 Okay, please take your seats. We are ready to start with the third course, and Stan Babak, who is now based in Paris, is going to give the first set of lectures about habitational ways. Yes, so let's start with my name. I will be talking about grotational waves, and it's a part one, and then next week, Vitor Cardozo will take over, and he gives second part of lectures. First of all, can you hear me well? Do you understand me? Okay, good. Right, so this literature where I took material from, and most importantly, figures, so I had to, since I used figures which are published, some of them, I had to put this up. So these are articles and the books which were used. Okay, so these two books on the general theory. There is a brilliant book, my McKelley majority on aggravitational waves, and these two books became classic on Bayesian data analysis. And before I start, I want to ask a quick question. Who had the course on general relativity or differential geometry? Can you please raise your hands? Beautiful. So first beginning of this lecture, it will be just a reminder, and since the majority of you did have it already, so I will go through this hopefully quite quickly. It will be just a way of introducing notations, and it will be the most mathematical part of the lectures. As we go further and further, there'll be less and less math, but I had to introduce at least something which I will be using later on. So in the first lecture, I want to cover very briefly, I call it reminder, general relativity, and then we'll go to gravitational waves. We start with gravitational waves in vacuum and look at the main properties, and then we will conclude today's lecture hopefully with generation of gravitational waves and giving the leading quadrupole expression. You're happy with this? Good. I'll also will give a quick overview of lecture two, three, and four, so you know what we'll try to cover, and I will repeat this outline at the beginning of each other lecture. So lecture two will consider binary systems. I'm mainly working on rotational waves from binary systems, various type of binary systems, and therefore I will concentrate on binary systems. We will touch modeling gravitational waves from binary black holes, and Vitor will give some more details. And then we will switch to detection of gravitational waves with ground-based observatories like Virgo, LIGO, and whatever is being built at the moment, not yet online. In lecture three, we will go to space and I will introduce ELISA interferometer space antenna, ELISA. This project does not exist. Well, it exists as a project, but it will fly in 2030-32. And then there will be two parts of data analysis. I will show how data is analyzed and how published results about found gravitational waves are obtained. I will touch very briefly, testing general relativity with gravitational waves, and I will try to conclude this possible timing array. All right, the first few slides will be very introductory, and the very first slide is basically showing the global effort to detect gravitational waves in three different bands. At the high frequency band between few hertz and few kilohertz, these bands were ground-based detectors like LIGO and Virgo operate. And as you all know, I presume they have detected first gravitational wave signals. Already we have a handful of gravitational wave signals from merging black holes and one gravitational wave signal from merging neutron stars. But there is a fundamental limit at low frequency beyond which we cannot go. This limit is a steeply rising seismic noise on the ground. And to detect the gravitational waves at lower frequency in millihertz, we need to go in space. And that's a project, ELISA. I will talk about this. And ELISA will be sensitive in the millihertz band between, let's say, 0.1 millihertz and 0.1 hertz. At the very low frequency in another hertz band, there is a third effort. It's, again, worldwide effort, so-called pulsar timing array, where we use pulsars, millisecond pulsars as a very stable clocks to detect gravitational waves. So we use nature-provided gravitational wave detector. I will also mention the main features of gravitational waves in general relativity. Well, actually, gravitational waves are not the only feature of general relativity, but they exist in every covariant theory of gravity. They might not be massless, it might be massive, but nevertheless, gravitational waves, we knew that they were, they are predicted by any, any, not only jar about any other theories. Specifically, in general relativity, gravitational waves propagate with speed of light, so graviton is massless. And the gravitational waves, they're acting as the time-varying, time-varying to other tidal forces. And we will show this. Gravitational waves, like electromagnetic waves, they are transverse, so they're acting in the plane orthogonal to its direction of propagation, and they have two polarization of state, usually referred as H plus and H cross. And the one key feature of gravitational waves is that they are very weakly interact with matter. On one side, it's a very good feature, because gravitational waves can propagate to almost unchanged from early universe up to us, or very, from very, very sources far away. It's good, and they are hardly absorbed or scattered into paris into electromagnetic waves, but the bad side of this is it's also because it's quite hard to detect them, because they weakly interact with the matter. And that's why we need devices which are of kilometer size. And the fundamental reason for this is the gravitational constant. It's very, it's a, it's coupling constant, and it's very small. I'll go very briefly, and I call this part reminder. So it's really basically to introduce the notations. So first of all, in special relativity, that's where we introduce local, sorry, inertial frames. Those are inertial, that's existence, the key point of special relativity is the existence of these inertial frames, and they're all related via Lorentz transformation. And the universality of speed of light, that's light travels with the same speed in all these inertial frames. Then I will be using throughout whole lecture, unit geometrical units, g equals c equals 1. So for me, mass, distance, and time are measured in seconds. To simplify your life, I can tell you that mass of sun, mass of sun is roughly 5, 10 to the minus 6 seconds. Well, then I want to introduce basically some other notations like Minkowski metric. This interval and Minkowski metric, I will use signature minus 1, 1, 1, 1, this written in Cartesian coordinates. Then I will assume summation of repeated indices. So if you see indices repeated, it's basically implies some like that. And while this is space like diagram, time usually is space there. There's null cones, it's