Here we have a rich TV live. And today we have a brand new IPO called Plant Vita. And I'm excited about this one. It's symbol is M-I-L-K on the CSC, the Canadian Securities Exchange. Put on your watch list, put on your radar and let's get right into it. All right. So first and foremost, I'm just gonna share this on social media. Hope everyone had a great weekend. Hope you're all ready for a big week. Brand new IPO for everybody to play with today. And we'd love to bring you winners and we'd love to bring them to you first. And this is just another brand new opportunity that you can potentially make money with. And if there's a new opportunity out there, we always wanna be the first to let you guys know. And once again, that's exactly what we're doing. It's Plant Vita Symbol Milk, M-I-L-K. And we're gonna get right into it right now. So milk on the bottle is called MYLK, Plant Based. That's the name of their product. But the symbol is M-I-L-K on the Canadian Securities Exchange. That's the symbol today, M-I-L-K. And it should be trading in literally minutes. And I will be the first one on the planet to bring you this pick. And I mean, that is our modus operandi. That has always been our goal is to bring you picks first. And once again, that's exactly what we're doing here with Plant Vita, bringing you this brand new IPO first before anybody else on the planet. And I just wanna remind you guys, if you like these videos, smash the like button, comment down below, share the video everywhere and subscribe if you're alive. I'm gonna go through the company overview. I'm gonna go through some information on the Canadian Securities Exchange that I think is important. And I'm gonna go through some articles and their websites, so you can learn more about this new IPO that's literally launching today. So Plant Vita producing a award-winning dairy alternatives and accelerating humanity shift to a plant-based lifestyle. We are focused on all natural low-processed and nutritional choices that are better for people, animals and the planet, the future is plant-based. Now, if you've been watching Rich TV Live, you know that I'm a huge fan of plant-based. We've been talking about plant-based foods for quite some time, probably been one of the leaders on a global level talking about brand new plant-based food opportunities and companies, and this company is absolutely no different. Put them on your watch list, put them on your radar. You won't even be able to find them anywhere because they're absolutely brand new. Absolutely brand new, so you won't be able to find this anywhere yet, but they are going to start trading today. Put them on your watches, put them on your radar. Milk, M-I-L-K on the Canadian Securities Exchange. Hold up dairy alternative, that's right. Dairy is a $490 billion market globally. Animal agriculture is one of the major causes for greenhouse gas emissions and loss of biodiversity on this planet. It takes 1,000 gallons of water to produce just one gallon of milk. Most people cannot properly digest dairy as approximately 65% of the world's population is intolerant of lactose, a sugar found in milk. The bottom line is that cow milk is really meant for baby cows, only not humans. In the last few years, we've seen a plant-based dairy boom with the likes of oat, almond, coconut, and cashew providing consumers with a great dairy alternative. Plant-based food and beverages are quickly becoming the next big and exclusive thing, giving nutritious choices that have delicious flavor profile and are created in ways that are better for our environment without harming any animals. This is the future, folks. And plant-veda milk could be the future. And if it is, you don't wanna miss it. And we love to make sure that everybody's aware of the next rocket. We'll show you why plant-based is the future. The market is still young, presenting incredible opportunities, not to be missed. Plant-based revolution, a large increase of consumers are choosing a hybrid diet that is more convenient, healthy, and ethical. 65% omnivore, I eat a wide variety of foods. 4% vegetarian, I do not eat meat, dairy, pork, chicken, et cetera. Vegan, I do not eat animal products at all. And 29% are flexitarian. I eat a wide variety of foods, but I'm trying to eat less meat, dairy, pork, chicken, et cetera. So the market demand 490 billion global dairy market, 39% plant-based yogurt year over year growth. And you can see the sales of the dairy alternative in the United States. Consumers are demanding healthier, whole, sustainable, and ethical source plant-based dairy alternatives as they become more healthy and environmentally conscious. While dairy-based yogurts continue to struggle, consumers are flocking to plant-based alternatives. The global plant-based dairy market is expected to reach 21.52 billion by 2024. The dairy alternative industry is still in its infancy within the 490 billion total addressable market with only a handful of brands in the space. Most products today are primarily based on a single plant source. Now, we really love to make sure we bring you winners and we bring them to you first. This is a brand new IPO and we're literally bringing it to you first. I mean, it's what we do. We do it better than anyone else. Let's see some of your guys' messages. Hey, Rich, I am back looking for something great to invest in. Well, I mean, you can definitely put this on your radar, put on your watch list. It's a brand new IPO. Symbol milk. Dylan Fraser says only 19 million shares. Yes, very tight float. And you guys know I love tight float stocks. Yeah, plant-based foods is the future. There's no doubt whatsoever. Yeah, put it on your watch list, put it on your radar. I think it has incredible potential and I'm gonna break down this deal for you guys even further as we get into it this morning. So let's go back to the investor deck. Actually, before we do that, I'm going to actually, because it's about to start trading. So let's do something here. I'm gonna show you something. Let's show you. Okay, so on the CSE, so this is the CSE website now. You guys can see the CSE website. I'll make that a little bigger. So this is PlantVeda on the CSE website, Canadian Securities Exchange. It says right here, PlantVeda Foods Limited is a plant-based beverage and food company that manufactures and produces for sale, low-processed, all-natural and natural nutritional dairy alternatives. So you can see here, PlantVeda and all their information, their website. And this is what I'm really interested in here, share structure, 25 million shares reserved for issuance and only 19 million shares issued and outstanding making it one of the tightest float stocks I talk about anywhere. So super, super tight float. And one of the reasons why I think everybody should put them on their watches, put them on their radar. This is the CSE exchange where it's listed. It is listed on the Canadian Securities Exchange and it will start trading today. As soon as it starts trading, I'll have a better idea of how and where it's trading and the market is now open. Hope everybody has a great trading day. So that is PlantVeda and now trading on the CSE on the symbol milk. Now, if any of you guys are watching PlantVeda and you know the price, please share it so we can talk about it while I'm live. I'm not looking at the live feed of the stock, but if you guys are looking at it, please share the price. Thank you. All right. Now, what I wanna do as well is we're gonna look at their website. Okay, so this is the website. So I wanna show you guys their website. I mean, this is how I do my due diligence and my research on all the companies to look at. I literally get into the company, right? So this is their website. I want you guys to be aware of it. PlantVeda website, symbol milk, M-I-L-K on the CSE, great symbol, right? And this is their website, PlantVeda, P-L-A-N-T-V-E-D-A dot com. You can go and subscribe to their list. I've already subscribed. And you can see here, we're accelerating humanity shift to a plant-based lifestyle at PlantVeda. We're using innovative technology to develop, produce and sell delicious dairy alternatives. And we love to make sure that you guys know about winners and you know about them first. The dairy alternative market is a huge market. There's more than 300 million dairy consumers in North America alone. That's more than 50% of the population. Animal agriculture is one of the leading causes for greenhouse gas emissions. Now, if we are consuming that amount of water at the cost of the environment to produce something that's not even meant for human consumption, that is a significant problem that PlantVeda is trying to address. Hi, I'm Michael. I'm the president of PlantVeda. The plant-based movement is still in infancy. It's like the electric car movement as well. The majority of products out there in the market are what we call them single-based products. Now, there's some companies out there who are trying to make them taste nicer by adding some artificial flavors to them. Now, the problem is, you know, they're no longer healthy. What PlantVeda has done and will continue to do is to create revolutionary products that are completely natural, healthy, but taste good at the same time. PlantVeda has been recognized three years in a row now. In 2019, we won the Product of the Year Award at the Veg Expo. In March 2021, we won another award, which is the Clean Eating Choice Award by the Clean Eating