Good morning and welcome to morning prayer. I want to welcome those of you online as well as in the room What a great day it is to be in the house of the Lord We're gonna open up today Reading from Psalm 13 and verse starting in verse 5 But I have trusted in your mercy My heart shall rejoice in your salvation. I Will sing to the Lord because he has dealt bountifully with me. Let's just enter in through prayer But we just come before you And we thank you that you're merciful That our heart can trust you That we can trust your character What our heart rejoices in your salvation your rejoice in relationship with you we rejoice And what you've done for us and God we just remind our souls today to sing Because you have dealt bountifully with us not once Have you let us down? Not once have you steered us wrong? The only truth God is that you have dealt bountifully with us You have loved us well. You have cared for us well and Lord it's it's our hearts desire to just continue to see How much you love us all the days of our lives?  So I just invite you now Stir yourself up start to offer the Lord your own voice He doesn't want to just hear mine or the worship teams, but he wants to hear yours Sing to the Lord because he has dealt bountifully with with you. This is a command from the scriptures Let's just offer up our own song to the Lord Reminding your soul the Lord has dealt bountifully Never be able to know the full measure We'll never know how much it costs when we look at you when we look at the cross we're at a loss for words at the beauty of Jesus Such an act of love Such humility who was God who is God come down to choose a human body To come and take our place who are we that we're worthy of that kind of love you took our stripes for us You took our beatings for us you paid our very debts And we don't forget that this morning We don't forget that we remember we remember Calvary Bros the blood that was shed on our behalf Lord we remember we won't talk about the cross if it's just some passive thing But we'll look at and we'll remember the act of love That you you did for us Love was full on display that day As you humbly chose to carry your cross up that hill for us you hung there suffering For us so that we could be with you So that we could commune with you there was a barrier between us got separated Our own sins separated us from you And you came down and you took our place to tear that veil And we don't forget that this morning We don't talk about it like it's something we've heard before talk about it like it's something we've heard before I just want to keep keep singing the song keep keep telling Jesus who he is or today we choose Not to talk about the cross as if it's something we've heard before Let's just go back into that song and keep singing Jesus this morning We respond in love To your sacrifice on you this morning as we remember the sacrifice that you gave the sacrifice of God that can never be overstated worthiness of God that can never be overstated We respond with our love this morning Thank you, Jesus praise we could ever give and then some really of all of our worship all of our devotion It's because you're beautiful Jesus Because of your love for us, and we just thank you Lord that we get to worship you We get to praise you we get to show you love by pouring out our praise before just want to pray from this reality shifting into To intercession praying for unity in the church in Kalamazoo out of Romans 15 starting in verse 5 Now may the God of patience and comfort grant you to be like-minded toward one another according to Christ Jesus You may with one mind and one mouth glorify the God and father of our Lord Jesus Christ Therefore receive one another just as Christ also received us to the glory of God Jesus we just come before you and again. We say that you're worthy.  It's divided. It's it's sick with pride. It's It's sick Lord, so we just come before you and we ask God that you would unify your church That we would be like-minded toward one another Like-minded to glorify the God and father of our Lord Jesus Christ But would you unify us around the beauty of Jesus? Unify us Lord around the worthiness of Jesus that he is to be worshiped that he is to be praised Go but all of our other agendas fall flat would we put them aside and would we unify as one body and one church one bride Worshiping Jesus giving him the praise that he's do pouring our love on him Would your church not forget the cross?  And Jesus be lifted up of your blood God by the power of your Holy Spirit Moving the minds and hearts of church leaders Make us compassionate towards one another But only Jesus be lifted up Bring us back to a place of worship Lord bring us back to a place of prayer Bring us back to a place of adoration to see Jesus rightly To see the beauty of the Sun To remember the cross revelation of the cross Revelation of the love that Jesus the Jesus has for us only thing that matters Be lifted up Would you be lifted up from a unified voice of your bride the church Lord? Would you make us Compassionate towards one another and we would receive One another just as Christ also received us But I just pray for for church leaders everywhere God pastors elders Lord would you just would you just give them a fresh humility or to look past denominational differences But with the unifier on this one thing the one thing of worshiping Jesus The one thing of worship and prayer God, would you unify your church around worship and prayer? Jesus if you're not exalted nothing else matters If you are being worshiped nothing else matters all of our programs they fall away Every other work is done in vain Also, I asked them in the hearts of leaders that you would you would soften them over there may be Years of hurt from different denominations different relationships got specifically here in Kalamazoo I pray for just a unifying of church leaders God that the heads of the church would unify They would lay down their their differences or that There would be forgiveness and compassion in the hearts of the church leaders of Kalamazoo and God that with one voice All the churches in Kalamazoo would worship Jesus that they would worship the man Jesus Because regardless of how we feel towards one another Lord, you are worthy to be praised That is something we can always unify around So God in Jesus name unify the church leaders in Kalamazoo Unify us God, unify us to worship, unify us and you The one thing of worshiping Jesus Think of desiring to see you the one thing of desiring to know you The one thing of seeing others come to know you Counter the presence of God and be praised from every altar here in Kalamazoo The altar from every church Would you be praised be exalted Kalamazoo would sing with one voice In our hearts one another think of worshiping Jesus God I just asked that you would humble us that you would humble the church of Kalamazoo God where there's there's hurt where there's division where there's pride But you humble us Would you humble us God your word says that you give grace to the humble that you oppose the proud Would you not would you not oppose us Lord we as a church big-sea church come before you humbly Worshiping you Denominational difference is falling God I'm asking for Catholics to to lock arms with Lutherans and Methodists to lock arms with Baptists And for the non-denominally non-denominals the non-denominally that word church to Lock arms to Lord Would you just would you bring us together what our differences be overlooked for God and would we unify?  Unify your church We need you to humble us To look past each other's differences and to unify Around worship around prayer Around the things that matter The closed-handed issues Unify us God another Link arms and worship Receive one another well Just as Christ has received us would we love our neighbor or Just as Christ has loved us Would we love our church neighbor just as Christ has loved us Pour out your grace Or out your grace to do it Lord Would you grace wash over our hearts today? Removing all bitterness all hurt all insecurity Pour out your grace Draw near to us Lord. Yes, God. We take these last few minutes And Lord we we pray that you would unify us around your word That no other doctrine would be preached But the word of God Lord unify us around the Bible Unify us around your scripture that it will be taught Taught on Sunday mornings not our own agendas our own agendas Lord Not our personal preferences When we humbly come and unify around the word of God around the scriptures Around the lessons that you taught through your word that you're still teaching today God would know other doctrine be taught except the word of God Except the truth of your word God where we've got it wrong where we've let Our own agendas come into the church Where we've let hurt and bitterness into our hearts But I just asked that you would Humble us and bring us back to a love for your word And Lord that as the word is preached as it's unified that the sword of the spirit It's just the word of God that it would go and then it would cut Then it would set people free is your your word is preached to God That there would be testimonies rising up from messages When your Holy Spirit encounters people and they're set free set free from addictions Unify us together around your word Why don't you say there's a blessing when brothers dwell together in unity? It's got to unify us around your word and pour out your blessing in Jesus name nothing else Just your word our agendas fall our preferences fall.  Not a doctrine that is sinking sin But we not build in vain God. We build on the cornerstone Put your house Build your house on the foundation of your word But unless you build the house we build in vain, but so we just ask you this morning That your church would be unified And built on the cornerstone of Jesus on the cornerstone of the word of God We thank you that this is on your heart It's been on your heart for Kalamazoo Well, we just close in a posture of thanks We'll pray all these things in your almighty and holy name. Amen. Amen. Thank you so much for joining us here for morning prayer I want to invite you back. We're back here at noon and 6 30 tonight. We hope to see you one of our weekend services Thank you


