The Many Ways of the Characteristic Cauchy Problem
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Title 
The Many Ways of the Characteristic Cauchy Problem

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2014

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English

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14:09
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00:08
of England In the end it is the greater and also honored to be able to reassure you birthday and I had the great again provision 102 work receive circular during the last 5 years the story about the characteristic of the problem so what I would like to do today is to no 1 has she was I would like to report on some work we did together and work which evolved out of our joint cooperation recent results in going to talk about all in cooperation with my PhD students team pets I think this is for the bottom line of my while the punchline of my talk which has the following universe slight :colon we saw .period Cervantes findings said angling to explain what this means that and this is the
01:25
formula for the something called the Troutman Bondi mass and so on well this is the last this is a 5 3 1 with clean air well this is the next something I'm going to explain to you what it is the lending to explain what this is an interesting with this is an industry with this uh wise it interesting when you will notice that everything here is positive and so it seemed like you say you want to prove that the Monday is positive as well it's clear that that Europe proved so let me start To tell you it's em now being there 58 Troutman and in 62 Bondi published papers where they tell you how to calculate the mass simple you have some spacetime an industry standard the surface which is an all angles all the way up to infinity then they told you what is the right expression for calculating the mass of the subject and the antics any of the men to that and so false amendment that was his masters In this setting an editorial it's going to be in the setting for jumped out of the light and that's going to renewals of symbols of but in any case there is this notion of mosque which has been defined by Feldman and bonding and it is some kind of fully meeting into role as you go to infinity so this is an easily meet this all goes to infinity .period 60 by of you to roll over its 2 of some expression beat out of the metric which they tell you what it is and and this is this Trotman Bondi so this is something you have assisted to any also what I'm going to work with any all surface I'm going to be interested in small services you directory light cones so they can you take a point in space time and you I should like to rays from this point and this is the most of his I'm going to OK so what is known about this it's a Trautman Monday well 1st as it is already known that is positive and he's positive he delights come to the surface while I know this is a little more general you don't need to have this light from here on in the setting so they contesting of this asymptotic region of this article artistic surface so this must be positive if the lights go on if and asymptotic 2 I would behave Titus In other words you have space psychiatrists space like at the service in here which has the same asymptotic so that this thing is positive OK so this is something which has been can approved by using go there we denigration so we can type of proof so this statement is much more general than the minefield because mine is just to to like so there is no issue uh uh that uh this needs to the the positivity needs to be established that this simply gives you a very simple enter elementary proof of well there's 1 other thing which is known is that this drug mass decaying the further if you move the services the future right so if you take another surface which is the future to this 1 then its mass would be smaller this also means that you go backwards the mass would be larger citing factors you have proved that matters but he did on scale you know it's positive for everybody before they sell you get lot more than just the proof want to go well this is a question mark but it is expected that the need as you go we is this and so suppose that do you can think of this surface is being curious equality miners are and it's you go to 0 you push this surface down so go West Newton minus infinity a seller of this
06:44
Trotman body mass is the ADM while this hasn't been proved to my knowledge there is a paper by Astor ,comma which proves something they that assuming a lot of things about doesn't studies of the spacetime a lot of things which we do not know whether they're true so presumably and I think it's physically result expected this would be true but then of course there as I told you I what if you go backwards the mass only increases so this limited if it is positive that Blake Stein is going to be positive always in the fall you get an elementary proof of the positivity of must as well In this set for any smooth In a spacetime were heaviest Moose Lake along the south wanted it so this is this for my formula all our formula full the Totman Monday there while this 1 other property which I could mention here that this problem but must actually unique unique when there's so unique functional the field assuming which satisfies 2 that is fine too enter and and invariant under something called Super translation so so so this is an interesting object by Beijing Municipal 1 no less common here is that well since this thing decay uh and all these things all positive you get explicit balance on this