 and that will take a few seconds to kick in. Let me check. I believe we might be live at the moment. Okay. Welcome, everyone. Thank you for joining us for today's low physics webinar. My name is Alejandro and I'm going to be your host. Today we're presenting Graviton Magnetic, Tidal Driving and Affordating Neutron Stars by Eric Poisson. Eric obtained a bachelor in physics from Laval University and a PhD in theoretical physics from the University of Alberta, where he worked under the supervision of the great Werner Israel. Before joining the Department of Physics at the University of Gold, he spent three years in Keystone's group at Caltech and a year with Clifford Wheel at Washington State in St. Louis. Professor Poisson works in many areas of general relativity, including black holes, neutron stars, and gravitational waves. He's also a fellow of the American Physical Society and affiliate member of the Premier Institute for Theoretical Physics, also in Canada. I'm a member of the editorial board of Physical Review D and he served as the president of the International Society of General Relativity and Gravitation. He has published two extremely beautiful textbooks, Relativistic Toolkit and Gravity, Newtonian, Post-Newtonian Relativistic, and that one was co-authored by Cliff Wheel with Cliff Wheel. So remember, you can ask questions over email through our YouTube channel or Twitter, and then the questions will be read at the end of the talk. We are very happy to have you, Eddie. So please take it away. Thanks for joining us. Yeah, well, thank you very much, Alejandro. Thank you very much to all the organizers for the very nice invitation. I'm very honored to be speaking in this speaker series. So let me share my screen and hope that everything goes well. So do we see, do we see that? Yeah, very good. Okay, so... Okay, very good. All right, so lots of very strange words in this title, so I will try to explain all of this. So my intention here is to have a fairly short talk, introduce all the stuff with a minimum of technical details, and hopefully convince you that there are some very interesting physics going on here and physics that can soon be revealed through the measurement of gravitational waves. In case you're interested in the details, I have two papers here that details everything that I've, that I'm talking about today. So, all right. So the context for this work is gravitational waves. So we've, you know, about five years ago, we've entered this very exciting era of gravitational wave astronomy. So we're all very excited about this and the results have been coming. So it's already been a very exciting, you know, discovery path. And we uncovered, you know, a class of black holes we didn't know about. But my focus here will be on gravitational waves that might be coming from a binary system involving at least one neutron star. So we could have two neutron stars. We could have one neutron star and a black hole, but one of the objects has to be a neutron star. And I'm interested in the late stages of the inspiring motion of that binary system just before the system is about to merge. So as all of this is ongoing, the system emits gravitational waves and those waves are now being, you know, measured by a network of gravitational wave detectors. We have the two LIGO detectors in the US. We have the Virgo detector in Europe. And I think the Kaggle detector is just about to go online. And so far there's been a few detections of gravitational wave events that, you know, probably contain two neutron stars. This one over here, that one is for sure a neutron star event that was observed not just in gravitational waves, but also in gamma rays and, you know, the whole band of electromagnetic radiation, a very spectacular event that we've, you know, been very excited about. This one here is likely to involve at least one neutron star, but there's been no electromagnetic counterpart to this. So we cannot say for sure. And this one may contain a neutron star, but it would be a very large neutron star, larger than anything that we've seen so far. So that one is a bit uncertain. But in any case, we already have had measurements of gravitational waves coming from neutron star systems. And as things keep going, when LIGO goes back to, you know, observations, there'll be more of them as the detectors go more and more sensitive. This is a picture of the Virgo facility in Italy. So you have the two vacuum tubes. And as you know, LIGO works with laser interferometry, gravitation wave detections work with laser interferometry, where you have laser beams bouncing back and forth and being recombined here. And you hope to reveal the very tiny motion of the end masses at the end of the vacuum tubes here as the laser beams recombine over here. All right. So why neutron stars? I mean, black holes are pretty exciting objects. They're very fascinating, but neutron stars are, well, you know, sometimes in my mind, even more interesting because black holes have nothing, right? Black holes are just boundaries in space-time, whereas neutron stars have all of this very complicated, very poorly understood nuclear physics that goes on deep inside these bodies. And what really excites me and excites a whole bunch of people is the perhaps ability to use gravitational waves of observations to learn something about the poorly understood nuclear physics that takes place inside neutron stars. And the way that this is going to be done, at least in what I'm going to be talking about here, is through the tidal interaction between the one neutron star and whatever the companion is. So I'll focus on the neutron star and we'll, you know, think of the companion going around it. That companion could be another neutron star, or it could be a black hole. It doesn't really matter for what I'll talk about here. So the idea is that the neutron star will experience the tidal field exerted by the other one. So the tidal field is the henomogeneous gravitational field exerted by the other one. It's the fact that, you know, the field on one face of the neutron star is going to be stronger than on the opposite face. So that creates a differential gravitational field across the volume occupied by the neutron star. And that henomogeneous gravitational field will tend to produce a tidal deformation. And because the object, the neutron star deforms, well that changes everything because the tidal deformation means that what you have is no longer a point mass orbiting another point mass. Now you have deformed objects and that means that the orbital motion is going to be slightly affected. And because the orbital motion is slightly affected, that will be manifested in the gravitational waves. So there's, you know, a subtle effect in the gravitational wave form that reveals this tidal interaction. And