for all the particles which travel with speed of light, like light, like a graviton. And I quite often will use the notation of u. It's a tangent vector for a work line of observer. Again, I will try to go quick. You don't have to write. Hopefully you know this. And the slides will be online. But when we come to something more important, gravitational waves, I'll slow down a lot. No? That was a previous slide? Yes. A little bit of differential geometry. And I want to introduce very important knowledge of a local flatness theorem, which I will be using a lot. So it means that, well, there's an example physically or mathematically. If you have a curve like that, this is not straight line. Nevertheless, in some small region of this curved line, you can choose the part of it where it's approximately straight line. Local flatness theorem tells us that we can do transformation of coordinates in such a way that at a given point, this is an important part, at a given point, we can put metric in a Minkowski form. And all first derivative can put to zero, but not the second derivatives. This will be important. And what it means for us is exactly this. At a given point, we can put these quantities to metric Minkowski form and derivatives to zero. But in vicinity of this point, it's not true. Nevertheless, for our experiments, we can demand some accuracy. We can say, well, it's not exactly zero, but 10 to the minus 15 is good enough for me. And this number, which you can tolerate, defines region of the curved space time, where your part of space time is approximately flat. And therefore, in that region, usually where distance or much less than radius of curvature, we can introduce local inertial frame. And this is important thing which I will be using all the time. I'll call it LIF, local inertial. And I will return several times to local flatness theorem as we go along. We will consider space time which is smooth, so we can introduce vector field on it. And so if you introduce vector field on it, which is smoothly changing from one point to another, we try to compare the vectors at different point. And we need to get the rule how to compare vectors at different points. And well, if points are really close to each other, we can take a derivative of a vector field. And we need to take into account that not only components of the vector field decomposed on some basis changing this part, but also we need to take into account that the basis vectors could change from going from one point to another. This happens not even in curved space time, but if you take polar coordinates on a plane, you will get similar thing. And this also the vector, we can decompose this vector into the same basis. And these coefficients which comes from these decompositions are called Christopher symbols. I want to emphasize that Christopher symbols is not a sign or signature of a curved space time. It could exist only even in flat space time where you have curved linear coordinates. And we introduce covariant derivative. Covariant derivative, generalization of ordinary derivative. Comma means partial derivative like that. And semicolon will go for covariant derivative. And these are Christopher symbols. I also will be using Greek letters, alpha, beta, gamma, delta, etc., etc. For those indices which are going from zero to three, so which take time, zero component, and spatial components one to three. And I will use Latin indices to only for the special part. So x, y, z if you wish. Another fact is the covariant derivative of metric is equal to zero and that's how we can relate Christopher symbols to the metric. This really reminds me if I'm going a bit too fast because I really want to come to gravitational waves. Next thing which I want to introduce is parallel transport. So parallel transport is a way of transporting vector from one point to another. This is a particular way of doing it. So we're trying to transport vector along the curve which is parameterized with some parameter lambda. It could be proper time of a observer or it could be something you made up but smoothly changing as you go from one point along the curve to another. And we want to preserve the size of the vector and we want to preserve the angle. Angle in this respect defined with respect to the tangent, to the curve which we're considering. And so this condition goes into that in local inertial frame. Again I'm using local inertial frame. It's very simple. Derivative of my vector field should be equal to zero. And then I use basically local flatness theorem to extend from a normal derivative to the covariant derivative in this way. You can start right away and you can show that in any coordinates you can get this but it's quite easier to start with local inertial frame and then extend it to the general coordinate frame. Geodesics. It's very important notions and I cannot skip this. I want to remind what it is. So geodesics between two points is a curve which is basically minimizing the distance between this point. It's the shortest distance between short curve which is shortest in a given space time which connects two points. In a flat space time it's very simple. It's a straight line. Now let's try to generalize straight line to the non-flat space time. What would it be? So it's again shortest we're trying to find the curve which is as straight in some sense as possible. How would you define straight line in a coordinate invariant way? One way of defining it's a tangent to straight line is basically straight line. So this is the idea also you can transform it to geodesics. You're trying to construct the curve which parallel transport its own tangent vector. And that's basically what is written here. So this kind of generalization non-flat non-straight line generalization of geodesics. Of course you can go other way around, try to elementary interval between two points, trying to integrate, minimize it and you get the same equations. But you can try also to get a shortcut. As I said you're trying to introduce curve which is as straight as possible. So this equation for the geodesic use a tangent vector to the curve. If you take a proper time then it will be of the observer then it will be for velocity if you wish. Now curvature tensor. Again I will introduce it for the hand wavy and just to remind you what it is. One of the classical example if you take the vector on the sphere at the point A and start transported parallel transport of this vector along the loop from A to north pole to B and back to A. So keeping the rules for the parallel transport you will find that you know the final state of your vector V is not the same. So there is an angle which is non-zero. This could be used as one of the definition of the actually of curvature. If you move the vector along the loop and angle is