John drives a 2013 Honda Accord with a lot of character. It features Honda's K24W2 engine, which produces 185 horsepower at peak operating conditions, which is about 6,400 rpm. Full of stuff and with John at the wheel, the car weighs 3,600 pounds. If it were able to output peak horsepower all the time and everything else were ideal, that is, there are no mechanical losses, no air resistance, perfect combustion all the time, whatever. How long would it take for John's car to accelerate from rest to 60 miles per hour? Well, our system is going to be John and his car, and we are considering an energy conversion process. That process involves an increase in kinetic energy. I'm going to define the process that we are analyzing as being from state 1 to state 2. Those are different points in time for our analysis here. Again, state 1 and state 2, different points in time. I'm saying that the mass of the car is going to be the same, mass 2 is equal to mass 1, and I'm going to include the given information, v1, the velocity at state 1 is 0 miles per hour, and velocity 2 is 60 miles per hour. Furthermore, I was told that the car weighs 3,600 pounds, that would be pounds of force, and I was told that the engine produces 185 horsepower. So I'm going to call that an E.in term, E.in, the rate of energy entering the car. For the purposes of this analysis, we're saying that the energy is entering the car as power, even though in reality it's really entering as chemical energy, but again, depending on how you define your scope, you could consider lots of different starting points. Eventually, it all goes back to the sun, and I guess before that it goes back to the big bang, but we can't start every analysis to the big bang, or we wouldn't get anything done. So E.in is 185 horsepower. Now I'm going to write out my energy balance, and that energy balance has to be performed on something. I am defining my something, my system, as the car, and as a result of having a constant mass, that system is going to be a closed system. The other thing that we have to consider here is whether or not to treat the analysis as a steady process or a transient process. Is time important is really what that comes down to. In a steady process, all effects of time are neglected. The situation is the same regardless of when you look at it. In a transient process, time is important. Is this a steady process or transient process? It is transient, that's right. We know time is important because the problem is asking us for a duration, trying to figure out how long it would take to accelerate from rest to 60 miles per hour. A steady state analysis would be something like if the engine were sitting on a dyno and had been running long enough that it was as hot as it was going to get, that would be a steady state analysis of the engine. Does that distinction make sense? It has completely warmed up. Or it might be steady state two if we were considering a situation like how hot does the engine get when it's running? If we were considering a situation where it was just constantly running at the same rate, regardless of when you looked at it, and we were considering the temperature increasing over time. As the temperature is increasing, the amount of heat rejected by the car is increasing as well. And eventually, it reaches a point where the temperature levels out. So before that point, we would have transient analysis. After that point, it would be steady analysis because time doesn't matter anymore. Again, I think that's one of those things that makes more sense the more practice you get. So this is a closed system undergoing a transient process. Good habit to get into. Just think through that every time. Now that we have a system defined, we can perform our energy balance. By the way, I abbreviate energy balances EB in my analyses. I will write this out for your sake. The first couple of times energy balance. We are performing an energy balance on the car. And as all energy balances is going to begin with delta E is equal to E in minus E out. The energy change of the vehicle is equal to the energy entering the vehicle minus the energy exiting the vehicle. For our purposes right now, I'm saying that the energy could enter as heat transfer or work. But it isn't important that we distinguish that. Because we know the energy is entering at a rate of 185 horsepower. We don't really care if it's heat or work. I mean, we know it's work because it's mechanical power. But we don't have to break this apart. All we have to do is recognize that we have a rate of energy in. We can consider any opportunities for energy exiting, like maybe the car is hotter than its surroundings and is emitting some energy to its surroundings, things like that. But for our purposes here, we are considering ideal operation. So we're saying all of the power of the engine is going into the kinetic energy of the car. So the proper thing to do here would be to assume that no energy was exiting our system. In fact, I will go as far as to start a list of formal assumptions. Generally speaking, you don't have to establish assumptions that are already listed. So I don't have to say there are no mechanical losses as an assumption because it was already given to me in the problem. I don't have to assume that there's no air drag because that's already given to me in the problem, etc. So my first assumption, I guess, is that I have a closed system. The mass of the car doesn't change throughout this process. Two, I'm saying energy out is zero. Then this term disappears and I'm left with just energy in. Now, is the energy in 185 horsepower? No, it's not. Because 185 horsepower was E dot in. And E dot in is the rate of change of the energy entering with respect to time. That means it is energy entering over duration divided by that duration. If we have three kilojoules entering the system per second over 30 seconds, then the total amount of energy entering is going to be three kilojoules per second multiplied by 30 seconds. We are considering the entire process when we run it out as E in over delta T as opposed to an instantaneous point. Does that make sense? Whatever the case, we can say E in is E dot in multiplied by duration. If I can make that substitution into my energy balance and write delta E, the static energy change of the system as being equal to the rate of energy entering the system multiplied by a duration. Okay, the right hand side of the equation has been considered. The left hand side we know is important because we have a transient process and I will go as far as to say this is the energy of our system at state two minus the energy of our system at state one. The energy of our system at state two would be the internal energy of our system at state two plus the total kinetic energy of our system at state two plus the potential energy of our system at state two minus the total energy of our system at state one which would be the total energy internal energy of our system at state one plus the total kinetic energy of our system at state one and I guess that minus is distributed so it really should be minus the total kinetic energy of our system at state one minus the total potential energy of our system at state one. This, I'm more commonly right in terms of u2 minus u1 plus k2 minus k1 plus p2 minus p1. I grouped them together, in which case I would be left with delta u of our system plus delta ke of our system plus delta pe of our system. That just means it's a little bit easier to write. But that doesn't change the meaning. Anyway, it's still equal to e dot in times the duration on the right. So we are simplifying our energy balance and we are making progress. We are left with five terms for now. Are there any other terms that you think it's reasonable to get rid of? Well, we can make a case for there being no internal energy change of our car. Presumably the car hasn't changed phase, presumably the charge, presumably the car hasn't undergone any sort of nuclear fission or nuclear fusion. I mean, we could make an argument that the temperature changed, but for the purposes of this analysis, we have ideal conversion from power into kinetic energy. So we can get away with assuming the change in internal energy of our system is zero. And while we're at it, I'm going to assume that the change in potential energy of our car is also zero. That is reasonable, because I am accelerating horizontally. How do I know that I'm accelerating horizontally because I'm in South Dakota and there's no hills? Whatever the case though, I'm assuming perfect ideal conversion of energy from power into kinetic energy anyway, so this is all part of that ideal umbrella. Anyway, change in kinetic energy of the car is equal to E dot N times delta T. I'm looking for duration, so at this point, I'm going to solve for duration and I'm going to write the change in kinetic energy of the car divided by the rate of energy entering the car. And delta KE is just KE2 minus KE1. Remember that one and two are referring to state points here. That's still divided by E dot N. And then total kinetic energy of the car can be written as mass times velocity squared times one half. So I could say one half times mass two times V2 squared minus one half times mass one times V1 squared divided by E dot N. And then I can recognize that because we are treating this as a closed system, m dot two, excuse me, m2 is equal to m1. And I could factor out one half in mass and write this as one half times mass divided by E dot N times the quantity V2 squared minus V1 squared. So so far, we have written out a full energy balance. We have neglected terms until we ended up with just the relevant energy changes in our system. And then we are rewriting that to solve for duration. While we're here, let me point out that a very common mistake among thermal one students is to simplify this change in velocity. That's V2 squared minus V1 squared as V2 minus V1 quantity squared. But that's not true. V2 squared minus V1 squared is not the same as V2 minus V1 quantity squared. I feel like there's a joke in there about you foiling your own analysis, but I'll leave that aside. We have E dot N, we have V2, we have V1. In fact, V1 was zero, was it not? It started from rest and is accelerated to 60 miles an hour. So I can get rid of V1 altogether. And we don't know the mass, but we know the weight of the car. And we could describe the mass in terms of the weight by recognizing that weight is going to equal the mass times gravity. Therefore, the mass of the car is going to be the weight of the car divided by gravity. And the cool thing about pounds of mass and pounds of force, by the way, is that the amount of mass in one pound of force under standard gravitational acceleration is one pound of mass. That's how we define it. So if we were to say 3200 pounds of force, 3,600 pounds of force, excuse me, and if we were to assume standard gravitational acceleration, 9.81, excuse me, be more logical for me to work this in a previous, 32.2 feet per second squared. And let me list that as a formal assumption. G is equal to 32.2 feet per second squared. And if we were then to go to our conversion factor sheet and recognize that one pound of force is 32.174 pounds of mass times feet per second squared, that actually should be what we're using for gravity, but rounded and infrapenny and propound that I'm saying one pound of force is equal to 32.174 pounds of mass feet per second squared. Then pounds of force cancels pound of force, feet cancels feet, second squared cancels second squared. And I'm left with theoretically 3,600 pounds of mass. Now it's slightly different as a result of my