the reason why this is interesting and important is because, well, objects of different internal structure will deform differently. So the amount of deformation that can be revealed in gravitational waves will depend in a very sensitive way on the internal structure of the body and it's that internal structure that is governed by poorly understood, poorly known properties of nuclear matter at those huge densities that we find inside neutron stars. So what I'm talking about here is the prospect that a measurement of, you know, tidal deformation in gravitational waves can tell us something about nuclear physics. So just to give you an idea of how poorly understood the nuclear physics of neutron stars is, I mean, we have some ideas. I mean, we know that we're dealing with neutron, which matter. We know that, you know, at fairly low densities near the surface of the neutron star, we have some pretty good understanding. But as we go deeper into the star, as we increase the density, well, the properties of nuclear matter are poorly known because those densities far exceed what we can produce in the lab. So the nuclear matter at those densities can be very different from what we can measure here in the lab. And people have formulated all sorts of, you know, tentative, you know, candidate equations of state to describe this. And what I'm plotting here is, as a function of those equations of state that have been compiled by various groups doing nuclear physics, what I'm plotting here for each one of those model equations of state is the mass of the neutron star in solar masses. Hopefully you can read the numbers here. And radius of the neutron star in kilometers. And what you should see here is the fact that, well, if you pick one equation of state, you have one relationship between the mass and the radius that might look like this. But if you pick another equation of state based on different assumed physics for the nuclear matter, well, you might trace this curve or you might trace this curve. And the point is that, you know, these all being plausible models of equations of state, those curves are all over the place. So for a two solar mass neutron star, for example, you might have a radius that ranges from 10 kilometers, more or less, to 16 kilometers. And that really means that there's a ton of uncertainty about the you know, the details of internal structure for a neutron star. And what we're hoping to achieve with gravitational waves is constraints. We want to constraint this range of possibilities. That's one way of doing it. There are other ways of doing it. There are currently a very nice apparatus going on board of the International Space Station. It's called NYSER that measures X-ray emissions from neutron stars. And that gives you a handle on the radius of the neutron stars. So that's a different way of putting constraints on the equation of the state. Here I'm talking about something slightly different using gravitational waves. So that's the context of the work. What I want to do next is to mix in two ingredients that are going to be the essential things that will give us all the physics that I want to describe in this talk. And those two ingredients are what I call gravitational magnetism. I'll describe what this is. And stellar rotation. So stellar rotation is just the property that the neutron star might be spinning on its axis. And if it does that, it basically opens up some new physics that you wouldn't get if you didn't have stellar rotation. And if you mix in stellar rotation with gravitational magnetism, then you have something that's really fabulous. And that's what I'll be describing in the rest of the talk. So in principle, I'm working in full general relativity because if I want to do justice to gravitational waves, if I want to do justice to neutron stars, I have to work in the full framework of general relativity. But here I'll allow myself some leeway and I'll say let's work in an approximation to that. Let's assume even though it's not a very good approximation, but let's assume that velocities are small compared to the speed of light. And let's assume that the gravitational field is not that strong so that this combination of mass and distance is small compared with one. So not a very good approximation for neutron stars, but it allows me to do something that otherwise would be too complicated to do in full general relativity. If I adopt this approximation to general relativity, then I find that gravity is described in terms of two ingredients that are going to be familiar if you remember your graduate work in electromagnetism. In this approximation, we have that gravity is described in terms of the scalar potential that I would call gravity electric. That's basically the Newtonian potential that's familiar from Newtonian gravity. But on top of that, we have a vector potential that's very analogous to the vector potential in electromagnetism. And that I call gravity magnetic. So we're talking about gravity, but we have a scalar part that is analogous to electricity in Maxwell's theory. And we have a vector part that's analogous to magnetism in Maxwell's theory. So basically we have a theory of gravity in this approximation that really resembles Maxwell's electrodynamics very closely. For example, the scalar potential here that is familiar from Newtonian physics is produced by mass. So rho is the mass density and we have this familiar looking Poisson equation relating the mass density to the scalar potential. That's familiar Newtonian gravity. But in addition to that, we also have this vector potential that is now produced by mass currents. So if I take the mass density rho and multiply by the velocity of whatever is moving in my system, then I have a mass current that's analogous to a charge current. And that is producing a vector potential in the same way that charge currents produce a vector potential in E and M. So we have this close analogy at the level of the field equations, but in addition to that, we have a close analogy in the force that fluid elements, for example, if I'm looking at the fluid body in my neutron star, if I'm looking at the fluid inside the neutron star, each fluid element will undergo a force that's gravitational in origin. And that force will have, well, a Newtonian component, that's the gradient of the scalar potential that's analogous to an electric field. But it will also have this force, V cross B type force, where my B field here is not a magnetic field, but it's produced, it's the gravitational analog of this associated with that vector potential. So the curl of the vector potential produces a gravitational field that's very analogous to a magnetic field in gravity, and that B field in gravity couples to the velocity in the same way as in the Lorentz force. So we have a