non-zero then this space time is curved. More precisely you can derive it in a strictly way if you're considering coordinate lines. Imagine that you have x y in the Cartesian way but you curve them because your space time is not flat and you construct the loop between two nearby points A B C D and you transfer parallel transport this vector V from A to B back to C D and then to A and make it properly you will find the difference between two vectors is a vector. It's proportional to the distance between nearby points. It's normal because if you difference is zero it's zero. It's proportional to vector itself so if magnitude of the vector is zero difference is zero and there is something else staying there which also saying that if this is zero the difference is zero as well and this something is a tensor and this is curvature tensor and this basically tells us that curvature is zero means the vector returns to its own state and if it's non-zero it will be different. Again it's a bit hunt wavy but nevertheless I try to remind you what it is and the Riemann tensor could be written in terms of crystalline symbols or in terms of the metric I will do it as well and the important part is this crystalline symbols entering also as the first derivatives and the crystalline symbol itself depends on the first derivative of the metric so Riemann tensor depends on the second derivatives of the metric and local flatness again I return back to the local flatness theorem tells us that I can kill first derivative but not second. I don't have enough freedom in choosing coordinate frame to kill second derivatives and it means that if Riemann tensor is non-zero in one coordinate frame is non-zero anywhere. Okey dokey let's move on. Again so let me discuss a bit Riemann curvature Riemann tensor it's an important notion and its properties as I said first of all it depends on the second derivatives of a metric and there is also it's also tells us that space time is flat if and only if Riemann tensor is equal to zero so it's both ways straight. Another thing which also where Riemann tensor enters it's if you take second derivatives also of vector field and try to commute them they will you will see that the difference is non-zero and they're proportional to the Riemann tensor. Not all components of this so you see quite many indices here alpha beta menu each of them goes from zero to three but not all of them independent and so first of all there are symmetries which the easiest to see if you consider again local inertial frame all first derivative equal to zero so all this term goes to zero and we have only second derivatives and all the symmetric properties can be easily read from this simple expression and they will be they will be also general of course. For instance if you switch a pair of indices alpha beta and menu make R menu alpha beta tensor previous nearly means the same and other properties you can switch alpha beta and see how it changes etc etc. Another non-independence comes from the so-called Bianchi identity it's a differential identity so if you take a first derivative covariant derivative of Riemann tensor and make a linear combination of these first derivatives you will see that they are zero identically. There's quite important properties which we'll play in Einstein equations which we'll discuss a bit later today and if you contract Riemann tensor once with a metric you will get Ricci tensor. Ricci tensor could be zero Ricci flat spacetime but spacetime not necessarily flat for instance black hole solution comes from Ricci flat but it is a known by far known flat spacetime so again it is important that Riemann tensor is equal to zero. You can also form Ricci's color by contracting metric with Ricci tensor and another important tensor which we have is Einstein tensor and this will be on the left-hand side of Einstein equations so as you see they're all relatives of Riemann tensor of tensor of curvature. I'm sorry again I'm going a bit too fast I want to regegrivitational ways. This is really a reminder. Okay let's move on. Yes important thing geodesic deviation. In flat spacetime we know that two parallel lines remain parallel. What happens if spacetime is not flat? We can still put two observers each of us have a clock ticking clock this let's say will be our parameter lambda which we parameterize our world lines and at some point where we were really close to each other we could synchronize our clock and start parameterization of the curves at the same time I mean the same way so lambda node is my initial time if you wish or some other parameter if whatever you prefer and we're measuring a small distance between two of us and then day months year later we start we you know for instance we're sending laser to each other and we're trying to measure distance between us of course we need to measure distance at the same parameter of lambda we're trying to synchronize I mean as it goes and we find that in a flat spacetime this distance changes and how the distance changes is governed by tensor of curvature remain tensor so here what is written is a covariant derivative along the curve basically it is this one u is a tangent vector to the curve at each for each of these geodesic psi is a separation as it changes and this is remain tensor again this is reminder I keep repeating this um finally we are done with general relativity and we're coming to Einstein equations so there are several very important postulates which are entering the Einstein equations in general you can get Einstein equations simply by trying by assuming these postulates and trying to get Newtonian limit in the weak field regime so the first one is that no particles actually are neutral to gravitational interaction and this allows us to associate gravitational field with a with a geometry it would also happen in an electromagnetic field if we didn't have neutral particles then the weak equivalence principle saying that it will be about three falling particles they move on time like geodesics and this one of the consequence of this is equivalence of the gravitational and inertial mass of the particle and the third one is an Einstein principle is that if you take a local inertial frame and you conduct their experiment which doesn't go outside validity of local inertial frame then the output or outcome of your experiment should be similar or the same as in a flat spacetime important part that the experiment should not involve gravity. Einstein equations in more general form are written like that kappa is a constant this you recognize probably from the previous page is Einstein tensor this lambda term which appears and disappears from this equation so you I put it here for now but later on I will remove lambda term because I will look at the isolated sources and I