assumption that gravity is 32.2 as opposed to 32.174 which is what our conversion factor sheet is using. So since I'm doing that already, I might as well determine the mass of the car. Instead of 3,600 pounds of mass, it is 3,597.09 pounds of mass. That's the result of assuming gravity is 32.2. Anyway, even though I don't know mass, I have mass and that means I have everything I need to finish the question. So I will write this as horizontal line and then one half multiplied by the mass of the car, which we know is 3,597.09 pounds of mass. And then we are dividing by 185 horsepower. We are multiplying by V2 squared, which is 60 miles per hour. And then it's a squared term, so I square everything 60 squared miles an hour squared. And we are looking for a duration presumably in seconds. I mean, my car isn't exactly a sports car, but surely it doesn't take a number of minutes to accelerate from zero to 60. So let's shoot for seconds as our unit that we're targeting for an answer. In order to get to seconds, I'm going to take my secondary units here and break them apart into primary units, or rather my units of secondary dimension and break them apart into units of primary dimension. So I'm going to say that a mile per hour is a mile per hour. And I square everything. And I recognize that a mile is 5,280 feet. And again, I square everything. Then mile squared cancels an hour squared, excuse me, mile squared cancels a mile squared. And then I will break apart the horsepower. A horsepower as per the conversion factor sheet can be written as 2,545 BTUs per hour or 550 feet pounds of force per second or 0.7457 kilowatts. Most convenient one for me is 550 feet times pounds of force per second. So one horsepower is 550 feet times pounds of force per second. Then horsepower cancels horsepower, feet cancels one of the feet in the numerator. The other feet would come from our pound of force. So I'll break them apart as well. A pound of force, you'll remember, is defined as 32.174 pounds of mass per feet second squared. And again, that came from our conversion factor sheet specifically under force here. One pound of force is 32.174 pound of mass times feet per second squared. Okay, now feet and guys, it's pound of mass times feet per second squared. Now pound of mass divided by feet times second squared. So that should be in the denominator. What am I doing? Okay, let's try that again. One last time for all the means. Feet squared cancels feet and feet. Aha, we did it. Then pounds of mass is going to cancel pound of mass and pound of force cancels pound of force. I need to get rid of my hours and seconds. So I will recognize that one hour is 60 minutes. And 60 minutes, excuse me, one minute is 60 seconds. So far minute cancels minute and hour will cancel one of the hours to get rid of both of them. Then I need to square everything. One squared is boring. One squared is boring. Hour squared cancels hour squared. Minute squared cancels minute squared. Second squared cancels second square leaving me with an answer in seconds.  So calculator, if you would join us, please. We have one times 3597.09 times 60 squared 60 carat two times 5,280 squared two times one squared, which is one times one squared, which is one divided by two times 185 times 550 times 32.171, which gets rid of the error introduced by dividing by 32.2 and then multiplying by 32.174 in the previous step of our analysis. Up here. Up here. Isn't that neat? And then we are multiplying by, oh calculator, what are you doing? That's not at all what I said. 550 times 32.174. No. 32.174. Right? 174, not 171. 174, not 171. Told you, calculator. Okay, I'll go fix that. And then we multiply by 60 carat two times 60 carat two. Let me just double check that I typed all of those correctly.  Neat. We get 4.255 seconds. Not too shabby car. Now that we have an answer, let's have a conversation about it. Do you think that my car actually goes from zero to 60 in 4.255 seconds? No. Definitely not. This was an ideal analysis. We were ignoring mechanical losses. We were ignoring air resistance. We were ignoring every opportunity for energy to go anywhere else other than the place that we want to do. Does that mean this number is not useful? It doesn't. This number is still useful. It is useful because it represents a best case scenario. We can look at how long it actually takes my car to get from zero to 60 and compare that to 4.255, for example. Like if you were the design engineer working on this Honda Accord and you tested it and you determined that it took five seconds to go from zero to 60 and you know that the theoretical maximum is 4.255, that gives you an indication of the headroom left to improve performance. You could recognize how much of an effect those losses are having and see where it might make sense to try to optimize. You could reduce the mechanical losses associated with the gear train, the drive train. You could figure out how much losses were associated with air resistance, et cetera. But you know that all of those losses are making up the difference between 5 and 4.255. It gives you an idea of how much headroom there is for performance. You can think of it like if you took an advanced engineering math test and you got 45 points, that isn't necessarily a useful parameter of performance in and of itself. In order to evaluate how well you performed, you have to know how many points were available. If you got 45 out of 50 points, then that's a pretty good score. If you got 45 out of 100 points, that is not so good of a score, although maybe average.


So this is one of these, um, chisels. It's called a flanger, um, dongle. Do I have any broccoli in my teeth? No. You can't beat a flanger, donger. Your teeth are pretty large. Oh, David, this is a fine vintage. Then come on over and have a coffee. That's so irritatingly cute, or what are you, charismaticly, challenging?  All right, we're back for a clock, for the clock, uh, part. There we are. 3, 4, 3, 4. What are we going to do today? That's what I want to know. Yeah, good question.  And then resin, hopefully that's the final resin coat. Yep. And that's in preparation for putting in the clock mechanism in the back that we're going to do next time after it cures. Yeah, I'm really excited to see how these turn out. Our, uh, 12 o'clock, 3 o'clock, 6 o'clock, and 9 o'clock squares. Yeah. Okay. So, we got our belt sander, we're plugged in. Where's the button? Trigger. Yeah, okay. Beautiful. What do you think, Dave? Looks good, eh? I think it's great. All right, so what have you done here, Joanne? We've taped off the bottom, because we don't want any, uh, pesky drips. So that's going to help with that. Yeah, it'll make our life so much easier. Yes. And we belt sanded down the face. We also pre-leveled everything. That'll do it. We did the math too. Dave did the math. Yes. Face it, Dave, I can't do math. Face it, face it. We, uh, one square foot is four ounces of art resin. So that's two of resin and two of hardener. This is more than one square foot, but we're just going to do a square foot, because... We can. We can. Give her. We're going to stir for three minutes. Oh, that's so nice and liquidy. I know. Be sure to scrape the sides and bottoms as you go. I'm going to stop doing clock puns. I feel like people are getting annoyed. Really? Yeah, look at them. For you? I'm sorry. All right, I think that's good. Okay. You say so. Yeah. So we're going to attempt to just push the resin right to the edge, instead of just letting it run over the side. It might run over the side, but we'll just do our best. All righty. No time like now. That wasn't, I didn't mean to do that. They were just saying. Okay, so we're ready to torch now, but I just wanted to point out, we are getting a few drips. We are totally not even worried about this edge here, because Dave's going to saw it off, right? You're going to... Yep, just run it through the table saw when everything's done. And then we're just kind of rubbing drips in here. If there are any, all the curves and everything in the wood will hide any imperfections in our resin job on the sides. We're not worried about that. Yeah, and they're probably just going to run right down anyway, right? Yep. And level out. We're not going to have big pools of them or anything, so... That's right. Okay, girl. Okay, here we go. This is always the best part. Look at that grain. Oh my lord. Awesome, girl. Is that enough? Yeah, let's cover it and get out of here. Okay. Okay. Here we go. Here we go. Okay. See you next time, little clock. So what do we have left? All we have to do is... We're so close now, can you believe it? Cut the bottom, drill out the middle part and put in the hardware and then just enjoy it. And then tell the time. Thanks for joining us on this adventure.


Yeah, my name is C.K. Chris is... You know, there's too many quick Chris's in the world. I'm too original for that. Right? And that's why I named my son, my son's 12 now, but we named him Christian. And I'm like, dude, just make sure no one calls you Chris. And so for the last 12 years, so far, no one's called him Chris. Because I don't really like Chris, right? But his name's Christian. And so, yeah, dude, sometimes like short names or nicknames or names that you don't even like.  Right? Yeah, totally. You know, I compose reggae music and sing reggae music. No way, dude.  Yeah, I sent you a song, actually. It's a link there. But I'm like, so my name on stage is C.K., you know what I mean? It's been like that for years. And then I've just been writing with it and now I use it in business. Because it's so easy to understand. You're not going to hear another C.K. You're going to hear a thousand Christians. But if you hear C.K., you're going to know exactly what you're talking about. Heck yeah, dude. That's really, really cool. Really cool, man. I love music, man. I love music. I love reggae music. Like I grew up on all that shit, man. Are you from Hawaii? No, bro. I'm from, I mean, well, I'm not from. I was born here in San Diego my whole life, but my mom is from Guam. That's why I am Islander. Yeah, for sure. My mom's from Guam and my dad's from Mexico. Okay. So I'm just, I'm like literally 50, 50, but I got that Islander blood more than anything in me, you know? Yeah, yeah, yeah. Yeah. Yeah, I'm from Hawaii and I saw Mahalo. I was like, oh, stop.  And I'm like, dude, let me just turn this into something. And so I ended up on the video. I kept it recording and I ended up just showing people how I share a sequence. So that whole walkthrough of me getting the sequence out of my, out of my account and then creating the share code and then emailing it to you. I was recording all that. And then I was even telling the people, I'm like, I know he's from Hawaii. So let me, let me put a key word that he's going to really appreciate. Yeah, dude, you're going to see that video come. I think it released like in a few days, but, but yeah, I'm actually recording that part when I'm typing it in.  Because I, I, I know that word because not only, you know, that word, but also aloha because when I was a loan originator, cause I did mortgages like you too. And I, and I did it for, for six years. I was an aloh licensed for the state of California and licensed for Washington state of Washington.  And we would do a lot of business, bro. A lot of business in Hawaii.  