interim roles on Net services units based on so this is something which could be useful for studies of the beating of black also on things like that what I have read in this formula in the case where service starts from a point but of course if it started from initial data you could do you get a longerterm view coming from usually do to control their gets some energy type and quantity which perhaps might be useful say something more about the dynamics of the spacetime it's certainly bonded for the times OK good have this kind of explain you work and 16 bodies the violence is so maybe I should start with the same and today we would sing my is that we just go where I can tell you something about the because she problem In general and while it's David simplest equation possible not less fight was 0 when the space like problem you have to give up the angel data on the surface and a it's time derivative on the surface you get a solution if you on a light ,comma on going through assume From on for these services if you want to solve the equation so you only proof providers initially defied and 0 OK so that's how useful the wave equation that's the difference between the space like a few problem and the problem of life ,comma you just didn't give the function and derivative young function and for the past some regularity and that takes reaches were understood good service is set for where integration Fleischer so well and this is something that Iran has taught us you have a space flights the 1st what we can do is just give the format tree on the surface all the time derivatives of the at the surface well this things are not quite free because the reserved for relations call the constraints which especially and if you don't know there is no much .period write them letters for relations between this space thought of the metric and they times well well to Bradley speaking between this optics that quite true but that's essential the might up yeah all the governor elementary all the time derivatives they have to satisfy for full relations called the constraint equations you get a solution now is there on the lifeguard you get if you give me Jimmy anew a on this guy and 1 relation which I'm going to explain shortly and regularity the verdicts when you get a solution so so the claim is that so here is right :colon you get initial data and there is the subset of weather is a neighborhood of a specific subset of the late kung would you get a solution this is so the teacher From a PDA fighter here now if wants to do the geometry teacher here L well what often think that rather than giving this to 1 gives GI Joe and something called K which is related to these derivatives and for relations please us constraints answer in other words while the extra data here which you're free to prescribe or just the gage artifacts so you choose them with you want them to be there and 4 different data going to get the same solution of up to to up to 0 summitry now hear the job matching data I will be something adhering to God being and as connection confusion Kupper and so this constraint equations let's see Selvin g body bees what are those well if you look at event have an
14:09
answer for that I have I can endure from this spacetime entry that the field on this and so the stance a few agree call John
14:23
Jason score Jia the and this is so eager to loom took a variant symmetric offensive field of signature 0 last class and this year stemming from the fact that this is an all surface so there is in all directions involved so this is my G born here now he added value scored United's politics a adapted to this small direction than we have no idea square component if you have no Dr yet a component and utility have ongoing components so this is there the GE and what is that bad weather in the sporting its system if you want to look at the succeeded spacetime this is a connection the efficient yeah instead you the following words the overdue or isn't all that the small on this all hydrocephalus the sale based on the result is a lot of our rights said not moving along itself is going to be "quotation mark itself so copper is this a connection to efficiently here and the gun to integration is later during equation which says that Dr cannot bomb blasts Kyle couple plus onehalf Mosquera last this is in line it's just minus capital plus 1 have seen square block its blessing Musgrove 1 have just worry was era said well you know where you would cap is this is this connection can efficiently which by the way you can said stews you by choosing Europe all appropriately that would have to by the end so this is the equation now what is seeing my wife I have to define something which was called I think a couple chai plus in last lecture and it is it's in my like it in my invitation just cry this is going to be the radial derivative of this 1 half of the oil the DAB so then seeing my wet towel is something which was called trace guy blasts in the previous lecture and it's often called Tito Paul teed up blasts and this is the trace of this sink with respect to this twodimensional metric as they said most of the future uses a similar pattern to turn all that blast while has told me that I have to use style when he started logging on this time following what had been told and as so seeing my be that's the trace free part of so seeing my view would be it's called this year it's so a B minus 1 has to go indeed so this is the right the equation and this is the only constraining the fury and why well and there is a full liable because it is very sad things start of useful observation is that