did not I do not consider a very big cosmological distance even if I do consider cosmological distance for me lambda appears only in the propagation effect it's a relationship between the luminosity distance and the redshift and nowhere else and indeed if you assume a big gravitational field you can reduce this equation to the Einstein equations I'm not going to show that but I'm just telling you I hope you believe me and the right-hand side is a stress energy tensor of matter any possible matter it could be even energy like electromagnetic energy and it could be particles gas liquids whatever what this equation tells us that matter creates gravity the distribution of the matter will tell us what is the metric what the gravitational field and here we have association of gravitational field with the geometry it's not necessary but that's how it was introduced and geometry tells us how matter moves so there is this coupling okay matter creates gravity and the gravity acts on the matter yes it is arbitrary so you can start with the ah yes so the question is how why lambda constant was introduced is it for consistency or arbitrary so in general you can ask you can start with the action and you can ask what the most general Lagrangian if you wish you can put in the in the system and with few demands first is that it's equation field equations are no higher than the second order and second you can say that I wanted to be covariant and then there is no much freedom it's r this one or times constant so there is not much freedom you can go to higher to our higher derivatives or higher orders of r etc etc but these terms will create higher order field equations so from very general principle what I said you know you want it to be covariant and you want to produce a field equation of second order no but no higher that's what you will get months of what yeah well yes yeah well I don't know actually I don't know I know that yes it is true and it's funny it's interesting because it was also in my thesis you know at some point I looked at this because I was looking at the mass of gravity on I know but I don't know that if there is any deeper reason for that I mean I don't know or it just coincidence I don't know yes an important part of this equation also that this covariant derivative for h time tensor is equal to zero identically these segments of ibebe anki identities again it's identity and it's basically gives us covariant conservation law for the matter you can reduce this to equation of motion of course because it's a covariant derivative it includes a metric and in some sense even though we call it matter part it's not separable from gravitational field and this freedom which you have here not only reduces number of independent components in my Einstein tensor but it has a more deeper meaning that it should be like that because we always we should have freedom in our hands to choose coordinate frame we prefer and then different problems it will be one or another frame I am we'll be using quite often two frames one is local inertial frame and another it's transfer transferor striceless gauge and the harmonic gauge depending on the problem okay is it the next slide really yes it is okay yes gauge transformation so since we start to speak about freedom to choose the coordinate frame I want to introduce a gauge transformation or gauge freedom and why and how is it related why is it called gauge transformation and how is it related to coordinate freedom well it's actually come from very artificial splitting of the metric into two parts I just want to split metric into it a menu which is Minkowski form and I will be using even more than Minkowski form I will be using specific form of Minkowski metric as minus one one one one and then I introduce whatever remained h menu and therefore this problem I will demand that normal or at least each element of h menu is much less than one I want it to be a very weak gravitational field on top of the flood space time if you wish so I'm now switching in some sense to field theory field theoretical approach to gravity I'm not what I'm talking about Jiminy I will be talking about metric but here I'm introducing some field on top of the flood geometry of course all the equations which I will derive which I will be using all which gives me measurements they include both these terms and they combine into the metric nevertheless it's kind of sometimes it's very useful especially with when we talk about gravitational wave experiment to speak about fields rather than metric or we can jump between each other it's you know making our point a few more reach in local inertial frame in leading order in these quantities I can write it this way and what is a gauge transformation so gauge transformation is applied only to h menu so if we introduce h new which is related to h old like that where psi is arbitrary vector arbitrary up to the point that it should preserve this that the new h menu must be much less than one then if I substitute this transformation to my Riemann tensor it will remain the same so this is a freedom in which in electromagnetic theory we often referred as a gauge to freedom gauge transformation and we know that electromagnetic potentials also have gauge freedom and this is a complete analog of it but in terms of the full metric so now I want if you want if you want you can start from the full metric and make a very small infinitesimal coordinate transformation from all coordinates to the new with this little vector field psi and then you apply it to the metric we know how metric transforms under coordinate transformation you use the fact that psi is small and then you get a new metric and here you again decompose it into the part which is it a menu and h menu new so we won't always wanted to keep it a menu as is and we want to put all the transformation which comes from the infinitesimal transformation of the metric into the new field so it's a slightly different point of view going away from geometry to the field theory the important part is this we have this gauge transfer freedom and we can if we choose psi and I want to show us to say that this gauge transformation gauge freedom transforms into the coordinate transformation for the full metric is it clear now I want to introduce gravitational waves why there is orange there in reality we have a mixture of gravitational fields okay of the metric if you wish or curvature tensor part of it will be related to the gravitational field created by let's say solar system by galaxies or if you want to go into cosmological scale by universe itself and we want to separate this part of the metric this part of the curvature from other which is created by which is due to gravitational waves and gravitational waves could be seen as a small ripple small