Jorge Pullin: Our speaker today is much is called an upscale who will speak about gravitational
Jorge Pullin: Waves in the sitter universe.
Maciej Kolanowski: And thank you for the introduction and for the opportunity to
Maciej Kolanowski: Speak here. So I would I want to tell you something about
Maciej Kolanowski: Some very recent progress in description of gravitational waves in the presence of positive cosmological constant. Well, I know that it is not a look. Quantum Gravity topic, but I hope that you're still going to be interested in this.
Maciej Kolanowski: And so I will start with some motivation. Why actually in 2020 usual, we should care about gravitational waves and the I will try to convince you that there's still a lot of stuff we don't know. We don't understand
Maciej Kolanowski: And then I will present our free mainly results from this paper that was mentioned on the previous slide. So first of all,
Maciej Kolanowski: Muslims for Emily and it's cry. And so we derived
Maciej Kolanowski: Formula for the changes of energy in financial period in final interviews intervals of time what what amount of energy was emitted.
Maciej Kolanowski: In contrast with previous work when only the total energy emitted was calculated and then then we'll also show you similar energy flux for the horizon and
Maciej Kolanowski: For a thing that for the first time, and we're going to see that.
Maciej Kolanowski: You can take limit with lambda going to zero at the customer at a cosmological horizon, actually, because we have a lot of those in this universe.
Maciej Kolanowski: And you can go with them that zero and actually see horizon and becoming cry, not only on the picture on the background about actually from the point of view of the dynamic of fields.
Maciej Kolanowski: And
Maciej Kolanowski: Last but not least, will also discuss some issues concerned with the fact that time and spatial translations in the Citigroup, and do not commute and what does it mean
Maciej Kolanowski: For the definition of energy and whether does it make sense or not.
Maciej Kolanowski: Okay. Actually, I was told that I talk too fast. So if I talk too fast, don't hesitate to tell me this, or as question or whatever you want. So moving on the what, why should we care about the gravitational waves. Well, first of all, because we actually know that there's some real fundamental
Maciej Kolanowski: And it's kinda surprisingly new and that we really know that they're real.
Maciej Kolanowski: So it's definitely worth pushing better understanding of gravitational waves because, well, it's physics, it's not so this topic is somehow connected to experiment.
Maciej Kolanowski: And not only to some abstract mathematical structures.
Qiaoyin Pan: And so
Maciej Kolanowski: This will. This is the first part of our title gravitational waves. But we also have the sitter universe and obviously what Oh, when I say this, either. I mean, that lambda is
Maciej Kolanowski: Typically positive. So, and no no anti testator things no holography whatsoever here.
Maciej Kolanowski: And we know that, again, this is physical and because
Maciej Kolanowski: Our observation shows us that acceleration is the dead expansion of the universe is accelerating. So we have positive logical construct of us, we will have some public pension here and it's not so clear, but I think the, just the fact
Maciej Kolanowski: That it is positive is nobody doubt that was it. We don't understand nature of cosmological constant, whether it is some fundamental Konstanz of nature or it is simply potential energy of some background field. We didn't know it it
Maciej Kolanowski: For simplicity we assume the to our background is simply disappear universe. So we have
Maciej Kolanowski: So this constant is really a physical constant
Maciej Kolanowski: Okay movie gun.
Maciej Kolanowski: Is
OK.
Maciej Kolanowski: OK, so the problem because actually both those topics are present from the very beginning of
Maciej Kolanowski: Our present from the very beginning of general relativity. So, in
Maciej Kolanowski: Moral more than 100 years ago. Einstein was concerned with positive with cosmological radiation. He do he derived is in linear is gravity, he derived his
Maciej Kolanowski: Quadruple formula. And then there were literally decades of work on this topic and also the Seekers solution to answer any questions is one of the oldest oceans ever found. So one could think that we should understand right now, everything about it.
Maciej Kolanowski: But unfortunately, it's not true because apparently those two topics are not exactly compatible.
Maciej Kolanowski: And because as
Maciej Kolanowski: Usually when we think about radiation. For example, in a course on electrodynamics, we say that radiation, something that an observer at infinity can see. So this infinity in the sitter universe is totally different than in demean coughs Q1.
Maciej Kolanowski: Because in Assam politically flat space times infinity on the which
Maciej Kolanowski: One measures.
Maciej Kolanowski: Edition is now infinitely
Maciej Kolanowski: Worse. In this case, as you can see from this over 10 rows diagram and cry is actually space like
Maciej Kolanowski: So it changes everything because all those decades of work of Troutman and Bondi and Zach's and Iranian pencils and so on.
Maciej Kolanowski: It was what in for us in particular that space times with now sky, which is equipped with at tone of universal structures. So you have universal metric in youth on this cry and you have BMS group and now everything is lost, and we need to start from scratch.
Maciej Kolanowski: Moreover, it is natural to define energy as something connected with some killing vector as some density
Maciej Kolanowski: Some charts of a killing vector, but here in this case all vectors tangent prescribed so all King vectors are spaceflight be because the sky is a space like so it means that the charges could be either positive or negative. And they could be actually arbitrary negative
Maciej Kolanowski: Is so we don't have any. And so we could a priori have negative energy, which seems kinda it is disturbing because we think about energy even even this very small energy of gravitational waves we can detect as some physical thing that we could use for example to warm up a cup of tea.
Maciej Kolanowski: And here apparently there is a risk that if and if the genre additions calm then actually our T is going to be colder instead of warmer.
Maciej Kolanowski: And more over
Maciej Kolanowski: Anybody if if we have some physical observer and you were then he or she cannot
Maciej Kolanowski: See everything in the universe, and they are bounded by the cosmological horizon.