if power is positive then I can just get rid of this equation in it OK this is but story that have yeah if Dallas positives then I can just sold equation trivially for cover and didn't have the constraints so up would be you have to find at this time and the cost not a day that I don't care if I lived here here but I think the just men may be right no I think it's on come on whatever it is as longest always positive I can just sell liquidation in a stupid way by writing that this is 1 of a toe the auto last Sigma "quotation mark turned to the maybe I have as a sign here right things again well it if it was as if those Nunzio whatever the sign and confusion there I can always do this .period and then have sold and my own constraint and then in sounds in this sense that this is the same as served as giving the base of the initial late I can freely prescribed dispense G square ABA which is the geometric objects and this turns out is good enough to determine the whole solution In the stole laden 0 Arkansas now of course the assumption .period zeal is restrictive and but when you have a smooth flight you necessarily have told me about her music because his system has called the this is the In Augusta spacetime cut by 0 and so whether it's Plasil miners doesn't matter tallies to overhaul but cut by 0 . place the acceleration up by as he how these 2 overall argument go skiing entered in this town that privatization couple would be 0 no why does this have to be true if you have smooth flight cut while you should take a general spacetime and start shooting geodesic some .period
22:06
this city's you study intersected so even tho the thing was smooth at some stage I had I don't think they know how to draw this but in general I've come to expect while in general In many situation expect light comes to start being smokers because geodesic starting the form of any Surinam of this for any Smith like Cone who sold conjured .period is going to be restored invite you can ask other spacetimes at all which are reasonably well controlled and each have smooth like guns and hammered explained to me once that this is true and follows by a kind of codeine oceangoing construction is small data space spacetime and so there isn't any issue but it is a byproduct of what doing here that there is a lot of spacetimes here anyway just directly can be constructed using these days so now what is the story about stolidly because 0 well this is actually rather easy to see from this equation if you take that once you have well I suppose when you know that whenever might raise you can always find tries and so the topic was the rights of the stamp is bigger than anything you can only get rid of it regardless of of this story of the safe do this that I see from the liquidation that style is their Madonna's the decreasing function so this is positive so the derivative of negative so decreasing now you have a light coming on bloody flat spacetime for large businesses style will be behave like In Gusky 2 over all would be positive South whenever you have a possibility that spacetime light cone then necessarily Dow would always be positive ever said this hypothesis that that is not in I get it for free if I know that there are smooth flight comes spacetime and then I can call soul couple reasons equation and then once I know what I can change my alltime so that this couple becomes you so after during this change our and get out equation right the equation which and so ghetto Rachel during risk up like this good well said at this stage it is to know what's in Mosquera is now here I am assuming that the parties that I am in this topic would you day so I started resigning Jr square AB I I am assuming that this derivative so that this function here the G square DRG score AB is positive I guess soulful Compaq and go to a geodesic parameterization work GOP i 0 so that this feud is manifestly judges no no uh let's see so I can tell you know what side is I should have written before subside is just an explicit integral today but what so you know what size now provide services this year .period this is to offer which is coming from like continent of spacetime and so this is expression here now what is this fight the fires only thing I can I don't know right off hand held to give you an explicit expression for all but I can't tell you how I can construct this what you have to do is to solve the equation the OS qualify minors seeking Mosquera fight over confusion here I think that maybe a to hear something like that equals 0 With the condition 5 equal 0 at 2 0 and defy over the it or equals 0 is 1 and then this big 5 would be be game this all goes to infinity of filed off set when you play a with the creation is rather simple you can see that this thing it exists and it is Ladakh was 0 so the only thing you need to worry about is fight with you because if got it all 5 0 I have a problem so but when saying this to obtain this formula not too restrictive conditions anything close 1st that always positive and as been seconded fire which could perhaps vanishing some direction is positive if I take any initially denied that which I asymptotic E flat in the you're all Sensoji square Jabari B this is the ongoing part of the Matrix site has to Decatur Minkowski spacetime noting the meeting mystical sense that as soon as I have this to properties I have this formula for the Trotman bonding now well for the experts a just a comment on on this wait wait wait to at effected excited should excite but was so if you think that in terms of Yukon formal completion so you ascribe last year and you have right call emanating from a point underestimated that there's life produced nobody's losing spacetimes about no candidate good points and most of crossing anything like that then when you get into scribe last this thing which could perhaps happened is that you get out going to get .period sitting exactly on Scriber said you don't use clothing is not an infinitive colorful icon anymore but if the final outcome a distance and you could have a conjugate .period Ridge seats precisely here and this would correspond to to fight capital flight was from so so I have to exclude this possibility I guess I have assume the most likely sluice