Magazine. We already have a recognized brand in the marketplace. Now, as we expand geographically, we're gonna see growth. As we create new products, we're gonna see growth. As a consumer brand, PlantVeda is gonna benefit tremendously from the going public transaction. PlantVeda is going to use the capital to complete the facility upgrade, expand our e-commerce business, and expand geographically. What PlantVeda is today, it's an award-winning plant-based food and beverage company that is currently focused on their alternative market. What PlantVeda is going to be tomorrow is going to be a driving force. The hind humanity is shipped to plant-based lifestyle. And you can see that video on their website. I thought that was a good video to share with you guys, and that's right on their website. Once again, the website is PlantVeda.com. Now, you won't miss dairy a bit. PlantVeda is a blend of, we're breaking it down right here. PlantVeda is a blend of multiple plant proteins, natural ingredients, probiotic, vegan, sustainable, great tasting, no refined sugars, and no additives. This is all on their website. And you can see the current product categories. So it's like a yogurt that's healthy, with healthy ingredients. I've actually interviewed with the president that you just heard from. So stay tuned to that interview. It should be coming out shortly. Probiotic Lassi Plant-Based Drinkable Yogurt, a plant-based yogurt drink contains 10 billion probiotics per cup, promoting good gut health, features real fruit and contains no cane sugar, unlike anything on the market, a variety of flavors, mango, turmeric, ginger, saffron, cardamom, strawberry, and blueberry. The milk plant-based milk, plant-based coffee creamer, cashews, coconuts, and oats differs from the competition, doesn't contain any oils, sweetened with only one gram of maple syrup and a variety of flavors, cane sugar, sweetened, agave, and sweetened. Coffee creamer, plant-based coffee creamer, plant-based milk made from whole food ingredients, healthy, dairy-free, cholesterol-free beverage with no added oils, available in convenient, smaller to go, or large formats, variety of flavors, original vanilla, hazelnut, and chai spice. And you can see lots of reviews, customers giving very positive reviews on these products so far. So already very positive reviews and these guys are literally just getting started and you can download the investor presentation, which I've been going through right on the website. So all that information's on the website, you can submit your information to learn more, find out about store locations, investor information, their team, products. This is a great way to learn, you can follow them on Instagram, Facebook, Twitter, and YouTube. Actually, let's just do that. So let's go to their YouTube. Only 13 subscribers and that's 14 now. All right, and let's go to their Twitter. 55 followers, now that's 56. I'll hit the bell for notifications. Let's turn on notifications. All right, all right. So we get some notifications every time they tweet and we'll follow them on Facebook. Well, I'm already following them on Facebook, so that's positive. Okay, and let's see if we can follow them on Instagram. All right, so 1,600 followers on Instagram and we are now following them on Instagram. So this is how I research my companies, guys. People always ask me, how do you do the due diligence? How do you do the research? How do you find out about these companies? It's a lot of work. So on their website, here as well, you will find an investor section and you'll find a news section. If you click on news, you'll see some of the most trending news for the company. So I clicked on the most recent news right here. Canadian company, Planveda, is game-changing the dairy-free milk industry. So I'm gonna show you that article real quickly here. I'm gonna show you that article. All right, so here's the article that has been published on June 3rd. Canadian company, Planveda, is game-changing the dairy-free milk industry. And this talks about the dairy-free milk aisles maybe getting crowded, so how can one competitor stand out amongst the rest? Simple, do something that hasn't been done before. So this kind of breaks down what they're doing. Planveda is a Canadian company that creates unique plant-based milks and beverages that are the first of their kind after a family health tragedy, Planveda's founders launched their company looking for healthy, plant-based options, like the world's first vegan lassi made from a multi-plant-based blend of healthy and delicious ingredients. Planveda focuses on not only making an alternative to dairy, but making it better than the original in taste and health benefits. Unique products. While many dairy-free milk brands have gone the route of finding a simple alternative, Planveda has created products that are one of a kind. By creating the first vegan lassi that is creamier than real dairy, they've landed awards such as Veg Expo's Product of the Year 2019 Award and a Clean Choice Award by Cleanating Magazine in 2021. Planveda has formulated their dairy-free cashews milk to include unique ingredients such as pistachio and saffron. They've created a line of products from their lassi range to creamers that are unique and rich in flavor in order for not only stand out among competitors but give consumers varieties to choose from. So what are the health benefits? When creating the Planveda line, the goal was not to find a replacement for dairy but to find something 10x better in flavor and higher in health benefits. What we decided to do is create a revolutionary product that's healthy blend of multiple ingredients, said Planveda President Michael Yang, who I interviewed, stay tuned to that interview. That doesn't just taste nice but is also natural and provides total nutritional value. This innovative approach ensures your meals and snacks don't miss the health mark. For example, their cashew lassi boasts a high 10 billion live probiotics per cup to promote a healthy gut, making this dairy-free replacement healthier than the original when switching to a more plant-based lifestyle. Many dairy-free products miss out when it comes to adding probiotics which is often why consumers still choose dairy, knowing that 70% of the immune system is housed in the gut. Planveda ensures the level of probiotics in their drinks surpasses expectations. Among, along with promoting a healthy gut, Planveda incorporates plant proteins into their drinks, making the need for traditional dairy even more obsolete. Whether transitioning to a plant-based lifestyle or incorporating more diverse products into a diet, Planveda makes the process smooth. How they continue to change the industry. Planveda continues to change the plant-based industry with providing unique alternatives that stand out in the milk aisle by creating products that taste better, are healthier than dairy and are unique in flavor and nutrients. Planveda has shaken up the plant-based milk market. So that's a press release that came out about Planveda and you can find out more right on their website. So I just wanted to share that with you guys because I think that's pertinent information. So it's currently trading at $1.50, not a bad start. Not a bad start whatsoever. If you guys have any questions, please put it in the chat. Love to know what you guys think about this brand new IPO. I think there's a chance for it to be very special and we love to bring you guys winners and we love to bring them to you first. We've been doing it for four years.  Now I'm gonna go through some more investment highlights. So this is an award-winning multi-plant recipe, an award-winning proprietary blends of multiple plant sources mean that our products not only offer all around nutritional value but also are great tasting. Our customers don't miss dairy a bit. Established distribution, product placement in over 100 retailers, some big names including Whole Foods, Save On Foods and with our premium grocery partners. So some big names that they're partnering with and that could definitely be huge for them short-term and long-term, multi-channel revenue model reaching consumers through wholesale business and our direct-to-consumer e-commerce platform. We're meeting the market where it's growing. Growing market opportunity, the plant-based dairy industry is in its early stage within the 490 billion total dairy consumer market. It is growing faster as consumers demand healthier options and become more environmentally conscious. PlantVeta Innovation Center, our 25,000-square-foot state-of-the-art facility is a springboard for new product development and large-scale production. It will serve as an end-to-end plant-based go-to-market hub right from innovation to production and distribution, national and global expansion. Foundations are being put in place to distribute products nationally and beyond and are ready to scale up. Proven management team, driven by a powerful team with years of experience, our knowledge, expertise and passion places as a leader in the plant-based market. State-of-the-art distribution and R&D center. So you guys can see this a little better. This is the investor presentation. Well, yeah, of course it's interesting. It's a brand new IPO. Doesn't get better than that. It's a new toy to play with. And of course, we're bringing it to you first. What do I think about the OTC Play fill? Well, I think it's a very risky. And anytime you have an OTC play with billions of shares, I would say proceed with caution. But it's done well.  