Maciej Kolanowski: So we that another issue is something different that we are not able to see everything on the part of our universe.
Maciej Kolanowski: And last but not least, there is also a thing called just noun for 60s 60s, I think, which is called the origin problem.
Maciej Kolanowski: Usually when you describe radiation, either in electrodynamics on in
Maciej Kolanowski: On linear is gravity. You start with someone over enter our expansion and I differ identify appropriate term as radiation. And then you say that some higher order times in one of our, our ARE NOT A NATION bar code on big mode.
Maciej Kolanowski: But unfortunately it was pointed out in 60s by Penrose that you, you cannot do this in a reasonable way on the decision if you have cosmological concept of any sign either plus or minus
Maciej Kolanowski: And because then it is dependent on the region around which you have this are coordinate
Maciej Kolanowski: In the inverse of which you expand. So if you move your origin as we actually seen and later on in this topic in the stock if you move the origin, you mix radiative and Colombia modes and you also mix their energies, even at their level of linear is the gravity.
Maciej Kolanowski: Oh, last but not least, it's the question of
Maciej Kolanowski: The second last but not least, is the question of boundary conditions, one should impose. So for example, in a DS. You can impose at infinity.
Maciej Kolanowski: Reflective boundary conditions which simply means that magnetic part. Leading. Leading order magnetic part of the violence or vanishes and the boundary. But here you as well. Sean, I think I should buy those outdoors, you cannot
Maciej Kolanowski: Put any such restriction and on vital tensile, you need to work with the whole object both electric and magnetic
Maciej Kolanowski: Part
It's crazy.
Maciej Kolanowski: Excuse me. Was that the question or
Seth K Asante: Better. Yeah.
Jorge Pullin: No, go ahead.
Maciej Kolanowski: OK. OK.
Maciej Kolanowski: So moving on.
Maciej Kolanowski: And there was a lot of activity in this topic in the last few years. So there were different proposals of energy.
Maciej Kolanowski: And and using different formulas and also different proposal of boundary conditions, different
Maciej Kolanowski: Gauge choices which were supposed to be better or worse for whatever
Maciej Kolanowski: And then finally, some people actually remember that they should be
Maciej Kolanowski: Up and that they should be doing physics and applied this formalism to some physical astrophysical say astrophysical situations.
Maciej Kolanowski: And so here's the good news. Good news is that we are not going to propose a new definition. So we believe that a number of proposal is enough, who are going to work and with what those people
Maciej Kolanowski: Need to appear with what
Maciej Kolanowski: Those people proposed and also. And recently, a few months ago also proposal by Khrushchev Haka and smoker. And so we want to combine as simplistic formalities of our sticker bongha and cassava with
Maciej Kolanowski: With light cone and definitions of energy of who show hawk and smoker and
Maciej Kolanowski: Let us see, where can we get, what can we get from this
Maciej Kolanowski: Okay.
Maciej Kolanowski: Okay, and
Maciej Kolanowski: So what is our strategy.
Maciej Kolanowski: We start with the sitter universe, who would she can all see here and we're going to Linares gravity around it.
Maciej Kolanowski: So it's very preliminary thing at something and Einstein did more than 100 years ago.
Maciej Kolanowski: So it's not very and but we are still not really able to do anything interesting at the nonlinear level. So it is Wakefield approximation first time.
Maciej Kolanowski: We work with those so called burden coordinates, because
Maciej Kolanowski: There's worse suited for taking a min costs can limit, as you can see when lambda goes to zero, we simply good mink of system. It's nice and those coordinates are well defined.
Maciej Kolanowski: And also because they were surprisingly well at scribe, of course, using appropriate as you're using appropriate confirm or factor, which in our case is given by
Maciej Kolanowski: And Hubble constants minus one times are
Maciej Kolanowski: Healthy
Maciej Kolanowski: And I'm going to use frequently Hubble Constant in the case of the sitter. It's simply square root of lambda over free
Maciej Kolanowski: And okay so this will be useful, and we want to define our charges of the digital group just as
Maciej Kolanowski: A speaker mongan seven did using simplicity definitions and actually
Maciej Kolanowski: Everybody knows that those definitions are and nice and up to boundary terms. And of course, if you're interested in the whole charge at those bundy terms should not contribute. So it doesn't matter.
Maciej Kolanowski: And bad if you are, if you want to derive something like Feldman Bondi Maslow's formula then you actually care about and bank details. So our strategy is
Maciej Kolanowski: As such, and that we don't care right now about bank details we simply ignore them and we wanted to fix them somehow by some other methods at the end of our calculations.
Maciej Kolanowski: And show that our result is the only one which which has the following nice properties. And this is why those boundary terms are. I'm the way they are. And so
Maciej Kolanowski: I mentioned already,
Maciej Kolanowski: This paper from this year by who's Jehovah and smoker, in which they were very careful in deriving boundary towels and only to find out at the end that
Maciej Kolanowski: Those bandy terms and produce some infinities which they need to regularize. So, which basically means that they needed to change somehow a talk.
Maciej Kolanowski: Those bandy terms. They just the right so we decided that it doesn't seem like fun, and we we just don't care about Mandy terms at the spot.
Abhay Ashtekar: And ask question. I mean, is your control factor correct or is it the inverse of the confounding factor.
Abhay Ashtekar: I think a confounding factors should go to is blows up at infinity.
Maciej Kolanowski: Excuse me, I have. Okay, so this is confirming factor.
Maciej Kolanowski: Come from our factor is at times are
Maciej Kolanowski: Okay.
Abhay Ashtekar: So, it blows up at infinity.
Maciej Kolanowski: Okay, so I can put minus one.