V cross B term in the force that is given to us in this approximation to general relativity. So what I want to do here is to take those two ingredients, gravitational magnetism that produces this vector potential in this B field, and velocity. And I want to actually really focus on this Lorentz force acting on the fluid inside the neutron star. That's going to be the force that governs the physics that I want to describe next. So there are two V's in this slide. V here, you should think as the orbital velocity of the companion, of the companion. So I'm focusing on my neutron star here, the fluid inside of it, but I have this companion in orbit around it and that V here is the orbital velocity of that companion. That's the origin of the mass currents that gives me the origin, you know, that gives me this gravitational magnetic field. That gravitational magnetic field is going to be stronger here and weaker here, so it's not going to be homogeneous. And the inhomogeneity in that gravitational magnetic field is going to produce a tidal force inside my neutron star. That tidal force is going to be created by the orbital velocity of the companion. That's the first V. The second V is in this equation over here. And that V is going to be the velocity of the fluid elements inside my neutron star. And that is given to us by the stellar rotation. That's why I need both. I need the B field coming from the orbital motion of the companion and I need stellar rotation to produce that V over here. And combining the two, that means that I have this magnetic type force acting on the fluid element. That's the two ingredients that are coming together to produce that force. And what we have here, therefore, is a new type of tidal force that is not this one. That's the familiar Newtonian tidal force. That's something extra that comes from general relativity and described in that approximation. So stellar rotation, gravitational magnetism, and this extra term in the equations governing the fluid mechanics inside my neutron star. All right. So those are the two main ingredients and that's where all the physics is going to be coming from. And what I want to do next for the next few minutes is just to go a little bit deeper into the fluid mechanics inside a rotating neutron star. And I'll do this with as little technical details as I can manage, but I just wanted to give you an idea of what people do when they do fluid mechanics, especially when you want to perturb a fluid configuration from equilibrium because you have this tidal force acting on it. So the idea is that the tidal force that we've talked about just before, well, that's going to create a deformation of the fluid distribution. So the tidal force will change in direction. It will change in intensity as the companion orbit around the neutron star. And that's the thing that's going to describe, deform the fluid. And that's the thing that I want to then eventually incorporate in a model of emission of gravitational waves and all of that. So we'll come back to those aspects later. But now, excuse me, I want to focus on the physics of the fluid mechanics inside the neutron star. And what people always do in physics, and that's true also in fluid mechanics, is to try to decompose everything in terms of normal modes of vibration in this case. You want to form a basis of basic deformations and you want to analyze whatever is produced by the tidal forces in terms of this basis of basic vibrations. So I'm going to decompose everything in terms of normal modes. And the idea is that the fluid variables collect together, they form collective variables that can be grouped into collective behaviors. And that's the sort of thing that we call normal modes. So what do you do? Well, you say, well, let's consider the perturbation in density. Let's consider the perturbation in pressure. And let's consider all the relevant perturbation variables. And we're going to assume that they vary in time as e to the minus i omega t. So there's going to be some frequency associated with those modes. And there's going to be some profiles associated with each mode, how it produces a deformation in density of deformation and pressure and all of that. And when you work out the fluid equations, Euler's equation, continued equation, all of those equations from the fluid mechanics, and you make this assumption that the deformation you want to describe is of that form, what you end up with here is an eigenvalue problem for all of those modes. So the collective behavior is going to be captured in terms of eigenvalues for the frequencies and eigenfunctions for the collection of fluid variables. So what's going to be important for us here is the fact that as you solve this eigenvalue problem, you get basically a whole set of eigenvalues, eigenfrequencies for the modes. I'm going to label them with a k here. And corresponding eigenfunctions, we won't focus on the eigenfunctions, but the eigenvalues here, the eigenfrequencies will be important. And then what's even more important is the fact that conceptually, this collective behavior boils down to each mode behaving as a driven harmonic oscillator. So what you can imagine here is that instead of a star jiggling about because of fluid deformations, you can replace all of this by a whole set of harmonic oscillators that are going to be driven by that tidal force created by the orbiting companion. So conceptually at this level, what you have is a whole bunch of harmonic oscillators driven by an external tidal force. So it's a very convenient way of thinking about the problem. And in fact, it corresponds to all the mathematical stuff that has to go behind the scene here. All right, so what are those modes? So what do they describe? So let me focus for the time being on a non-rotating star, and then I'll switch on rotation in the next slide. So if you do this fluid dynamics for a non-rotating star, what you find is when you do this eigenvalue problem business, what you find is that there are two classes of modes that emerge and they have slightly different physical properties. And people call them p-modes and g-modes. And the p and the g have to do with what is restoring the mode to equilibrium in the fluid mechanics. So the energy here is a little pendulum. So if you push a pendulum, there's going to be a restoring force excuse me, that's going to take it back to toward equilibrium if you push it. I'm sorry, my voice is losing my voice. Sorry about that. So if you have a pendulum like this and you push it forward, there's going to be a gravitational force that's going to restore it back to equilibrium. And then, you know, the balance between the motion, the inertia and the restoring force will produce this, you know, oscillating motion. Well, the same is pretty much true for the fluid motion