That's all up to you. I have talked about it within our club and there are members that are buying the stock, but it's the ultimate in risk. And you need to be aware when you're dealing with penny stocks, especially sub-penny stocks with billions of shares that you are literally buying a lottery ticket and you have a very good chance of losing your investment. So it's got a state-of-the-art distribution and R&D center, a 25,000 square foot state-of-the-art facility in Delta BC. Phase one upgrade underway that will support 10 million a year in sales with plenty of room to grow, capability of housing approximately 200 staff. So this is a huge facility. Very, very, very amazing facility, state-of-the-art facility, impressive. And you know what? I'm gonna try to do a tour of this facility. So if I do, I'll try to make some videos and give you guys a chance to see the facility even closer. Established distribution. We're in over 100 stores, including nationally positioned retailers, whole food markets, choices markets, save-on-foods, spud.ca, fresh prep, luneo mall, mine heart, natures, food markets, fruity canna, and palm natural markets. So plan VEDA is the first of its kind offering award-winning multi-planet source products with explosive growth. So these are the products right here. The original Oatly, Oat Bay's Dairy Alternative Company recently went IPO, and you can see how some of these IPOs have done extremely well. Very good butchers. We also brought you guys very good butchers. They've been a huge success. Good food. I've talked to you guys about good food. They've been a huge success. So a lot of these companies I already talked to you about have already been a huge success. And FoodX, another company that's recently gone public and had a huge success. Plan VEDA is an award-winning multi-food technology company ready to scale. A recognized brand in the market, one of a kind of multiple plant source products combining the best of cashew, oats, coconut, real fruits, et cetera. Three current product categories, 25,000 square feet facility under upgrade, e-commerce shop. So, and only a 16.5 million market cap. Okay. So when you compare them to Oats, O-T-L-Y, Oatly, which is a 13 billion market cap, we compare them to Vary, which is a 450 million market cap, which I also brought you guys first. We compare them to food, which I've been hearing their ads on the radio here in Vancouver, 582 million market cap, and now Plan VEDA, only a 16.5 million market cap of baby. So timeline and key milestones. So in 2019, they launched the world's first vegan probiotic lassi, one product of the year award for the Veg Expo, launched the cashew milk product category. In 2020, launched the coffee creamer product category, probiotic lassi named a finalist on Vancouver Magazine, product sold in 100 plus online and offline channels, launched e-commerce subscription service in BC. Q1 2021 secured 25,000 square foot, which I just showed you, state of the art facility in Delta, British Columbia, won clean choice award by clean eating magazine, increased distribution, and launched on LumineuMall.com, the largest Asian online grocery delivery service in British Columbia. Wow, that's impressive. Q2 2021 Plan VEDA Innovation Center Development, new product development, yogurts and dips, and secure national distribution partners, welcome innovation partners in the innovation center, and launching their IPO today. In Q3, which is starting next month, launch vegan yogurt line, national rollout in major retail chains coming up, fulfillment and delivery partnerships, significant push on direct to consumer subscription business on planveda.com. Q4 2021, product and licensing requirements ready for international expansion. So we're talking about international expansion by the end of this year, folks, expand production capabilities outside of BC, plan VEDA Innovation Center completion, and additional revenue streams. So this is their current products, the award-winning lassi, the cashew creamer, and the cashew milk. So we're gonna go through it. Cashew creamer is the plant-based coffee creamer made from whole food ingredients, cashews, coconuts and oats, differs from the competition, doesn't contain any oils, sweetened with only one gram of maple syrup, and a variety of flavors. So they're putting maple syrup in their drinks to give it that sweetness. Their award-winning lassi is a plant-based lassi, plant-based yogurt drink combining all natural ingredients such as cashew, gluten-free oats, herbs, spices, and a real fruit of your choice. Contains 10 billion probiotics per cup, promoting good gut health. Worlds first, unlike anything on the market, unsweetened or sweetened by maple syrup. So you have your choice. And the cashew milk with spelt MYLK, plant-based milk made from whole food ingredients, healthy, dairy-free cholesterol-free beverage with no added oils, available in convenience, smaller to go, or large formats, variety of flavors, and those are their three. And the R&B Research and Development is working on Innovation Center Serves as a springboard for new products. So they are working on a wide variety of new products. So they're working on plant-based yogurts and dips, plant-based dips. Here's the team that makes it work. So this is the team. Michael Yang is the president that I've interviewed. Stay tuned to that interview. The chief executive officer is Sunny, Gurnani, and you can see the director of product innovation, Venita Gurnani, and the chief revenue officer, Mayor Sajnani, and the director and advisor is Claire Smith. So that's plan Vita, guys. Brand new IPO, hope you guys like it. We are the first to bring you this pick. And that's what we seem to do here better than anybody else's. We bring you the winners and we bring them to you first. We put them on your radar, we put them on your watch list, and it's up to you, the public, to decide what you're gonna do with that information. I'm not here to tell you to buy anything. Remember, Rich TV Live is strictly for information and education purposes. Please do your due diligence, do your research before you invest in anything that we talk about here in Rich TV Live. What do I think about Pharma there? Well, I brought you guys Pharma there first. So clearly I'm a fan of Pharma there and what is there not to be a fan of? They've been a huge success. Congratulations to members who got into Pharma there. I brought it to you guys first and they've been a huge success. I believe they're up about 200% in the last month alone. So what's not to like?  Love to know what you guys think about this new IPO. If you liked the video, please smash the like button, comment down below, share the video everywhere and subscribe. Brand new IPO, new toy to play with, new opportunity for investors, symbol milk, M-I-L-K on the Canadian Security Exchange. Company name is Plant Veda, their website. Let me show you guys their website once again. Please guys, go to the website. This is their website, plantveda.com where you can learn more about the company. This is how I do my research. This is how I do my due diligence. This is how I'm able to understand more about what the company does, what makes them unique and get a chance to really get to know the company intrinsically is by doing the research. So type flow, growing industry, strong growing management team in a growing sector with huge revenue growth potential and the chance to grow internationally and this is the first day it's trading. So typically with IPOs, we've seen that I would say at least 60% of IPOs do well right out of the gate but some don't. So I'm not suggesting you should buy it right away but what I am suggesting is put it on your watch list, put it on your radar and let's have a discussion about it, right? Let's talk about it, let's have a discussion about it. You can go to richpixdaily.com and we've got a forum where we can go and talk about picks and the community can talk about picks and share ideas on picks like this brand new IPOs. So that's the whole idea here with Rich TV Live is we're gonna bring you the picks, talk about them, break them down and then as a community come to a consensus. So I'd love to know what you guys think, comment on the video, let me know, do you like this pick? Are you planning on buying this pick? Are you waiting for an entry? Are you buying in today? Are you putting them on your watch list? Let me know, is your boy Rich from Rich TV Live and stay tuned to the interview with the president, should be coming out any day now. Get your popcorn ready if you're not winning, probably not watching, we bring you the winners and we bring them to you first. What do you think about Plan VEDA? Symbol Milk, M-I-L-K on the Canadian Securities Exchange is your boy Rich from Rich TV Live, bringing this brand new IPO and we're the first ever bringing it to you.  Happy trading everybody, hope you win big. Let's have a big week. And it's these types of picks that are helping the community win because we just keep bringing winner after winner after winner and new IPO. It seems like almost every week now we're launching a new IPO for the community and for the world to learn about first. So here's a new one for you guys, hope you like it, comment, let me know what you think. This is Rich from Rich TV Live, signing off. Happy trading everybody, we'll talk to you soon.