Abhay Ashtekar: Minus
Maciej Kolanowski: Good. Yes, there was also another minus one missing. Thank you.
Maciej Kolanowski: Oh, and actually we use this definition of conformal factor because and now it's dimension less and metric induced on Sky has proper has the same dimension as the metric. We start
Maciej Kolanowski: Okay.
Maciej Kolanowski: So, let us move on.
Maciej Kolanowski: So we have a killing vector of the decider universe, and we can define a chart of line calm using the following formula formula, as I said, it's well defined up to boundary charge up to boundary terms and we hope.
Maciej Kolanowski: We hope that we would need to care about them right now.
Maciej Kolanowski: Okay.
Maciej Kolanowski: As I said, we need to establish boundary terms later on I will focus mainly on the decider G which is simply given by this vector d up and do you because as you could see, as you can see here, this vector becomes a time
Maciej Kolanowski: Generator of time translation when lambda goes to do so is the most natural choice.
Maciej Kolanowski: Okay. And so this is an area of the one of one calm and our plan is following. We can fall. Yay.
Maciej Kolanowski: At least part of this faith in by light comes emitted from a single line so we
Maciej Kolanowski: We have a line and we have different now comes emitted from this
Maciej Kolanowski: And we can calculate and their gene.
Maciej Kolanowski: In each now calm and copper them.
Maciej Kolanowski: Of course we should introduce some cut off. So let's say that we have
Maciej Kolanowski: It looks like this. And we on the integrate up to some
Maciej Kolanowski: Some are. And then when we go with our to infinity, we should obtain difference of energies but when to infinity. Now cons, which would identify with.
Maciej Kolanowski: And then emitted by the system.
Maciej Kolanowski: Here, this, this picture is kind of misleading because it seems to those are surfaces are equal constants
Maciej Kolanowski: And it seemed that their timeline, but in fact it. But in fact, the correct answer would be that it depends. As you can again see from this metric, if you have some if our small, then those surfaces, our timeline at are equal equal one of our Hubble Constant and it is
Maciej Kolanowski: Now surface. In fact, it is a cosmological horizon and
Maciej Kolanowski: And the future.
Maciej Kolanowski: Magical horizon. And then when we go with our to infinity. It's going to be space like and this space like surface in the limit becomes cry.
Maciej Kolanowski: Okay, so let me know.
Maciej Kolanowski: And move on.
Maciej Kolanowski: Okay, so that was our strategy and now we need to make it happen. So obviously we need to introduce them gage some expand some boundary conditions at infinity. And so we use Bondi
Maciej Kolanowski: And gauge and assume the following
Maciej Kolanowski: Some topics as our excuse me, it's obviously a typo here it's are going to infinity.
Maciej Kolanowski: Okay.
Maciej Kolanowski: We assume the following as competitive.
Maciej Kolanowski: And and those boundary conditions here.
Maciej Kolanowski: As you can see, and there's no one over r squared term here.
Maciej Kolanowski: And because it zero and it follows ultimate in the case of positive cosmological constant, a fact that this term is zero flows automatically from Einstein equations, assuming the to have smooth structure as described it's it was shown, I think by Friday in 80s.
Maciej Kolanowski: And by contrast in synthetically flood space time you all you have the term and it produces some logarithmic corrections, but they are totally missing in
Maciej Kolanowski: One lump that lambda is positive.
Maciej Kolanowski: And
Maciej Kolanowski: The two main differences between us, in particular space times are such that this age zero is not zero.
Maciej Kolanowski: Although, as we'll see, it is of order.
Maciej Kolanowski: Lambda for small lambda
Maciej Kolanowski: And more over tense those terms here are free data which needs to satisfy some constraints at scribe bought, but we can prescribe them. They are not
Maciej Kolanowski: Yeah. And so our free data here are actually this A H zero and those terms you
Maciej Kolanowski: H minus one and here it is actually
Maciej Kolanowski: And it's actually determined by h zero
Maciej Kolanowski: And this is kind of strange choice of
Maciej Kolanowski: Gauge because usually, I think that people work with so called radio gauge in which you know which H. A. Your starts with minus one and a HIV starts with zero
Maciej Kolanowski: We decided on this gauge, simply because who show kaka and smoke at it and they they chose this gauge because their expressions. Well, we're less divergent using this gauge. Then there are the radar one
Maciej Kolanowski: And and we wanted to compare our results to theirs.
Maciej Kolanowski: Okay, so
Maciej Kolanowski: Those are our assumptions around infinity and let us calculate and this energy emitted.
Maciej Kolanowski: It's kind of tedious calculation. But at the end of the day, using Stokes Durham. Obviously, up in the following
Maciej Kolanowski: Expression.
Maciej Kolanowski: And no natural quick, let me let me emphasize that this is energy emitted between two now cons, not necessarily infinite. Infinite times. So it means that we have really energy emitted during some finite period of time.
Maciej Kolanowski: And we have some boundary terms to which simply come from our definitions. The question is whether we should discard them.
Maciej Kolanowski: Well, we definitely should defer
Maciej Kolanowski: This plan because it's always the limit lambda goes to zero.
Maciej Kolanowski: If we discard this term which simply means that we added to the definition of the energy or with with opposite sign we added to the definition of energy of anelka little 50 correct lambda zero limits, but we have this them and it's not obvious, whether we should also discard it or not.
Maciej Kolanowski: So because it actually goes to zero as as as lambda goes to zero.
Maciej Kolanowski: So it doesn't matter in the limit. So we are going to, as we mentioned in our strategy, we just don't care about branded terms. So we discard it. And we perhaps we will restart. Later on, we'll see.
Abhay Ashtekar: What is gamma not
Maciej Kolanowski: A gamma not is metric on a unit spirit.