and you have different types of restoring forces that give that give different properties to the modes. And p-modes are are modes that are primarily restored by pressure. So the pressure is acting back to restore the equilibrium of the mode. And that's very much like the physics of sound waves. So p-modes are like sound waves, but there are, you know, modes inside a spherical star instead of being, you know, modes in a three-dimensional, you know, homogeneous fluid. G-modes are primarily restored by gravity and those are like gravity waves that are familiar if you're, you know, studying the ocean or something like that. So two broad classes of modes and when you work out the eigen frequencies for those modes, well, typically you find a mode frequency that will have an order of magnitude that's going to be governed by the mass of the neutron star and its radius. And if you put in the appropriate scaling here, a 1.4 solar mass for the neutron star and a 10 kilometer radius for the neutron star, what you find is that typically the mode frequency will be in the kilohertz and the kilohertz, you know, two kilohertz or so happens to be quite a bit larger than the gravitational wave frequency that you get during the in spiral of a binary system of neutron stars. And that fact will become important when I talk about resonances a little bit later. So typically the mode frequency that you get is in the kilohertz range and that is larger than the typical gravitational wave frequency that you get during an in spiral through the LIGO Virgo band. Because of that, because of that mismatch in frequency what we find is that P modes are relevant but not that relevant because they never have a chance to come to resonance during an in spiral. G modes tend to have frequencies that are numerically quite a bit smaller than this order of magnitude. And because of that, they in fact can come to resonance during an in spiral and that could be exciting except for the fact that unfortunately their coupling with the tidal field is very weak and when you work through the details you find that the G modes never do very much dynamically. They're not that significant. So what I'll be you know telling you about in a few moments is that well with rotation you can identify new modes not these ones but new modes where in fact you can establish not just a resonance but a strong resonance that will have measurable impact. All right, so let's switch on rotation. So what happens when you switch on rotation? Well two things happen. When you switch on rotation first of all you know the spin of the neutron star will produce a deformation so you will have a centrifugal deformation of the star it will tend to bulge at the equator it will flatten at the pole and that will affect the modes that we've talked about so the P modes and the G modes that we've talked about in the case of a non-rotating star will acquire corrections because the deformation of the star will certainly affect the description of the modes and those centrifugal corrections will tend to scale in this way so they will be you know quadratic in the angular velocity of the star that's my omega here and the size of the correction tends to be you know of the order of magnitude of the ratio of omega squared to that gm over r cubed something that I've talked about in the previous slide and when you put in you know relevant scalings here a spinning frequency of 100 hertz for the neutron star solar mass you know 1.4 solar mass radius of 10 kilometers when you work out the scalings you find that the centrifugal corrections are fairly small so you would have to spin up the star to a large value in order to generate centrifugal corrections that are important because of that and because of the smallest of this number here I'm going to be happy to take my rotation to be such that you know I can neglect all centrifugal effects so for me I have slow rotation in the sense that this ratio is small and I will be neglecting centrifugal effects so my rotating star will you know to this approximation retain its spherical shape what I will do though and that's the second aspect of including rotation I will retain the Coriolis force which is linear not quadratic in the angular velocity and adding the Coriolis force has you know an impact on modes not on those modes if you include the Coriolis effect in the description of the fluid mechanics what you find together with those people here is that a new class of modes emerges you have new modes that come out they are restored by the Coriolis force instead of gravity or pressure and they're called inertial modes because well they you know they are associated with with that fictitious force that inertial force that comes linearly in the angular velocity those new modes are going to be behind the physics that I'll be talking about for the rest of the talk those inertial modes have some very interesting properties so they are mostly perturbations in the velocity field so before I was focusing on perturbations in density perturbation and pressure those turn out to be small in this case what really comes out strong for inertial modes are perturbations in the velocity inside the neutron star the velocity of the fluid inside the neutron star another interesting property is that the eigen frequencies of inertial modes are proportional to the angular frequency of rotation of the star up to an numerical factor of order unity that number is interesting because it depends on the mass distribution and the mass distribution is determined by the equation of a state so if you have access somehow to those frequencies and you know the spinning rate of your neutron star that number in front tells you something about the equation of a state and remember this is what we're after we're after trying to pin down the nuclear physics of neutron stars through things that we can observe and perhaps those frequencies can be observed in which case we have further constraints on the nuclear nuclear physics because this number is tied to the mass distribution now two very important things the frequencies here being proportional to the spinning rate of the neutron star can be in the LIGO Virgo Kagra band you know between 100 hertz or so you know 10 hertz to 1000 hertz if the neutron star spins at about that rate so if the neutron neutron star spin is around 100 hertz then that means that this frequency over here will also be of the order of 100 hertz and that's right in the middle of the LIGO Virgo Kagra frequency band and that means that you have the possibility of creating a resonance and that's good now if you try to work out the properties of those resonances what you find that seems a bit discouraging at first is that those inertial modes in fact couple very weakly to a Newtonian tidal field but as those people realized