What's good Josh your boros back again with another video, so there's a lot going on right now in AEW with CM Punk and the the EVPs from like Kenny Omega to the Young Bucks Tony Khan has a lot going on in the backstage scene. It's like chaos. It's just straight chaos. I think the backstage Antics are more important now and like being more talked about Then what's happening with what's going on on the main show? So we got to talk about this This is just insane. Let's get right into this one. This is brought to you by cagesideseats.com Let's get right into this one Question persists about who started CM Punk slash elite fight police involvement updated man The latest wrestling observer radio has the latest on the biggest stories coming out of the busy weekend The backstage bra between the young Bucks and Kenny Omega and CM Punk and a steel it took place After all out as every wrestling fan who checked the the wrestling website since Sunday September 4th knows the fight took place after press the drinks Dressed the lingering rumors about his role in coke cabana being moved off AEW television And into behind-the-scenes role with ring of honor at the post pay-per-view media scrum His comments including sold in remarks about hangman page whose promos alluding to The cabana situation back in May is a trigger point for every everything that followed and the elite trio Who he referred as people were already into our third or fourth round of reports on the melee and there There's a lot of agreements about the details first and foremost despite what portion of the internet thinks and Will probably always think the fight was not a word police were involved at the time of the incident and none of the Principles are talking about the possible pending legal action Among people Dave Meltzer and Brian Alvarez spoke to no one disputed that punk punch Matt Jackson or that steal through a chair which hit Nick Jackson and pulled Kenny Omega's hair and bit him Geez where there's where there is an agreement per Alvarez The disagreement between the two sides is that one side claims that see a punk through the first punks The young bucks walked in and he started punching the other side claims that the young bucks stomp Stormed in and where the aggressors and that punk and ace were defending themselves Those are the two sides of the stories and depending on who you talk to who was there who heard from Who heard what from who you're gonna hear one side of the story or the other Which sounds about right in an altercation? Sorting out which side of the story is truthful will fall to Tony Khan whose leadership is already in question Since the issues between the two sides were apparently well known backstage for months and Because he took no steps to stop or redirect punk at the strong strong Multiple believes an official decision on addressing the entire entire embarrassing situation won't come quickly That's partly because of the involvement of police and lawyers But also because that's how Tony Khan handled a similar crisis when Jeff Hardy was arrested earlier this summer And as if all of that wasn't enough of a disaster a disaster Meltzer says punk was in the trainers room at the end of the night So he may have been injured in the brawl just some stuff to look forward Forward in the six in the seventh reports. We get on the brawl probably coming later today or tomorrow And I've been seeing some reports saying that he may have torn a muscle in this melee of sorts This is just insane. This is truly insane. What's going on right now? I think a lot of people have criticism for seeing punk. There's a lot of people have criticism for the EVP's in In Kenny Omega and the Young Bucks and there are people are giving criticism to Tony con for not really stepping in and taking leadership leadership and saying, hey Let's let's talk about something else. He didn't he didn't step in really. He just kind of let see a punk just riffed and I'm I'm I'm gonna give my perspective on everything that's going on here Some people are saying this is very unprofessional of seeing punk and then there's some people saying well, you know I understand where seeing punk is coming from. He's frustrated. I see both sides of the argument. I Have no problem with seeing punk may be defending himself saying I didn't get coca-banna fired or what not I have no problems with that What what I guess you could say the issue I may have is The product is the fact that he started burying people. I'm talking about the EVP's had Adam Page He started burying them now the Adam Page situation I think he kind of already took care of that live on television after he gave Adam Page his receipt in the promo form But then to continuously bury him in the EVPs You only had to know there was gonna be some issues like there's no way seeing punk didn't know There was gonna be some altercation when he walked in the back. I mean he pretty much said it We got a problem with me. What's up? So I'm pretty sure when he walked back there They had a problem with it and it went to fisticuffs. Now the question is who throw through the first punch You see him punk through the first punch. I Got it. Hey, that's not what's up. See him punk That's definitely not cool and he should be reprimanded if they ended up throwing the first punch They should be reprimanded but at the end of the day See him punk What he said may have been true and he may be justified in saying that but there's always a time and a place and Really, you're supposed to be talking about the show No, one even really is talking about MJF coming back. That should be the biggest thing for AEW this weekend MJF making this return that should be what everyone's talking about but everyone is talking about now What the hell see him punk said that the media scrum and now this bra that is supposedly have happened That's what everyone's talking about. No one's talking about MJF coming back What he's gonna do potentially how they're gonna start this feud up again That should be the main point. So this media scrum you kind of took away what happened on the show No one no one's talking about what happened on the show Everyone's talking about the media scrum and everyone's talking about the backstage bra and that takes away from your product People are gonna want to know hell. What the hell's going on backstage instead of what's going on in front of them now. So Yeah, I think see him punk in the hindsight He let I do feel like he let his motions get the best of them and started going on Going on this tangent about how he felt about the people in the back and that's cool But it's a time and a place you should have did that somewhere else You don't do that after the meet on the media scrum and now it's I Expect nothing to happen Like he had to know as soon as he got out of there There was gonna be some issues and it wasn't gonna be on a talking basis. It was gonna be on some what's up You got a problem with me.  I'm not gonna really buy into that Maybe he did maybe he didn't but either way if he injured if seeing punk is injured again.  I this is just Who this falls on more than anybody else is Tony Khan? Sorry Tony Khan should have nipped this in the bud. You need to nip this in the bud, bro You need to not be as passive when it comes to the backstage stuff and nipped us in the bud And if see them punk did throw the first punch. We got to be fair I know it's not gonna happen But he probably should get suspended you saw what happened with Eddie kinks in Eddie kinks And they'll get suspended and his altercation was a little bit less as severe Doing what we got going on here his altercation with Sammy Garza Sammy Guevara was a little bit I know but the report saying he pie faced him, but you know I'm saying it is still less of intense as what's going on here you strong bunches, so Fair is fair. You got to suspend them. But once again, this is wrestling that this is a business Remember when Brock Lesnar stormed out supposedly and wasn't gonna be on Smackdown when they found out Vince Was I have retired or left remember what he did there and then they brought him back Nothing happened to him.  I'm sorry. It's messed up.  That wasn't cool. I get your frustration, but I got to suspend you And then what this is do for the overall locker room perception like this this is not good your top guy Is getting into it with your EVPs and it's just a whole it's just a whole mess of things This is I don't even know what to say. This is really really wild Yeah, bro, I'm thinking it's just gonna you know I did hear about the bra yesterday, but I didn't know the exact details and now we're getting exact details supposedly This is just insane Tony con. You got to take control of your company, bro if you don't More stuff like this is gonna happen and you don't want this to spill over into your Television product to the point where maybe somebody goes out goes into business for themselves and they actually start throwing like You know stiff punches at someone that's not prepared for it like it can get dangerous So I don't know Tony come you got to figure out something But comment down below. Let me know. What do y'all feel like? How do y'all feel about this whole situation? Do you feel like seeing punches in the right? Do you feel like Adam Adam page is in the right? Do you feel like? The the EVPs are in the right for how they feel do you like what do y'all feel about this whole situation? How do you guys think things will play out and do you guys think Tony con should take some of the blame here by not Stepping up as the guy that owns the goddamn place and getting involved Let me know down below, but I appreciate all love and support road to honey. Okay, appreciate y'all kicking me See y'all next one


And we wrote down a form of the metric that took the form and now I'll stick to Landau-Liefschitz's notation. So, in general we could write it as h of rt dr squared plus t of rt sin squared theta d phi squared d theta squared. Plus l of rt dt squared plus a of rt dr dt. Okay? We argued this last class to change the names of the functions that this matter. Okay. Now to proceed, you see theta and phi have physical meaning, meaning they are the coordinates on a round two-sphere. We've used the spherical symmetry to distinguish coordinates on the two-sphere. However, the coordinates r and t are so far completely arbitrary. This tells, yes? There are four functions because ten in the metric, four are two from the symmetric. Two from? Two from the symmetric. Okay. Let's see how we would come. Suppose we looked at rt theta and phi. The ten functions that you have are the various elements of this matrix. Since it's symmetric, we'll just look at this and, you know, not have things here. Okay? Now what elements do we have? We have dr squared, we have dt squared. Now, we have a d theta squared and a d phi squared. We have no d theta d phi. All of these three have been replaced by one. That is the coefficient behind this guy. We're just not going to allow anything that is crossed between r and theta, sorry, r and t and phi inside. So that kills four here. Those are the cross terms between r and theta, r and phi, t and theta, t and phi. And we've killed two additional ones here by saying that of the three possibilities in the theta phi, in the theta phi space, that's just one. Okay? So that's the ten minus four minus two. But you see, so far what we've done is not work with any particularly distinguished coordinate in the rt sense. Now, you can ask what coordinate redefinition freedom do we have in our hand? Because we've got full coordinate redefinition freedom. If we do any redefining of either involving theta and phi, that will change this distinguished part of the metric to something else. We'll be using some different coordinates on this sphere. We don't want to do that. We want to use particular coordinates on this sphere. And any coordinate freedom that if we redefine r and t in a theta and phi dependent way, it will give you spurious dependencies on theta and phi. That's not convenient for our problem. So we want to keep this form of the metric. So what we are going to try to do is to ask what coordinate freedom do we have left that preserves this form of the metric? And the answer to that question is obvious. Can somebody tell me what it is? Rotations just don't change the metric at all. We want to look at more generally coordinate transformations that will preserve this form of the metric but change the values of these functions a, k, l and r. Scaling