Abhay Ashtekar: Okay, so it is it there's no automated just magically
Maciej Kolanowski: Because those are the coefficients in one over our expansion, so it is somehow already. This is as our goes to zero, it goes to infinity.
Maciej Kolanowski: And there's a need
Maciej Kolanowski: And now come
Abhay Ashtekar: Thank you.
Maciej Kolanowski: Thank you for the question. And also that let
Abhay Ashtekar: Me mention that for the
Maciej Kolanowski: Future use so
Maciej Kolanowski: The whole immediate energy. We're going to call duty, because, well, after all, it is a chart.
Maciej Kolanowski: Okay, by in the same way we could define radiated angular momentum, either from the minus two plus infinity or for some
Maciej Kolanowski: For some finite
Maciej Kolanowski: Oh, I think I have a question on top.
Maciej Kolanowski: Oh,
Maciej Kolanowski: Yeah, it can we discuss it later on.
Jorge Pullin: Good.
Maciej Kolanowski: Because he is definitely there is the connection. Okay, great. Thank you.
Maciej Kolanowski: And as I said, we could also and design eradicated angular momentum, either at finite or infinite
Maciej Kolanowski: Time.
Maciej Kolanowski: We could also define and spatial momentum, but it was
Maciej Kolanowski: Actually too complicated for us.
Maciej Kolanowski: And we're somehow not able to
Maciej Kolanowski: To produce finance limited because there was a lot of terms which should cancel on on the show, but there's too many of them so will derive this charge in a different manner.
Maciej Kolanowski: Okay, as I mentioned in the introduction, and
Abhay Ashtekar: Because we're going to the next topic now.
Maciej Kolanowski: I'm not. It's still about this term just said infinity.
Abhay Ashtekar: Okay, but I still this white answer now so I'm just understanding the phone if you can go back to your Penrose diagram on page five
Abhay Ashtekar: And just, I think it looks wrong. Yeah. Yeah. Yeah, about that vendors like I'm just draw what you're doing.
Abhay Ashtekar: Yet I I do look at the left hand straight line and two points on it and light comes going from there. Yeah. Can you draw the light codes.
Maciej Kolanowski: It's of course
Abhay Ashtekar: Yes.
Maciej Kolanowski: Thank you. We we calculate flux actually density not flux at this part.
Yeah.
Abhay Ashtekar: Thank you.
Maciej Kolanowski: Okay, so you mentioned actually at the spindle diagram. It is natural to consider violence or
Maciej Kolanowski: Its cry so
Maciej Kolanowski: We introduced it's electric part. And of course, of the linear eyes advice answer and and to one contract that it vanishes as our goes to infinity. So, we
Maciej Kolanowski: Just want to correct. And so we can introduce rescheduled electric part of the winner is vital. So, simply by multiplying by
Maciej Kolanowski: One over confirm the factor. And then the limit a particular one contract.
Maciej Kolanowski: That this is
Maciej Kolanowski: All one get from the definition
Maciej Kolanowski: And okay, why does it matter. It matters because we can rewrite our charges use those are now those charges between minus and plus infinity using those two equations.
Maciej Kolanowski: And we obtain the following very nice form and this h minus two is because and the
Maciej Kolanowski: Metric induced by perturbation on Sky is not actually a zero but public Constance minus two times ages zero. So here it is actually
Maciej Kolanowski: Um, this is metric
Maciej Kolanowski: Metric induced unsubscribe.
Maciej Kolanowski: And they look surprisingly similar to once obtained before and Professor ash Tucker and his guru. Namely, we have
Maciej Kolanowski: So,
Maciej Kolanowski: This is, this is the result. You can see some differences. A letter P here stands for one correct because they calculated in in Concord.
Maciej Kolanowski: And you can check
Maciej Kolanowski: Okay I think now, and you can check that by
Maciej Kolanowski: That if you perform the following come from a transformation from the data and defined in the pond carapace to the data in the sky, Allah Bondi you obtain exactly and lower result of course for time translation and for angular momentum.
Maciej Kolanowski: And. Okay. And so, it suggests that
Maciej Kolanowski: This that also momentum should should be obtained by such a transformation. And then in our annotation. It looks like this.
Maciej Kolanowski: And a translation generator in Bondi gauges. It's not very nice. It is given by equation.
Maciej Kolanowski: And of course all boundary can which we could derive from some cryptic formulation are totally lost because those expressions here are our global expressions.
Maciej Kolanowski: Okay. So we see that we have some sort of compatibility
Maciej Kolanowski: Compatibility and between reviews results previously found in the literature and this nice and let us
Maciej Kolanowski: But what are perhaps some differences. So you can check what kind of data is
Maciej Kolanowski: What kind of data and in those a BK charges are allowed, and you have some smoothness conditions at at the origin of your scribe
Maciej Kolanowski: Which translates into the following conditions in our gauge as you goes to infinity that this metric induced on Sky goes to a constant plus some exponentially small terms and electric part of the right answer should vanish exponentially.
Maciej Kolanowski: But actual we can check that we don't need such strict boundary conditions as you go through minus a plus infinity.
Maciej Kolanowski: To have final energy. We only need that electric part of the very tense or is constant and then it is a little bit stronger condition to have finite angular momentum and to have finite
Maciej Kolanowski: To have finite a spatial momento, it kind of stronger, but still, it's much weaker come it is still a weaker condition, then the one obtained in the quantity
Maciej Kolanowski: And let us also note that should what should the sitter is actually what in the first category.