But he, but we, we, we had the whole mindset of crushing the phones and he's a lead guy and an online marketer. So we would have like literally bro, like, like I would literally dial anywhere from 700 to 800 numbers a day. And so when we get into the office, let, no, no, actually not in the morning, but, but the grinders, the ones that get into the office early, right? Like I would always get in the office early. I'd crush everybody in that damn office and we killed it, but I would also stay late. And so staying late gave us the ability to call Hawaii because they were three hours behind. Yeah, exactly. So I'd be in the office at 9pm and it's only six o'clock over there, you know, remember lower my bills.com. Oh yeah, bro. For sure. Basically turning into lower my bills with this. You know what I'm saying? Dude, it's freaking awesome, bro. Awesome. Yeah. Awesome. I love it. So this is the last segment. I'm really excited to launch tomorrow. Like I said, I'm only all I'm doing is scheduling callbacks for no cost, no obligation estimates. It's an air purifier, $1,300 value that they receive. They receive a coupon, their first email opt in. And then with any new AC or furnace installation, they get this air purifier. And then I go on about financing because it makes sense. You know what I'm saying? So anyway, that's, that's what I'm doing here. And so on the sequence, the email sequence, really what I'm looking for is either they schedule an appointment and then the sequence deceased or stops after the schedule callback or they don't. And then they receive emails going through. Okay. So I don't have any this and that pretty much once they schedule a callback. Then. Okay. Perfect. So then let's look at everything. Let's look at the. The pages. Let's look at the communications for sure. And then I'll switch the names just yet. Okay. I'll switch it when, when we're done here. Yes. Because if it doesn't make sense to do it, then I'm, I have another one that I pre-did, but this will give me an idea. Yes, exactly. That's what I wanted you to do is, is look at this real quick, just as a, as a template. Okay. So the way I have it set up in here is this is based on, this is based on a tag. And this is, this tag is based on if the person watched the video. So it could be the same thing on scheduling the call. It could be the same thing. Same, same exact concept. Can we switch that tag right now? Yeah, for sure, dude. We're going to, we're going to do everything right now. Okay. So. This right here is, is, is what it's all about. So for example, the starting rules. Would be. The tag. Do they have this tag? So let's, let's go to. Let's go to, yeah, let's edit. Now it could be based on a tag or it could be based on an opt-in form. Now. So that's what's the opt-in form to mine. You can, but that's the starting rule. So is it, is it based on whoever fills out this form? Or is it based on whoever schedules the call? It's a, oh yeah, you're right. So it's actually whoever scheduled the call. Yeah. So let's go with tag. Let's go with tag. And actually, you know what? Go back to opt-in form and X that went out. Let's, let's get rid of the lead. Yeah. Yeah. Let's get rid of that one. And then put in tag. And then category hit the dropdown. And then I don't know if you put it, did you put it in, in one of these categories? I think it was. The only opt-in form that I had was email opt-in only. But that's not correct. So I'd have to bring in a, oh no. This cat. Yeah. Go with that category. And then, and then hit the select tag to the right, hit that dropdown. And if you don't see one there, this will create one. Let's create one. So schedule appointments. Is that the one? I don't think I did it. Correct. No, you did. It's there. That's the tag. Okay. So let's base everything around that tag scheduled appointments. Got it. So go ahead and just put add. Yeah, let's hit add. Okay. Okay. So then now let's hit save. So that's the starting rule. This. Now, you know what? Here's, here's the deal. We have, we have two sides of the email sequence, right? So either they, either they scheduled an appointment or they didn't. Right. Is that how you have your setup? Um, the, the very first opt-in is me collecting her email. Yeah. So the first page, like how you had it, um, on, on, uh, the Facebook advertising for the female, um, obtaining needs is the first form is an email opt-in name, email, and you get a coupon sent to you. And then after they hit learn more, they immediately go to the page where they can schedule a callback. But some people, they put in your email, but they don't end up calling them. Give them a callback. So the email that they get is here's the coupon for your free air purifier. If you haven't scheduled a callback already, please do. Okay. So then hit that red tab to the left, right there. Let's, let's minimize that one real quick and then scroll down right here. Yeah. Just minimize it. Yeah, there you go. And so let's, let's go down. So then in your, in, and we'll look at your sequence as well, but, but is it, is it set up this way where there's a, there, because the way I have it set up here in this one is, it's based on, it's based on the opt-in form. So the starting rule is everyone who fills out the opt-in form. And then the split is who has the tag? Did they schedule the appointment or did they not? That's, that's exactly what this is. Right. And so if they did not, they go to the fall side. If they do, then they go to the true side. Yup. Okay. So then let's go back up real quick. Let's base it on the form. The starting role is the opt-in form. And then, and then the split is going to be the tag. Do they have that tag or do they not? That's going to determine which side they go to. Left or right. Okay. Get it? So let's, let's, yeah, let's go back to that one. So anyone who fills out that form, that's the starting rule. Yeah. Yeah. Perfect. Let's go with that. That is the starting rule. Yeah. That's the starting rule. And then let's get, let's hit add and then let's get rid of the tag. Okay. Perfect. And then save. Yeah. So we're good. Okay. So that's the starting rule. That's the starting rule. And then let's go to the split. And we're going to simply just take out, hit the three dots and this, this, so the split. So they, they opt-in right now, right? We can set the split up to check if they have the tag in an hour, in two hours, in a day. So hit the delay. If you want to change that one hour to the right, you could change that one hour. But I think in your case, one hour is fine. So in one hour is going to check if that person, an hour ago scheduled or they did it. Great. Right. So let's put the conditions. Let's put the conditions. Edit that one. And that condition is going to be. The scheduled appointments. So let's go to the. So, so lead has tag. We're going to go to, we're going to go to hashtag. It might be up. Right here. Is there a certain active to the calendar? No, let's not based on that. We're going to base everything on tags. Got it. So has tag right there. Has tag. Has tag. Has tag. Has tag. So has tag right there. Has tag. And then hit that. Yeah. Hit the dropdown. And then let's go to the category, which is the had an appointment. Campaign. Or is it the email opt-in only? I think it's the had an appointment's campaign, right? Yeah. Cause one does it goes after the, the first one. The only, the first one was the email opt-in. Okay. And then the tag. Is the appointments. So boom, that right there. It is going to, that's our split. So now what we're going to do is hit edit. Okay. And then you'll hit close. And then now all you got to do is just model. The true and false side. So for example, hit the three dots on the true side, that first email. Attached to your coupon. Yeah. Hit that one. Okay. So right here. The recipient. So the split is, do they have the tag or do they not?  The recipients right there. Yeah. It was, um, Shit. What was it before it was, um, If anything. Okay. I think let me, let me share my screen real quick. Okay. Because I got to remember what that, um, I don't believe I've put anything in. Yeah. I think you, I think you might have changed edited, edited, but it's all good. I can, I can, I can show you right here. Hashtag club ever is not in California. Yeah. So my communications. And then let's go to my sequence real quick. And this is, this is a sequence that I'm actually rolling out on the first as well. So this is, um, This is powerful, man. This is super powerful. So. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah.  If you're into marketing or creating anything, like, Right. Because if it's not this deal, we got to juggle multiple services.  Cause I'm like, Anyone trying to build a dam online business, you're going to need five different softwares, five monthly fees. And that shit's expensive. Right. This one dam does it all. And that's, that's all we need. Right. Yeah. So that's my tag. So in my case, it's do they have the tag, which is, I think finished. Web class. Yeah. Watch. Yeah. Finished web class. So that's my tag. So based on that, that's going to determine, you know, what side they go to. So over here. This is what I wanted you to, to, to change on your end. Was the, the simple, um, recipient. So look at my recipient. The recipients. Even though we determined it on the split, we told, we told the split. We said, Hey, if they have it true side, or if they don't have it false side, but over here on the recipient, I still have to put that right here. So, so the true side is. Does not have tag course. So meaning they didn't buy the course yet. They haven't bought the course yet. That's going to make sure that they get this email. So right here, your tag is going to be, um, um, Um, it's not have tagged schedule email or schedule callback. Does not have exactly. And so then you just simply open up each one. And, and, and just simply edit that, that one part. And then as it scrolls down. And let's say, for example, they do schedule a call then. As the next as, because I have mine spread out one day. As that next one comes out the next day. If that recipient, if that tag isn't there, then they won't receive the next email. Right. Cause yeah, once, once they schedule a call. That's it. That's it. Yeah. So then, and then, so if, as I take the person down the automation, let's open up this, this is telling the system, Hey, if they still don't have that tag, subscribe them to a different list. Okay. Because now I want, I want to know who hasn't scheduled a call. And as they go through my email sequence, if they finish every email and they still haven't scheduled, I need them on a separate list. So I can then eventually email them again in the future. Got it. Right. So this is the automation. If the lead reaches this step, then subscribe them to this list, finished web class, but didn't buy. Okay. Okay. So you'll create another list. You don't got to create another sequence just yet, but you could just create another list and call it maybe opt-ins who never scheduled call. Okay. I believe I have it. Okay. Perfect. And then on the fall side. So the split, if they, if they, if they don't match that tag, then they go to the fall side. And in my case, I'm talking about a webinar. Hey, CK, I noticed you registered for the webinar, but you haven't watched it. Right. That's that whole email. And then the recipient is, does not have tag course students. So that means that they haven't bought. They haven't bought. They haven't bought. They haven't bought. They haven't bought. They haven't bought. They haven't bought. Actually, you know what? No, I take the back. It's anyone who reaches this step because it now, now I'm still, now I'm basing it. I'm basing everything on the fact that they don't have that tag. So I didn't even have to put a tag in. I just put anyone who reaches this step. And then the automation, the automation is going to check if they have that tag. So if I open up the automation, we'll see the tag right here. So a day later. This automation is going to take it into effect. And the recipient. Let's let this thing load real quick. This is the automation. So if they reached this step. And they finished the web class, meaning they actually scheduled an appointment.  