of r. Scaling of r. That's one possibility, but that's a very minor possibility. Function of r. Right. Any field redefinition which is r is equal to some function of r prime and t prime. And t is equal to another function of r prime and t prime. Well, that will change what this particular function is just by putting what r is as function of it. And we'll change this part of the metric. But we'll present this form clearly. So by adopting this form we have essentially broken our difumofism invariance down to difumofism invariance in two dimensions. Okay. So now we have this coordinate definition freedom of r and t and we can use that to further simplify a metric. Okay. So now we're going to do two things. The first thing we're going to do is to... So since we've got two functions in our hand, two coordinate freedoms in our hand, we can use it to set two of these functions to something nice. Okay. The first thing we're going to do is to set this guy to zero. The second thing we're going to do is after having done that adopt the convenient definition of what r is. So we will just set k is equal to r square. So we use coordinate redefinition ambiguity to say that our new r coordinate is whatever came behind this thing. The new r coordinate square is whatever came behind this. So that has now given a physical interpretation to this r coordinate. The r coordinate for pi times r square is the physical size of the two-sphere at that location of r. Physical area of the two-sphere at that location of r. Okay. So we use a physical criterion to give meaning to r and then we use the further field redefinition ambiguity, the t field redefinition ambiguity after that to get rid of this term. Okay. Now you could ask when you always do that, postpone such questions. I mean certainly the local patching. There are topological obstructions with this concept. Okay. So what this leaves us with is a metric of the form ds squared is equal to and then we do some renaming again to stick to Landau-Liefschitz conventions. ds squared is equal to e to the power nu dt squared minus r squared dt theta squared plus sine squared pi d phi squared. Minus e to the power lambda dr squared. Well we've just taken sine squared and we've just renamed these functions. Okay. Whatever comes behind dt squared we call e to the power nu. Whatever comes behind dr squared we call e to the power lambda. And here we are committing to implicitly committing to the assumption that this coefficient will be positive whereas this coefficient will be negative because an exponential can't change sine. Please. The RT term is not there because it was translation. Where? The what? RT and r squared. R. R and dr dt. Dr dt. That's just a choice of coordinates. Coordinates. Coordinates. We've got two functions. We've got two coordinate redefinitions left. Here we had a dr dt. We're going from here and here we get rid of the dr dt by using our two coordinate freedoms to set two conditions. Okay. The two conditions are a, the radius is whatever appears behind dt theta squared plus d phi squared and b is the dr dt term is given. Clear? Okay. So now this is our form of the metric. Now can somebody tell me is there any further field redefinition ambiguity left here? A coordinate redefinition ambiguity left here. There is one. What? e to the power lambda is equal to what? Well first how do you say e to the power lambda equals one? Lambda in general is a function of r and t. So you have to tell me what coordinate transform I can do. Can I redefine r? No. Because r has physical meaning. Can I redefine t? Yes, nobody can stop me from redefining t. If I redefine it in a way that depends on r, that will generate a dt dr term. Okay. However, if I redefine it in a way that depends only on t, all it will do is change this new here. Okay. So we've got some coordinate freedom left. And the coordinate freedom effectively allows us to take nu to nu plus some function of t. Where that function is the logarithm of the derivative of the times field redefinition. Is this clear? Okay. So two metrics that differ only by nu, differing by function of t, but otherwise equal are the same metrics. Just differ by a coordinate redefinition. We're not going to actually fix a coordinate choice for t just yet. We will do that in the solution as we get to it. Okay. But it's important to remember this at this point. Is this clear? Okay. With this kept in mind, we've now completely fixed coordinate freedom. This is a lesson of how we work with general relativity. The formalism of general relativity is completely coordinate invariable. But to solve a given problem, you want to set up equations. Equations have to be set up in some coordinates. Sometimes mathematicians manage to solve problems without writing down equations in any coordinates. I don't understand how they do it. Okay. But physicists like equations that you can see.  So what you usually tend to do is tend to try to search for coordinates that are as well adapted to the problem as possible. Okay. Suppose you worked in a completely coordinate invariant way. You said I'll work with these arbitrary coordinates. Then when you try to solve equations, you will find more solutions than physical solutions. Because in addition to having the solutions that you want, you will also have coordinate transforms of those solutions. Okay. And then you'll have to disentangle which parts are physical and which parts have trivial. Okay. So it's often convenient to fix coordinate freedom in as good a way as possible at the beginning of a problem. Okay. Now we've done that. Now we have to solve Einstein's equations. Okay. So now this is the painful thing. Just because it's the first time, we're going to do this quite explicitly in class. We're going to try to solve these equations quite explicitly in class. The painful part of this is computing all the crystals. But let's do it anyway. Okay. So we'll roll up our sleeves and start computing crystals. Okay. So let's first start with... Okay. Let's use Landau-Liefschitz for guidance. Let's see which are the easiest ones. Let's first do the Christoffel symbols that are purely in the sphere directions. Okay. So let's do gamma, theta, theta, theta. Okay. The metric is completely diagonal. So this is equal to g, theta, theta times theta derivatives of g, theta, theta without zero. Because g, theta, theta is not theta. Okay. So this is equal to zero. What about gamma, theta, theta, phi? Okay. This involves either phi derivatives of g, theta, theta or theta derivatives of g, theta, theta. There's no g, theta, phi. There's no phi derivative of g, theta, theta. So that's zero. So let's... All the zeros will put... So we've got clear that theta, theta, phi is equal to zero. Okay. Now let's look at gamma, theta, phi, phi. Is this going to be zero? No. Because? Exactly. There is a term which is... Now let's be careful. Minus half... Oh, sorry. I'll put the minus later. This is theta, theta times the piece that has the theta derivative of phi, phi. That comes with a minus sign. Del theta of g, phi, phi. Okay. g, phi, phi was sine squared theta times r squared. So that's equal to two... So the two cancels this. So sine, theta, cos, theta, r squared with a minus sign. Okay. Let's record it. Let's see if we can get this whole thing right ourselves. Okay. So we'll record this. Do you agree with me? If you spot an error, please tell me. We'll also compare with lambda. Okay. And so this was gamma, theta, phi, phi was equal to this. Okay. So now we've dealt. Okay. What about... Let's do everything with the theta. Okay. So what about gamma, theta, t, t? This is either involves... There's no t, theta. This either involves g, t, t with respect to theta, which is zero. Or... Or nothing else. So that's zero. Okay. What about gamma, theta, t, r? Amma, theta... Okay. The only non... Okay. Let's speed it up. What could be non-zero? With the theta, what could be non-zero? We've dealt with one thing here. Okay. There's one more thing that could be non-zero. What could it be? Somebody said, theta, r. You see, suppose we take gamma, theta, theta, r. Then that will involve the term that's an r derivative of g, theta, theta. This will be equal to... Oh, and I forgot here the g, theta, theta up. Oh, I put that. No, no, no. g, theta, theta up. I messed up. Sorry. Let me just do this again. Sorry. Let me do this again. This one was g, theta, theta times minus half times del theta of g55. Ah. So the r square cancelled. There's an r square in here, an r square in here, and that cancels. So sorry. Okay. Now what about this guy? This guy has a g, theta, theta up. So that's a 1 by r square. So we get 1 by 2 r square times the r derivative of g, theta, theta, which is minus 2 r. Another thing we messed up under the sign. That was a minus, because g, theta, phi was... So minus 2 r. So that is equal to minus 1 by r. Okay. So g, theta, theta, r is minus 1 by r. Any other things with theta on top that could be non-zero? I don't think so. Because the only metric components involving theta that are non-zero are g, theta, theta. And then you need an r derivative. Or you could have a theta derivative and only phi, phi. Okay. Great. So let's rub these out. We've dealt with everything with theta on the top. Now let's deal with everything with phi on the top. Which gamma is with phi at the top could be non-zero. So firstly, anything involving a phi derivative is zero, because phi doesn't appear in the metric. Okay. So that phi has to be one of the metric components. The only metric component involving a phi is g, phi, phi. What does g, phi, phi depend on? It depends on theta and an r. So there are two things that could be non-zero. Okay. So there's gamma, phi. There's phi, r. And gamma, phi, phi, theta. Okay. Now let's compute this guy. So this guy, we need... Firstly, we have a g, phi, phi. Then we have a plus half. Aren't we messing up a lot? I'm going to come back and look at the signs of this. Remember that the g inverses also have signs. We were messing up on that. I'm going to come back and look at the signs of the top two ones again, but let's do this correctly. So there's g, phi, phi. There's a half. And then there is del r of g, phi, phi. Signs cancel between these two. Okay. This thing is like r square. So the two factors says that this is one by r. Okay. Now I'm going to go back to the earlier ones and this is one by r. Here it was minus and here it was plus. Everything else cancels. Right, because we had g, phi, phi at the top and then g, phi, phi at the bottom differentiated with respect to r. So everything here doesn't depend on r cancels. So it's basically a logarithmic derivative of g, phi, phi with respect to r. A derivative of log g, phi, phi with respect to r. So that's a one by r. Okay. Now what about this guy? Here it's very similar except that what we're differentiating is the sine square theta. Right?  With no sine, I believe. Tell me if you use cot theta. Let me now just check against Landau-Liffith. 3, 2, 3. 3 is presumably phi. 3, 2, 3 is cot theta. Good. But he doesn't have a 3. Sorry, just wondering. 3, 3, 3. I suppose it's 1. 3, 3, 1. He doesn't report 3, 3, 1. I think he's just messed up. Okay, he doesn't report it. He says all the rest are 0. This is not 0. Okay. It's probably messed up. We'll see. Okay. And let's check these two. So this is what he calls 2. So 2, 3, 3. Does he have 2, 3, 3 at least? 