Maciej Kolanowski: Of those, but it has finally old charges because sometimes it cancels out. So perhaps it is possible that even somehow we care about the conditions are needed. This is and this is the easiest the version one can get it simply by looking at formulas and require that they're finally
Abhay Ashtekar: I guess or to say some couple things. First of all, you call him Uncle patches okay with me, but it's slightly misleading because you're also working upon correct patch. That's why I STARTED TO LOOK AT THE LOOK AT THE Penrose diagram.
Abhay Ashtekar: All you know khonsari within that one correct patch. I think so. The difference is really that we are using
Abhay Ashtekar: Kind of
Abhay Ashtekar: Cosmological coordinates adapted to conquer a patch, whereas you're using the Bondi coordinates. Is that correct, that is a difference. You're talking about
Maciej Kolanowski: Yes, indeed.
Abhay Ashtekar: But OK, then I think that if I just actually solve the linear equations in the
Abhay Ashtekar: Spawn carry patch, then the fall off that we assume is the fall of you get from solutions. So you might assume Lord follow up, that's fine, but the solutions actually on the sitter.
Abhay Ashtekar: Will have the fall of that we assume we give the solutions explicitly. I mean, it's not as many other people that I worked on this already. So I just wanted to point out that the assumption that we get made was really based on the exquisite fall of the solutions themselves.
Maciej Kolanowski: And yes, sure.
Maciej Kolanowski: I totally agree with in this comparison, will we rather had in mind some next thing, one could make so introduce actually some black
Maciej Kolanowski: I don't know. What about fall off of because I think you also derived solutions from
Maciej Kolanowski: From the source. Do they also have the same decay is
Maciej Kolanowski: The CNN. Okay. Oh.
Abhay Ashtekar: Yes, yes. I think that was that was that something that follows from the Greens function that other people have actually calculated before. Yeah, okay. But I just want to clarify this so that I'm sure that I'm not missing something. Thank you very much for you.
Maciej Kolanowski: Know,
Maciej Kolanowski: Okay, I mentioned a few times that we want to fix somehow boundary conditions, and we are going to do this by imposing some sort of gauge invariance.
Maciej Kolanowski: And he transformations are those are somehow large gauge transformations, because they act action inscribed on the initial data can act as falls in 18 and
Maciej Kolanowski: 19 and actually not all, we're still missing some large image transformations, but it doesn't make any difference in what I'm going to say
Maciej Kolanowski: Since the sitter is confirming flat, it's very tense or is going to always to be engaged independent so it doesn't transform at all.
Maciej Kolanowski: Okay, using those we can calculate how this difference of energy changes. Okay, we obtained such expression and to check that actually there are no boundary terms which would be an and
Maciej Kolanowski: Try linear in in violence or in metric in use. And in this attack vector, which would killed this entirely so you cannot actually demand that it is totally
Maciej Kolanowski: gauging very despite the fact that this expression is very nice geometrically and but we can make something
Maciej Kolanowski: We should establish at least the residual gauge transformations which do not change our initial choices. So they're of this form, it's not very nice and
Maciej Kolanowski: And okay, and there's still too many of them because they are parameters by one
Maciej Kolanowski: Function on a spirit which is you independent and also by you dependent confirm are killing vectors. So it's definitely too many of them, we need to, but it was proposed how to kill at least partially this gauge freedom.
Maciej Kolanowski: We can impose some additional gauge condition and which roughly speaking, means that cones do not rotate with respect to each other.
Maciej Kolanowski: And and as a result we opt in, much simpler as a jewel gauge transformations mainly only those generated by dysfunction on a sphere which is you independent plus killing vectors of the whole met
Maciej Kolanowski: But obviously clean vectors of the whole metric at the gallery on PR to native data so they can change it and those
Maciej Kolanowski: a gauge transformation is generated by this function and
Maciej Kolanowski: Also, does not change our energy loss formula.
Maciej Kolanowski: Okay, and now it would be tempting to call such a residual gauge transformations super translations, because they look exactly the same as in the case of lambda in assembly in BMS group, but I think it's a misnomer because spatial translations are not have this form.
Maciej Kolanowski: So we decided to go for much catch your name there, namely super pseudo translations and let us also know that they do not commute and we're killing vectors of our metric just actually just as in the case of mean cosmetic when it is a source for them.
Maciej Kolanowski: Angular momentum problem.
Maciej Kolanowski: Okay, so we have different muscles formula without any bounded tense whatsoever.
Maciej Kolanowski: And which is the only expression which has the correct correct lambda to go to zero limit. And there's also invariant with with respect to all super pseudo translations. So it is a difference, I think, and
Maciej Kolanowski: And physical result as a result in this presentation. And this was also obtained by the means of network CRM and by kusa Hawkins smoker, but then they had some ad hoc regularization, and now we understand that this this way of regular rising was justified exactly by gauging variance
Maciej Kolanowski: Okay, so what we could also do we could try to drive us formula for the rest of charges but then they do not Camille killing vector do not commit with super translations.
Maciej Kolanowski: We could ask about non small cry and there are some solutions, actually. And we could also generalize it to the nonlinear theory.
Maciej Kolanowski: And then it's hard to mention a super translations and not also mentioned stuff to get it done. So it is natural, natural question whether we have some soft modes here. And what does it actually mean. And since we have totally different boundary conditions for our
Maciej Kolanowski: For our as metrics.
Maciej Kolanowski: Okay, so it is different. It was the first part of our talk. The second part is, is concerned with the horizon, you only work with flux, it will to calculate it again in Bondi coordinates in Bondi gauge. It's not very nice expression. As you can see,
Maciej Kolanowski: But let us try to check what is going on in the limit as lambda goes to zero. So we have the following expansion of our fields.