In your case, it's, they actually scheduled the appointment. So if they have that tag, then move them to step two. Step two is now. On this other side. So, so, so in your case. Because they're going to go to this side. If they actually scheduled. Yeah. Yeah. Yeah. So then if they actually, yeah, yeah. So then if they go to the false side. And, and they, and they, and they, and they match, they actually scheduled an appointment. So let's say on the first email, they actually scheduled an appointment. Right. When it comes down a day later to the automation. It's going to send them over here. This over here is where you're emailing the coupon. So you can change the one day to one hour. And say, okay, if they got this email. And they scheduled an appointment an hour later, that automation is going to check and then send them the actual coupon because they actually scheduled an appointment. Okay. So it's going to send them straight to step two. And then again, if they still haven't scheduled an appointment, this one right here, anyone receiving it, and then the automation is going to take effect again. So it's after every other email, that automation is going to actually check if they have the tag scheduled appointment. If they don't. If they don't have the tag scheduled appointment, they can leave them in that false site. If they do send them to the top of true, which is step two. So if I open up this, there it is there. Lead reaches this step. And they have the tag scheduled appointment. Then move them step two. Now, once it's scheduled an appointment, they don't really need to go to the two side. It just stopped from there. Okay. Then it could stop. It could stop. Cool. And what you can do in the automations. I'm glad you, I'm glad you mentioned that. So in your case, what you can do is you can, you can, you can change this or just simply edit this automation. And yours is going to look like this. Lead reaches this step and has scheduled appointment. Then schedule them. I mean, then subscribe them. To this list. Right. So now we want to set, we want to put them on a whole different list and subscribe them to that schedule appointment list. So if I hit edit. I'm going to go ahead and add this. This move lead to it's down here. So I can X that out. And then just put. Up here at the top lead reaches this step. They have the tax schedule appointment. Then. Subscribe them. To another list. Cool, dude. Good. Yep. Got it. So, um, So that's, that's the sequence. That's the sequence. And you can always delete these. You can always. Right. So it's, it's making so much more sense to you. Like this is the hardest part. I think for myself. Well, reach out. I got you, man. I got you. Your screen here. Yeah. You cannot share a screen while other participants sharing my bed. Go ahead. Bro, you could pass as my cousin, dude. You're my cousin. I'm some guy. Yeah. Hell yeah. All right. So let's, let's go to the automation really quickly here. The first one. Do you want to look at your other one that you had set up? Honestly, honestly, this is allowing me to. Understand. Because if it's already done. Yeah. You know, mine is mine looks like the. Remember the sequence I sent to you yesterday here. Yeah. So what mine looks like is yours is more basic than even that one that I sent you that that one, like yours is more basic right here. So if you ignore that, yeah, it's this right here. Watch video, watch and schedule. Stop. Did not watch, but did not schedule. That's video. So I have these splits right here. Yeah. Yeah. Perfect, man. So all that is based on a tag. And I figured the automations. Yeah. Yeah. So that's exactly what I figured. So I was like, I'm just going to see what he has to say about it. And then kind of go from there. You know, it's important for me to understand what I'm doing. Yes, dude. You know what I'm saying? Yes. Reaches this step. Add. Damn, I already forgot what the hell mine looked like. Then it goes to. Did you already edit the emails?  Yeah. Okay. Shit.  It was. Cause I forgot. Let me. This step and then it was fricking. Let me. Yeah. We've, we've been, we've all been too damn. Let me. 35 days learning this thing. And then it's like a college course. Recording this meeting. Okay. Let me. Okay. Here we go. Perfect. Okay. So let me. Let me look at my sequence real quick. Okay. I'm going to go back to it.  Yeah, I'm doing on my end.  Let me just. So we're on the true side email number one, right? We are on the false side email. Oh, the false side email. Okay. Yeah. Yeah. Yeah. Let me look at that. And I'm pretty sure it is anyone reaching this step. So the false side. Did you change that? Yeah, we changed the split tag, right? So the false, that first email, the recipient. Is. Anyone reaching this step. Yeah. Perfect. So that right there. Okay. Let me just get into that. Okay. Anyone reaching this step. Boom. Perfect. Perfect. We're good. Okay. And then also, also, also. You know what I would also do. Here's what you haven't done. You haven't. You haven't changed your gateway. You haven't changed that email.  I'm thinking that's a cartoon on my bad. Okay, cool. You're good. You're good with that then. So obviously the, the, the title. Hey, John, what happened. You know, unless you want that title, but you can always, you know, make sure you change that, but let's switch that out. We'll focus on the recipients. So yeah, that one's good then. That was good. So then go down to, let's exit out of that one. So go to the very top and then let's just close that. And then scroll down to the automations. Now again, you can either choose it one day or, or based on, you know, shit, what you're trying to do, maybe, maybe an hour later. You know, one day is cool. Okay. One day is cool. Okay. Good. So then this automation, let me pull back up mine. This automation is going to be. Okay. So. Okay. So lead reaches this step. And has tag. Scheduled appointments. So let's, let's do that. So, so lead, lead reaches this step. So scroll up. Yeah. Lead reaches this step. And hit the, hit the select, hit that dropdown. And has tag and lead has tag. Yes. And they have scheduled appointments. Okay. Perfect. Add. Then. But see, and so we don't need to move into the true side after this, right? So if they have perfect, we're done. Okay. Then hit the select. Subscribe them. Bam. Right there. And then, and then also in the automations, which we'll, we'll, we'll get there after we finish this part. But what we're going to also do is we're going to, we're going to set up another automation. We're going to set up in a, it just, it's just to back ourselves up. It's just to back ourselves up. I don't think we, I don't think we actually technically, I don't think we need to. But it's just to back ourselves up. So for example, I'm sick. Lead reaches this step. Hold on real quick. Let me walk you through this part. And it's just to back ourselves up. Lead reaches this step. And they have that tag. Subscribe them to that list. But what I also want to back myself up on is we'll set up a separate automation where we'll set it up where if they, if for anyone who subscribes to that list. Unsubscribe them. From the sequence. Okay. Does that make sense right now? We'll do it. We'll do it after, because we have to get out of this part. But you know what? Let's see if you can set it up right here. See if you can set up right here, which I would, which you, you might be able to, but there's another part where I normally do the automations, not just right here. And I was going to show you that back in part. Actually, you, yeah, you can. So then leave that part there, go to the bottom. Then. And then we got the and unsubscribe them to the sequence. Unsubscribe them. From the sequence to bam. Whoa, whoa. Game time, bro. Cool. And then all sequences. Right. Yeah, yeah, you can, or, or this specific one. That's cool. All right. Perfect. And then edit. So, so remember how this is set up because we're going to set up all the automations on the fall side under, under that. Which are only a few more automations. I think there's like two more automations. Yeah. So what I'm going to do, what I'll do on my end, or, or you can do it on your end, but I got my phone right here. It's no big deal. I'll just take a picture of this. Let me take a picture of, of. Of this automation. Bam. Okay. Cool. Nice. So we'll close this one out. And then let's just. This, this program was like the sickest thing for any loan officer ever. Really? Bro. I swear to God, this is like a loan officer dream. And I haven't seen it. I have an idea of, I've had this idea for like 12 years now and finally I'm like, Oh, it can happen. And no, no, like, dude, it's crazy.  Dude, that's my next step. That's my, I put it all together the other night. I couldn't sleep. Yeah, dude. Like, that's the thing about all these share codes. Like you can package it all up and sell it, you know.  But my whole thing is, is just, that's why I emailed you the sequence. Cause I, what I try to do is I try to get people on my dam on it. And as long as I can help you succeed with it, chances are you'll stay with it. And then I'm just kind of, I'm trying to bank on the monthly residual, you know what I mean? But I could easily charge people.  So I forgot it, but already. Okay. Let me pull it up. Okay. So lead reaches this step. And has tag. So select. No up. Yeah. Yeah. And is already there. So select drop down right there. And lead has tag. Up. Yeah. Okay. Lead has tag. And then they're going to have the tag. The schedule appointments. Wait, wait, wait. Yeah. Okay. Perfect. And then, and then down. We'll put then. And then we're going to add one more down. And then we're going to hit. Subscribe. Subscribe to list. And that list is the. Yeah, the list. The schedule callback. And we're going to hit add. Actually. Okay. And then we're going to add one more down then. And select. Unsubscribe from sequence. Bam. There it is there. And then we also got to make sure that the recipients on the emails on the false site are correct. So hey, close. We're good with that. Yep. And then let's go to the next automation down there, down right there, and let's do the same thing. Okay. So lead reaches this step. And then it has tag. And has tag. And it's like a lead reaches this step. Yep. Okay. Cool. Then. Subscribe to list. Okay. This is really, really making sense right now.  