Thank you. So I've redrawn the pictures from last time. We'll come back to them later. But first, I wanted to say a couple of things. So I must apologize for an error in the second lecture. In the second lecture, I put down, which was called a number 3i. That's the scalar curvature of the horizon would be constant as a consequence of the Gauss-Goudatz equations. This was part of exercise 7. And I got carried away. This is true for the photon surfaces I'm going to talk about today. But it's not true for the black hole horizon that this follows from Gauss-Goudatz. So it's, of course, true after we proved any of these theorems that we proved in the last two classes, but it's not true a priori. Not so easy to show a priori. So I hope no one wasted enormous amounts of time on this exercise. OK. This being said, let's go back to very, very briefly reviewing the long proof that we did last class by Banting and Masudo Lalam of the static black hole uniqueness in three dimensions because we'll generalize it to high dimensions today. So what we did is we took our static vacuum system and we doubled it through its horizon. Then we made a conformal transformation and inserted a point and infinity and convinced ourselves that all the assumptions of a weak regularity version of the rigidity case of the positive mass theorem were satisfied, which told us that everything was actually R3 in this case. And then I very briefly mentioned to you a method using the cotton York tensor developed by Robinson and his collaborators that allowed us to recover Schwarzschild.  And this was actually hidden in the recovery procedure with the cotton York tensor implicitly, OK? Somebody else asked me after the lecture, well, but this argument with the cotton York tensor even requires third derivatives of the metric. And our metric regularity along these doubling horizons and also at this point was only such that we had C11. And so why does the Eisenhardt theorem and all these things even apply in the recovery with the cotton York tensor? The answer is very simple. We're only interested in showing that this original manifold was Schwarzschild. So we just forget everything we did. Now we know this upper part is conformally flat. And we can perform the whole argument in this original manifold. And this will be a recurring theme in the different arguments we'll give today in higher dimensions, where the only difference from the procedure we studied will be in this recovery part anyway. So every time that we arrived at this point where we know we're globally conformally flat, maybe we'll do topological arguments on this level. But then whenever we do any smooth arguments, we'll do them in the original manifold so smoothness is not an issue. Then somebody said that in the Bartnick paper there was a condition on the Ritchie tensor. No, there's not. I read up. So the Bartnick positive mass theorem is a higher dimensional version of Witton spinproof positive mass theorem. And it is under the same conditions as the original theorem. So no extra conditions there, so we're good to go there. And it's actually, I read it up. It's theorem 6.3 if you want to check it up. And one last thing I wanted to mention because it came up in a question yesterday after the lecture is, in fact, yes, the Banting Masoudou-Lalam proof is strictly speaking stronger than the Israel proof. So the Banting Masoudou-Lalam theorem implies the Israel theorem. And I'll write that down. Theorem 2 implies theorem 1. So if you know that we potentially start with multiple black holes and we show there is only one and it's Schwarzschild, that implies we start with one black hole and we show it's Schwarzschild. And the technical condition in this theorem that the gradient of the laps doesn't vanish, it doesn't occur here. So even from that perspective, it's stronger. Okay, so today we're going to talk about generalizations. In generalizations, of course, you could think in many directions. One could be to include matter fields. And this has been done for electrovacuum and for many other fields. I'm not going to say much about them because I don't know much about them. Of course, I know the electrovacuum proof and, in fact, there has been an electrovacuum proof in the style of Israel by Israel and collaborators and also by Masoud al-Alam with the Banting Masoud al-Alam approach, which also works in the electrostatic realm. And then there have been dilaton fields and in high dimensions even other models like brains and whatever to which this has been generalized. And if you're interested in this, see Heusler's book that I gave the reference in the first class and for newer developments, Robinson's overview article. I also gave that reference in the first class. So they list, Robinson even has an update on his website of his article having a small section on higher dimensions with matter. Okay, so that, we're not going to discuss that very much. Another direction of generalization, of course, one could do and we will talk about is higher dimensions. And as I just said, of course, higher dimensions could include matter as well, but we again are going to ignore this today. So we're going to only do it here, only in vacuum and see all those for non-vacuum. One paper I want to mention though is an upcoming paper by my student, Sofia Jans on archive shortly. This is a Bunting Masoodu Lalam style proof in an electro vacuum in all dimensions, starting at three. So we'll talk about that in a second. Other generalizations we'll talk about today is to photon surfaces and photon spheres. And I'll say it more about this later. And of course, and there's been lingering at the background of my lecture a lot. You could talk about non-static situations and most interesting at the moment, I guess, is stationary space times. And there's lots of open questions there. It's known in real analytic situations, but the full case is open. Okay, but I'm not an expert on that, so I'm not going to say anything on that stationary, except that there is a living review article on this topic by Piotr Kuchel and other authors. If you don't know, the Living Reviews is an online journal which you can access through the library of your institution, which is a review journal that posts updates on a regular basis, so they're always very up to date. So they change authors also a long time, so I'm not putting authors. But it's very good and very easily, well-readable living review article on stationary black hole uniqueness there. But before we go to higher dimensions or to photo on surfaces, let's go and look at other proofs of the three-dimensional case. And of course, if you're a physicist, you may wonder why it's even interesting to study different types of proofs of the same theorem if you already know the result by bunting in my pseudo-lalarmina as much generality as we like. And the answer is if we want to go and generalize through different scenarios, of course the more proofs we have, the better it is, because maybe other proofs will generalize better. So, and also we want to do a little bit of justice to the people who did the work. So there's one proof by Robinson from 1975 and the article again is in the book and in Robinson's review article. And it generalizes the Millard-Tomhagen, Robinson, Cyford paper I already cited was 1972. And it's for connected horizon. So a single horizon in our language. And it is not, and this is important, assuming that the differential of n is nowhere vanishing, okay? But still it's assuming that there's only one black hole horizon. And so if you remember Israel's proof, which was then generalized by Millard-Tomhagen, et cetera, to lower regularity, what we did is we wrote down the Einstein equations in a constraint and evolution equation system and integrated some of them on the level sets and in n direction from the horizon to infinity. And the idea in the Robinson paper is exactly the same, but instead of integrating inequalities as we did in the Israel case for densities, we will integrate a magical identity that I won't write down because it's very long. So he finds a magical identity, coordinate independent between new of n and the cotton York tensor, which is completely geometric and contains the cotton York tensor as norm c squared. And of course, this is non-negative, so this gives us a magical inequality. And then he integrates this inequality again over the surfaces and out to infinity. And again, like in the Israel proof, he gets inequalities between m and the area of the horizon and the normal derivative of n at the horizon. These are algebraic, algebraic inequalities. The mass, the normal derivative of the left function at the horizon and the area of the horizon. And again, these together with the information that we have that m is one on four pi the normal derivative at the horizon times the area. These are sharp, so they are saturated. So all the inequalities were identities. In particular, the cotton tensor had to be zero everywhere and we're back in the situation where Masoud al-Alam took us and bunting. So all saturated. And unfortunately, no one has found a way to write a magical identity involving the vial tensor instead of the cotton tensor in high dimensions. So no one knows how to generalize this proof to high dimensions either. But it's conceivable that the magical identity can be used for other things as well. So I still think it's worthwhile studying this proof. There's very recent work by Agostiniani and Masieri. Of course, this is Lorenzo who is here from 2017. And because this is after the review, I have to give the reference on the geometry, the level sets of bounded static potentials. And this appeared in CMP in volume 335, issue one. The page is 261 through 301. And their approach is in some sense reminding me of the Israel approach and of this Robinson idea. Only they don't do a magical identity and then integrate it out on the surfaces and all the way to infinity. Instead, they write down magical functionals on the surfaces, I'll write them down in a second, and show that they are non-increasing from the horizon to infinity. Which also then gives them inequalities between all these numbers, which also can be seen to be saturated. And then their functionals are such that if they're not non-increasing but constant, they are characterizing Schwarzschild. So it's a similar idea, but instead of integrating something from the horizon all the way up to infinity, you construct quantities where you do integrate over the level sets, but then you look at them and you discover that they're non-increasing. So, study integrals for numbers P bigger than three. And these integrals are called UP, and they go from zero one to R, and they go like this. You take a number T and you assign to it two M over one minus T squared to the power two P minus one in three dimensions. And you integrate over the level set where N is T and what you integrate, maybe I should write it consistently like this, with the area measure that we've always looked at, you integrate this, which is the normal derivative also, if you want, to the power P. So for P equals one, this is just one, and this is just the integral over the normal derivative that we've studied that gave us this identity down here by the divergence theorem. But now we introduce this additional parameter with a normalizing factor, and in fact I should mention that the proof assumes that the mass is positive. To begin with, it even makes sense of this factor, and to get a definite sign on this thing. And then they work really, really hard with method from potential theory and geometric measure theory to show that these integrals as a function of T, so as a function of the level set, value N are non-increasing. And also, U P equals constant, and I'm glossing over a lot of details here, of course, if and only if we are in Schwarzschild of mass M. So now again, instead of having something that we integrate all the way out from zero to, from zero, which is the horizon, to one, which represents infinity, we look at functionals that are now non-increasing. So, most likely decreasing unless we are in Schwarzschild. And this, of course, then gives us, again, inequalities between the normal derivative of the lapse at the horizon and the area of the horizon. And these are saturated as, again, M is one on four pi, nu of N, N equals zero times this area.  