29:46
and and perverse thing which happens precisely OK we don't know what's exciting is what makes Lindsay government's stance it's pretty excited nope side he whatever you need more to have a complete connection in the geology or direction so you're right D a dealer DIR where there is a finding nominees nation we tree which we use so happy together with fine are just the connection on the In turn on their air space on the bundle of the deal at Yale sigh here and now this site you can calculated on the initial data there's an equation for you to it and I'm going to write down what should well I can visit waited for for you but it's 6 something that Dr plus style signed a well modular confusions you get something like that last studio hello plus DA property was so vacuum that immigration like that which if you didn't properly I wish we had and if you believe in God which we have because this is the Straits of the I and if against the map which we have because of the trace free bottles of wine began disintegrating liquidation starting from the people of the Red Cross and get overexcited so so now you know the ingredients and while and we show you how you prove this formula because it elementary get more blackboard fullbodied supporting with right at the start of the the diminution of them would you take indeed textbook which doesn't bond expansions and you get a formula which I told you but I'm going to give you the former lineman in when you translated to this former "quotation mark something and so on and I mean this is so simple and I can't believe that this is isn't already somewhere in the literature of that say someone has seen this former that decision because how could people have overlooked it OK so well there's calculation which you take out as I said take but the definition writes about this definition and it's going to get the mosque is equal 1 of the 16 fine and very sad about a new symbol which now is the time coming in and this is you can best believe me that is a calculation of today do you don't stand lower Zeke II itself it probably can't be wrong that but I'm going to tell you what the and what is felt to well if you asymptotic if not then tell the sequel Vinning guns scan at 4 large distances behave like this the expansion of likening cost you just because of all classic correction term which decays like us were glass I ordered and so this make correction jumped out to In the same which appears here and what is Zita Wells it's again 1 of these is immigration for the Dow which is important and that will lead to see the positivity coming out of it right away which says that DIR glass tower the last the curvature of these 2 dimensional matrix GAB minus onehalf my size glove and this is exactly the stem 1 over To coming there and prosody virgins over this CSI easier so this is 1 of the equations you get out of that I should equations we determined my functions it up by the way should say that the guys actually following while I was in the right column and you have the lights the cross section of this light :colon that I've told you that prowled is the day virgins Of the likening his direction the guys actually miners of evidence this to conduct must be wide and in this notation CSI dies starting from our joint work with the governments of stressful and keeping it for historical reasons maybe if I had to rewrite everything I would use the limitations that but in any case this is 1 of the highest immigrations into going through to get vacuum said and will desecration where is 1 with the right boundary value said that origins and in fact for small you know that is the guy sequel minus to of all plus 0 1 and 4 large or because when I the quotation say once you know these are you going to do this you can get it there as you know the twodimensional alleging that it has no place disintegrate radially and you get and soulful large on the values equal minus 2 of our last and square the term class of ominous so far the got to his escape efficient which appears in the extension of the debt for large so let's start again Europe you give me a G square maybe in a couple so that positive and this is a necessary condition for a smooth like coming in and I eventually got of which injured probably erased and then go to breakup equals 0 then I can solve the equation for the dog which where is it all about in somewhere erased it it was intent of folks a fullsize on sides the I think you're right consult integrates I was a notes I can integrate the I can calculate destined the expansion of the things so I can find this confusion here I know the confusion for of expansion of power here
38:13