Hit add. Finally. And one more. The unsubscribe. Yep. So down. No. No. Down. There you go. Unsubscribe from sequence. Automations are complete. Cool. So hit edit. Close that. Son of a gun out. And then was that the last one? Yeah, I believe so. Cool. So we're good with that. So it's cool. Yeah. Close that out and just scroll down. Is that one more? Yeah, there's one more. Yeah. Okay. And so in these, in these emails. You know, that's where your link goes for the calendar, obviously. Yeah. So. And then emails. So instead of the link to the calendar, I have a schedule. Schedule callback button. Perfect. Perfect. Yeah. So lead has tag. Yeah. And then I have like scheduled date appointment too. I'm just kidding. Yeah. Hell yeah. I just got a cart to get dates, bro. Oh yeah. You saw that knowledge too. Dude. Package that up and sell it. Look how many leads I got. Oh my God. All right. One more. And then here. And then. Unsubscribe. Unsubscribe. Yeah. From sequence. Okay. Actually wait, I don't think it, I don't think it went through. Okay. Very good. Let's. Do some automation. Yeah. Let's go back to it. Edit it again. Yeah. So subscribe to the list. And then hit the select and then unsubscribe from sequence. Perfect. Add. Okay. Cool. All you lead generators out there. This is the video to watch, boy. Yes, man. Yes. Okay. So now, now let's go back up to the top of the fall side. And let's just, let's just make sure, let's click on the dots for the emails. And let's just make sure that those emails. Are, are, are proper. So the first one, what happened? Yeah. Open up that one real quick. And what did we put there? I think we. The recipients. Let me check mine again. Anyone reaching the step. Okay. Perfect. And then let me check my email number two. Get out of this. No, you can stay there. Yeah. Actually, yeah, you can get out of there. So go ahead and open up the second email. So you got that first email, then below its automation, and then open up the next one. So the next one is also the recipients. Anyone reaching the step. And then I'm pretty sure all of them, all the, all the rest are as well. So let me open up third one. Yeah. Okay. Yeah. The third one. Anyone, anyone reaching this step. Yeah, they're all, they're all anyone reaching this step. Okay. Because everything, and the reason for it is because everything is based on the automation, the automation is checking for everything. Got it. Perfect. Okay. All right. Okay. So then, so then now. So the true side then to the true side. Hit the three dots and let's look at the recipients. On the true side. Let me open up mine again. Okay. So mine was, was who, who, who does not have the purchase tag. So in your case is going to be, is going to be lead does not have scheduled appointment tag. Okay. Yeah. That's going to be your, your recipient. So let me leave it. What's that? Lead does not have scheduled appointment tag. Yeah. So if you hit the three dots and then you go to. Recipients. Okay. Let me, and then so that, that, that recipient on my end does not have tag. So you're going to put. Does not have. Yeah. So let's hit the recipient. It has none. So let's change that to. Does not have tag. So let's hit the select. And then lead. Yes. Does not have. Boom. And then. Yeah. And then the appointment one. This are correct. Yeah. Yeah. Bam. That just goes on.  Test run the bitch, you know, like really put your email in there and, and, and what I always do too. Do you have a Gmail email? Or you have your, your company one, right? I have it. Oh yeah, yeah, yeah, yeah. You have a Gmail as well though. Test it. Test it with the Gmail. Because sometimes when you put the same email into the system, multiple times, sometimes it's down. Yeah. Yeah. Right. So, so you want to test a different email, but with your Gmail, like for example, if it's CK. At gmail.com. The way you're going to put it into the opt-in form is CK. The plus sign. And then you could put any word. So CK plus test at Gmail. Now after this, I don't even need all these right here. Right. You don't, huh? You don't because. No. I'm done. Perfect.  So hit. So the second one. Delete. Okay.  Right. We've got this campaign. $25. Yeah. Yeah. The code today. Yeah. The opt-in. I'll send you the coupon right now, boy. Yeah. Yeah. Oh yeah. Perfect. And then we'll get rid of the emails, but the actual complete. Okay, great. I'll tell you, because a lot of culture is surrounded by purchase now, up sale, selling actual physical products, you know what I'm saying? Physical and digital. Yeah. Yeah, exactly. But with me, and I'm sure a lot of people out there, to generate just appointments, this is the type of funnel that you're looking for. You know what I'm saying? It's appointments and digital products. That's what this is really for. Physical products is more like Shopify and Amazon. For digital products and for appointments, that's where the culture shines. Yeah. So anything digital, if I want to sell you a digital book, if I want to sell you my digital services, if I want to sell you my service, I can run all that through cartra.  And once you learn it, it's pretty damn basic, too, man. It's not as crazy technical. It is like a college course, though. Every single thing to learn here. You don't have to learn lighting of green screen, and then editing videos, and then going into cartra. It's like putting a pizza together. You know what I'm saying? Right. Yup. Yup. I don't know why, okay. Anyway, I don't know why. There it is. Should I just go like that? Yeah, you could drag it down.  Okay, so, okay, connected there, and then hit that button at the top, to the very top, no, very, very top, top, top, top, top, in the black, all the way to the top, right there, to the right. That one, hit that one. Okay. That arranges everything. So now scroll up, you'll see it right there. Man. Dude, there it is. There it is.  David, you are the man, bro. Hey, bro, you're the man. I'm the man who knows the man, and you're the man. So hit me up anytime, bro. Is that, test your automations, obviously, and then let me know if you need anything else on the pages. What you're, are you going to be running traffic or are you going to be turning around Facebook ads or what? Yeah, I'm running traffic. I launched tomorrow. I'm launching 17 different cities. I'm only putting five bucks in per city for the week. Perfect.  Do you know which ad to run? Do you know which, which campaign to set up? Yes. Conversions. I got that all set up. Yep. And you set up a custom conversion around the thank you page. Yep. And then I've had my friends the last two days go through it with me and do it and it works like a charm, bro. Hell yeah, bro. Keep me posted on that, man. Keep me posted on that. I will. So I'm going to run it for a week. We're testing it for the week. Today is Wednesday. I'll call you next Wednesday. We can schedule something like this. Hell yeah. And then our next Thursday for the full week and then we can, because that way it'll make me. Like really fricking do it and not be super scared because this is scary. You know what I mean? It's like making a new boat. Like, oh my God, I hope this thing. Yeah. And remember it's all as long as the back end is covered, which is the email sequences, the landing page, our Facebook pixel is on every page. Our custom conversion is set up around the, around the confirmation or the thank you page. As long as that part's all set up, then it just comes down to the front end. It comes down to these ads. And it's not that you, it's not that you can, you, there's really no such thing as, actually no, there is a such thing as running the wrong ad. So for example, if you're trying to capture leads and you're running a video view ad, you're not going to capture leads. If you're trying to capture leads and you're running an engagement Facebook ad, you're not going to capture leads. So there is, there is a such thing as running the wrong Facebook ad. But once you have the right Facebook ad, which is conversions. Yeah. That's all you need to know. As long as you run the conversions ad and, and, and, and you're running it to a good audience, then, then all it comes down to is your, is your content. As long as you're putting out multiple ads, multiple variations, multiple images or multiple videos, that, that's where you're going to win.  It's like, dude, it doesn't work that way. Like we got to put out a shitload and, and you start small, right? Right. Start small. But, but if we're running, let's just say a total, let's just, let's, let's call it 50 bucks a day or 25 bucks a day. Like you want to be at at least, at least 20 bucks a day to start. But with the 20 bucks a day, you're running four different ads. Yeah. Right. Just like you already know. So, so that's what it's all about is, is the different ads, the content pieces. Sometimes this image might attract more people versus that image. Sometimes this video might be better than the other video. Right.  You got to just do the fun. I have, I have four different videos that I made. Awesome man. Pretty much the same information, but each video sparks on something else. This one sparks on financing. This one sparks on the home, home purification system. Awesome. This one sparks on my services free. You know what I'm saying? And then I mimicked it literally exactly how, you know, the Facebook, the one that you did with the, the female real estate agent. Oh yeah. Bam. She killed it right here. Yeah. Thank you, Paige. I literally mimicked it to exactly how you did that. Amazing.  Awesome, bro. Awesome. How do I turn? How do I turn my monthly subscription over to you? It ain't going to happen, bro. Hit done and exit.  Hit done and exit. But yeah, it all, it all, I tried, I tried, you know, really just being that part of my pitch in the very beginning. Hey guys. Yeah. And this is all through YouTube.  As long as you go through me, I'll give you my time. Right. And so sometimes people didn't go through me.  As long as I can record it and put it on my damn YouTube channel, that's cool too. Right. So I don't even care, bro.  That's kind of all I care about. So I'll give you my time.  We'll put it up and freaking create something out of it. All right, bro. Thank you so much. You owe me a blunt. That's a butt. Oh yeah. All right. All right, brother. Thanks for watching. Thanks for watching. We'll see you next time. All right, take care, brother.


In occupied territories, population turned against Russians. Kremlin uses harsh punitive measures. The influence of the Russians in the temporarily captured territories of the south of Ukraine is rapidly decreasing. The invaders resort to punitive measures. The influence of the Russian occupation administrations in the temporarily occupied territory of Ukraine is rapidly decreasing due to their inability to maintain appropriate living conditions of the local population. Deputy Minister of Defense of Ukraine Hannah Malia posted on Telegram. According to her, residents of many temporarily captured settlements of Zaporizia and Kursan regions demonstratively disregard the orders of the administrations and express distrust of Russian propaganda. Malia emphasised that truthful information about the situation at the front is spreading more actively among citizens and the share of the pro-Ukrainian population is growing. To counter these trends, the command of the Russian troops resorted to harsh, massive punitive actions. In particular, last week, Kirillivka village in the temporarily occupied territory of Zaporizia region was encircled by Russian troops for several days, during which total filtration of the local population raids on local infrastructure facilities and citizens' homes were carried out. Representatives of Russia's FSB checked especially thoroughly the citizens who were reported to have pro-Ukrainian views, including based on denunciations. Malia added, in most cases, such people were immediately detained by the invaders and taken away for interrogation. Some of them did not return to their places of permanent residence after interrogation. The fate is currently unknown.


We have food as a word to describe sustenance. No difference between garlic and chocolate, eggs and watermelon. In the same sense, from a learning perspective, we need tools of precision that allow us, as educators, as learners, as designers, to meaningfully start to make an argument between the machine and the computer. I have a lot to grapple with, and especially with the changing nature of learning and the institutions of learning, and also with some of the new technologies that are coming. Access to education, it enables them to change their own lives.