here Flanagan and Racine realized a while back is that those modes in fact couple strongly to a gravitational magnetic tidal field so that's why I needed those two ingredients I needed stellar rotation to produce those inertial modes and I needed the gravitational magnetic tidal field in order to excite those modes in potentially a strong resonance that will be what I'm trying to do next describe the impact of this so just to recap a little bit we're interested in tidal effects within a rotating neutron star so that's my first ingredient rotation we are interested in the effects associated with a gravitational magnetic tidal field that produces this V cross B type force that we've talked about so we have the rotation entering into that V and we have gravitational magnetism entering through that curl of U here and what I want to basically do next is to use this force as a driving force for my inertial modes and see what that gives me in terms of physics and in fact I can show that if I have a force of this type the only modes that will care about this force are the inertial modes the only modes that we've talked about P modes and G modes in fact are you know transparent to that force the only modes that participate here are the inertial modes that that I've just introduced so let's look at some consequences of this and one consequence it has to do with the fact that if I want to describe the tidal deformation of my object given the existence of that gravitational magnetic tidal force I have to do it not through what is called the love number that would be familiar from Newtonian physics but I have to do it through a love tensor and love here just to to tell you it's not just the name that we picked it's you know the name of Auguste of Love who was a geophysicist who worked out the theory of the tidal deformation of the earth and introduced a bunch of numbers that characterized this tidal deformation here we're doing the same thing for a neutron star so the usual story of a love number is you know something that you can infer in Newtonian physics so it's a relationship between two tensors and the first tensor is something that characterizes the strength of the tidal field let me describe this so u is the scalar potential created by the companion that's the familiar Newtonian potential coming from the orbiting companion if I take a first derivative I get the force associated with the scalar potential and if I take a second derivative I get the homogeneous part of that force that's the force field that varies across the volume occupied by the neutron star and that's the tidal force and I can characterize this tidal force in terms of that two that you know that tensor with two indices corresponding to the two differentiations that I've made on the scalar potential if I want to describe the tidal deformation I can do it through the quadruple moment of the mass distribution and that's the integral of the density times two occurrences of the position vector and then for technical reasons I remove the trace of that that tensor over here tells me how deformed the body is and if the body is spherical that qab would be zero so that's the deformation that's the source of the deformation there's a relationship between the two that's governed by this number k that's called the love number and in addition to that I have you know a scaling quantity g and a scaling with five powers of the radius of the of the body k is the interesting part here because that's the thing that depends on the details of the mass distribution inside the star and that's something that's determined by the equation of the state that's the familiar story in Newtonian physics one number does it all but when we do it for a rotating star and when we do it for gravitational magnetic gravity instead of you know Newtonian gravity then we end up with something more complicated and that number has to be replaced by a tensor and I'll explain that in the next slide so I'm taking my gravitational magnetic potential here and I'm expanding it you know in in time as a four-year series omega will be the frequency of that title field and I'm also expanding it in you know a four-year series in phi the the angle around my neutron star with the familiar quantum number m that comes with it if I want to describe the gravitational magnetic title field and characterize it in the same way I did before well I have to take my field so that involves one derivative through that curl here but then to capture the inhomogeneities I have to take a second derivative that's the extra derivative I have here and when I do two things those two derivatives then I have a characterization of the gravitational magnetic title field through that tensor if I want to describe the deformation of the body in that context well now I'm not talking about the mass distribution I'm talking about a deformation of the mass currents inside my star I have mass but I also have a velocity perturbation inside my star and that produces therefore a deformation in the velocity field inside the neutron star and that has to be captured by something called a current quadruple moment that I define here what is that well I have my mass current over here rho times v and I have two occurrences of the position vector here and here and that combination here you know involving that cross product is the thing that gives me a meaningful deformation of the neutron star given that the nature of the deformation now is in the mass currents not in the mass density itself another way to read this is to recognize this as the density of angular momentum inside my star and then I add an extra you know occurrence of the position vector and do the integral the love quantity the love tensor is going to be a relationship between the source of the tidal field and the consequence of the tidal field measured by those two tensors and what happens here for you know various technical reasons is the fact that that relationship between the deformation and the tidal field cannot be captured by a single number it has to be captured in fact by a collection of three different numbers because I have a tidal field that depends on M that azimutal quantum number that I've talked about before and each value of M produces different tidal effects and each of those tidal effects has to be captured separately with a love number that depends on M1 for M equals 0 M equals 1 M equals 2 and I stop here because I'm describing everything here at you know quarter polar order and you can think of that tidal tensor that occurs here as something that is a collection a meaningful you know packaging of those three love numbers and I'm showing you here what the you know love number for M equals 1 what it looks like and what you should see here is well you have the love number in some you