Maciej Kolanowski: And so it might seem that going with lambda zero is like going to with our to infinity. So then this term would be
Maciej Kolanowski: So then
Maciej Kolanowski: This time time would be dominating, but it is actually not the case.
Maciej Kolanowski: And because all those coefficients can also depend upon lambda. And in fact, when we go with lambda zero we have that H zero is proportional to lambda, whereas H A B minus one is a constant. Hmm. So we should also include this in our limiting procedure.
Maciej Kolanowski: And
Maciej Kolanowski: And then you can check that, in fact, the only term that contribute is
Maciej Kolanowski: This term.
Maciej Kolanowski: Which which happened to be
Maciej Kolanowski: Done.
Maciej Kolanowski: So in the limit on it this time contribute and this thermal exactly again reproduces flocks of the energy Angelina is gravity.
Maciej Kolanowski: In a simplistic this flat space times. And again, we have some boundary terms which one should should fix somehow
Maciej Kolanowski: It. Okay. So as I mentioned, it is not a simple large our expansion because also somehow coefficients also depend upon it.
Maciej Kolanowski: And the leading form of those coefficients Islam that goes to zero, is as follows. This that HIV goes to infinity as it should actually and HIV goes to a constant, constant again as it should in mink of suspects.
Maciej Kolanowski: Okay. And again, we have some open questions. First of all, can we show this on the horizon. So
Maciej Kolanowski: Because here to show this limit we needed to use some additional structure for from scribe. So it's kind of like cheating, could we make it interesting and then
Maciej Kolanowski: There's a question about our assumption because evidently we assumed and that radius of convergence in this one of our expansions reaches cosmological horizon and
Maciej Kolanowski: Is it true, and also we somehow expand coefficients in lambda. So then the question would be whether this expansion is also convergent, in some sense, or our adult those only formal
Maciej Kolanowski: Formal writings and then again. And this question of what happens in the non Smith's at an unsubscribe.
Maciej Kolanowski: And because when lambda is going. It's positive and this notion of structure at Sky is totally different much more complicated than India simplistically flat space time
Maciej Kolanowski: When it is simply logarithmic term. So it would, I think it would be also interesting to check what's going on at the horizon, whether we have some logarithmic term or something different when we have some sort of neurons movements and what one would also think, what does it mean, actually.
Maciej Kolanowski: And again, this is only the whole flux and not mass loss formula and can one could ask whether we can use the same argument will use before to fix bounded terms and it's somehow
Maciej Kolanowski: Hard because differently. We shouldn't use to pursue the translations it because they are not tangent to the horizon.
Maciej Kolanowski: So they are totally artificial from this point of view. So once asked about some natural is what kind of transform engagements permissions are natural from the point of horizon and require gauging variance with respect to them.
Maciej Kolanowski: Okay, and then let us go to the last topic of today and namely this very basic observation that generators of the Citigroup dinner commute.
Maciej Kolanowski: And so it follows the day charges are not very young and their, their actions. So if we have some solution if we transfer to linear is Einstein equations then. And if we translate this solution we obtain solution with different energy and we actually tracked
Maciej Kolanowski: It is really the case.
Maciej Kolanowski: So no natural question will be, whether this is a problem or not. We expected or not from a physical energy and
Maciej Kolanowski: So geometrical meaning is quite clear, namely that this T vector would you defense energy is fixed fixed by a plus is. So roughly speaking, by this point where source hits cry.
Maciej Kolanowski: And translations, not at this point. So,
Maciej Kolanowski: And this is a geometrical meaning and. But the question still remains about physical meaning of this cutie. Especially if we had more sources or so on and then it's this i plus is not uniquely sound is not somehow uniquely defined
Maciej Kolanowski: So, let us start with totally trivial situation that we have perturbation, on the whole, the sitter, which is topology. They are times as free
Maciej Kolanowski: And we have perturbations, which are global defined without any singular just whatsoever. Then since we have a compact cushy surface all such perturbations ma muscle vanishing all the sitter charges. Oh, extremely few it should be data of view of God.
Maciej Kolanowski: So it is not a surprise at all that it's true. So, so, so this is true because our both energy and momentum is zero, not very interesting either say
Maciej Kolanowski: So now, from this we restrict ourselves to one Hubble or one carapace and we don't care about universe outside. So thanks to this, we can have some finite energy and financial momentum.
Maciej Kolanowski: So the first
Maciej Kolanowski: Simple scenario is such to do have some a God given waste which comes from the fraud. The cosmological horizon. They are not produced by any sort within our patch.
Maciej Kolanowski: And they have some momentum say to the right. And then if we move this solution to the right is actually cool kind of look
Maciej Kolanowski: Like we move them in time, not in
Maciej Kolanowski: And not in space, but it simply means that it would means that the same solution entered expanding universe at a different point of time.
Maciej Kolanowski: And but it is an expanding universe. So we don't expect that we've pockets entering a different times has have the same energy. Actually, I think it would be much weird. The other ones from this point of view, it's totally natural and there's no surprise here.
Maciej Kolanowski: Then we have some astrophysical gravitational waves.
Maciej Kolanowski: And and obviously our previous segments don't work.
Maciej Kolanowski: Because now they were produced here at some the visit some physical moment.
Maciej Kolanowski: However, the whole system's a binary black horse and repetition ways admitted her husband, she momentum. If
Maciej Kolanowski: We are in their breasts time. And so the energy isn't very so we somehow rediscovered in this way this origin problem of Penrose that by this simple but by translating sources and also waves we mixed Columbia. Columbia, there are additive modes and also their energies.
Maciej Kolanowski: So it would mean
Maciej Kolanowski: So it would mean that we have
Maciej Kolanowski: And that
Maciej Kolanowski: This cutie is
Maciej Kolanowski: It is somehow problematic in such a sense
Maciej Kolanowski: That from the point of view of different origin.