I plugged it in this into wall and I'm going to visit my mask so by the way this is obtained by transforming the boundaries an expression into expressing blunders expressions From a break so Bundy's assuming various party if conditions that you have to be the main and the satisfied and that this expression and now this expression involves only things which are intrinsic to the Cyprus service because it on it and everything has been defined in terms of G urge got AB everything I did here starting from Gibraltar can obtain so I don't care what's happening was courting itself the surface of anything like that and I could just say Well this is the definition of the mass reduce the thing to do about I could do this with salt but in fact it is exactly the same as bond his mask when bond is most defined so for example and they spent that's enough to want to know how to prove this that should be at it's the helm finding this Stein OK yeah but I don't need to reassert probably just need that ministers Paulista to most of them I need the reach relinquishment of all races well you see it this the declaration produces positivity when integrated of us here for a very simple reason the evidence given because if you into written evidence goes away and you can do but the integrate are bad and the good something simple and this is a positive step now this pal c . is exactly what you need when you're differentiating an integral indeed With respect to the vehicle measure because you're going to get it which is a derivative of the that and there during the eve of the measure produces exactly the styles don't a few years the equation you're going to find that this is minus integral of all block which is a 2 prideful play it but I did it on this year 8 because not the ghost curvature but this skillet contributors so this produces 8 pie wow minus place good becomes plus onehalf interlocks his club and that evidence goes away so I have this very clever idea to write this is development 2 of science class 8 by are the sequel 1 of In 2 books at school now this thing Turns out 2 extract this confusion of this in the grant when you integrate goading Cheney integrate and 0 by the body condition the verdict circuits here at infinity this something amusing happens which says I don't give a shit that if they integrate limits or goes to infinity of integral part of the government side off is well you think that this is integral this as of the 2 buttering up y you get minus 2 true and this is easy to see how well or at the heart of his full knowledge the dollar behaves like minus durable citing disintegrate when you look at the leading order behavior where indicating minus2 overall this year you get for from the area the turnover the year 8 5 with the miners so and from there get full by squad which glances out 1 some of the leading role attorneys minus 8 by all the function of this Council this 1 now there is a subject order which coming from the fact that integrating 1 over or against not displace the flat
43:21
spacetime measure but the physical measures as you do the calculation this producers so in other words 2004 if I call this 1 then my is 1 of with 16 times Y which is positive the indeed given that this is my 1st injury although the class and now while minus because they had In this formula fall L vanished somewhere it was the right side have in said the event too which is here I minus 2 . 0 2 and blast out too and I got mine integral of so now it's a question of I think I have they should do it in 3 minutes but you can't just but knew that if you take the creation here misstep Lakewood and you integrated In clever way then this definition style to which is starting from this integral is exact so that's all there is and this is how you prove that Trotman body mass is positive using a lot of ideas which he won't help me understand and relate to all of the along but we we developed together this formerly worked understand the lake house of and live their faith and that he had his hands the the where when or if it doesn't the that this I don't want to do that at the brink of such things he wanted his desire to is is also said to be interested volunteering those from yes this is so I've tried to because obviously well 1st if you have just and also offers coming out of the initial data that you get a body temperature control by the initial data and get this formula was quoted at the time he is obvious because it is you integrating those then you're going to get this at the initial value the other 1 is a little more messy but you can work it out and I wish I understood it because if I did that could be perhaps as some kind of pandas inequality floating around but I felt that something I'm thinking about certainly this these phone calls all of the things I call the was idea as compared to the look at the money must propose a dividend think the notice they assume small at all because I was thinking this is not a lot of good quality if you as I said I mean I'm what I'm so astonished that this has been overlooked nursing it before I'm interested enough bringing Monday masked political mass is usually a big deal and complicated things and I think I know you'd of innocent people the he has actually have to do with reference to recent paper by showing art was as you on the these things are so bad so easy on the side of the square so this is defined this the issue further 48 finished fixed the other regions hit by a welfare state of the art of Bonnie Teresa defined around like and do you fixing the possibility because of the metric and the so yes so that there is some freedom left and about well what it is in this case is a physicist choices yes I am assuming that the matching goes however he says that this is the square AB is intriguing Constantine usual sense for us a visitor means a lot of normalization here and some irrelevant 1 of disease should be no drove for those intervals 4
48:29
Minn