And then this implies that we Schwarzschild. So we got rid of the cotton tensor. So in principle, you could think that this method works in higher dimensions if you put their dimensional power in the correct way. And in fact, I think their monogenicity properties are very nice also in higher dimensions. However, you cannot, as far as I understand, get these things to be saturated for the same reason that the Israel proof doesn't generalize to higher dimensions and that is that you would need a Gauss-Boune on the way. And if you're in higher dimensions, you can't do a Gauss-Boune and then you cannot get saturation and then you cannot conclude. So the problem again is not in the analysis but in the geometry. But I think it's a very, very nice proof and it's also a nice new aspect to this theme of looking at the level sets, integrating something over the level sets and then getting inequalities between the quantities on the horizon and the quantities at infinity, now not by integration, but instead by monotonicity properties of some functionals. So the next other proof for N equals three I want to discuss is a proof by myself which is not quite yet on the archive. Maybe only for baby reasons. She doesn't like me to sit on the computer, okay? So it's in progress but that's not a problem with the proof but just with the typing. And it's a Banting Masulu-Lalam argument up to here and then it offers a different suggestions of how to recover Schwarzschild without using the cotton tensor, without using anything that's inherently three dimensional. So in particular it will allow everything to be done n-dimensionally later, okay? And here's the idea. We found, so we found, I'd write it three dimensional to be consistent with last class that this manifold M3 wedge union P infinity. So M3 wedge was this guy. No, there wasn't any wedge. P wedge was actually the same thing as R3 delta by the positive mass theorem. But I mentioned yesterday that we also found that the Laplacian with respect to this added metric of the conformal factor phi was zero on M3 wedge. Be careful, it's not true that this is vanishing at this point that we glue in because the conformal factor has a singularity at this point. And recall also that the definition of phi was nothing else but two over one plus n hat where n hat was the lapse function that we defined here just to be the lapse function and here to be minus the lapse function. And now we argue as follows. One, we think about the topology, meaning we prove that I is one. This goes like this. We know this guy is R3 globally, globally isometric to R3. So if there were several connected components of the horizon to begin with, then this would have non-trivial topology. For example, second fundamental group or something, whatever you like. And this has trivial topology but they're isometric globally, no. So just for simple topology reasons, there has to be just one black hole. Then the second aspect is the geometry of the horizon and now it's a singular horizon. So I mean a single horizon, not a singular horizon, luckily. So the argument goes like this. We showed in the beginning, or I claim that we could show in the beginning that this horizon is ambilic. This trace phi part of the second fundamental form vanishes. Then we perform the conformal change. It's well known that being ambilic is invariant under conformal changes. So it's also ambilic in this manifold. But this manifold is R3. So now we know the horizon is ambilic in R3. So sigma two, ambilic in M3. Sigma two is ambilic in R3. But it's a very well known fact which can be verified by a computation. And if you've never seen it, I suggest you do that. That any conformal, any ambilic surface which is closed in R3 is nothing else but a round sphere. There are no other ambilic surfaces. The only closed ambilic surfaces in R3 are spheres and spheres of the form radius S0 and center C. So C is in R3 and S0 is a radius. I'm calling it S again recall because we call the coordinates in R3S, the radial coordinate S and not R to remember that we are in the isotropic coordinates in Schwarzschild. Okay, so again, we were ambilic to begin with. So we are ambilic in R3 and the ambilic surfaces in R3 are classified and they're all round spheres. So that means that topologically, now we know we are round sphere horizon and we know more. We know in R3, the image of the surface and I'll suggestively draw it like this is a round sphere intrinsically. If it's a round sphere intrinsically, we can verify that it has to be been a round sphere in here because the conformal factor, and I'll write this down in a second, the conformal factor on the horizon. If you look at the formula and had a zero on the horizon, so the conformal factor on the horizon is a constant. So the metric on the horizon induced in the original manifold and induced in R3 are related by a power of, by two, by a power of four, so by a constant. So it's a round sphere. So phi on sigma two is constantly two. That means that sigma two in M3 was already intrinsically round. We knew it was extrinsically totally geodesic. So round, but also now we know it's intrinsically round because the conformal factor is a constant. And three, it's in fact, the whole thing is four-chilled by the following argument, which is much simpler than the argument using the cotton tensor or the explicit computation that Israel gave. And the argument goes like this. We know that phi is a harmonic function outside the ball of radius S0 around the center C in R3 because it was harmonic with respect to the new metric, but the new metric turned out to be Euclidean space. And if we stay away from this one point, which is inside this ball, we get this. We now just computed that phi on the boundary of this ball is two, which is a constant. We also know from the asymptotics that phi goes to one as S goes to infinity, S being the radial coordinate in R3. Well, I can solve that, right? So that means that phi of some X in R3 is nothing else but one plus, we check S0 over X minus C. So here's the picture for any point X out here. I compute, and I'll now write this as C. I compute the distance from X to C, plug it down here, S0 is the radius of that ball and one is the value at infinity. So there's, the only thing I don't know is the center and the radius. But now I know more than the asymptotics. I know that N goes like one minus M over two S, M over S. So I can do a compare asymptotics N being one minus M over S plus lower order terms. And hopefully not too surprisingly, get that S0 is M over two, which is the radius of the horizon in the isotropic coordinates. So this is indeed one plus M over two X minus C, which is almost the conformal factor from Schwarzschild except that we have the distance to some constant point C instead of just having S there. But that's a matter of translating the coordinates on R3. So translating coordinates on R3 says this is Schwarzschild and we're done. So that's a method of recovery that doesn't ever switch into the Schwarzschild coordinate picture like the others do, which is why it seems so much simpler. And instead of using anything about the geometry of the original manifold, we now stay entirely in the R3 and exploit this equation that we just computed as a side computation for computing that the scalar curvature transforms to something non-negative. And then we forgot about it in the other proof. And if we don't forget about it, it gives us the answer immediately because it really is already the conformal factor for Schwarzschild. And this clearly generalizes to higher dimensions and explains why there's an N minus two here in higher dimensions because that's the seed solution of the Laplace equation in that dimension. It also gives you a very nice insight why there's no Schwarzschild for N equals two because then you would have a logarithm and fundamental solution which would grow and you could never get something asymptotically flat. Questions? The C is the center of mass. That's a good question. So if you think about your ordinary Schwarzschild solution and you just translate your coordinate, then that's what's gonna happen in the isotropic coordinate picture. It's not so easy to do that in the Schwarzschild coordinate picture because it exploits the spherical symmetry much more. And of course, geometrically, you can't distinguish the two Schwarzschilds because they're just written in translated coordinates. That's why in this proof that exploits the geometry very much, it comes up naturally. Because I'm not exploiting the spherical symmetry, instead I'm exploiting this conformal picture which still has this translational freedom there. So somehow you could have anticipated that you catch the center of mass term, right? So usually we center our Schwarzschild at the origin even in the isotropic coordinates, but in the isotropic coordinates there's no reason to do so except that it's simpler. So this shows how this works. So let's move to higher dimensions explicitly. So as I said, it's not known how to, or if it's possible to, to generalize to higher dimensions any of the proofs, Israel, Robinson, and Agostiniani and Masieri. Agostiniani, Masieri, plus their matter versions because they use Gauss-Bohne. In 1989, 1998, one, I already mentioned this yesterday, gave a generalization of this part of the Banting Masoud-Ullalam proof and then a recovery mechanism using the vial tensor instead of the cotton tensor for higher dimensions. So this is, and you find the reference in the Heuser book and, or in the Robinson paper, n bigger equal to four, generalize Banting Masoud-Ullalam up to Schwarzschild recovery and then produces an involved argument using methods including the Bohner identity, Bohner formulas, the Moritz theory, and the vial tensor to be construct, to recover Schwarzschild. And as they use the vial tensor, these methods only work for n bigger or equal to four and not for the case already covered by Banting Masoud with the cotton York tensor. Then in 2002, Gary Gibbons and Iida and Shiro Mizo, letter two from Japan, I believe, in 2002. And again, the reference is in the Robinson article, citation 1112 in Robinson because there are several ones by these authors in that year. They do the same as Fouang, so meaning they generalize the Banting Masoud procedure all the way up to the Schwarzschild recovery to higher dimensions. And then they try but recover Schwarzschild maybe a little bit a-historically under the condition that Dn is nowhere vanishing by a rather explicit method, similar to that of Israel