Maciej Kolanowski: It would be different. But what would be a invariant is combined energy of radiative and Columbia mode. And then in real astrophysical situations like Lego
Maciej Kolanowski: It is actually not an energy that is measured, but much more observational so perhaps it is much we need to look into this more, but perhaps
Maciej Kolanowski: Somehow, one, one, have no better way of fixing this origin by actually knowing when when and where this happened.
Maciej Kolanowski: Okay, so let me conclude now.
Maciej Kolanowski: And we have derived muscles formula scribe for the linear is gravity, we checked that it is the only gauging variant familiar with the proper lambda to zero limit.
Maciej Kolanowski: We calculated the flux of energy for all the cosmological horizon and we tracked that it is it has correct lambda zero limit. And we also check, at least at leading all the, what is going on with fields defined nearby and in this limit.
Maciej Kolanowski: And we and we discussed
Maciej Kolanowski: What the what are consequences of the fact that moment and that translation and time translation in the City University not commute.
Maciej Kolanowski: Okay, so thank you for your attention.
Jorge Pullin: Any questions.
Abhay Ashtekar: I have a couple of questions if there is nobody else. I mean, if somebody else has questions, please go ahead before me.
Abhay Ashtekar: Yeah okay so
Abhay Ashtekar: I took lessons. I mean, the
Abhay Ashtekar: Oh, there's a there's a chatting somebody saying that. How about the question I asked or something like that. So do you want to answer that first
Maciej Kolanowski: Yeah, yeah. Yes, of course. So hungry asked, it seems that you are using the definition of Hamiltonian proposed by Waltons lupus.
Maciej Kolanowski: In 2000 in their construction is independent of the boundary. We also need to consider the integrated problem in the construction and
Maciej Kolanowski: I'm sorry I'm not
Maciej Kolanowski: So, so actually we we use
Maciej Kolanowski: That there was a few works actually by word on this topic of Hamiltonian description and we actually used him a little bit newer one by one and how lots
Maciej Kolanowski: And in which they actually don't care at all about integral billet at
Maciej Kolanowski: At infinity, I guess, in contrast to this old proposal where where you had a few terms and you need. I think you needed to show that one of them is exit is the track.
Maciej Kolanowski: They remember it correctly.
Hongwei Tan: Or ish
Hongwei Tan: I mean, the
Hongwei Tan: Problem and
Hongwei Tan: Who
Hongwei Tan: Knew
Hongwei Tan: And then
Hongwei Tan: They can get
Hongwei Tan: They can get there.
Hongwei Tan: You go
Hongwei Tan: Well,
Hongwei Tan: The problem I mean
Maciej Kolanowski: I'm sorry, I'm afraid I didn't exactly understand what you just said.
Maciej Kolanowski: Perhaps you could repeat or right
Okay.
Jorge Pullin: So perhaps while he writes, you can address a base question.
Yeah, sure.
Abhay Ashtekar: Did you I, my question was is the first question is about the flux through the horizon and
Abhay Ashtekar: Yes, it was whether
Abhay Ashtekar: What you're talking about is on Page 24 the formula you give is that the final result or
Maciej Kolanowski: This one, yeah.
Abhay Ashtekar: Is that your dessert for the, I mean, is this supposed to be. I suppose this is supposed to the flux of the energy because you say to edge. Lots of
Maciej Kolanowski: Energy after integrating after the integration of course
Abhay Ashtekar: Of the integration or the final touch you will get this this answer or
Abhay Ashtekar: You get
Maciej Kolanowski: It at finite are it is a cosmological horizon.
Abhay Ashtekar: Right, right. But in terms of the
Abhay Ashtekar: It's still a finite no code that is going on. So you're looking at finite region of the of the cosmological horizon. Yeah.
Maciej Kolanowski: Well, a priori. Yes, but we actually know that boundary terms are
Maciej Kolanowski: Not correct if you look
Maciej Kolanowski: On the
Maciej Kolanowski: Interval
Abhay Ashtekar: Ice. So therefore, for the boundary. So therefore, the statement is that you are looking at the flux through entire cosmological horizon. Is that correct, yes.
Abhay Ashtekar: Okay, but then is this flux positive, we can just check looking at. I mean, we know it should be positive because the killing vector field is future directed. Now there. So, can you check. I mean, I can't tell. Look at our formula, whether it is manifesting positive definite or not.
Maciej Kolanowski: Check it. But it's definitely a good security.
Abhay Ashtekar: Sorry, I can't hear you.
Maciej Kolanowski: We didn't do it, but it seems like a good security check
Maciej Kolanowski: Can you hear me.
Abhay Ashtekar: Yeah okay yeah I understand now. Okay. And then the first the question. The other question was about your first part, which had to do with scribe and you talk about the pseudo super translations.
Abhay Ashtekar: Yeah, so, so they do include time translation.
Maciej Kolanowski: Years of
Abhay Ashtekar: Times. Okay, so what, why, at once they do say something like that. We know that this is gauge and therefore energy should not depend on them or something.
Maciej Kolanowski: Well,
Abhay Ashtekar: But I mean time constellation.
Abhay Ashtekar: Is the generator of this right so it's not gauge
Maciej Kolanowski: Yes, of course. So it is
Maciej Kolanowski: It is this, those are awesome topic symmetries in such a sense that for Molly. They look like gauge like gauge transformations and bad such that
Maciej Kolanowski: Do not vanish at infinity.
