yes. Right. Jacobson's papers literally titled 'Einstein as an equation of state' or something like that, which is a thermodynamic term. So why don't you spell out for the audience the difference between thermodynamics and statistical mechanics, why you see the latter as having an advantage here?"
"So, um, thermodynamics has, has been fundamental. And, you know, it arose from the study of heat engines. So practically, the efficiency of heat engines, it, it arose from very practical considerations during the industrial revolution. And of course, you know, there is this second law of thermodynamics, which is quite fundamental. But at the level of Clausius's result, we don't have any understanding of where it comes from. And this was the genius of Boltzmann, which describes the H theorem. And the H theorem shows that H, which is his definition of entropy, really is an increasing function. And so this is, is shown in a particular setting of what is called the ideal gas. So practically, you have particles bouncing around with given velocities, and they have an interaction that is hard-core interaction. So when, when they bounce to each other, they scatter around. But then there is no interaction at all. And, and this ideal gas is, is our most profound understanding of the second principle of, of thermodynamics. And, and this entropy is the number of microscopic configurations, compatible with the macroscopic configuration of this gas. Right? That's the beauty and the universality of statistical mechanics. So practically, you want to describe microscopic, uh, laws from a microscopic understanding of the degrees of freedom. You don't need to know all the details of, of your system. You need to capture the important information theory content of the microscopic degrees of freedom. And this has been fundamental and, and pervasive in all physics, I would say. And also behind. So practically, our understanding of phase transitions or phases of matter comes from statistical mechanics. Our understanding of, you know, classical and quantum phase transitions. Our understanding of also, you know, uh, the building blocks of quantum information and quantum computation comes from this rich interplay between statistical mechanics and information theory. And without mentioning AI, of course. And so this, this idea is that the, the idea is that information theory is such a pervasive, um, concept that as is already a common language across many different, uh, disciplines. And with gravity from entropy, this, this idea, this information idea are reflected in an alternative action for, for gravity. And which is the gravity from entropy action. And it, it's, it's interesting. And I think it's, it's quite stimulating because, you know, in statistical mechanics, there are possibly two points of view, right? Or even three points of view. So one point of view is the one of emergence. So it is the one that, of course, has been possibly the most fundamental, uh, approach to statistical mechanics, is that from the microscopic degrees of freedom, you explain what happens macroscopically. And this has been fundamental, coming from, you know, a pivotal paper by Phil Anderson, 'More is Different.' But then there is another thought, another line of idea, is that nature, the fundamental aspect of nature is information theory. And actually, this has been put forward by, by John Wheeler. So, you know, gravity from entropy comes from this kind of point of view that actually maybe we can have a statistical mechanics theory for the fundamental degrees of freedom of, of geometry and matter fields. Of course, it might be that, you know, there is a next theory that builds on gravity from entropy and finds that it is an emergent theory. But for the moment, it is conceived as a fundamental theory of geometry. So instead, and here I come back to what we, we were saying before. So gravity, instead of being a fundamental interaction in this reductionist approach, in which you look at the, you know, at what happens at the interaction, uh, when two particles interact. Gravity is a reductionist theory in which the object is the geometry itself. So in this sense, it is, is, is similar also to, you know, network approaches, in which you want to study the interplay between geometry and, and dynamics."
"What's the difference between entropy and information?"
"Um, so entropy can be used to quantify information. So in your theory, should it actually be gravity from information, because information is the more fundamental quantity or substance, or what?"
"No, no, it's gravity from entropy because the action is an entropy, but the entropy captures the information content of the microscopic degrees of freedom."
"Ah, okay. So what do you suppose is actually existing then? Because entropy usually counts something. So what, in your mind, is going on?"
"Yes. So in, so in my mind is that the metric, this, this true metric are kind of encoding the degrees of freedom of geometry and matter fields. Does that mean that information here means the degrees of freedom of the matter field, or or what?"
"Information is, is both. The, the idea of gravity from entropy is that it captured the information content present in the true metric that can be codified by the metric induced by the matter field and curvature. So it is the interplay between the, the two kinds of degrees of freedom, the, the geometrical and the matter field degrees of freedom are described at the same time."
"So what would you say is the primary difference between your approach and Erik Verlinde's approach?"
"Well, my approach stems from an action, and I don't use concepts related to holographic screens. So I, I want to capture the microscopic degrees of freedom of geometry and matter fields and their interplay. So I think in, in Berlin, the approach, there is no focus at all in the interplay between structure, between geometry and matter fields. It's very limited."
"You mentioned early on that this interplay between structure and then dynamics is important to you. Can you please outline what is structure, what is dynamics, and what does it mean for them to interact?"