actually. And in fact, they argue that in principle, at this point of the proof, we should actually be able to show that there is no zero in the gradient of the lapse, which is one of the things that Wang actually does. He shows that using the Bohner formulas in the Moritz theory that actually this doesn't happen after you already know you're globally isometric to globally conformally Schwarzschild, globally conformally flat. So they're right, but they don't show it. Both these proofs are for spin only, for spin manifolds only, because at the time, the only available positive mass theorem in higher dimensions was Wittens, and Wittens proof with spin manifolds, and then the Bartnick's regularization thereof, or low regularly version thereof. Using Bartnick's weak regularity. Neither of the papers actually gives an explicit explanation why the fact that a manifold is spin, it survives this construction. If you think about it, what the assumption would be is that this northern manifold be spin to begin with, and then you would have to verify then doubling preserved spin structure. And I thought about it and I'm convinced myself, but I can't reproduce the argument of the top of my head. And then that this conformal change doesn't affect the spin structure, but then you glue in one point and that could very well affect the spin structure again, and they don't say anything why, but I convinced myself that yes, but that's something, if you're interested in spin structures, I think you should think about why you get a spin structure here. Okay, and neither of them says anything about it. So then, what do I raise? I raise this. The argument that I'm currently erasing goes to higher dimensions easily because all the arguments were completely geometric or about fundamental solutions of the Laplace equation on R3 outside of ball. All those go easily to higher dimensions. There was no Gauss-Ponnet. And in fact, now we know that there is a positive mass theorem in higher dimensions, also without the spin condition by Shane and Yao, and I'll give you the references. So, to higher dimensions, dimensions, and of course, this applies to these other theorems too. Well, I'll say that later. And the positive mass theorem story, I also already said this yesterday, but I still need to give you the reference. Is that Shane and Yao in 2017 in an article called Positive Scalar Curvature, Minimal Hypersurface Regularity, The Singularities, and I couldn't find it published anywhere yet, but on the archive, it's 17.04, 05.04, version one. Together with this paper I mentioned yesterday that my Mac Farron and the Hedy from 2012, which is called On the Positive Mass Theorem for Manifolds with Corners, and appeared also in CMP, Volume 313, number two, pages 425 through 443. So, Shane and Yao prove Positive Mass Theorem under smooth conditions in all dimensions. Sorry, without a spin condition, and Mac Farron, Shackle, Hedy in all dimensions prove that if you have a non-regular, non-smooth manifold, which is smooth enough to start a Ricci flow, then, and has zero mass, then it keeps zero mass under the Ricci flow and becomes instantaneously regular in which case then the Shane and Yao rigidity case tells you you're smooth and thus you're globally isometric to RN. So using this, instead of the Bartnick version of Witten's proof with spinners, we get away from the spinner condition by now. And of course, this can also be inserted into Huang's argument and also into Given's Ida Shiromitsu and removes the spin condition in those works as well, a posteriori. So that means we now have proofs and several versions of proofs for a higher dimensional, static vacuum black hole uniqueness. So we know in all dimensions, and bigger or equal to three, if we start with potentially several static vacuum black holes, which are asymptotically, sorry, Schwartz Hildian. So suitably isolated, then there is only one black hole and it is actually Schwartz Hildian of the same mass we started with. And this mass has to be positive. None of these theorems that I just mentioned today in the higher dimensional section needs to assume positive mass, they all get it out as a consequence, as I explained yesterday. Sorry. But however, if you think about the proof again, let's think about the proof again under the aspect of have we ever used the part of the static equations which involves the Hessian? We ever used this equation. We doubled, okay, first we derived properties of the horizon and when we derived properties of the horizon, we did use this equation and I didn't show this to you, so I have to tell you. So in order to show it's ambilic and to show that the normal derivative of the laps is a constant, you needed this equation, okay. But after that, we doubled, didn't need this equation. We did a conformal transformation, we didn't need any equations for that. Then we computed the scalar curvature of this new guy. That involved only the Laplace of, so we used the Laplace with respect to g of n is zero, for sure, in the computation that the scalar curvature be non-negative and we used that the scalar curvature is zero to conclude that the scalar curvature of this new guy is zero. But really, for the positive mass theorem, we only needed that the scalar curvature of this guy is non-negative and to show that, we would only have needed that the scalar curvature of the original guy be non-negative and we could have used only that the scalar curvature of the original metric is non-negative. Then we did a positive mass theorem that didn't, of course, need the static vacuum equation for the Hessian at all. Because it's just the positive mass theorem. And then the recovery method I erased over only used Laplace phi is zero. And Laplace phi is zero was a disguised version of Laplace n is zero, okay? So we didn't use this either. So we didn't ever use this, except when we derived that the horizon was ambilic and had constant normal derivative of the labs. For the proof, I put Zenerbaum in progress so you know which proof I'm talking about. Except that we use Hessian n equals n Ritchie, which then implies scalar curvature is zero to derive that the trace phi part of the second fundamental form vanishes and that nu of n is constant, which we actually never used in this proof anyway. We only use that it's non-zero. Okay, so that's the only point where we use this. So we actually proved a much more general theorem. We proved a theorem saying that if, if m and g is Romanian with smooth boundary and the boundary has vanishing mean curvature and vanishing trace free part of the second fundamental form. So the boundary is totally geodesic. And if n on m3 to our extent smoothly to the boundary vanishes on the boundary and the normal derivative of n does not vanish on the boundary and n is positive in the interior and m and g n is asymptotic to Schwarzschild or asymptotically isotropic as we always also call it of mass m. And for those who are interested in the details, the error term is Sn minus one and such that the scalar curvature is non-negative, which is also known as the dominant energy condition in this context. And the Laplacian of n vanishes in m3, mn, then Schwarzschild of mass m. Now this could be rather surprising because in the beginning we started with all this rationale, yes? Oh, it's the three, thank you, more. So in the beginning we started with all this rationale that we have several black holes and they gravitationally attract each other and therefore they should accelerate and shouldn't sustain in static equilibrium. And now we got rid of the static in the thing, right? Instead we said let's have things that are black holes and have more properties like black holes in static space times, but let's forget about the static other than that and let's just think of a function n that takes the role of the lapse if it was static, but doesn't have to be static and the whole result sustains. There really, there wasn't so much statisticity in the proof and at least if you modify the last bit of the proof in the way I showed you, there isn't much statisticity except that the horizon components are priori, they're potentially multiple components, have the properties that they would have in a static vacuum environment, right? So from a PDE perspective, this says instead of looking at an old highly over determined system of elliptic PDEs with highly over determined boundary conditions, you don't need the highly over determined PDE anymore. Actually it's sufficient to look at a harmonic equation with so many boundary conditions. Why are there so many boundary conditions? On the inner boundary we put the directly condition of vanishing laps plus non vanishing normal derivative which is almost a Neumann condition but at infinity we impose that the function goes to one that's a Dirichlet condition with two derivatives. So the first derivative is also prescribed which is another Neumann condition and then the second derivative is also prescribed which is even stronger, okay? So we have hugely over determined boundary conditions for this lapse function. So it's not a good idea to think if I have a metric that on MN which has non-negative scalar curvature and I'm totally geodesic boundary, then I'll just find some N like this and then everything is Schwarz yield. That would be too much to hope for because of course there are manifolds with boundary that is totally geodesic that are not Schwarz yield. But if you can solve this highly over determined system of boundary conditions for the lapse then you'll be able to solve boundary conditions for the lapse then you're Schwarz yield which just means that it always never happens. Okay? So this could be slightly surprising, hopefully. So the error, the lower order term in the definition of asymptotically isotropic I never said how fast the lower order term needs to decay. And this is saying in dimension N the lower order term would be at least of the order 1 over S to the N minus 1 and then the first derivative like 1 over S to the N and the second derivative S to the N plus 1. It's the same condition appearing in the Banting Massoudan. I'm not claiming that this is the lowest assumption that you can make it's just an assumption under which this works and under which the gluing of the P infinity point is straightforward. I didn't want to glue it in I think, but I didn't want to go in that direction. So let me spend the last 15 minutes on explaining a little bit about this photon sphere business and I'll continue writing this theorem later so leave some gap in your notes if you're taking notes. Oh, in fact maybe let's do it immediately. You can generalize this theorem also remains true if additional components of the boundary are allowed to be of the following form. N is constant so the components like yesterday we will call them sigma 2i. The lapse is constant the N there exists a constant C i bigger than N minus 2 over N minus 1. I'll explain what this means in a second. And the scalar curvature of the induced metric sigma i is C i times h i squared and h i is the induced mean curvature of course is constant and 2 times the normal derivative on sigma 2i is C i minus the special N minus 2 over N minus 1 times h i times N i is constant. Let me say this in words now we are allowed to start with the northern manifold so a Romanian manifold with the lapse function that's harmonic and non-negative scalar curvature with some components that have mean curvature 0 lapse