2, 3, 3. Minus sine theta cos theta. Excellent. And 2, 2, 1. 2, 2, 1 is 1 by r. Excellent. Okay. So apart from hopefully his rather than r messed up. With this guy, we agree with that. Now let's continue. Okay. So we've done everything with an angular coordinate on top. Now let's take the things with r on top. So gamma r. Now it's more complicated because everything depends on r. Okay. So r could be either the derivative or it could be the metric. So suppose we want the terms that are non-zero because r is the derivative. There are many such things. There's gamma r, r, r. Okay. There's gamma r, t, t. There's gamma r, pi, pi. There's gamma r, theta, theta. Let's get all of these. So let's look at this first. So this thing is minus half, sorry, it's plus half. g r, r. Del r, g r, r. Right? So that is equal to now g r, r was minus e to the power, lambda, the minus cancels between the two. So this is equal to lambda prime by two. By prime, I will mean r direct. Lambda prime by two. Okay. When we do our next problem in class, I will just write down first-off examples. Okay. Okay, but once we'll calculate them. Okay. What about gamma? Then there's t, t, for instance. Okay. In this case, it's minus, this is equal to minus half, g r, r times del r of g, t, t. Okay. Now let's get the signs carefully. This guy comes with a minus, but this guy has a plus. So the minus goes away. Okay. So this is equal to e to the power, nu minus lambda into a nu dash. Into the power nu minus lambda into nu dash by two. Dash is derivative with respect to r. Okay. What about gamma r, t, t, t? Okay.  So that is equal to e to the power minus lambda, the minus signs between these cancels.  The minus sign goes away? No, I don't think so. There were three minus signs. There was a minus sign for the inverse metric, the minus sign for the metric, and the minus sign because we're using the minus part of the gamma. Okay. And there was a gamma r phi phi. This is extremely similar. It's just that you multiply the space signs with it. Okay. But we're not even through with the, are we through? Let's see. I think we're through, maybe. So all that remains, we've taken into account all terms that come about because we're taking r derivatives with an r on top. But there are also the terms that could come about because we're differentiating a part of the metric which has an r index. The only part of the metric that has an r index is g r, r. Ah, so there's one more thing. So we could have, so there's one more thing. What could it be? No, but yeah. So it could be r, r, t because g r, r depends on t. Right? What could that be? So that could be plus half g r, r into del t of g r, r. Times cancel. And so this is equal to half lambda dot. But dot means derivative to this vector time. Okay. Great. So we've dealt with everything involving an upper r. All that remains are things that involve an upper t. That's actually easier than upper r because the angular coordinates, the coefficient of that depends only on r square and not t square. Okay? So let's get the remaining guys. Okay? So there's gamma t, t, t of course, which was equal to plus half g t, t, del t, g t, t, which is equal to, signs cancel, so it's equal to nu dot by 2. Full symmetry. Lambda prime by 2, nu dot by 2. Okay. Then there was gamma t, r, r, which was equal to minus half g t, t into del t, g, r, r, okay, which is equal to what? Exactly. So first the signs, this minus cancels a minus sign here. So it's plus. And the derivative is t derivative of r, r. So it's lambda dot. So we get half e to the power lambda minus nu into lambda dot. Okay? Again, look at the symmetry between this and this. Flip under r goes to t and nu goes to lambda. Okay? What about gamma t theta theta? Zero. Because g theta theta doesn't depend on t. Okay? Only remaining guy, there's one remaining guy. What could it be? Now, okay, we've dealt with all that come about because we're taking t derivatives. But what about the term with the t component? Exactly. This is the last nonzero guy. So what is it? It's plus half g t, t into del r, g t, t. Okay? So that's equal to nu prime by 2. Okay? Yeah, I suppose it's worth checking all these. Yeah, it's probably worth checking all these. Okay. Fine. So let's see. So firstly, the guy is with one on top. So gamma 11 is lambda prime by 2. Correct? Gamma 110. 110 is lambda dot by 2. No, right. Gamma 110 is lambda dot by 2. Correct? Gamma 122 is minus r e to the power minus lambda. Gamma 133 is minus r sine squared theta e to the power minus lambda. Correct? And gamma 100 is nu prime by 2 e to the power nu minus lambda. Correct? Okay, so all that remains is this. So gamma 011 lambda dot e to the power lambda minus nu. Correct? Gamma 000, nu dot by 2. Correct? And gamma t r t, so 010 we want. 010 is nu dot by 2. Perfect. So apart from the fact that lambda issues is one less gamma than we do, we've agreed with him completely. Okay, excellent. So now we know how we've computed all our... all our... Christoffels. Okay, but now in order to get Einstein's equations, we're not yet done. We need to compute curvature. Okay. Now, if you were really working out a physics problem, you know what you would do is to go to Mathematica. You'd find a package that takes a metric and computes curvatures for you. And that's what you would do. In your life, you should see how Mathematica does it. Okay. Yeah. Okay. Yeah. In this one, deciding the sine of what is the sine of dt square and dr square, how do we do this fix? Okay. So we're thinking of t as a time-like coordinate. Okay? So in flat space, we would have had minus dt... sorry, plus dt set. So we're looking at that by continuity. However, it's a very good question because as you will see in the black hole metric, this goes wrong at some point. So if you're near enough to being flat-spaced, it can't go wrong. You know, because this was plus one. These were minus ones. Okay? So you have to be very far from the flat-space metric in order for it to possibly go wrong. In dozen some cases, but we'll ignore that possibility for now and correct for it when we do. It's a good question. So I think we fixed another choice of coordinates. By another choice of coordinates? But you see, it could be that this... the coefficient of dr square, for instance, will become positive. Science cannot be changed.  It's the thing behind the two-sphere. And how do you know that that's the coefficient of that? In this case, but in general, if your dt term has a negative sign, then it can be changed by a charge of 30 seconds. The signature of spacetime cannot be changed. The fact is that one minus and three pluses cannot be changed. But of course, which one is minus and which one is plus? You can do just by renaming, right? If you call R, T and T are... In the blackboard case, inside, doesn't the signature change? No, it doesn't change. But the choice of coordinates can add to the other side. Well, you can choose to call what is R. You can choose to call it T. Okay? Except that then you'll have T squared behind the two-sphere metric. So once we've fixed coordinates like this, we've removed much of our coordinate redefinition of the gate. Then the question of what the sign behind this term is and what the sign behind this term is is a physical question. These things. These things can flip physically. Good. Coming from there to here. Yes. We have fixed K and A. Yes. And then we are also fixing the sign of K and R. So is it not like that gamma goes to gamma plus and T goes to gamma? Gamma goes to gamma as well? Mu. The sign fixing... No, no, no. This doesn't change the sign. But in an exponential, if you change a function, it doesn't change the sign. But then putting it to the exponential somehow, how many of these are negative? Now, we've made an assumption by putting it to the exponential with this sign. You see, the most general thing was some function that could have been positive or negative. And they've given up some freedom by saying that this is an exponential because that fixes it to be positive. That freedom... So that freedom, like I said, is true in flat space. I mean, function-wise, is it not true something? Like, we had only four functions. No, no. That's a discrete thing. It doesn't count as a function. It's a... But this is, you know, more... It's an assumption. It's not a coordinate definition kind of freedom. We're assuming the metric is of the form that this thing is positive and these three are negative. And this is an assumption that is either true or false. You see, when we solve equations, at some point, the assumption will turn out to be false. In which case, we'll have to do better. Okay? But we're not using some coordinate definition. Physical assumption. Okay. Okay, great. Other questions, comments? Now... Now we want to set and compute Einstein's equations. Okay. So... Basically, in order to do that, we need the components of the curvature that are non-zero. Okay? Now I'm not going to compute the zero ones for you. Okay? I'll leave that as an exercise. It's too exhausting to do that. Okay. But what I'm going to do is compute all the non-zero ones, all the ones that will give us our equations out. Okay. So, first I'll just list them and I'll ask you to verify that you get nothing else. The ones that give you something, you know, are r11. So, what we call rr. Now, of course, once you know rrt, you can compute rtr because once you lower the curvature, the Ricci tensor is symmetric. So, there's also rtr, but I'm not going to write that because it's not new. Okay?  Okay? These are the only non-zero curvature Ricci components. This is the claim. Okay? Maybe we'll check one that's not of this form at the end. But at the moment, I'm first going to sit and calculate these five components. Do you have an interpretation for these components? I was like, are we a new alpha, beta? No. I know how to interpret this, but... There's probably something to say about this, but I don't know. I don't have... There's no way I think of it like that. There probably is someone who has a good way of thinking of it. I don't know. So, now, let me start by computing, for instance, rr, rr. Okay? Now, let's remember what this thing is. Okay? So, it was, in terms of gammas, it was gamma firstly r, let's... So, firstly, let's write it in terms of rs. Riemann. So, it was gamma. We contract first and third and get second and fourth. Or maybe we contract second and fourth. Okay. So, suppose we contract second and fourth. So, we have a and then you've got... So, this is... We're going to contract second and fourth. Or... That's what rr is. You can choose to contract either first and second or fourth. Okay. So, now, this is equal to r a r b g b a. Now, we want to know what this guy is. So, what was it? This was b, r. We got the sign right this time. Minus gamma r a r, b.  Just a minute. Yeah, r b doesn't matter. Just one minute. Let me... Possibly, let me just remind myself first. a, b, c, d is what? The other terms are it's gamma times gamma. So, this gamma so gamma c a, b gamma d sorry. Hello? Hi. Yes? Yes, yes, yes. Sorry, this one. Hi, definitely, I'm teaching a class now. I'll call you in the afternoon. Okay. Yeah. So, this is how it goes, right? Sorry. So, it's gamma c a gamma c a theta gamma d theta b minus gamma d a theta gamma c a theta b. So, yeah. So, that's the rule. Let me write this down again now. So, it's... This was... Sorry about this, people. This is r a. What was this? Okay. So, we get gamma b times gamma a gamma a it was this thing.  