know it's a dimensionless thing as a function of the ratio of the external frequency that's the frequency of the tidal field or that's the orbital frequency of the companion divided by the rotational frequency of the star and what we have here are four different curves corresponding to different density models for the star and what you should see here even though the plot is a bit busy is that each model produces a different distribution of mass inside the star and that produces a different love number as a function of that ratio of frequencies and what you should see on top of that is the fact that at some frequencies here this one over here or that one over there we have a jump from minus infinity to plus infinity and here a jump to you know from plus infinity to minus infinity those are those are the resonances that I've talked about when you have a frequency omega here an external frequency that matches one of your mode frequencies you get a huge response from the fluid that's a resonance and that resonance produces a huge spike in the love number that is designed to measure that response so that's an interesting thing and what I want to do is to wrap this up by telling you what observational impact those resonances can have so let's go back to the original context we have this binary system involving at least one neutron star we have a tidal field created by the orbiting body around that you know reference neutron star and now we see that that tidal field especially its gravitational magnetic component can introduce a resonance sorry I have my 40 minute warning here so we have that the gravitational magnetic part of the tidal field can exert those can drive those inertial modes inside the star and if we have a match between a mode frequency and the orbital frequency of the companion star we can produce those resonances and in fact when you look at you know the fact that you have more than one mode active here but you have a whole collection of modes what you discover is that in the course of an inspiral there can be a succession of four you know different resonances because four different modes will undergo resonance one after the other so this is a cartoon here of the frequency evolution during an inspiral starting at low frequency evolving to large frequencies so that's frequency you know orbital frequency as a function of time during an inspiral merger occurs right here at t equals zero what you see is that when the frequency when the frequency matches something a little bit under 50 Hertz or a little bit above 100 a little bit you know before you know 130 or so and 140 what you find is that you have a match between a mode frequency and the orbital frequency at that time and at that point you have a resonance and when you have a resonance that means that you you know you create a very large response of your fluid the fluid goes crazy and that means that you're putting in a lot of energy into the internal motions of the fluid at the expense of orbital energy and that means that during each one of those resonances here you're taking away suddenly energy from the orbital energy and that means that you're changing the properties of the orbit the orbit will jump from one orbital radius to another very suddenly and that will happen once twice three times four times during the inspiral and this sudden change in the orbital and in the orbital radius is going to be manifested in the in the emitted gravitational waves and that's something that can be you know measured you know by LIGO Virgo Kagra maybe not right now but you know in a few years when the sensitivity of the detectors have become a little bit better so the point is that these sudden changes in the orbital motion has a manifestation in the emitted gravitational waves and that in time will produce a defacing of the waves that can be measured in gravitational wave detectors so what I've done in one of my papers is to calculate this defacing and what I'm showing here is the end result how it scales with say stellar radius you know the mass of the neutron star the mass of the companion the total mass of the system how it scales with the spin frequency of the neutron star and what remains after you've taken all the scaling away is a numerical coefficient that's going to be a function of the inclination orbit and that's the angle between the spin axis of the neutron star and the axis of the orbital plane you know the normal to the orbital plane you know in which the companion is is going so that's gamma the numerical factor in terms of inclination angle as a function of different models of neutron stars different mass distributions come you know corresponding to different assumed equations of state for for the neutron star material and what you should see here is that well that numerical factor here tends to be fairly small unless you know nature is kind and produces for you misaligned situations where the spin axis points one way and the orbital angular momentum points in the opposite direction that's the angle here being equal to pi producing the large values here and values that can be you know up to something like point 20 or point 10 or something like that and that's going to be almost my last point here that means that well you can you know if nature is kind gives you those spinning neutron stars and give you those misaligned orbits that means that you have a phase shift of the order of you know point 10 or so that can be measured in a gravitation wave detection if your signal to noise ratio is larger than say 20 or so and not possible right now but possible in in your future and certainly routine when we're talking about you know next generation third generation gravitation wave detectors so those resonances produced by gravity magnetic tidal effects are real and eventually they will be detected so let me just stop here and just point out that while the physics I think is very interesting and rich in terms of consequences and and some of them are I think interesting from theoretical you know from a theoretical perspective but some of them I think are important for gravitation wave of astronomy and I think they open up a new way of constraining constraining nuclear matter the equation of state of nuclear matter because those resonances give us a different handle a different way of observing those tidal phenomena that are ultimately governed by all this nuclear physics so I think I will stop here thank you very much for your attention and I broke my promise of keeping under 40 minutes so I apologize for that no worry thank you very much for this wonderful webinar let me check the YouTube channel okay there's a question on the YouTube we have to wait a little bit there might be perhaps a delay one is talking about it says the following is it possible that once a certain mass