function 0 and are ambilic and some components that have this other quantity I forgot to write that they're also ambilic so the lapse function is a positive constant the normal derivative of the lapse is a constant the scalar curvature is a constant the mean curvature is a constant and they're ambilic and all these constants are related via this parameter constant C i and of course h naught needs to be 0 as well and we could have some components of one kind and some components of the other kind if you take C to infinity then you can think of the horizon components as arising as limits of those anyway and I'll say a few words later what the C i means and then everything is also Schwarzschild so the boundary conditions you can make them different from just horizons conditions this way and before I'll say what this has to do with photon surfaces and photon spheres let me explain the only two pieces where this makes a difference in the proof we cannot double because if we have okay let's have this a horizon component so h equals 0 and so on and so forth but the other components we cannot double through them because they have positive mean curvature and so if we doubled they would get a crack but we need C 1 1 so instead what we do and we first did this with Greg Galloway in a special case of this and then I did it on my own we glue something in and we glue something in which has a horizon and it's end and then we double and the thing we glue in is in the work that I did with Greg Galloway and I'll give the references later to whoever is interested and write them in the lecture notes and they're all on my website in the case with Greg Galloway we glue in a piece of a schwarzschild neck so therefore the static equations hold on this glue in piece also and then all the rest just continues the same way in higher dimensions we only have that the surface has positive constant scalar curvature if you look over there and in higher dimensions this doesn't tell us even a sphere which was the whole problem why Israel's proofs and so on didn't generalize so we glue something in which looks like a schwarzschild but the base is not a round sphere the base is something else and this means that the static equations won't necessarily be satisfied on this neck which is why it's great that we didn't need the static equations anymore and in fact we can do that in a way that the scalar curvature remains non-negative and the lapse function we will just continue to call the lapse function will be harmonic on the sky so that then the rest of the argument goes through because we didn't need the Hessian equation anymore okay and I can like it will be on the archive soon with the details and then the recovery just works exactly the same only you have to first forget about the glued in regions and do the whole argument I gave just in the original manifold as before okay so much for the strategy for the proof now what's the role of the CI if you think about so this is positive if you think about a surface of this type of course a spherically symmetric cross section in schwarzschild would be of this type and CI would tell you that it fits into schwarzschild anywhere between strictly speaking the horizon and infinity the threshold is exactly what you get if you try to fit it into Euclidean space and if you go below the threshold you would be in a round surface in a negative mass schwarzschild so this characterizes the round surfaces in positive mass schwarzschild again this is the threshold for Euclidean space and if you go below you would be in a negative mass schwarzschild so it tells you you're one of the surfaces that could a priori sit in a schwarzschild as a round surface so here's the story with photon spheres and photon surfaces if you're in a space time a photon surface is a time like hypersurface in any space time which is ambilic and you can think about this for a while and the reference will be in the lecture notes and this ambilic turns out to mean nothing else than if you take a photon so a null geodesic started tangentially to the surface somewhere then it will remain in the surface okay that's why it's a photon surface it catches photons and then a photon sphere we only know to define that in a static manifold static space time it's a photon surface with n equals constant along it photon surface p with pn with n equals constant along pn so what does that mean if you think about it for a while it means the color of the photon stays the same all the time from the perspective of the static observers so no redshift or blueshift the color of the photon or the energy level of the photon stays the same it's a way of saying that the photon is trapped and an example of such a photon surface is the cylinder excuse the stupid language this is a cylinder in Schwarzschild with spherical base r equals 3m and then up in time direction a cylinder and this is such a photon surface and in particular also such a photon sphere because n only depends on r so n will be constant meaning that if you stand if you were able to have a rocket that allows you to stand on this cylinder you could see the back of your head okay and it wouldn't be color shifted okay so that's a photon sphere by definition and excuse the language it could be in different spacetimes also of a different topology than just a cylinder over a sphere so now Greg Galloway and I showed that in Schwarzschild in all dimensions you have a funny diagram so this is Schwarzschild n bigger or equal to 3 and this is r equals 3m the photon sphere that's very well known and we found a whole family of photon surfaces in Schwarzschild which came to a surprise to us at least and I'll just draw a diagram so there are some which look almost like hyperboloids there are one sheet of hyperboloids in Minkowski space I'll also photo on spheres then there are ones that come in from infinity and asymptote to this 3m then there are ones that come in from infinity and asymptote to the black hole horizon which in our language is 2m then there are some that come from the horizon go to the photon sphere and return to the horizon then there's no more color but there's still one family coming in from the photon sphere and going to the horizon all of these if you take them and rotate you get surfaces in Schwarzschild spacetime which are time like ambilic so they're photon surfaces and they're spherically symmetric so these are all photon surfaces and just the one with constant radius is a photon sphere but they all have spherical cross section and you can if you look at the intersection with t equals 0 each of them is a sphere of some radius and each of these spheres we can show our priori if they are in a static vacuum spacetime we can specify all these conditions so as a consequence a theorem by Greg Galloway and myself which applies this abstract theorem is saying that if you have a static vacuum spacetime which is asymptotically Schwarzschildian and it has an inner boundary which arises as a photon surface such that at each cross section the level it's a level set of the laps and I won't have time to write this down it's Schwarzschild in particular if you start with the photon sphere that's a result we had for a while in three dimensions you'll have to be Schwarzschild so not only are black holes in static vacuum spacetimes unique and have to be Schwarzschild also photon spheres and now with the new theorem that allows for non-hashian solutions to show that also in high dimensions this holds true for photon surfaces in three dimensions actually you don't even need to drop the static equations so now instead of just knowing static black hole uniqueness we know something which is in some sense nicer and not clear whether this is a stronger or weaker statement we know this photon sphere or photon surface uniqueness why is it not clear that it's a weaker well on the horizon we have all these singularities of things where we need to be super careful that we don't compute anything that's not well defined especially if we work in low regularity in all spacetimes once we got out that far that we are at the photon sphere everything is super regular in static yeah so everything is super regular here and we don't need to look at static problems that there's no problems that any vector fields become null that we would like to work with so we're in the super regular regime of the spacetime not anywhere near the horizon and still we have a uniqueness and you need to be careful we only have the uniqueness to the exterior that means if you had you could potentially and this was something people in astrophysics believed and we're quite surprised when I first worked on this you could have had a star which is not which is static and it has non-homogeneous structure so it could potentially be non-round and it was expected in astrophysics that if the star is suitably compact it would form a photon sphere around it of trapped light this is saying well if it does then the photon sphere and everything exterior to it is spherical symmetric anyway but the static equations are real analytics so it should be then spherical symmetric all the way down to the star so this is saying if you have a non-sphere symmetric star which is static it won't form a photon sphere which is a surface it can still have individual trap light rays but they won't form a nice smooth surface they will do something more complicated maybe something even as complicated as incur we don't know but the naive picture that it'll form a surface is too naive and has to be given up and the same thing is true for these photon surfaces which I don't know if they have any physical significance beyond being pretty thank you


Our main time stakeholders in the political and educational sectors agree that federalism is the solution to ethnic agitations and other security challenges facing the country. They noted this at the Oshun M.U.J. Madean Edition of Guest Platform by the State Chapter of the Nigerian Union of Journalists. This event marks the Madean Edition of Guest Platform, a quarterly program organized by the Oshun State Council of the Nigerian Union of Journalists. It is claimed appraisal of the state of security in Nigeria yesterday, today, and tomorrow, the way forward. In his speech, Professor of Political Science at Obafemi Awanawa University in Leifere, Olufemi Miko, says one of the challenges is over centralization of powers and the solution is the adoption of federalism. So every structure of governance is faulty. Every structure of governance is saved inappropriately cast. Then it will not be a surprise that such a structure will not be able to deliver on the functions given to it. It has been said that the primary purpose of governance is security and welfare. This is the basis of the social contract. If we address the issue of the constitution, the greatest advantage is for us to be together as a nation. But where one is not even respected in the unity, then there will be the tendency to ask whoever is in charge that let's go the way of Albert Lutuli who says let my people go. Media as a platform, we have given opportunity to experts to come and discuss the way forward. We have praised yesterday we are looking forward to tomorrow and we believe that with the recommendation of the experts in the house today we're able to send a good message out there and we serve as a guideline for government a workable document for them to find solution to the issue of insecurity in the country.