So, that's r c c c a Okay. I think I finally got it right. You agree? So, it's gamma r gamma b minus gamma b gamma r in the sense of matrix multiplication and these are the matrix indices. So, r and a r and a and c c c c. Okay. Fine. So, this is the formula for gamma r sorry for for this thing and then, of course, we've got to take this whole thing and contract it with GBA. Hello. Okay. It's the insurance for my car. Okay. Okay. So, okay. Excellent. So, so, so now we let's compute this guy. Okay. So, firstly, let's look at these two terms.  This one is gamma r and we have an r derivative of it. Okay. And we've got a b so these these two have to be the same. So, what gamma r depend on r? Okay. Well, all of these depend on r. But, these two have to be the same. So, all four of these contribute. Okay. Painful, painful. Okay. Right. So, okay. So, let's write it down. So, from here, what do we get?  That's this term. That's it. We've completely dealt with this term. We'll put in what these are. Okay. What about this term? Now, this term has a gamma r r and the derivative has to be with respect to my neighbors here. So, that's much fewer terms. Uh, what it could be is gamma r r and an r derivative.  And the other term that it could be is uh, this guy. Okay. Uh, so if we had that guy, this would be t, so this would have to be t. Okay. So, we would have plus g t t uh, gamma r um, t r comma t. Okay. So, we've dealt with this guy and this guy. Okay. What remains are these two guys. Okay. So, now let's look at this guy.  Okay. Um, so the only option is either c is r or c is t. Let's look at the case c is r. If c is r, then this guy is gamma r and we have a equals b and that's a lot of possibilities. Okay.  Let's keep gamma r r out common. Okay. Let's, I'll rewrite this.  That was this term where, that was this term when c was r. The other possibility is that c is t. Okay. If c is t, then uh, uh, and we want both equal, then the only thing that it could be is t t or r r. Right? So we get, no, it's just a little bit of a plus.  That's dealt with this term. Now we want this term. Let's look at that term. Um, in this term what do we have? We've got one r at the top and one r, uh, down there. And then, these two are the same as these two. Okay. So now we're going to ask, what are the possibilities for these pairs? The, yeah, because b and a are contracted. Right? Yeah, so b is the same as a. Okay. Sorry about this. We're almost getting there. Maybe we won't. Wow. Okay. I think we will possibly have this trend to compute one of these. Let's, let's see. Okay. So um, um, now b c and b c have to be the same. Um, now that restricts possibilities a lot. Um, let's see. One possibility, of course, is b c, b c and uh, uh, c a, they're both r's. Okay. So we get minus gamma r, r, r, gamma r, r, r, r. And then the up, the g r, r is upstairs. What are the possibilities do we have? Let's see. One possibility, let's exhaust the r t's first. What about r t? That's a possibility, right? So, suppose b is t and c is r. Okay. Um, so we get minus, uh, it be, so g t, t, gamma, r, t, r, gamma, r, b is, we did b is t and c is r, but whatever b is r and t is, c is t. That's also a possibility. Um, and that will give us minus, so, gamma r, r, gamma r, r, gamma t, r, r. What about b c above t's? B and c above t, that's also possible. So minus, minus, gamma r, b and c above t is a t t, gamma r, t, r, t, g t, t. Uh, say more clearly, I don't understand this. C term. This term? Yeah, I understand, but we're looking at cases one by one. We've looked at the possibilities for b and c being between r and t. Okay. Uh, now, uh, now, now what remains? Um, that involves at least one angular coordinate. But, there's never, there's never anything that has one angular coordinate. You have at least two angles. If you have one angular coordinate, you have two. Okay. So, b and c could take various combinations of the angles. Okay. So, uh, wait, wait, what are you saying? That, uh, r, what are you saying? In terms of, no, are you saying we missed some terms? Which are these? Just, just say in terms of c. This is r, what is b going to be? R r. I see that. Which is theta? Now c is, these two are equal. C is theta. Then b is also theta? I see. b and a are r, and c is theta. This is r, r, theta. How is that non-zero? I don't think there's anything left. Yeah. Okay. So, all that remains is that, I'm sorry about this, people. This is a very boring class, I know. Okay. But, let's finish this. Okay. Right. So, all that's left is b, c, and c, are both angular coordinates. Okay. So, what do we get? Let's, gamma, r, let's say, theta, theta, gamma, theta, r, theta, g, theta, theta. Okay. Now, what are the other possibilities? The other possibilities are, gamma, suppose b is theta, c is phi. Is that non-zero? That's not non-zero. So, we can forget about it. Suppose, b is phi, c is theta. That's not non-zero. So, the other possibility is, b and c are both phi's. So, we get phi, phi, gamma, phi, r, phi, g, phi, phi. It will be absolutely a miracle if we get this right. I'm going to try for five minutes. Let's, let's, let's, let's, let's, let's, let's, if we may have to, this one? Okay, good. Firstly, what's this g, t, t? And this had a minus sign, let me see where it comes from. It comes from the, the second term from here. Yeah. Yeah. And therefore, term from the second term had a minus sign. Now the first thing we are going to try to do is to cancel terms if we can. So these two we cancel and out of the terms with derivatives involving gamma R. What about this one? T R, I suppose this was R, R T R, no okay. So now in this line this, this, this and this nothing cancels right? Okay. What remains is all these product terms. So now let's look at the product terms carefully. Firstly, gamma R R, gamma R R, R, G R R. This term cancels with this term. What about G R R, gamma R, T R, gamma T R R? G R R, gamma R, T R, gamma T R R. Beautiful. This cancels this. I suppose this was G T T. So what about G T T, gamma R, T R, gamma T T T? G T T, gamma R, T R, gamma T T T. G T T, gamma R, T R. It doesn't look like there's anything to cancel that. Okay. Anything else that could potentially cancel here? Okay, let's organize terms. Let's write down the terms by organizing them. And then we'll see if they, if they cancel and on. Okay. So first the second derivative terms. Let's put up this board. Okay, so the second derivative terms are G T T, gamma R, T T, gamma R, plus G theta theta, gamma R, theta theta, gamma R, plus G phi phi, gamma R, phi phi, gamma R, minus G T T, gamma R, T R, gamma T. Okay. So those are the second derivative terms. Now we get the product first derivative terms. Anything with a gamma R R R? Anything else with a gamma R R R anywhere? Right. So all terms with a gamma R R R will write.  Great. So that's dealt with this term. Now let's see. Put this term there. Yes. Okay. Okay. So now what remains is this term and the remaining terms here. Okay. So we'll do the ones without angular coordinates first. So all of these have G T T's, right? G T T, G T T, G T T. So plus G T T into gamma R R T, gamma T T T, minus G gamma R, T R, gamma R, R T, minus gamma R T T, gamma T R T. Okay. And then what remains? That's this and this.  So we've got at least a one blackboard expression for this one thing. I should have computed it easier. This is the toughest one because R appears everywhere. Okay. So I should have started with an angular one. It's a bit easier. Okay. So now we have to collect these things. Okay. Is it worth it? On the other hand, what was the point of doing this if we don't do it? Okay. Let's try. Okay. I'm sorry. Ten minutes more. Okay. So now I'm just going to try to write down what these things are. So that people help me with this. Gamma R T T, gamma R. Okay. So that's e to the power nu minus lambda. Okay. Maybe it's not worth it. Okay. It looks like I'll take till the end of the class and how about I ask you to compute this guy. Oh, the problem sets by the way already. We'll make sure to email it to you. The first problem set is 13 problems. That sounds like a lot but they're not difficult. They're not difficult. So they'll be emailed to you today. Okay. How about I ask you to complete this computation to correct and complete this computation in a problem set. Okay. From now on either we've done it wrong. In which case we'll never be able to fix it here. Or we've done it right and okay. Then you'll have an easy time in your problem set. Okay. Okay. I'm going to retreat to that. I'm sorry. Maybe there was some value to this because you see what there is to do. You also see it's tough to do. Okay. Yeah. So sorry. It's just too time consuming. I hadn't realized it would take this long. We just retreat. We'll retreat past this to say somebody has done this calculation. Mathematica can easily do it. I'm going to give you the answers. Okay. It's a mindless algebraic procedure. I'll give you the answers. Okay. Okay. So the answers go this way. Okay. So Einstein's equations when you work them out. Okay. Actually maybe there's an easier one. Just one note. You know what's really irritating is that Landau-Lieschitz has a simple calculation gives. Okay. Maybe they have clever ways of doing it. Okay. But fine. So the Einstein equations work out to the following. Eighth, yk divided by, okay, c we're not keeping.  This is one equation. Okay. The second equation is pi k t 22 is equal to 8 pi k t 22. That's theta, theta. The 55 is equal to minus half e to the power minus lambda into nu double prime plus nu prime squared by 2 plus nu prime minus lambda prime by r minus nu prime lambda prime by 2 plus half e to the power minus nu lambda double dot minus lambda dot squared by 2 minus lambda dot nu dot. Okay. This is two of the equations. Then we also get an equation from t 8 pi k t 00 is equal to minus e to the power minus lambda 1 by r squared minus lambda prime by r plus 1 by r squared. And the fourth, the last equation is that 8 pi k t 10 is equal to minus e to the power minus lambda dot by r. What I could do if you want, what I could do is email all of you a package of Mathematica. Okay. I'll do this. This is actually more useful than actually we get it. And email your package of Mathematica and ask you to use that to compute these equations and you submit the Mathematica, print out along with your. Okay. So this will be part of your, this will be your, this will be in your homework set. Okay. The package, the package I'll email you is written by a guy called Matthew Hedrick who was my first PhD student. That's a package I'm particularly familiar with. Okay. Excellent. So now we've got these four equations and they look like big complicated equations and the first question is what are we going to do with them? Okay. So I'm sorry, we're hitting lunchtime. I know it's 10, 15 minutes. We can finish this. Okay. No, no, we want the equations on the board. So we want a nest. Okay. I'm clearing this part. So Mesh, we promised, I don't think you were there. We promised to email them Mathtense. One of the problems in the assignment would be to get these equations using Mathtense. Okay. Okay. Great. Now, now what do we do? You see, what we want to do is take these equations and solve it in the simplest situation. The simplest