is achieved the observed event horizon might be something akin to superconductivity but for gravity over a small locality so I'm not sure I understand the question so if we're talking about a black hole then we don't have for black holes the same sort of you know superconducting property of materials for example that they would expel magnetic fields so it's still possible for black holes to you know absorb magnetic fields you can have magnetic fields that are you know threaded by you know magnetic field threaded by magnetic fields so you still have you know magnetic field lines inside the black holes so you don't have this you know the this this phenomenon associated with superconductivity where the magnetic field lines would be expelled by black holes so I guess the answer to the question to the extent I understand it is no so so there's no phase transition for black holes that would be analogous to a superconducting phase transition for you know for matter okay and I think I I have a question which is a little bit related to what you were just mentioning which is could you please comment a little bit on the connection between the question of state and measuring these properties like because I can imagine I don't know maybe we can have a a neutron star that is so crazy inside that none of this will matter I don't know because the pressure or because of superconductivity such that the fluid just moves around and and then we don't get is it does it depend and how does it depend on the question of state yeah so it depends very strongly on the equation of the state so basically all those so so the main idea is this so in gravitational waves you can see the fact that you've hit a resonance you know maybe once maybe twice maybe four times during an inspiral and hopefully in the gravitational wave measurement you can measure the frequency of those resonances now those frequencies probably will not match the predictions that I've made based on my models of neutron stars because they incorporate all the relevant physics but the point is that if you measure the frequencies you can perhaps build a better model for for neutron stars with more exact information concerning the equation of state more information about possible phase transitions inside the neutron star in a way that eventually the model will hopefully match you know the frequency the resonant frequencies measured in one detection another detection the third detection hundreds of detections eventually so that through the you know through the measurement of the resonant frequencies you can start saying something meaningful about you know the nature of the neutron star its equation of state its mass distribution whatever phase transitions might occur all of that so that's really the hope here is that you know through observations you can infer enough properties of neutron stars to be able to pin down the nature of nuclear matter and of course you know that works in this context it works in the context of measuring x-ray emissions it works in you know other ways too so you hope to bring all of this together all those strands together and and eventually say something that you know measurements in the lab on earth won't be able to tell you thank you okay well i'll check the youtube channel someone here in the audience in the local regulators have a question I have a question so so all this analysis has been assuming that general activity is is general is what what is describing this correctly right right how sensitive is all this analysis to modifications of gravity so for example if if these frequencies don't match the model that you have or the equation of state that you have for a for a neutron star how is it is to discern if this is because we don't understand well the neutron star so maybe it is sensitive to some modification to the formalism right so that's a very important question and you know I'll give you my personal opinion on this and I observed that testing GR has become a bit of an obsession everybody's trying to test GR and I think there are good ways of testing GR when you have a very clean astronomical system like a binary black hole and you for example observe gravitational waves coming from a binary black hole you can measure inner things in the gravitational waves that allow you to perform you know good tests of GR I think that's a very good way of doing it but when you're dealing with a messiest physical system where you know there's a lot more uncertainty in the matter physics than the gravitational physics I think it's probably not likely that you can use those systems to test GR you can use those systems assuming GR to test the unknown astrophysics of your system the unknown nuclear physics of your system that's where all the uncertainties are they're they're far dominant compared to the uncertainties associated with perhaps GR breaking down or not being applicable in strong fields that's a personal opinion I think that kind of you know in my mind this kind of proviso applies to for example trying to test GR through observations of uh M87 I mean the shadow of M87 people have tried to you know take those observations and you know read them into tests of GR well I don't buy that because the astrophysics is so messy so far more uncertain that I think you have to assume GR and do your best to try to pin down the astrophysics maybe one day the astrophysics will be so well understood that you can use those sources as ways of testing GR but I don't think that will happen anytime soon thank you I'd like to comment on that but let's let's go on the youtube channel we have a question from Emanuele Blue which is asking which is the order of importance of these gravito-magnetic modes compared to the usual modes of neutron stars like the t-mode and that depends on what is driving the mode so the so in the context that I was talking about here if you so the the initial modes are so if you if you look at the driving in terms of a Newtonian tidal field what you find is that the initial modes are completely irrelevant so the the physics of Newtonian tides is completely governed by p-modes and g-modes mostly p-modes but if you forget about this that's that's true if you forget about this and now look at the influence of the gravito-magnetic part of the tidal field then it's the other way around the p-modes and g-modes are completely irrelevant and it's the inertial mode that that that dominates now the Newtonian tidal forces of course dominate everything so the the gravito-magnetic tidal forces that I focused on are smaller but uh but it doesn't mean that the impact of this force is not important because even though the tidal force is smaller gravito-magnetic tidal forces smaller they can still produce resonances whereas the you know stronger Newtonian tidal force doesn't produce