Thank you. OK, hear me? Move this up a little bit. Hello? There we go.  And I'm just going to zoom into one of the eggs, OK? So if I zoom into one of the eggs, here's my black hole. I said, this is the horizon. Imagine this is some sort of horizon, marginal trap surface, something like that. And what I want to do is I want to, oh, and way over there on the other side of the board, there's another black hole, OK, really far away. And now I drop you, or me, right here, crazy hair, because I'm very close to a black hole. But I'm happy because I can make measurements. OK, so the question is, what will I measure if I just sit there?  So there's going to be a tidal perturbation that's being generated by the companion. Now, we cannot really study this, at least not easily, within post-Newtonian theory. Because post-Newtonian theory, when we're dealing with black holes, it typically uses a stress energy tensor as that of a point particle that's distributional and has all these issues. So typically, remember, we had this region around each of the black holes that you sort of need to excise from your manifold. This is a whirl tube, it's like a cylinder, if you want. Because if I try to use my post-Newtonian expressions here, I'm just too close to this black hole so my fields are not going to be weak anymore. So my expansions in G are not going to be valid. So you typically eliminate this, and then the question is, OK. But sometimes you sort of need the metric everywhere, right? You don't just need the metric in between the two black holes. My other black hole would have been over here, right? So sometimes you need the metric everywhere. If you're trying to prescribe initial data for a numerical relativity for a 3 plus 1 evolution, you can't just tell the numerical relativists, oh, well, you have initial data everywhere, except in a two-spheres surrounding each of the black holes where the rays of the two-spheres larger on the horizon. Because they're going to be like, well, what do I do with that? How do I evolve forward in here? Moreover, if you're studying, for example, accretion disks, you need to know what the metric is very close to the black hole. If you want to study how light bends around black holes and understand, say, the image that the event horizon telescope took, do you guys see that picture that they took with very long baseline interferometry? You need to understand this based on here.  So what we do is we start a new calculation, and we say, well, I want to calculate the metric here. So if the other object weren't present, if this black hole is isolated, then the metric here would just be the metric of Kerr or the metric of Schwarzschild, the metric of your isolated black hole. But if I'm in the presence of an external perturbation, I don't care what that is. It's something. I have in mind always my happy black hole over there, but it could be anything. It's some external universe. I can parameterize the external universe in terms of the remand tensor generated by that external universe. And I can do an expansion in that remand tensor, assuming that the external universe is producing a small perturbation of the gravitational field right here. That is, the gravitational field here is dominated by the black hole with an external perturbation that's changing slowly. This is the field of tidally perturbed black holes. And as far as I know, the person that pushed this forward to the death was Poisson. And you can look at how he's constructed tidally perturbed metrics of Schwarzschild. So you have a Schwarzschild black hole here with an external perturbation. And he carries out this perturbation to higher and higher order. So I'm not going to go into detail for the construction of this, because I don't have time. But if you're interested in knowing what exactly do we do in this Poissonian region, you'll go there. Essentially, what you say is g is equal to, I'm just going to sketch it here, because I just can't resist, g is equal to g of your black hole. So say Schwarzschild plus some h perturbation. And this h perturbation is going to be expressed in terms of remand tensors and derivatives of remand tensors and second derivatives of remand tensors and so on and so forth of the external universe. And of course, if you just do that and you put it in here with the right index contractions, for example, see the metric is a rank 2 tensor. The remand is a rank 4 tensor. If you're an intimate contract, the remand tensor will have two vectors to have the number of indices work out.  So you can assume you can contract this thing with two powers of the distance from the center to this object. And if you do that, this becomes a rank 2 object. And then it begins to look a lot like a tidal field, like a tidal force. Remand tensor, remember, is mass divided by separation cubed. You multiply by x square becomes dimensionless. And so on and so forth. The problem is, if you plug that into here, now at least it has the right number of indices, it won't satisfy the Einstein equations. So what you need to do is you need to multiply each of these things with some functions, f1, f2, and so on. And then you have to, which you don't know. These are unknown functions that have the property of going to 0 as you approach the horizon. Because as you get closer and closer to the horizon, then these perturbations should vanish. You'll be completely dominated by the black hole.  And then you solve those differential equations for those f's with the right boundary conditions. And you have a metric for a tidally perturbed black hole. That's in a neighborhood around each black hole. Of course, I cannot take this metric and evaluate it here in between the two objects. Because now my expansion, that this distance is small, breaks down. But I don't have to use that metric in between. Because in between the two objects, I can use a post-Newtonian metric.  So you do asymptotic matching or match asymptotic expansions. And ta-da, you have a metric that's valid in the entire domain. And that was my thesis, like 15 years ago. But the back in Einstein equations. Yes. So if you insert this into the Einstein tensor, you set it to 0. And then you'll get the different components of the Einstein tensor. And that defines differential equations for these f's. You also need an angular basis to this expansion.  I'm going to talk about that in a second. So this works all very good, except when that black hole, because this black hole, a little bit later in time, it's going to spiral in. These things are going to be moving around. It's going to be losing gravitational waves. So it's going to get closer. This one is going to get closer over here. Then a little bit later, it's going to be over here. And then that one's going to be over here. And eventually, they're going to sort of coalesce. So eventually, I'm going to have some sort of thing that merged. And what do I do now? I cannot use post-Newtonian theory, because I just be silly. I'm totally, absolutely wrong. I cannot really use this type of structure, because the objects that we're going around each other have already merged. So the remnant here doesn't have an external universe that's perturbing it. It got perturbed because of the merger. And now I want to understand the forward evolution of this. So what do I do? So this is what I'm going to talk about today. So I call this topic four.  And it's a little bit misleading, because when I talked about self-force, that's also black hole perturbation theory.  But what I mean by black hole perturbation theory now is understanding the behavior of the peanut after the object has merged. Let's begin. So here are some good references for you to look over, because unavoidingly, when you're trying to make an omelet, you're going to break some x, which means I'm going to make mistakes on the board. And sometimes it's so to check that you're paying attention, but most of the time it's not. It just happens. So the good references that I've been going to be following are Birdie, Cardoso, and Starry Nets, so Birdie et al. This was an app J185. It's one of them, another one that. So that's like a review paper. It's like many, many, many, many, many pages long, but it's actually quite clear. Then, of course, you can go to Tukolsky's paper, which is not bad, but very, very mathematical. Oh, sorry. This one is app J185. This is not app J185. This is a review paper. So this is Birdie, Cardoso, Doso, and Starry Nets. It's a review paper. I forgot the journal, and I didn't write it down, but you can ask me later if you want. It's called Quasernormal Modes of Black Holes and Black Brains. Then Tukolsky's original paper was an app J in. This is Birdie. This is Tukolsky. Two, ah, two calls, deformable to that name. So this was in 1973. There's also Cocotas. He wrote a nice review as well with Schmidt. This is a Living Reviews in Relativity. And then there's Brian Nollert from Ireland, I think. CQG16 from 1998. So depending on which paper you look at, you'll get a slightly different take on this same topic. The first one, the third one, and the fourth one are more like review papers. The second one is just the classic Tukolsky paper that separates the equations. So with that said, let's begin with a simple example. The physicist's favorite example, Scalar Field. People have been asking me about like ADS and whatnot in this summer school. So let's consider a Schwarzschild ADS black hole because it doesn't really matter. Questions are going to be the same. So here's my Lagrangian. So you have some complex scalar field, because why not? And it has the standard kinetic term. You can add a mass term if you want. Not that it matters. For what I'm going to do, I'm actually going to drop this term because I don't like Klein-Gordon equations. They're bad. And then let's look at a d-dimensional. So this is going to be in d-dimensions. So d spacetime dimensions. So that's what the d stands for, d minus 2 divided by 4, d minus 1. Gamma is just some constant.  So that's a coupling between my scalar charge, my scalar field, and which I'm going to call it the curvature. Just consider that. Let's see what happens. So if we consider that, look at, oh, and we're going to consider these situations. We're going to consider this Lagrangian. So what we're going to do is we'll understand the motion of this scalar field. I'm going to derive the equations of motion in a second. But I want to consider evolution of a scalar field of phi, d-dimensional, Schwarzschild, AdS, Blackhole. So I'm guessing that you have seen the line element.  Need a bigger pen.  So l here is the ADM radius. r naught would be the mass in four dimensions. Gamma here is a coupling constant. Of course, constant. Thank you. This is Pewter's fault because he keeps on talking about ADM and it's in my head now. So the mass of this object, just in case you're wondering, is d minus 2 divided by 16 pi. The area of a d minus 2 sphere times this r naught to the d minus 3. And the area of the d minus 2 sphere is 2 pi to the d minus 1 divided by 2 divided by the gamma function evaluated at d minus 1 over 2. Now that it really matters, but those are all the sort of physical quantities. So this is the background. So I'm not going to consider the back reaction of the scalar field on the background. We're going to do just standard field theory where we just consider fields propagating on some background. Yes. That's to ensure that my equations look nice. That is irrelevant for what I'm talking about. So the equations of motion, you take that Lagrangian. You get box acting on phi equal d minus 2 divided by 4 d minus 1 times gamma r phi. Very simple. And I've set the mass to 0 because, as I said earlier, don't like Klein-Gordon equations. And of course, if you were to consider the back reaction of the field to the metric, then you would have to add to the Lagrangian, the Einstein-Hilbert Lagrangian. And there would be a variation that you have to do with respect to the metric that would give you the Einstein tensor with a cosmological constant term plus or equal to the stress energy tensor of that scalar field. I'm not going to worry about that part. So very good. So if I now take this and I evaluate it on this fixed background, background phi going to be equal to d times d minus 2 divided by 4. Now r is equal to just the cosmological constant term. So it's 1 over l squared and then gamma. And I have my phi term here. That's why I said, sure. So what we're going to now do is we're going to make an ansatz for the scalar field. So we're going to decompose this into plane waves. So we're going to do a sum on l and m of e to the i omega t for some constant omega times some psi of r divided by r to the d minus 2 divided by 2 times some ylm theta and phi and possibly other angles. So this is supposed to be the d-dimensional generalization of spherical harmonics, which is constructed such that it satisfies the angular sector of the Laplacian. And so if you do that and you plug these ansatz into here, magic happens.  