situation is when there is no stress tensor, no matter stress tensor around. It's the vacuum. So let's first solve it in that situation. Okay. So then in that situation, the left-hand side of all equations is zero. Now let's look to see if we can find a particularly simple equation. Okay. This equation and this equation are particularly simple because they involve only lambda. What we want to solve for is lambda as a function of r and t and nu as a function of r and t. Right? So we want a computer metric. Okay. Now, it's a good question. The initial data will in general be lambda and nu as a function at some initial point. Now r and t, we'll examine that in more detail. But so we're going to encounter something very interesting. That is that the solution is going to be unique in the absence of matter. So then it will not, you know, it's going to be like the fact that in a spherically symmetric potential. Yeah, yeah. There are no spherically symmetric waves basically. There are no spherically symmetric spin-two waves. Okay. It's something that you're familiar with in even electromagnetism. If you've got a completely spherically symmetric distribution of charge, no matter how it's falling in, no matter what the charge is doing, the outside is basically given by the good answer. If you preserve, if you preserve spherically symmetric, we're going to encounter something like this in this situation. So we will not have to answer your question until we start looking at infalling collapsing matter to form black holes, in which case we will return to your question. So what is wrong with having no matter and choosing to give it function use? Well, the only functions that will obey the spherically symmetric answers and solve the equations will be uniquely determined. Let's solve the equations that we see. Okay. The first equation we look at is this one. This tells us that lambda dot is here. Therefore, we conclude that lambda is equal to lambda of r. Okay. The next equation we look at is this one. This equation now is just an ordinary differential equation in lambda. First order ordinary differential equation and it's quite easily integrated. Okay. So let's write down what the equation is. So the equation is e to the power minus lambda 1 by r squared minus lambda prime by r plus 1 by r squared is equal to 0. What? Thank you. Okay. We will return to this equation in a moment. But even before that, let's also look at this equation. This is also a pretty simple equation and I'll write it below it. It's then minus e to the power minus lambda of prime by r plus 1 by r squared plus 1 by r squared is equal to 0. Okay. So even before integrating any particular equation, let me take these two equations and subtract them.  So what we get is e to the power minus lambda and that we can throw away since it's overall common outside is basically we get lambda prime minus nu prime is equal to 0. Is this clear? Okay. Now, since we get lambda prime minus nu prime equal to 0, we can integrate this equation to get nu is equal to lambda plus a function of t. But you remember we had a residual coordinate invariance that allowed us to shift nu by a function of t, t coordinate redefinition. We'll use that freedom to set this function to 0. So we'll choose our coordinates so that nu is equal to lambda. Our remaining field redefinition ambiguity, coordinate redefinition ambiguity is used up in this manner. Okay. So we conclude very simply that nu is equal to lambda and both of them are functions only of r. Notice we have not assumed that the problem is static. We've not assumed that things aren't moving. So what are we doing? We've got, we're solving this problem in a region where there's no stress tensor. But there could be stress tensor somewhere else. You could have a cloud of dust say that's collapsing. But as long as it's spherically symmetric, where there is no dust? We have concluded that nu is equal to lambda and both of them are functions only of r. So we've got an essentially static solution. This is the statement that there are no spherically symmetric gravitational waves basically. Okay. And of course we presume spherically symmetric. Okay. Now what remains is to solve the equation. So let's take this equation. Okay. So we've got an equation that we want to solve and that's easy to solve because we can, you know, linear first order differential equations can always be solved by integration. Now you might think this is not a linear first order differential equation because it's complicated dependence on lambda, but that's a fake. Let's move to the variable e to the power minus lambda. Okay. So this thing is minus e to the power minus lambda by r squared. And then we had minus and minus, but I'll make it, so minus e to the power minus lambda prime by r. Okay. Plus 1 by r squared is equal to 0. So that's pretty simple. So now such an equation can always be solved by integrating factors and it's actually a very simple solution here. So let's just multiply the whole equation through by r squared. So we get an r here. And now we recognize that this is simply r times e to the power minus lambda prime with a minus is equal to minus 1. So we conclude that r e to the power minus lambda prime is equal to 1. Okay. And therefore r times e to the power minus lambda is equal to r plus c by integration. And therefore e to the power minus lambda is equal 1 plus c by integration. I'll write minus c for reasons that we'll see. So the physical case will be c is positive. So what is our solution? Our solution, even though we had to cheat them, sorry to get it, is ds squared is equal to, now dr squared was multiplied by e to the power lambda.  Okay. Sorry, again minus. Sine squared it. Plus dT squared in 1 plus, 1 minus squared because nu is equal to minus. Solution. I mean there's one parameter ambiguity from this, from this, from this constant c. It is otherwise a unique solution of gender relativity in the absence of matter assuming spherical symmetry. Okay. This plays the role of the unique Newtonian potential of a spherically symmetric mass distribution. Outside the mass the potential falls off like 1 by r. In fact it reduces to that in the Newtonian limit because we wanted to identify this thing as 2 phi. So c by r should be identified as, twice c by r should be identified as m by r. Okay. Or maybe k m by r because it's Newton's question. Okay. So this tells us that it's natural to replace this c by k m by r. I did this wrong. 2 phi which is equal to 2 k m by r should be identified with c by r. Minus 2 phi which is 2 k m by r should be identified with c by r. So c is equal to 2 k. Okay.  In the sequence what went into the thing was that you see suppose you've got a situation like the sun where there's matter inside but nothing outside. What we've done is accurately solve for the metric outside where there's matter. Basically you aren't of very symmetric solutions in our total absence of mass. Yeah. That every spherical symmetric solution other than the vacuum has a mass sitting at the center. Somehow or the other. We turn to this question more as we go on. But one thing about this is that this we've identified as m by, as the mass only with analogy with Newtonian limit and that could be wrong. It could be a non-linear function of the same. Okay. However this analogy actually gave us a right answer.  One of the knowledge we can make is that we are flipping the p r sin is for the real radius. What? I don't know.  Huh? Flip when you. Go past a particular area. Go past a particular area. That's a good point. However in many real objects like the sun or the earth that radius happens well within where there's matter. So this doesn't have to happen. But just let me say that we'll start next time's class by do it over. We'll sometime in next time's class I will explain to you that general relativity does come equipped with a conserved energy and a conserved momentum. There's something we have not really seen yet. Okay. And that the energy and momentum of a space tank can be measured by doing some integral at infinity. It's called the ADM energy and the ADM momentum. Okay. And doing that integral at infinity for the space tank will identify the energy as m. So writing in this way correctly identifies m as the energy of the space tank. The conserved energy of the space tank. We will see that we haven't yet seen this. No gravitational waves are present. What? Prescription. Gravitational waves are present. Some energy? Yes. Yes. Yes. Yes. In general relativity there is no gauge invariant way no coordinate invariant way of localizing energy. However the question of what the energy of a space tank is is completely unambiguous. Okay. All time coordinates that reduce to the same thing so as we will see when we do this we will make the following assumptions. We are dealing with a space time that is completely arbitrary but an asymptote at infinity becomes flat. Now at infinity then we will adopt a canonical T coordinate. Any choice of coordinates which reduces to the same canonical choice at infinity will give you the same answer. So we will use a metric that at infinity reduces to the Minkowski space metric in standard Minkowski coordinates. Okay. That's the only thing we will assume about a coordinate system. And then with that assumption we will get a well-defined energy that will not depend on your choice of coordinates. Yes. The solutions we have found out for unit and lambda. Yes. Those are found out from first and final two equations. Yes. But those also should be consistent with that T theta delta d phi prime. Excellent question. Excellent question. So you say that this does not work out? Well, let's see. Firstly, everything here involves dots. So this is just zero. Now, new prime minus lambda prime is zero. Here, new prime lambda prime remains. And that cancels out with new prime. And that cancels out with new prime square. No, it adds because there's a minus. And new is minus lambda. No, new is lambda. No, new is lambda. New is lambda. Wait, wait, wait. I did something wrong somewhere. Our metric had one on top and one at the bottom. Let's subtract. I just wrote this wrong. New is minus lambda. Our metric had this thing and this thing inverse. Sorry, I just wrote that. Now let's check. What you want is that, so we had lambda. So where did we have our lambda? We had e to the power minus lambda was equal to 1 minus c by r. So lambda was equal to minus log of 1 minus c by r. And new was equal to plus log of 1 minus c by r. So these two terms here are just, for instance, lambda prime or new prime. Let's say new prime squared, those added up. And so we got 1 over 1 minus c by r into the derivative of minus c by r, which is equal to c by r squared. And this thing, the whole thing squared. What about this guy? Well, let's simplify this so that's equal to c squared by r to the 4 in 1 minus c by r, the whole thing squared. Now let's compute new double prime. So the first prime we had, so that was c by r squared over 1 minus c by r. And let's write this as c divided by r minus c, the whole by r. Now I'm going to differentiate. So what do we get? So differentiating gives 1 by r squared r minus c, the whole thing squared with a minus into the derivative of this guy. Which is 2r minus c. Doesn't look like the same.


And this is going to be about how Dutch be with your drawing and another 2,000 channel points well spent.