perturbations on top of the very smooth background curvature of the ground metric and the orange here it's an example so orange itself is its shape and it serves as a smooth background and the small things on top of it is you can see this as a gravitational waves so how we separate the smooth part from the small fluctuations small oscillations due to gravitational waves the easiest way is to doing averaging so we take a metric or we take Riemann tensor and we average these angular brackets over several wavelengths by several I mean so that it's not too large in the meaning we don't want to cover the space which is comparable to radius of curvature but it's significantly larger than one gravitational wavelength and we subtract the smooth part which we obtained from averaging from the total metric and what we get is what we will call gravitational waves the same you can do with the Riemann tensor after that you can work either with Riemann tensor and show that Riemann tensor propagates with speed of light and derive well derive gravitational Riemann tensor for gravitational wave in a frame independent way because we know that Riemann tensor is not is a tensor and basically it's non-zero you cannot put it to zero by coordinate transformation or you can deal with a metric and make further calculations in the metric I have chosen metric because it's a bit easier then we also considering several scales so we always considering already here we started I imposed that gravitational wavelength is much less than radius of curvature radius of curvature created by the ground the ground for me as a galaxy's solar system you know something smooth with the large radius of curvature and also there is another scale which usually enters the problem characteristic length on which this curvature changes so lambda is wavelength or gravitational wavelength much less than radius of background curvature in much less also than characteristic scale l on which this curvature changes okay and well let's wait for the second before I introduce the next step I also want to say that right gravitational wavelength much less than radius of curvature radius of curvature but not the curvature itself if you look at the individual elements of curvature which are inverse proportional to the radius of curvature square okay and if you look at the same as this for the ground let's say and if you look at the same way as gravitational waves it will be proportional to h divided by lambda gravitational wave square and that thing is much larger than actually this so elements of the curvature for gravitational waves significantly higher but radius of curvature bigger well r is significantly less okay that's that's inequality which I want to use which allows me to separate gravitational waves from the big ground there are few cases in which it is not true one of them is cosmology so the really gravitational waves those which were produced from the vacuum fluctuations in early universe there actually is a key point because there is interaction of gravitational waves with expanding universe and that's a way of pumping energy into the gravitational waves okay and there you cannot decouple so the basically gravitational wavelength is comparable sometimes even larger than Hubble radius and those gravitational waves which interact with the customer with the expanding universe the most efficient way it is not true and another way where it's not true if you look at the binary system and you're coming close to the binary system radius of curvature around the binary system is quite large and there is a lot of known a lot of interaction between let's say monopole field if you want Newtonian potential created by binary with gravitational waves another thing which I want to introduce is a local inertial frame but not for full space time but only for the ground only for that component of the metric so I can choose L characteristic length which is much less than radius of the ground curvature so my background curvature has a huge radius and I can choose you know always L which is usually quite large such that my background metric can be put in the form of Minkowski it's just for convenience and quite often this is true with a very high accuracy so in local inertial frame I can split as I said metric into two parts my big ground as I said because it's local inertial frame is reported with respect to the ground I can put it into the form of Minkowski metric and this part is what I will call gravitational waves and I will look for a specific solution in form of the waves in some sense and you will see that there is such a solution you can always try to search for it but it's not necessarily will be it exists right then I consider Einstein equations in vacuum so it's a richer flood space time so a richer is equal to a richer tensor is equal to zero I will make one more complicated well one more complication but it's just mathematical thing which helps me further on I introduce trace the reverse metric so I introduce h bar which is related to h menu while this transformation h is a trace so h is basically this it just simplifies equations a bit and I will as I said I have a freedom to choose whatever with you can switch between two different approaches either gauge freedom I can use a gauge specific gauge or I can equivalently choose in a more general sense coordinate transformation or coordinate frame such that h menu comma new there are four equations here equal to zero this is called harmonic gauge it's also has different names the donder gauge lauren's gauge depending on the authors but it doesn't change the meaning of the mathematical formula and then Einstein equations if I take this all to account I plug everything into this Einstein equations I will have box h menu equal to zero where box here in the usual sense because I neglect all the terms higher than h I'm working in linear in h so I'm neglecting all the terms which are of other h square and higher so this is wave equations and well any wave equation allows my wave like solution but it's not all I told you about the gauge freedom okay I'm not going to jump back but there was a gauge freedom applied to h menu I have written it do you want me to write it down again okay you can transform this gauge freedom into h bar it will you will have extra term function of psi and then you plug it in h new as a function of h new or h menu old and you will see that this frame this vector psi is not completely fixed it means that there is not a single harmonic coordinate frame there are many of them each vector psi which satisfies homogeneous wave equation will also satisfy this gauge basically that's what I'm saying so this does not completely fixing the gauge in the in case of the of the vacuum Einstein equations in the vacuum and therefore