Abhay Ashtekar: Yet what is confusing to me is really that. Do you have a natural affiliation of infinity of Scott
Maciej Kolanowski: We what we work in linear gravity. So yes, Cry is has natural color is natural, our times. So
Abhay Ashtekar: Then I agree, but it is it is that, but so you're using the foliage of sky, which is somehow given to you by the killing vector that you're using. Right. I'm not vertical line in the Penrose diner. Is that right,
Maciej Kolanowski: Yeah, okay.
Abhay Ashtekar: So so so you got this super translational super translations.
Abhay Ashtekar: And I bet generators of
Abhay Ashtekar: Audio have super moment today or what
Maciej Kolanowski: Well, definitely, you could define them.
Maciej Kolanowski: As they kind of tried to indicate, we don't know what's the status of those supersede the momentum and to whether they are associated with some sort of the gravitas, or what
Abhay Ashtekar: But do you have a formula for that.
Maciej Kolanowski: No, not yet.
Abhay Ashtekar: Okay, thank you.
Jorge Pullin: Judges there. Yeah.
Maciej Kolanowski: So they have some boundary terms to consider the non integrated part in the proposal.
Maciej Kolanowski: Oh, okay.
Maciej Kolanowski: All right, so I what I mentioned was the different work, namely one by Walt and I are
Maciej Kolanowski: And and
Maciej Kolanowski: Okay, so okay so so knowing this, since we did not follow this proposal by Walt and Zoo pass and because we simply decided at the beginning that we don't know, and that we don't care.
Maciej Kolanowski: That boundary boundary terms are problematic and to we want to look at them only at the very ending
Maciej Kolanowski: At the very end it by some different arguments. And so we did not consider at this world and to pass a Newtonian then it although it is possible that it is the one who still at our US. And if it is, so then they unfortunately obtain some infinities
Maciej Kolanowski: But I will need to check it.
Jorge Pullin: Any other questions.
Jorge Pullin: So I have one you started talking about, like, oh, so I guess what, what is your message to to, like, oh, I mean, it's sort of implicit in your equation 25 but this equation is kind of difficult to interpret. Like, I don't see what lambda is in it, for instance.
Maciej Kolanowski: Yes. Okay, so lambda
Maciej Kolanowski: Lambda is in this age zero. So, this is
Jorge Pullin: So that's the
Lambda
Jorge Pullin: So that's the extra term with respect to this and politically flaccus
Maciej Kolanowski: Yes, exactly. Okay.
Jorge Pullin: Any other questions.
Abhay Ashtekar: But I mean, it's definitely true that there is extra day. But on the other than the he is that he is writing down are also different right because of solutions to utilize the equations on the sitter background and not I'm in class case basis. So
Abhay Ashtekar: Every he is different.
Abhay Ashtekar: But in addition to that, there is that
Abhay Ashtekar: There's an implicit lambda in every edge.
Marriage.
Maciej Kolanowski: But, in a sense, this equation 25. It is almost written in terms of initial data.
Maciej Kolanowski: And so, in this sense.
Maciej Kolanowski: You could start with the same initial data in Minka you could consider solutions which has the same behavior. Behavior at sky and the differences are only in the bulk
Jerzy Lewandowski: Right, so you can consider age minus one to be independent of lambda and
Jerzy Lewandowski: And and h zero, you would be proportional to lambda
Abhay Ashtekar: Why, but on the other hand, if you just
Jerzy Lewandowski: You would be also independent
Abhay Ashtekar: But if you add some
Abhay Ashtekar: Source inside on that vertical line that
Abhay Ashtekar: That was drawn in the in the Penrose diagram. And if you calculate edge from that.
Abhay Ashtekar: That is what logo, people are talking about writing to this much more sophisticated things, but for the quadruple formula. For example, the leading water current contribution.
Abhay Ashtekar: That
Abhay Ashtekar: They want to understand how does it relate to the sources not abstract edge. Right. So if you calculated coming from the sources, which is what they do, then he would depend on lambda. And there he does not depend on lambda
Abhay Ashtekar: If you don't worry about property. The sources. Then I completely agree with what both of you are saying, but on the other hand,
Abhay Ashtekar: I mean, that's not what local people interested in
Maciej Kolanowski: Of course, rather. My point was that what is actually really measured is this part.
Maciej Kolanowski: Oh, sorry.
Maciej Kolanowski: What is actually measured is this part and and to not some bad behavior.
Maciej Kolanowski: You only see leading
Maciej Kolanowski: Lady, or at least I think so. Did legal you only see moving or the terms in one over our expansion in reality.
Abhay Ashtekar: Yeah but Horace. I think we're going in circles. So I will say one more thing in this stuff Horace question was, what are we telling to the library people
Maciej Kolanowski: Mm hmm.
Abhay Ashtekar: And if we're going to tell something in live with people, then they are interested in not just looking at that age abstractly, but H which is produced by some sources like such as compact binaries.
Abhay Ashtekar: And then to the approximation, you're working
Abhay Ashtekar: The that he that you circled would also depend on
Abhay Ashtekar: OOD also. Yeah, you said it also depends on this land up there right but they're also calculating the total energy
Abhay Ashtekar: And that total energy. They're calculating would only also depend on exactly the the edge that you are circled whereas your total energy depends on other things as well. Yeah, so that so that so that is a different that we can tell to the, I mean, you will tell to the live of paper. Oh.
Maciej Kolanowski: Yeah, thank you. It's very good point.
Maciej Kolanowski: I think we could actually
Maciej Kolanowski: Take your solutions to with surface those using and green functions and translate them to the Bondi frame to actually obtain flat mass loss formula and so obtain basically quadruple formula, but for financial times. I think this is was something one could do and to actually make contact.
Maciej Kolanowski: With observations. Thank you.
Jorge Pullin: Any other questions.
Jorge Pullin: thank the speaker again.