In this lecture, we are going to discuss on another topic of interest that is pumps and turbines. This is another class of machinery one will come across in major chemical process plants, steel plants, power plants, ships, aircrafts. So, we have to see how we can find out the major faults in such pumps and turbines through vibration signature analysis, through motor current signature analysis, maybe at sometimes through wear debris analysis as well. And many a times of course, the NDT inspections are done on such critical machine components like aircraft turbines and so on. So, I will classify these pumps and turbines based not on the principle of their operation, but rather on the type of fluid they pump or type of fluid they use. As you know pump is a mechanical device where when we given some sort of an energy in the form of an external energy to give certain output energy to the pump in terms of pressurizing the fluid, so that it can gain much potential. So, the pump capacity the power of the pump gamma q H wherein gamma is nothing but rho g, q is the discharge rate and H is the height through which this fluid is to be pumped up.  So, these pumps are usually driven by a motor through a coupling and then we have the pump. So, this pump can again be positive displacement or could be dynamic. So, in dynamic they could be axial, they could be centrifugal and so on. And the most common pump which you will see is the centrifugal pump and you must many of you would have seen this centrifugal pump used for pumping water in the fields in the they produce certain water at a particular flow rate and the pressure. So, that we can irrigate the lands lot of chemical process plants if talk about distillery, refinery, a lot of pumps I am imagine the pumps which we were putting it under the ground to pump out crude oil from the bed of the ocean, under the bed of the ocean, crude oil pumping, pumping of water, domestic supplies, pumping of water, firefighting etcetera, chemicals in refineries. So, they all require pumps and this pumps have to be powered. Another pump is the pumps which are submerged in liquid, the sewage pump, pump inside pipelines, large pipelines wherein we have axial flow pump pumps, pumping of crude oil etcetera. So, these pumps can have defects and then how do you rectify them? Another important application is in nuclear plants. Think of a nuclear plant. Think of a nuclear plant, how do I monitor the health of such machines or such pumps? So, the candidates this is what we are going to focus on the nuclear plants and then submersible pumps. I will pose a challenge to everybody right now. The fact that we have studied about vibration monitoring in details in this course, but can they be applied in nuclear plants and submersible pumps because to install a transducer I cannot go inside a nuclear plant. I cannot go under ground and mount a transducer in the submersible pump. So, these are issues and we will see how this can be done, the pump diagnosis in a nuclear plant and submersible, submersible sorry pumps. Another important machinery which we will study in this course are turbines. Usually turbines again by the type of fluid will have two types. One is the steam turbine which is there in almost all the land based power plants thermal power plants in our country. Of course, we have the hydro power plants where we have hydro turbines like the Kaplan or the Francis turbine or even some done the Pelton wheels. So, we will briefly talk about the hydro turbines as well and then of course, the gas turbines in aircrafts, gas turbines used to power novel vessels and even sometimes small gas turbines of 20 megawatt capacity are used in captive power plants in many industries run by gas turbines to for generating electrical power. So, wherever the turbines are used to generate power what happens the turbines rotate and they are coupled to an electric generator and if you talk about large power plants and steam power plants are 200 megawatt or 400 megawatt super thermal power plants etcetera. They generate they have turbines running on steam and this turbines rotate and the rotor of the electric generator and then AC current is produced. By the way I should tell you when we talk about the frequency of the current which is being generated in our country this is you know this time period corresponds to 1 by 50 seconds because our supply is 50 hertz. So, the speed of the rotation of the rotor of the electric generator is governed by the turbines. So, there are lot of control mechanisms as to so that I do not produce a current supply frequency less than 50 hertz or greater than 50 hertz. So, I have to be produce the electrical supply frequency of supply at 50 hertz and that is my primary requirement of power plant of course, continuously and so on. But these turbines you know when you talk about 200 or 400 megawatts they are very large in size and these rotors are so heavy they are supported on hydrodynamic bearings. There are lot of mechanisms so that we operate them at stable frequencies stable frequencies by frequencies I mean rotating speed. And so if I if I give in more steam it is going to generate at a rotate at high speeds, but then this has to be governed. So, I will not go into the controlling mechanism of such such power plants, but it is very complicated. We have discussed about the individual machine components like rotating shafts on bearings, shaft scaring gears, shaft scaring pulley, shaft having misalignment, but when I talk about a turbine or a pump maybe how do we monitor the conditions of a turbine. So, this is not a single component. So, if you think of a turbine let us talk about a turbine first and then I may be I will come to the pump later. So, in a turbine what are the important characteristics large rotor supported on general bearings. And usually all the phenomena of course, in the turbine there could be blade vibrations, there could be blade problems, there could be casing problem, casing defects because of high heat there could be no creep etcetera. There could be seal problem because we would not like the seals to leak and so that the steams pass out through the shafts. So, in fact, the seals are mounted something like this as a rotating component which is tightly and the seals are actually given here, they are spring loaded and then they can have another component here which will be having the shaft could be rotating it can be carrying compressor sets and so on. And of course, as I was telling you right in the beginning if I go back here this figure here. So, turbine the first place would be of course, to put vibration signal signals at all locations by all locations what do I mean of course, all the bearing locations all the foundation locations these are all the locations where you can mount trans resource and so on. But the problem in nuclear plant is that this is not accessible the problem in submersible pump is that pump is not accessible. So, that something we will discuss later on and the normal method of vibration signal monitoring is well understood by now. So, whenever we have a machine we just install the trans resource and find out the characteristics. First of all in the time domain itself based on the ISO standards we see whether the levels are within the allowable limits as per the standards as per the power of the machine that is number 1. And if you want to diagnose faults in the turbine what we do through vibration monitoring is install the sensors at all the bearing locations or the foundation locations and then do a signature analysis to find out unbalance misalignment gear defects blade missing blade etcetera. The techniques of FFT is substance analysis and OLAB detection can be used that is good enough. But I was telling you for the case when these turbine itself is not accessible. So, for example, a nuclear plant. So, how is the health monitoring in turbines in nuclear plant done? First of all because of high radioactivity we are not allowed to go inside the plant. So, they are permanently installed, installed sensors. By sensors I mean usually in such turbines at the journal bearing end usual practice is to put some sort of an on contacting type eddy current probe at this is actually at 45 degrees x y. And you can do lot of orbit analysis to defect and we are discussed about orbits in one of our earlier classes to find out the defects in the. But this sensors can be eddy current can also be vibration transducers like accelerometers. The question is suppose a nuclear plant is made and you are not supposed to go inside it and it has life of about 20 years. So, what do we do? They put lot of extra transducers extra or redundant transducers are installed suppose and then all you have is because if you look go to a nuclear plant they are actually in a containment building and then all the machines are there. So, all you can have is these sensors signals coming out from such a plant and then you can do the analysis. But another important technique of finding out faults in turbines is through what is known as motor current signature analysis. And this is very helpful for fault detection in submersible pump and then in even in gas turbines. I will give you an example I mean I believe we have talked about motor current signature analysis in submersible pumps. For the fact that in a submersible pump which is under the ground which is rotated by a motor and then all we require is a hall effect current sensor and then we can have an A to D process and then we can have a signature analysis and this could be under the ground. So, we did not worry about the same is true for turbines. In fact, if you think of the marine propulsion system they have what is known as the gas turbine generator. So, the turbine which is the gas turbine driving a generator and this generator is actually used to power devices in the ship and this is the generator and this is the. So, I will I will show you few pictures of we did a condition monitoring on a gas turbine. This is the gas turbine and this is coupled to a generator. I am sorry this is the gas turbine and here you can see the heavy exhaust here and this is the generator and if you will see here we have put a axillary meter on the bearing housing of the generator and this is required. So, that we can and this was a case study which we are doing for the from one of our sponsored projects by the DRDO for the Indian Navy wherein we are trying to see the feasibility of using motor current signature analysis for monitoring the health of a gas turbine. This small gas turbine in a test establishment this is a replica of the actual gas turbine. This is one of the actual gas turbines which is used in a marine propulsion unit and we have put an axillary meter here a charge type axillary meter here to measure the vibration. So, that we would like to see whether the defect characteristics of the generator vibration can be picked up by such a transducer and then I will come to the associated problem. These are another view of this wherein we are monitoring the axillary meter by a uniaxial acceleration by an uniaxial vibration monitoring. This is another view of it is pretty noisy. So, that we have put in a earmuffs here. It is a very noisy test environment and you will see these are the shoots for the exhaust of the gas turbine because this is a stationary unit otherwise you know this should have been normally used to power the ship, but then we are only coupled it to drive the generator because for own generation of power in the ships they have what is known as the gas turbine generators. So, the small gas turbines have power to weight ratio they are very convenient in ships and this was being used to drive a generator.  