there are four equations here there are four more freedoms and we can choose for more conditions and we can choose these conditions it looks like there are five but it's actually not true because one already fixed from this condition so there are actually only three independent conditions which says that trace of h equal to zero and all the time component of my h metric is equal to zero and in this case this called transverse traceless or TT gauge and for gravitational waves traveling in that direction the most important for us is that this metric takes very simple form and it has only two nonzero elements one along the trace and one of diagonal so again what I've done here I assumed a small perturbation on top of the smooth background for a smooth background I have chosen a local inertial frame so that I can put it equal to Minkowski metric I introduce this h bar quantity I introduce harmonic gauge I found out that actually this condition does not fully fix me coordinate frame because there is a whole family of the this vector psi which still satisfies this condition and from the flat Einstein or sorry not flat but richer flat for vacuum Einstein equations I have this wave equation I use extra freedom which comes from here and I can write solution this is not the most general it's for gravitational waves traveling in that direction in this form now I return make one step back and I want to eliminate these conditions of gravitational wave traveling in that direction so I will be looking for for solution in general like a plane wave this k is a four-dimensional wave vector which zero component is frequency of gravitational waves and special component is a usual three-dimensional wave vector which amplitude is two pi divided by gravitational wavelength and its direction is direction of propagation of gravitational wave okay the Einstein equations which saying that box h menu equal to zero gives me this equation which tells me that gravitational wave propagates with speed of light then the gauge condition which I have chosen actually yes it tells me that first of all trace equal to zero and that there are no components of amplitude of gravitational wave in the direction in the direction of gravitational wave propagation so they acting on the plane which is orthogonal to the gravitational wave propagation so gravitational waves is transverse it's acting in a plane orthogonal to its propagation in its traceless not that actually Einstein equations allow other solution of course it must be the case because we should have Newtonian potentials and other contributions if for instance body rotating I'm coming from the rotation of the star but quite often we want to separate a relative degree of freedom from static or stationary degrees of freedom and you can do that by simply computing the metric and taking tt part transverse traceless part of it because we know that traceless transverse part of the metric corresponds to gravitational wave and the easier way to do that is basically if you have a solution hlm and if you think that there is a gravitational wave in there you can take the projection and this is a projection operator and it's written in this form just mathematically very convenient if you have a solution to get only a relative part you just take a projection of it using this formula and the unit vector in direction of propagation of gravitational wave is it boring not yet okay I think I hope it will be more interesting as we go along there is as I said it's a first lesson there is quite a fair amount of math but as we go along there'll be less and less of it and another thing is a polarization so because gravitational wave is a transverse we can decompose it into the basis in the plane which is orthogonal to directional propagation case directional propagation of gravitational wave and you can choose any basis there vectors p and q there are two orthogonal vectors in the plane orthogonal to directional propagation of gravitational wave so it's very similar to electromagnetic wave the difference is electromagnetic theory is actually vector theory so there is a vector here we have a tensor so we have to create the basis tensor basis to decompose our gravitational wave into the tensor real basis and we create the basis using p and q in the following way these bases will correspond to plus and this to cross polarization and we can decompose h ij into this basis e ij epsilon ij and epsilon j cross and that part corresponds to plus and cross polarizations there is a big difference between a transformation of the basis or how polarization change in electromagnetic theory and in gravitation so there was a freedom to choose p and q right you can choose any two orthogonal vectors in the plane so they all of course transform via the vector sorry via angle so it's rotation angle and while polarization in electromagnetic theory transforms as a rotation itself here you can see that each p and q transform via rotation matrix okay but their product will create cosine square sine square and mixed cosine sine so the transformation of the basis tensors will be not a single angle but twice angle and that's a later we can re-associate with a spin i mean when i show you how h plus and cross looks like associated with a spin of graviton so spin of graviton is equal to whereas spin of photon is equal one and last note least is how Riemann tensor looks like if i plug their solution for gravitational wave this will be important for us i can even write it down i have time actually i don't have time but okay let's move on and that's where i gonna use this geodesic deviation gravitational wave so if you remember i have shown to you let me try to reproduce by memory i hope i won't make mistakes we'll make mistakes sorry let's me go back yep this one so what i want to do next is i think i'm too far from a laptop and then it doesn't yes what i want to look next is i want to look at this equation geodesic equation where curvature is created by gravitational wave this is the key point of gravitational wave experiment and i will not talk about i will talk about a bit more the physical sense about geodesic deviations i put two observer one at point a and another at point b they closely separated by this initial distance x i not this is the same x i which we saw there but they're not far away they're not supposed i don't want to be far away so that we can again synchronize time and more over i use the parameters along each geodesic to be proper time of the observer a because observer a i'm sitting there it's always it's a coordinate frame associated with observer a and i can use my time my it's also my proper time and i can use this as a parameter which characterize not only my word line but also word line of the observer b and i'm trying to measure