"So in gravity, in gravity, my assumption is that what we know about structure is geometry. So it is the metric. So of course, there can be theories or research meant to represent, you know, where this metric comes from. But, you know, I assume there is a metric, and that, you know, the structure is, in some sense, the structure, like in quotation marks, is in some sense captured by the geometry. And the dynamics is, is just the matter fields. You know, I use bosonic matter fields. Uh, I use more recently some, some fluids for cosmology. The idea is that you can put there all matter fields. So the idea is that what I'm working on is that to formulate a gravity from entropy approach in which you can put the Standard Model there."
"Are you planning on going back to the discrete case, or are you now happily living in the continuum?"
"You know, I have my research agenda in, in networks, but for the moment, in gravity from entropy, I would stay in the continuum."
"Well, what about complexity theory? You mentioned that briefly, but how does complexity theory enter into this?"
"Yeah. Yes. So, as I say, uh, you know, information theory is so fundamental and so pervasive, and we got such impressive results using information theory. And, you know, with quantum computation, we might see even even more. But actually, it is in many aspects perceived that information theory alone is not enough, and one needs to enrich that with geometry and topology. And so you, you really need, or many different fields, starting from the brain to AI, you really need an information theory that captures the degrees of freedom of geometry. So from this point of view, geometric quantum relative entropy could, could be used potentially to address questions also beyond gravity, and addressing this need, a need of an information theory of geometry."
"Kurt here. Note that if you'd rather listen to TOE, we're on Spotify, iTunes, everywhere with a podcast catcher. You can just search my name or Theories of Everything. And also remember to hit subscribe. If I recall correctly, in 2021, you were speaking to the Network Science Institute and you said, I think you were working on this, and at that time you said it wasn't physics yet. Let me know if I'm mistaken. And anyhow, the point, my my question was, well, I was wondering when, in your mind, did it then cross over into physics? When did it become physics?"
"Yes. So it became physics practically immediately when I decided to go in the continuum. Because it's really strange, but in, in complex systems, most of the problems are in the discrete. So if you want to go to the continuum, you know, gravity is, is, is the most, uh, likely way to go. And, and this came, you know, after a trip that I made in, in the States, and, you know, and also this seminar at the ICTP. I felt the need, right, to establish the relation between this theory that I was trying to formulate in complexity and something perceived as more fundamental. And this, this is the outcome. And I'm very happy because, you know, for me, I'm kind of new in, in gravity and, you know, traditional theoretical physics. And it, it is a, a fantastic playground. It's fantastic progress that's made, you know, in gravity. Somehow, there are so many different directions that you need to be guided by your physical understanding of, of, of the problem. But for me, it's a nice, uh, new direction at this point of my career, and I'm embracing it and enjoying it."
"So what's the reception been like?"
"Yeah, the, the reception has been, um, open. Yeah. So there are people, I think mostly in, in cosmology, are quite interested. And they feel the need to work on modified gravity approaches. You know, there is this Hubble tension, there are different problems there, of course, dark energy, uh, somehow dark matter. So cosmologists are now becoming very welcoming for, for this theory. And yes, I had a very nice conversation. Of course, you know, I'm not in the stream of thought that, uh, starts with holographic screens. So there is also some, um, some clash there of our understanding. Of course, I, I think that, you know, the area law is fundamental, but I think it's a microscopic field. So, and what I want to say is that actually gravity from entropy, to some extent, reproduces the area law from the microscopic degrees of freedom. So this is, so practically, the, the Lagrangian is, is defined over volume. So if you integrate over a black hole, you know, until the Schwarzschild radius, you find a dimensionality reduction. And this is essential because practically holography has been, you know, a kind of explanation of the area coming after Bekenstein. Bekenstein didn't use holography. And it's a very simple conceptual idea, is that just if you have the entropy, which is the logarithm of the number of degrees of freedom, typically the entropy is, scales like the volume. So in order to have it in such a way that it scales with the area, the degrees of freedom must be only on the surface, on the horizon. But actually, this assumption builds on the idea that, you know, the degrees of freedom are the same in every point, you know, inside the black hole. And in gravity from entropy, because gravity from entropy doesn't depend only on the Ricci tensor or Ricci scalar, but depends also on the Riemann tensor and so on, on the Weyl curvature. So what happens is that the degrees of freedom in a black hole, right, the, the degrees of freedom inside are not always the same, depends on the distance from the origin. And, and so when you integrate something that is not homogeneous, you can get a, a dimensionality reduction. So that's the beauty of gravity from entropy. And so for the moment, I, I don't feel the need to use a holographic screen. And I think that, you know, there are many important insights coming from the area law, but if we are looking for a fundamental theory, we need to, to go beyond the area law."