So, on to this we put a Hall effect sensor and this is the tectonics Hall effect sensors put on one of the phases of the load output load and then we can correlate what is the defect in the generator and then in the turbine because the principle idea behind MCSA is the current gets modulated by a load torque. In the case of a motor driving a mechanical unit if this mechanical unit has a defect it is going to load the motor and then the motor current drawn will be loaded up, but same as to here if the like we had done in the case of in the laboratory I had shown you earlier that we had a motor driving a gearbox which was in couple driving a generator and then there was a load. So, we had established that is there is a defect in the gearbox motor currents get modulated then we have measured it this way in Hall effect sensor we have demonstrated this earlier. Now, the other way also happens if there is a load defect in the gearbox the generator voltage which is generated by the generator will have ripples this is the time generator voltage and these ripples are at a frequency at the defect frequency of the mechanical unit and that is the turbine in our case. In fact, this is a powerful technique that any rotating machines I just showed you for the case of a turbine. So, any rotating mechanical device if it is coupled with an generator the ripple frequencies characteristics of the defect in the mechanical device. So, this is one of the important observations and in fact, our group at IIT Kharagpur has a patent on the same for the fact that any mechanical rotating device which is powering a generator, generator is going to generate a voltage and this voltage will not be clean voltage will have ripples. So, if somebody can understand and measure capture this ripples and understand the frequencies we will see that they will are characteristics of the defect in the mechanical device. So, this becomes a very very powerful technique to find out defects in gas turbines powering aircrafts powering marine systems. So, these are very very powerful technique. This is one of the examples of the view of one of the gas turbines which is used for the marine propulsion system. This the compressor the low pressure compressor on the front of the gas turbine this red cover is just a cover because this has just come to the bay. This is in one of the Naval establishment units where we are doing measurements for our project and this has a low pressure compressor high pressure compressor and the combustion and then the series of turbines and each one of them has sets of blades. And so, how do you do the health monitoring of such gas turbines. So, the best place would be to keep vibration transduces along this rings. You will see lot of this rings and braces are there because you obviously cannot access them. The challenges instrumentation challenges for monitoring gas turbines are such because of high temperature. And those of you we had studied that we mount accelerometers using magnets, but this magnets are very very soft and they lose their properties. In fact, they crack. In fact, I myself have lost lot of magnets mounting accelerometers at locations which were at high temperature. I realize later once I had they cracked magnets. So, they have to be withstand the high temperatures and nowadays we have transduces which can go up to 600 degree Celsius accelerometers, piezoelectric accelerometers and they can be mounted permanently also, but again the problem here is you know these temperatures could be even higher than 600 degree Celsius. So, certain that is one issue the temperature and then the the itself is how do you lay the cables and the location. Location becomes a problem because I cannot be close to the bearings because if you if you look at the this this shafts this bearings are somewhere close. So, bearings are not accessible, bearings are easily not accessible from bearings are not accessible from outside. So, this is again a challenge and as you know in all these machines the vibration reflects pretty well at the bearing locations and if I cannot access the bearings how do I measure them. So, to some of the and that is why MCSA has become a good candidate. Next is we can use lot of the NDT techniques of course, NDT techniques cannot be real time, but it is a good practice particularly for gas turbines or turbines we can do on quick ultrasonic inspection, the internals of the turbines can be looked through visual inspection, boroscoping and of course, another technique people nowadays are used are on laser based measurements and the optical rotary encoders for what is known as detecting the instantaneous angular speed. These techniques are over the conventional vibration analysis. We will study about MCSA and NDT in the later classes, but these are techniques wherein this can be used to measure and understand more about such gas turbines, but there are few other associated problem like in such turbines be it the particularly in hydraulic turbines there is a limit to the maximum flow rate. Because if the flow rate increases and the velocities become so high because to conserve energy the pressures of the liquid if they fall below the vapor pressure. So, what is going to happen is this vapors will be formed or bubbles will be formed in the pipeline inside the suction pipe and when this bubbles come up with time come up to the surface to air pockets they will burst. So, they are like hammers high frequency hammer and this is what is known as the cavitation. So, cavitations limits the operational pressure or flow rate of turbines and there has to be and this is the cavitation goes undetected it is as if the inner walls will get pitting will be there and this bubbles because when they burst they will release energy and as if they are hitting and if this goes unnoticed they will weaken the structure and then lot of the surface will become weak may eventually break. So, to monitor cavitations because they are high frequency we can always have the high frequency it could be a vibration monitoring. Now, it is under water piezoelectric accelerometers are available for the most important and this could be put on the pipeline and the most important is the cable the triboelectric low noise triboelectric cable and they have to be watertight with good silicon sealant. Sometimes you know this cables are more costlier than the piezoelectric accelerometers. So, we can monitor the vibration of because of cavitation on the pipeline because if this goes unnoticed they will damage they will pit the internals of the pipeline or the casing and then they will become weak and they will maybe start to leak and then damage. Another important aspect which will go is the leakage. So, leakage has to be provided with seals good seals I think I need more papers. So, this many things happen in chemical plants in the pump impeller if I have just draw the blades there could be particles which will get stuck sometimes not uniformly. So, these are particles which will get deposited because of this deposited particles they will be unbalanced and this unbalance will again get detected at the bearing locations. So, if I have a shaft and then either there can be unbalance and there can be because of pitting. So, all these will give rise to again the same problem which you see in the frequency in the viruses in a strong 1 x component. So, this can be one will lead to the other and many a times again in process plants they go for visual inspection and there are lot of particularly in process plant lot of sight windows glass the high pressure quartz glass sight windows are provided in such equipment to visually see from the outside what is the condition cause. And associated with this impellers getting defected bearing there will be load on the bearing and then missile elements eventually again they will rub on the casing. This rub I told you in the case of the pumps this also severe in the case of turbines particularly gas turbines and then this will again give rise to lot of problem and we discussed in few of the classes earlier how this rubs can be detected by what is known as the non stationary signal analysis like wavelets or FFT. And particularly associated not with hydraulic turbines, but with steam turbines and gas turbines a serious problem is the bearing temperature. And there are lot of control measures circuits available. So, that the gas turbine the bearing temperatures are below the rated temperature of usually 75 degree Celsius. And many of you those who would have gone to power plant there are certain very important parameters which are monitored. And one of them obviously is the generator output speed because that output speed would correspond to the frequency of electrical power generation. And that has to be strictly at the synchronous speed of 50 hertz. And you know the effect if the speed falls below that there are lot of circuits you know lot of protection circuits in equipment which will shut them off if the supply frequency is less than 50 hertz because of the fact that many devices are synchronously many timing circuits are dependent on that supply frequency. And so your machine would go haywire if these frequencies were less than the toileted levels. And you would have heard of the grid failure because suddenly the load the machines shut down automatically the loads would again decrease. And again that would lead to again high speed increase. So, there will be lot of frequency fluctuation. So, that is another the hunting of the machines and that has to be avoided. And because of this speed fluctuation because of the change in frequencies there will be lot of torsional vibrations associated torsional vibrations due to change in supply of reconstruction. Because the speed is changing speed is changing. And I have seen many times particularly crankshafts of large machines structures failing because of such speed fluctuations. So, many times in many plants input power frequency less in India has to be maintained constant and 50 hertz because of the load poor power generation sometime this fall. But then again if the increase decrease. So, this will give rise to speed fluctuations. And the speed fluctuations will introduce fatigue torque torsional loads and lead to premature shaft failure. So, in a power plant parameters monitored power plant from a machines health point of view is of course, the power generation frequency. Then the turbine rotational speed bearing vibration bearing temperature of course, the steam consumption rate and of course, the voltage etcetera is there. But these are the parameters which have to be monitored. So, that we will know whether the turbine is stable or not and so on. So, in conclusion pumps and turbines are important important machineries which are there in all power plants, in all nuclear power plants. Gas turbines per se are used for marine propulsion for captive small power plants for aircraft propulsion. So, they have their online condition mounting techniques by vibration, motor current signature analysis is another new candidate which can be used for analyzing and monitoring such systems. Of course, bearing temperature can be monitored as well and particularly for gas turbines which are used in aircrafts. Lot of NDT inspections are done in the maintenance workshop lot of onsite internal inspection through techniques of visual inspection using boroscopes etcetera are done. Some of the NDT techniques we will discuss in the next few classes down the road and all including MCSA. But this was just to introduce you there are techniques of other than vibration monitoring like motor current signature analysis and NDT techniques which can be used to find out defects in machinery. Thank you.


Karabakh separatists try to derail Washington negotiation process. The so-called Parliament of the separatist regime in Karabakh has appealed to the Armenian Foreign Ministry's delegation in Washington to immediately halt the talks with Azerbaijan until a full ceasefire is established under line of contact. The authors of the appeal demand from the Armenian delegation to secure documented guarantees of a ceasefire. Otherwise, according to the separatists, continuation of negotiations will mean encouragement of aggressive behavior of the Azerbaijani side and privilege at the international level. The separatists also appealed to Russia to stop the anti-human genocidal actions of Azerbaijan with the most stringent measures. The Parliament of the Illegal Regime appeals to the UN Security Council and the heads of the OSCE Minsk Group Co-Chair Countries in addition to expressions of sympathy, exhortations and advice to take concrete practical steps, in particular to apply sanctions against Azerbaijan, curbing its aggressive ambitions. Before this, the separatists reported the killing of four Armenian soldiers by the Azerbaijani army in Agdeir and Koyavand. Earlier in the evening of June 28, Armenians attacked Azerbaijani positions in the direction of Galaychila village of Agdam region as a result of which a soldier Bakirov Ali Shukur Oglu was wounded. Add that talks between the foreign ministers of Azerbaijan and Armenia have been launched in Washington.


Are you in a productivity slump? This is something that I've experienced so many times. If you are, honestly my best advice from the heart is to just take a break. We're often in a productivity slump because we've overworked ourselves, we've gotten to the point where we feel like we've got nothing left to give. And so taking a short break, doing something you enjoy, something you love, going for a run, doing exercise, seeing your friends, whatever it is, and then coming back to your work can help you feel so much more motivated and ready to go. Don't underestimate the power of breaks. Give yourself breaks when you need to. They really help.