distance between observer a and b so i'm observer a there is observer b at different moments of my time it's governed by this equation so as we saw before the only part of the Riemann tensor which is non-zero is this one and it's related to the gravitational wave metric second derivative of the gravitational wave in such a way if i take not the x i absolute value of x i but the difference of x i minus initial separation so i'm looking at the deviations from my initial separation between point a and b then and i'm working linear order in h and the same time linear order in dot x i there i can easily solve this equation and it will be have this form moreover i can write more explicitly if i assume x and y as in this picture it will have this equation of motion for you know the difference between a and b you can easily see that now it's not the single b observer but you know you can put particle ring of observers which are initially at the equal distance from the observer a so initially all this ring is non-deformed and on equal distances so there are many many many many observers or you can look at this as a dust particles on the equal distance from me and then i switch on gravitational wave of course i cannot do that but nevertheless let's assume that we do propagating in that direction and we will see that this ring will start deforming and it's going from ring to this ellipse squeezed in one side and elongated another then after a half period it goes back to ring and then it's squeezed and stretched in other direction in opposite directions this happens for blue plus polarization for cross polarizations picture is similar but rotated by 45 degrees this 45 degrees actually related to this twice angle transformation in the polarization which i mentioned so it's very simple we plug to monochromatic gravitational wave here we looked at the geodesic equation we can solve it and if you look at the ring of particle we will see how this ring will deform as time passes by and gravitational waves go through later on we will place one mirror here another mirror there and there two more mirrors and we start measuring the distances how they're shifting using the interferometry laser interferometry so that's the basic principle of detecting gravitational waves so by some more complications i will talk about this as well but that's basis you also can write a line of forces for this equation so basically this force this is force this is basically acceleration you can show that this acceleration this force is divergence free and this acts very similar way as a tidal forces acting on on earth due to sun moon system so it's you can see this gravitational waves as a time varying tidal forces let's look at the generation of gravitational waves and i make few assumptions here first of all i consider isolated source so it means that i can confine my i can choose i can i can find the sphere in a space which completely covers my source so there is no matter outside there is no source matter outside this sphere i can cover it completely by some sphere it could be large sphere but nevertheless it has a finite size and the characteristic size of my system is will be l and then i consider the observer which is far away far away i will define it a bit later on jumping a bit ahead this observer must be much larger than gravitational wavelength away from from the source so that's called far zone i assume that velocities here and the internal gravity of the source will be weak in a way it is not necessary but it's just easier for me to derive solution but it it appears that this solution actually does not require as quickly speaking field of the source to be weak so we can really apply it to black holes and neutral stars we will use harmonic gauge conditions harmonic coordinates and Einstein equations could be written in this form this h bar is again trace reversed part of the metric which we defined before this box operator is again the same operator as we saw before key menu is again stress energy tensor of matter but there is something else this red small key menu and this is all non-linear part which was on the left hand side of Einstein equations we just moved it to the right so we artificially linearized our equations these equations are non-linear because t menu contains terms of order h square so strictly speaking it doesn't if you want to solve it exactly it doesn't help us at all what it helps us is when i want to introduce or i want to say that h menu is weak in some sense in which sense i will tell you in a second and i want to neglect this term in a zero order approximation and then solve these equations iteratively so where h is small in which sense it is small it is small in the sense that main contribution to the right hand side coming from the source term and h itself much less than one it's usually true at least in the week in this regime over here h menu is very weak we're looking at gravitational wave which is of order 10 to the minus 22 21 23 so it's very weak and therefore for time being i will neglect this term but and i solve it in the leading order and then when i want to go to higher orders i will restore this term we neglect this red term for time being and then general solution could be written in terms in terms of the repeated potentials in this form now i will make a few more assumptions i assume slow motion i think i already said that i take observer far away so that distance from the source to observer is much much much larger than gravitational wavelength so i don't see any i see only basically gravitational waves there i don't see any interplay between potential Newtonian potential and other quadruple moment of the binary system etc acting these gravitational waves i don't want to look at this i just want to look at the gravitational wave and it is the solution of this equation far away this where gravitational waves will be pure i can separate it from other contributions and under these assumptions i can write a solution which was here so basically i factor out this term because i assume that gravitational wave does not change much while propagating inside the source that's basically this condition and i can write it in this form i also took tt part because i'm interested in radiative degrees of freedom we already talked about this so if you want to look at the gravitational waves like you can take transverse traceless part of this of this tensor next we need to use the conservation law again because it is i can use a not covariant but you know partial derivative because of my assumption that gravitational wave is weak and i can in first approximation far away from the from the source i can assume that my background is is flat again i'm using local invariance local inertial frame of the local inertial frame of the