"So is this an emergent theory of gravity then? So gravity from entropy assumes that the geometry exists. And so it builds on Einstein's insight that gravity is a theory of geometry, but it provides new understanding of the interplay between matter field and geometry, but it doesn't want to explain where geometry comes from. This is not the goal of gravity from entropy. So as long as Einstein theory might not be considered emergent gravity, but in some sense it is, because, you know, Newtonian forces come from geometry. Also gravity from entropy is, is at the same level, right? So it assumes that the metric exists and captures the information content of this metric. And of course, you know, Newtonian gravity comes from, from the geometry, exactly as in Einstein gravity. Of course, a problem could be, you know, where geometry comes from from the beginning, but this is not tackled in gravity from entropy. What is emergent is instead this dark energy term that is dynamical and is, is new because it emerged from the theory and is, is driven by this G field."
"Right. And it's positive."
"And it's positive."
"And that's a positive thing."
"Yes."
"So what are some of the open problems that you're working on right now, or some of your colleagues are working on, or maybe your students? What's bothering you right now about this theory and you're tackling it?"
"Yeah. So, of course, there are the cosmological implications and the relation with predictions and, you know, possible experimental validation of the theory. Then there are aspects related to the second quantization. And then there are aspects related to entanglement. So how this gravity from entropy action is related to entanglement and Iraqi entropy. And then, of course, maybe if one wants to go also going in the discrete. But what, you know, keeps me awake at night is the second quantization of this theory."
"What is second quantization and why is it so difficult or tricky in this case?"
"Yeah. So, I mean, this is the real problem in quantum gravity, right? So to combine gravity with our field theory second quantization approach, uh, of the other fundamental forces. It is notoriously difficult there. There, you know, this is where the community has explored different directions. And of course, there might be the need for defining, uh, the graviton, right, or which mediates the gravitational interaction. But the quest is open, right? There are many different approaches."
"The graviton would be emergent in your view?"
"Uh, probably not."
"It would be fundamental?"
"Yeah. You know, it's also to be questioned if, if the graviton really exists, right? Maybe there is something else, right? Is, you know, because the metric is, is described by vierbein, is described by, you know, spin connection. So there are different options. It's not that you need to start from the metric itself."
"I just spoke with Philip Mannheim, who doesn't believe the graviton exists. Actually, did I hear you correctly that you use Weyl curvature as well?"
"I, I used the entire Riemann tensor. So this includes the, all the components, also the ones that vanish, that, that do not contribute to the Ricci scalar. Yes. So this is an important aspect. And this is what makes, you know, uh, the entropy of the black hole non-trivial. Because practically, in empty space, in the empty space, the gravity from entropy action is non-zero. And so is also if the Ricci scalar is zero, right? The gravity from entropy action depends on all the components of the Riemann tensor. So it is non-zero. And so you can integrate over, over the volume, it is not homogeneous all over. And this is an important aspect. And another important aspect is actually that this theory provides also some corrections at the Planck scale, also in, in flat geometry."
"Interesting."
"So the theory, treating geometry together with matter fields, finds some corrections at the Planck scale already in flat, uh, in flat geometry. And what about what's going on at the singularity? Now, I know that's a question that's notoriously difficult, and I'm asking you at the beginning of developing a theory, but I'm curious."
"Yeah. So, the idea is that, you know, uh, possibly the singularity would be avoided by the gravity from entropy theory. Because you have this G field that enters into the action, and this G field is dynamical. So for instance, for the Schwarzschild solution, the Schwarzschild solution is, is a good approximate solution to the black hole, because, you know, it's a solution of the Einstein equations that are approximate equations of, of the gravity from entropy theory. But actually, you know, close to the singularity, this G field becomes dynamical. So maybe there is no single static solution of the black hole in gravity from entropy, and maybe, you know, this singularity is avoided."
"Professor, what advice do you give your PhD students consistently?"
"Um, one advice, which is I think easy to follow, is to read articles and to study. H. Another advice is to try to follow your, what you like, you know, your, try to express your vision of, of reality in what you do, and enjoy, enjoy what you do, you know, try to have fun. It's not always the case, but, you know, I think that that's part of why we do science. We do because we want to enjoy it, uh, you know."
"What's a piece of advice that you keep coming back to that someone gave to you?"
"Um, so for instance, one advice is, 'Why don't you go in the continuum?' Right? This is an advice coming from a random person that was at my seminar. We, we didn't talk, we didn't discuss, but it was very important for me. And another thing that I learned, that possibly was not an advice, but, you know, try to explore and to go in other communities and see what, what they do. So if you are in a big conference, you know, very interdisciplinary, try to pop up sometime in a session from another community, and you will find answers to your questions, possibly, if you think about your problem quite abstractly. So this is, has been very important for me to formulate gravity from entropy. I had this idea, and I was at the DPG meeting, you know, speaking about my topology in networks, and then I went to, to listen at a session about, uh, gravity and information, and it was inspiring and useful."
"So try to, to look at nature with surprise, and maybe things will turn nice."
"Professor, thank you for spending so much time with me."
"Thank you."
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