resonances or at least not strong ones so the beauty of this is that through the resonance effect you can have a mode that normally would be irrelevant become dynamically very important and that's that's what happened here thank you uh okay I think Roberto has a question yeah I have a very nice the webinar Eric so I have two two questions I mean maybe now three with the last question the first one when when you were making this analogy between gravity and electromagnetics let's say with the scalar potential of the vector potential and do you expect also to have I mean in this model you also have this mathematical properties like gauge transformation but for the gravity potential let's say I mean so yeah the formalism is very close to electromagnetism so so in gravity you have you know something that's often you know often called gauge transformations completely analogous to what you find in electromagnetism except that the gauge transformations in the context of gravity are just ways of talking about coordinate changes so what you can always do even if you work close to flat spacetime is to change the the mathematical expression of the gravitational field by introducing some slight coordinate deformations and a priority it's hard to distinguish what is a true gravitational effect from what is just a you know a coordinate effect and the formalism of gauge transformations allow you to sort of make sense of all of this and uh what I've shown you know what I've chosen to do in this work is to you know pick a gauge and work in this gauge and that makes everything gauge and variant you know as a result so so I think I'm safe from all this you know complexity of you know gauges different gauges producing different effects and things like that thank you the second question is about the tidal effect the one that you had with the rho v times the the rest of the so is it possible that this tidal effect can modify the I don't know kind of generate electrical movement of charges within the neutral star of the companion neutral star in that sense that modify the magnetic field I mean the the magnetic thing not the the part of the gravity like and also to my point is that these jets that some neutral stars they have like like the kind of signature for poles are so yeah that's a very interesting question and I I don't know the well in principle yes and in principle that can be a very interesting effect and that's something that I at some point want to get back to so what's the what's the effect of this velocity perturbation that I'm introducing inside a neutron star to a magnetic field that might be there and we know that you know that neutron stars are magnetized therefore this you know combination of magnetic physics and fluid physics is going to be present and we know that differential rotation inside a neutron star can produce an amplification of the magnetic field so I suspect that you know those modes can do the same so those modes especially at resonance can probably produce some sizable magnification of the magnetic field but I haven't worked out anything about this I don't know the orders of magnitude I haven't worked this out but I suspect that this is something that's going to be very interesting to look into in the future yes now because I was also I mean because I remember that when sometimes the neutron star they have this kind of kind of relaxation ways that and in which that the kind of the size I don't know the but they emit a pulse of electromagnetic radiation from time to time when they are the ticking for the pulsar so I was wondering if this effect could enhance this type of ticking time because you show that you have some resonance mode in the when you are expanding so I don't know it's possible so what you were talking about is the effect on the crust that sometimes there's a there's a little bit of a technonic shift in the in the crust and yeah if you have you know fluid motions under the crust that that get resonant like this it could certainly affect the crust as well so yeah there's a there's a ton of physics that can be incorporated in all of this and yeah who knows thanks thank you and I think we're doing pretty well on time so we have let me try to ask the last question that I see here could you please comment on the discussion about black holes falling in love that's in quotes to get do black holes get deformed could you please share your thoughts on that right so so black holes are strange in the sense that so the love number that would give a characterization of its quadruple deformation relative to the strength of the tidal field that love number goes to zero that number is zero for black holes now what does it mean well it doesn't mean that the black hole doesn't deform so the black hole in fact does deform if you were to take the black hole and measure the shape of its event horizon you would find that that shape is not spherical that you know deformation that there is a deformation of the event horizon so it doesn't mean that so what does it mean well what does it mean for non zero love number for a neutron star well it means that the matter inside the neutron star is shifted is you know is you know rearranging itself because of the existence of the tidal field so the k really measured this rearrangement of the mass and that's the thing that's measured in the in the quadruple moment of the mass distribution for black hole you don't have this rearrangement of mass because black holes are nothing black holes are just things that exist as boundaries in space time except maybe all the matter of the singularity but if the matter is at the singularity well it's kind of stuck there it cannot rearrange itself so that is the thing that is you know that lack of ability for the black hole mass to rearrange itself is the thing that is you know captured by the statement that the love number for black holes is is zero if you are not happy with you know these words that I've just used I don't blame you because those are just words and I don't like to speak of mass being stuck to the singularity I have no idea what this means but you know as a sort of an analogy as a sort of thing that you know sort of conveys the result I think it's okay but you know don't don't you know don't ask me to hold on to the to those very vague ideas thank you and I think that's it so thank you very much for this wonderful low physics webinar and thank you everyone for attending today to this session we'll give you a very tall applause that is very common these days and thank you Eric for sharing your research and for everybody in the audience let us keep in touch this will keep having more webinars in this season so thank you very much and goodbye to everyone yeah thank you and thanks again for the invitation okay thank you bye good we are not life