observer far away from the source now it's a little bit tricky you can try it yourself but you can say you can actually show that using this equation this conservation law you can transform this integral into this form where m jk is a mass quadruple moment given by this expression this quadruple formula quite often is referred to as landau-lifters formula and it's quite a few assumptions went into deriving it and later on people actually show that this formula appeared to be more general than people thought so for instance it does not require strictly speaking weak internal gravity of the source and it's actually it's not trivial thing to show but nevertheless you can show that this also valid where your system has a strong field internal field like as i said neutron stars or black holes even so this is the key formula which we want to use and we will use later this is what it basically we'll describe it later yes no what it what it what it gives us so first of all it tells us that system which has non-zero second derivative of mass quadruple moment should emit gravitational waves if you just have accelerated body you will not have gravitational waves if you have spherically symmetric body which rotates you don't have gravitational waves you really need quadruple moment to be non-zero moreover you have to have second derivative of quadruple moment time derivative to be non-zero to have gravitational waves what is this distance actually this distance is always a it's luminosity distance graviton is a massless particle and it's propagates in a similar way as as light and what we're measuring it's a luminosity distance the mass quadruple moment is already without trace so trace is removed here you might want to take transverse part of this as well okay we are approaching to the end and let me just see how much i have okay i think one more slide and then i will wrap it up and make a conclusion so besides the leading order mass quadruple moment there are other moments which actually radiate as well it's similar to electromagnetic radiation so you have dipolar and then you have higher orders and there are two types of moments which you can attach to the system similar to electric and magnetic here it's called mass moments and current moments and they scale roughly as a mass times characteristic size of the system power of l and as you see current moments are usually smaller in magnitude than mass moments by value v over c you remember c and g equal to one so wherever you see v here it means v over c and we're considering systems which are slowly motion i mean the slow motion slowly moving systems and therefore these moments will be a little smaller than mass moments by factor v over c it could happen that some systems do not have quadruple moment or quadruple moment is suppressed then you need to go to current moment and octopal moment and either other moments you know the again radiation might not be very strong because there is a pressure factor nonetheless they it it exists in general h i j oh sorry h plus and h cross could be written in terms of the multiple moments in this form this uh ampersand i'm using as a terms of the form it's not even proportional it's a terms of this kind of form that there will be of of this characteristic i don't want to write exactly expression there in the length and they're not needed here but what is important for me here that the leading quarter will be a second derivative of the mass quadruple moment then next will come a third moment but this third derivative etc etc and the same for the current multiple moments but every time you take a time derivative you're introducing speed velocity so each next term here will be smaller than previous by factor v over c that's what i try to express show here so that indeed leading order will be a quadruple moment and all others yes they're present but they usually suppressed by powers of v over c and then i want to return back a little bit and saying about this t-menu red this non-linear terms so when i'm going beyond leading order i have to be rigorous and i want to introduce them again and plug back linear equations and solve it iteratively so that's what is called post-newtonian expansion you can decompose your metric into series with small parameter small parameters here v over c here i showed c explicitly so that it is there you understand it small and this infinite series and you're trying to solve it iteratively so you try to solve for first for h1 the leading order and then you plug this h1 into right hand side where there was this red t-menu and you and also so actually it also affects equation of motion so you need to modify your motion etc etc i will show this very briefly on an example of binary system but what i want to say that it's this infinite series and we were trying to solve it for many many terms actually the current solution exists well depends on the metric or the speed of the equation of motion up to 3.27 epsilon to the 7 epsilon to the 7 and partially epsilon to the 8 and each order becomes extremely its growth in geometric complexity of solution solving interesting equations goals in geometrical progression number of terms and the way of solving it ok doki i think i will leave it for next lecture but i want to quickly summarize what we have done today there was a very quick and i don't know if it was very comprehensive reminder of basics of differential geometry and general activity the important part for us it was a geodesic equation and also geodesic deviation equation i mean this we'll be using later on a little bit more and then we introduced also Riemann curvature of course it's important part because this is the thing which is tells us that space time is not flat and it cannot be killed and it's a covariant quantity listen listen i didn't finish and then we looked at the Einstein equations more specifically we tried to find the wave like solution in the vacuum and indeed we have found it we found that the gravitational wave is transverse and traceless it means it's act only in the plane orthogonal to its propagation and it's also traceless and in a preferred way you can choose such a frame let's put it really implemented this way which is called transverse traceless where there are only two components of the metric of the gravitational wave survive then we looked at the geodesic deviation equations for the gravitational wave and we saw how actually observers what is moving with respect to each other under effect of under effect of propagation of gravitational wave and later on we looked at the system arbitrary system and have derived expression for gravitational wave which is here what is called quadruple formula this leading order even though there are higher orders but they suppressed usually by B or C thank you very much