So you're all physicists. And while I look for a chalk that I like, because I don't like any of these chalks, they're all very small, think about what that equation is. There we go. What's that equation? Physicists should recognize it like that. The Schrodinger equation. Good. Yeah, very good. It's the Schrodinger equation. And the way to see that is the Schrodinger equation is to go to a different coordinate system. So let me define the tortoise coordinate to be a star which satisfies the equation dr star by dr equals 1 over f. That's my new coordinate. And you can write it as an integral if you want. If you do that, this equation becomes d2 psi dr star squared plus omega squared minus v acting on psi is equal to 0. Kaboom. That equation I can solve. I could solve that equation a long time ago. OK, so it's a Schrodinger equation. So somehow what we're finding is that if you take a scalar field and you consider the evolution of the scalar field in a black hole background, then the evolution equation of that scalar field satisfies the radial part of the eigenfunctions here, if you want, satisfy a Schrodinger-like equation. And I didn't have to do it in ADS or in d dimensions. I'm just showing off. I mean, you can do it in d equal 4. And you can do it in asymptotically flat boundary conditions to take L2 infinity. Yeah, yeah, so the b here was just to indicate that this is the box associated with this background metric which I wrote down here. Well, so this was, if you just do it as a gen, if you don't consider a fixed background and you just allow the field to back react on the metric, then that one would have to be also enhanced for an equation for the metric tensor. This is what I was saying in words. I didn't write it down. OK, so this looks cool. This is how a wave scatters off H-R-Shoot block. So it's a scalar wave. But the metric is like a scalar field or a bunch of scalar fields. So you sort of suspect that if you do the calculation right and you consider what happens to the vibrations of spacetime after my peanut forms, they should satisfy some sort of Schrodinger type equation. Intuitively, you would want that to happen. And indeed, that's exactly what happens. So now let me derive for you how that comes about.  But you can definitely do this on a Kerr background. If you do it on a Kerr background, then you get something called the Tukolsky equation, which tells you how the perturbations or how psi 4, which is one of the Neumann-Penro scalars, what equation the psi 4 quantity satisfies in a decoupled way in the sense that how this psi 4 decouples from the other Neumann-Penro scalars and the other metric degrees of freedom. I'm not going to go into any of that. Instead, I'm going to talk about Schwarzschild. That's much easier. So Schwarzschild perturbation theory. So we begin by decomposing the metric into your background metric, Schwarzschild, plus a metric perturbation, H mu nu. Moreover, it's convenient to decompose H mu nu into two terms. Well, first of all, to decompose it into sums over lm's. And I'm going to tell you what those sums, why I've done this. So there are some quantity H mu nu lm with an a here. And then there's a quantity H mu nu lm with a p here. So what I am doing is I am decomposing the metric perturbation into a piece that's polar and a piece that's axial. And by what I mean by that is that if I act the party operator, meaning like I flip my triad, so I take x to minus x, y to minus y, z to minus c, I act this on H, lm, polar, or any quantity that's polar. Let me write it as quantity a, lm, that's polar. Then I'm going to get minus 1 to the l. Time my quantity back. Whereas if I act the party operator on a quantity a, lm, that's axial, then I'm going to get minus 1 to the l plus 1 of that axial quantity. That's how I define the axial and the polar parts of any quantity, OK? So I can always do that. And now what I need to do is I need to do something similar to what I did over here. I need to decompose my field in spherical harmonics. The little problem is that this thing here is not a scalar. It's a tensor. And so you cannot use just regular scalar spherical harmonics. But there's a generalization to tensorial quantities, which allows me then to, so there's a generalization, generalization spherical harmonics, called tensors spherical harmonics. So your scalar one is still ylm. Oh, and I mean four dimensions now, by the way. And I mean asymptotically flat spacing, simply because dear god, OK? So there's a scalar one. There's a vector one. So the vector quantity is yalm. There's two of them. And there's salm. And this yalm, this is just the yalms that you know and love. But you take a covariant derivative of the scalar spherical harmonics with respect to the metric on the two sphere. So that's what the column means. So definitions here. Column means covariant derivative with respect to omega AB. Omega AB means the metric on the two sphere. So like 1, 0, 0 sine square. So I can definitely compute or define Christoffel symbols associated with this metric. And from that, I can construct a covariant derivative just like I do in GR. And then once I have that, I just take this covariant derivative on the y. But of course, this symbol a, so a, b, c, dot, dot, dot, these are going to span only the angular sector. So theta and phi, because otherwise you cannot. Otherwise this doesn't make sense. So S a lm is defined as omega AB times yb lm. And so this quantity happens to be axial. This quantity happens to be polar. And these things are called vector spherical harmonics. And then there's a generalization for spatial tensors. So there's three quantities yab, which is ylm times the metric of the two sphere. And this quantity is polar. Then there is zab, which is the ylm with two covariant derivatives plus an extra term that you need to add in order for the equations to be satisfied, which is also polar. And then there's a last term, Sab, which is the covariant derivative of the vector spherical harmonic, column B, symmetrized. And this quantity is axial. So I define these quantities. Yeah, how is that? How is it racer lowered?  No, so if you raise, one of these quantities are going to be factors of sine square. Let me continue, and I'll tell you about it afterwards. So I always am not going to finish. So just as a note, tensor spherical harmonics, and I'm using this word to imply scalar vector and the spatial tensor ones here, satisfy orthogonality conditions that I'm not going to write down, but they're similar to the scalar ones. So things like, for example, integral on ylm, yl prime d omega is like delta of ll prime mm prime, things like this. Except that, of course, now you have indices, so you have to be a little bit more careful, but there are similar orthogonality relations. And what is nice about this is that these angular tensor spherical harmonics satisfy the angular sector of the Einstein equations when the Einstein equations are evaluated on a spherically symmetric background. So that suggests that I should use for my ansatz for the axial thing a matrix. I'm going to split it into two sectors. I'm going to split this into the tr sector and into the theta phi sector. And in the theta phi sector here, I'm going to use the axial. Did I label this backwards? Maybe I labeled this backwards. It doesn't really matter. So in this box, I'm going to use some halm times sa lm. And here I'm going to use some hlm times sa b lm. And the idea here is that this ha and this h are functions that depend on t and r that I haven't yet determined. And the capital index a here, just like the small ones here, span theta and phi, the capital a, b, c, dot, dot, span the tr part of the space. So for example, ht theta of this quantity is going to be equal to, so I take this box here and I set for the capital a I set t and for the little a I set theta. So I'm going to get ht lm, which is a function of t and r, times s theta lm function of theta and phi. So that's what the notation means. Sorry, I didn't invent it. So I'm going to do that for one of the terms. And for the other term, I'm going to do something similar, but I'm going to use the other terms. hlm mu nu p. I'm going to write as ha b of t and r, lm here, times the ylms. OK, maybe Nico should choose a different part of the board to write this. How about I write it right here? So h mu nu polar lm, I'm going to write it here, this, that. OK, and so over here I'm going to use some functions ha b lm of t and r, times the ylms. Over here I'm going to use some other functions, p a lm, times y a lm.  OK, so for example here, if I wanted the t theta part of this quantity, then I would go here and I would put for a t and for the other a theta. So I would get p t lm of t and r, and then y theta lm of theta. Hope the notation is somewhat clear. It's this thing. So this is, so h sub t here is the t component of this two-dimensional vector. OK, and this thing here, so this thing here so it's a theta component of this two-dimensional vector. Capital h, here, this is a scalar, this is a function of t and r. Yeah, so I don't need different scalars here for the different components. It's a bit of historical reasons why people did it this way. Remember, people were doing things without computers, everything was being done by hand, and they just chose this particular parameterization of the metric perturbation, and it has stuck with us, so. This is not the only way you could do this calculation. This is what we're going to do. OK, so now comes the remarkable part. And by that, I mean like, I never thought about it. Surely, you can come up with something better. Probably because Nikon made a mistake. Also, it doesn't really matter, right? You can absorb it into the h. It's just a matter of like how many terms you want to pick and keep it. But you just sort of, you know that that thing is going to scale as r squared. So if you want h just to go to 1 in flat space, then that's why you factor this part. Anyway, so then you take this, you plug it into here, and then you take this, and you plug it into here, and then you take this in threshold coordinates, and you plug it into the Einstein equations, and then you suffer a lot because you have to calculate the Einstein tensor. But because these angular functions are constructed such that the angular sector of the Einstein equations on a spherical background separate, you know that the Einstein tensor will also be proportional, different components of the Einstein tensor will also be proportional to these quantities. So if you do the calculation, you can verify this. And indeed, if you look at the TR part of the Einstein tensor, obviously to 0th order in the perturbation, the Einstein tensor is identically 0 because Schwarzschild, the Schwarzschild metric annihilates the Einstein tensor operator. So this is to first order in h. So you get some quantity gLm times yLm's here, for example. So this thing here is a 2 by 2 matrix, symmetric matrix, with derivatives. All of the functions is h, a, lm. So this is with time and r derivatives only of the h, a, the other h, the h, a, b, in principle, all of the metric functions that you're trying to solve for. So there's some complicated mess with t and r derivatives of that. But what's nice about it is that the Einstein tensor separates like this. The angular part can all be absorbed into this yLm. And then you just have some complicated differential system for the TR sector. Similarly, the off-diagonal piece of the Einstein tensor can be written as calligraphic ga, again, some other complicated mess, times yLm, plus some other complicated mess, times sLm. And finally, if you look at the spatial spatial part of the Einstein tensor, you get some gLm times yBLm, plus some hLm, calligraphic hLm, times zABLm, plus some calligraphic iLm times sABLm. Apologize for the notation again, but this calligraphic hLm is not the same as this hLm. This one's calligraphic and that one's not. This gLm without indices is not the same as gLm with indices, with one index, which is not the same as gLm with two indices. But all of this quantity is this one, and this one, and this one. Well, they're all not. OK, let me erase this. So this one is not a 2 by 2 matrix. Obviously, it's a two dimensional vector. So it's this one, but both the components of these quantities are all some expression that depends on only time and radial derivatives of these metric functions. And similarly, g and h and i here are scalars. They're not vectors or matrices. And they're, again, some function of derivatives in t and derivatives of r of these metric functions. So that's what's nice of using this angular basis to expand this metric perturbation. I mean, it's constructed such that this will happen, such that you can separate the tr sector from the theta phi sector. And I'm sure it is related to SO3 and other mathematics that you can probably tell me more about. What I care about is that that separation looked really cool. So there's two miracles. You see, one miracle is that I managed to separate the tr sector from the theta phi sector. That's not, well, I mean, maybe it's obvious, but I mean, it's not trivial. But this is still a couple system of equations because each of these horrendous quantities that depend on the derivatives of tr, t and r of these quantities, they contain the derivatives of all these quantities mixed together. And it's a mess to solve. You still have, in principle, 10 equations for 10 unknowns, which are these metric functions that are over here. And you can impose a gauge if you want. Not that you have to. So I'm not going to. But there's a gauge called the Regi-Wheeler gauge that people sometimes speak, where you essentially, see, you have more than you don't need to have all of these degrees of freedom present. At the end of the day, you know that you have, for the metric tensor, 10 degrees of freedom. Well, we have 10 degrees of freedom. 4 can be eliminated through our coordinate. The pendulum can be eliminated via coordinate transformations. 4 are constrained by your constraint equations. And only 2 are really free or propagating. So you can impose a gauge to eliminate, say, 4 of these quantities. And the Regi-Wheeler gauge is one of these choices. But we're not going to do that. Instead, what we're going to do is we're going to note that since I'm in vacuum, each of these things has to vanish. Each component of the n-centers has to vanish. Moreover, for example, since these components have to vanish, that means since the YLMs can't vanish, because that just would be silly, then each component of this differential operator has to vanish. Similarly, since these components have to vanish, then this sum has to vanish. But remember that YA is orthogonal to SA, in the sense that if I integrate them, I get 0. That means that this has to vanish independently of this. And the same with this. This has to vanish independently of this, which has to vanish independently of that. Which means each of these things have to be 0, which means I can linearly combine them, if I want, because 0 plus 0 is still 0. So if you take F, so I'm going to define F just to be here. F is equal to 1 minus 2m over r for Schwarzschild in Schwarzschild coordinates. So if I take F divided by r and I multiply it by 2 over r times 1 minus 3m over r times this calligraphic ILM plus 2f times the calligraphic h, the r component, lm minus f times the radial derivative of ILM, then you would agree that because each of these three terms depends on these quantities, each of which has to be 0 independently, this whole thing has to be 0.  And I have to tell you what these things are.  And psi lm rw is minus f over r times the r component of the hlm minus the radial derivative of the hlm function divided by 2 plus the hlm function divided by r. So what have I done here? How did I get this equation? Magic. So what I did was I said, look, each of these things have to be 0. And I want to decouple the system of equations to find an evolution equation for the two propagating degrees of freedom in my theory. There's only two degrees of freedom that are not constrained, the plus and cross gravitational waves, effectively. And what I have found is that one way to decouple these equations for one of these degrees of freedom is through this combination. I'm not claiming this is the only way to decouple the equations. It just happens to be one of the ways to decouple the equations without imposing any gauge. And so by combining things like that, all of the gauge degrees of freedom cancel, exactly. And what you're left with is a wave equation with a potential, like Schrodinger equation, acting on some combination of the metric functions. It is a linear combination of the components of the instant sensor. A very clever combination with very clever, like if you put here a 4, then things on the couple, for example. And so this thing here is called the Regi Wheeler. Italians make correct my spelling here. Tulio, right? Tulio Regi. Regi. Sorry. And Wheeler equation. This thing is called the Regi Wheeler potential. It's completely specified by the background and by the LM mode of your spacetime. This thing is called the Regi Wheeler function, or master function. And you can prove that this thing is gauge invariant. So you can pick a different gauge, you get the same thing. This is truly one of the combinations of the two propagating degrees. One of the combinations of metric functions that gives you a true propagating degree of freedom regardless of gauge. And if you look, it depends on big h and little h. So it depends on these functions. OK? So it's contracted completely from this part of the metric. So you would expect that there must be some other function that can be constructed entirely from this part of the metric. And indeed, you can do that and you find, do you want me to write the other linear combination, or do you want to just write down the equation? Choose your own adventure. Well, I was going. Yes, so there's another combination that I can write down that decouples for the other. So do you want me to write that down, or should I? OK, I'll write that down. OK, so this one's more difficult. You take 4 times f squared divided. By the way, this is not written down anywhere. I do not understand why. It's left as an exercise for the student, which I had to do as a student. And it sucked. But we have maple now, so you can do it in maple. I don't know what it is. Or maybe it is written. I just didn't know where it was.  That's correct, 84. m squared divided by r squared. And all of that times calligraphic mlm. Close that, OK? Yes. And now let me do some definitions over here.    So what I've done here is I've constructed some mlm that is just a combination of gtt, of gtt and grr with factors of little f. That's what this m is. Then lambda sub l, the little lambda sub l, that's just a constant that depends on the l number of your harmonic. Capital lambda l is like little lambda l, but it also has a 3m over r dependence in it. And then you just put this thing together. Oh. And that's also equal to 0, right? It's also equal to 0 because it's a linear combination of 0s. So that in my life also equals 0. So if you do that, well, you have to decouple the equation. So you know that you've constructed some sort of differential equation for some combination of these quantities that combination is gauging variant. So one way to do it is to find what other combination can you make of these quantities that would also be gauging variant. So you can do that. Once you have that combination, you can rewrite the different components of the Einstein tensor in terms of those combinations. The equations will simplify a little bit. And then you have to try to figure out, how do I combine different components of the Einstein tensor to the coupled equations? How exactly did Sir William and Griff do this? Or Regi and Wheeler? I mean, the Regi-Wheeler one is easy compared to that one. I have no idea. They didn't even have computers back then.  This must be 0.  I'm just writing it out.   So very similar, right? Very similar, very similar to this one. And so this was done by Cerulean-Moncriff. So this thing is called the Cerulean-Moncriff equation. This thing is called the Cerulean-Moncriff potential. This thing is called the Cerulean-Moncriff master function. And what you notice is the Cerulean-Moncriff master function is made of combinations of components that appear here. Furthermore, you notice that it satisfies the same type of wave equation as the Regi Wheeler function does, except that the potential has changed. Furthermore, you notice that the potential only depends on properties of the background and the l number of your mode. And if you say that psi lm, cm, or psi lm, or w goes as e to the i omega t, then you can convert these wave equations into a Schrodinger equation for your radial eigenfunctions, just like I did for the scalar field. So I have done this for you guys in an arbitrary gauge. If you put in a gauge, then things simplify like a lot. But I just wanted to give you the expressions in general. And this is cool, because if you solve those equations, what are you solving for? What are we solving for? So you remember what we are solving for? We are solving for the pinot. We have this pinot thingy that got produced after the merger of two black holes. And it's in pure vacuum.  So maybe let's take the pinot to not be so deformed so that I can treat it as a small deformation. But still, some sort of form black hole is going to vibrate. It's going to produce gravitational waves. This is what this is telling us, that the perturbations of the background satisfy wave type equations. And those wave type equations will radiate gravitational waves. So you expect that psi Cm and psi Rw should be related to h plus and h cross, the two metric perturbations.  OK, I'm going to introduce a new one. Psi lm Cpm times ylm minus 2 of theta and phi. So this thing here are spin minus 2 or spin weighted spherical harmonics. OK, so spin weighted spherical harmonics are functions of theta and phi that satisfy equations that look a lot like the equations that spherical harmonics satisfy, but they have in generalized by the addition of a spin parameter that's related to something I don't want to get into. So let's not talk about it. But it's just some function of theta and phi.  But this is not the only way to do it. You can construct another master function if you want.  And this psi Cpm equation, if you combine the right components of the instant tensor, will still satisfy this equation with the same potential, but it will be acting on psi Cpm. So the equation is the same. It's just the metric degrees of room that you play with are slightly different. But bottom line is, if you go and you solve for those wave equations, you're learning something about how the spacetime is vibrating in order for the spinnet to relax to its final state, to its final configuration. And so price, and others have shown, that's what the pistons were priced, Cis, Cunningham, and Ms. Moncriff, again, show that the end state of this vibrating black hole will indeed be Schwarzschild, for example. And then the modes are going to be radiated away in quiescent normal modes. So these are the quiescent normal modes. So this is the expression you have for the quiescent normal part of the gravitational wave spectrum in vacuum. And from that, you can go and you can calculate fluxes if you want, and whatnot. So this ties back to the self-force problem.  I can still say G is Schwarzschild plus a metric perturbation. I can go through the same procedure. And now, when I combine these things, I'm not going to say that's equal to 0. I'm going to have to say now that's equal to some function of the stress energy tensor. But that's the same equation with a source. So an Emory, if you want the gravitational waves emitted by the black hole due to an Emory to leading order in the dissipative part of the self-force, satisfies sourced wave equations like that that you can then solve numerically to find the gravitational waves emitted for a given geodesic, which you can then use this gravitational wave. You can take the time derivative of this, square them, and compute the flux carried away by gravitational waves. So now you have E dot, and you have L dot. And in principle, you can try to construct something like Q dot. And now you can correct your geodesic, the geodesic you used to solve this equation. This is one of the clutch methods. In fact, that's exactly what Scott Hughes proposed in the 2000s. Right. Good. So I've tried to cover a lot of material, because Paolo asked me to talk about black holes. And I know I probably didn't do the best job providing all of the details of the calculations that I presented. That's why I'm showing you these references. But I did that because I really, truly believe that as grad students, what you should try to do is learn as many new techniques as you can possibly find. There's a lot to physics and to gravitational waves and to geometry beyond just specifically post-Newtonian or just specifically, you know, quasi-normal modes. And you never know what techniques you're going to need in your research later on. And when you get older, like me and us, you'll find that the amount of time you have to learn new things, it shrinks exponentially with time. It's like a decaying function. So this is really your opportunity to learn new stuff. So just absorb as much as you can. Take these references that we have all been sending you and look them over. Try to learn the basics so that in the future, when you run into problems where you need to use things that are a little bit outside of your expertise, you know where to go, you know what to do. And you're a bit broader than just a pure specialist in that one specific calculation. So it's been my pleasure to teach you all. Thank you very much for staying awake. And we'll end there.