Yo, what's going on, Leafrins? It's Salih for back again and welcome back to the channel. Today, I will be teaching you guys the fastest way to get the free 94 master set player that EA has given us. So, the first thing I want to show you guys is all these objectives EA has given us to do to get the free 94. So, as you can see on the screen, we have shots on goal, we have win 40 games, play 82 games, 286 hits, 475 goals, goals again, shutouts, and then shots on goal as well. So, I'm going to show you guys the easiest way to get all these objectives done as soon as possible so you can get that free 94 ASAP. So, what you're going to want to do is load into a squad battles game. You're going to put it on rookie on any of the teams you see here. So, now once you load into the game, I'm going to show you guys how to get goals done as soon as possible. This is super easy. So, as you can see here, I recorded some footage. What you're going to want to do since it's on rookie, it's easy to do anything against these AI. So, pretty much once the puck drops, you're going to do the walk face-off play. You're going to walk the sentiment on the other team. You're going to go straight down the middle between both defensemen, hold it to your forehand and then shoot it and the puck will go in 99.9% of the time. This is the easiest way to get goals done 100%. Now, if you guys don't know how to do the walk play or the walk face-off, I made a video a while back how to win face-offs in NHL 20, I believe. I'll leave a link in the description. I'll also show you guys how to do it now while we're here anyway. So, what you're going to want to do with your controller, you are going to hold down the left bumper. You're going to time it obviously when the puck drops. Hold the left bumper down and then you are going to go up on an angle with both sticks depending on what handiness your player is. Make sure you put it on an angle so you can get a clean face-off when you don't bump into the other center. Once that happens, it's smooth sailing from there. You go right up the middle, hold it to your forehand once you get close to the goalie and snipe it top shelf. Super easy. Now, as you can see here, you can see some of the games I won. I scored, the highest I've scored in the game is 99 goals. And obviously, that will get your shots done as well. I didn't go perfect on shots, so I think that game I had like 124 shots on goal. Now, the one thing about this is squad battles is going to take a long time, but you're guaranteed to get a lot of goals done, a lot of shots done, all at once instead of playing online games where, listen, I know you guys are all gods and stuff, but you got to think about it this way. You're not going to get that many goals in an online game. So you might as well just grind it on the squad battles if you want to get this card as soon as possible. Now, for hits, again, you can get this while doing the goals, but it's probably a little bit harder. But for hits, if you're done the goals and the shots, what you're going to want to do for hits, leave it on the same difficulty, win that face off back, dump it into the zone and start running guys over. It's not a lot of hits. I believe it's 286 hits. So literally just run some guys over. You'll get that done within, I think I got it done within three, maybe five games, something like that. So you'll get that done. No worries. That's one of the easier ones to do. Now for shots on goal, what I'm doing now, because I'm done all the goals currently the time I'm recording this, I would hop into a game, play it out normally. Now when I say game again, squad battles, put it on rookie, head into squad battles, you're going to enter the zone as soon as you enter the zone, you're going to rip a shot. Try to aim low so you can get that rebound and generate two, three shots of play. Now I think the highest shot count I've got doing that is around 95, 100. Again, that's me not really trying. It's just spamming shots everywhere from anywhere and everywhere and making sure it gets on goal. But these shots are super easy to get done. Again, I got this done within five, maybe six games max. It's super easy to do. And again, when you're scoring goals, when you're getting the goals objective done, you're also getting shots done as well. So it's just speeding up everything all at once, which is perfect. And you can only do this in squad battles because rivals, you're not going to get this amount of quantity of shots or goals in an online game. Now for the shutouts, again, you're going into squad battles on rookie.  So you play 11 squad battle games, that's your shutouts there done and over with. That's one of the simple ones to do. Now for the win 40 games again, you're going to be winning games when you're doing these goals. But when you get your goals done and you need to get around, I don't know, 20, maybe 15 wins. What I like to do again, go into squad battles, put it on rookie, score the first goal. And then what you're going to want to do with your player is you're going to skate into the offensive zone.  Couldn't get that word out for some reason.  They'll skate up to you, but they will not touch the puck. Now what I like to do is when I'm doing this, when I was grinding for the wins, I sit my computer there after scoring the first goal of the game. I turn on Netflix, watch Netflix, turn my head to my screen every five minutes and reset it up for the second and third period. And you get your wins done like that. Again, you're getting this 94 very quickly as opposed to playing online games and having that potential to lose a game. These are guaranteed wins you're getting, hopping into squad battles on rookie. Now to play 82 games, I mean, this is fairly easy. You can do the same thing with squad battles and just sitting on that corner, getting it done. I know some other people literally just hop into rivals as soon as the puck drops, they quit out of the game and it counts as a game completed. Now, the only bad thing about that is, if you're in a high division, you're going to be demoting yourself.  So now do that at your own will. I'm not recommending you guys do that because once you get into those lower divisions, the packs or the higher, yeah, the lower divisions, the pack rewards get worse and worse. And I know it's going to be easy to work your way up back into the division you were in, but for that weak arrivals, you're not going to have the greatest rewards. Now, I don't recommend doing this method, but if you want to get the 94 as soon as possible, I guess go for it. But if not, just play all your rivals games, play all your champs games. You'll get that done within a week, maybe even less depending on how much you grind the game. But yeah, guys, that is it for this video. I just thought I'd share ways to get the 94 as soon as possible. I thought it would make sense to do this. Very good stuff. But like I said, if you guys want to get this 94 as soon as possible, follow my tips and tricks from this video. You can get this card within three or four days if you really grinded it out using my squad battles on rookie methods. So yeah, that is it for today's video. If I did help you guys out, drop a like on it. If you guys are new around here, make sure to hit the subscribe button. We're on the road to 4,000 subscribers. And drop a comment down below. I love talking to you guys in the comments down below. If I did help you out, or if I missed any methods for this, again, leave them in the comment section below. Let's help out the community get these 94s as soon as possible. But yeah, guys, that is it for today's video. Thank you guys very much for watching and I'll talk to you guys later. Peace.


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Thank you. We have time for a couple of questions. Super. Thank you. A couple of questions. One in the back. Hi. Thank you so much for this fantastic panel. My name is Summer Lopez. I'm with Penn America. And I'm thinking a bit about the panel we had yesterday as well about hateful content online. And of course we all know much of that is targeted to silence individuals of color and particularly reporters of color. So that's something we're thinking and working on. I'm wondering how you're all thinking about that and addressing it in your work. My God. So I just went through an experience actually this weekend. Maybe I'll just use it as an example. I'm not a contributor to MSNBC, but I'm often asked to be on. So I'm not a paid contributor. It's an important distinction for me. But I did happen to, when Nevada was happening, I did happen to throw a question out to Bernie Sanders. It was a journalistic question, but it was like. And on Twitter. And, you know, I think what I tried to do is I was like, OK, so how am I going to use this and actually flip it and try to create dialogue so that whatever is perceived as anything hateful can somehow I can model how to say, talk to me. I'm listening to you.  But I'm going to try to listen. So that's one way, a tiny way in which, and again, as we've been saying, a lot of this is just human to human, right? So I'm trying to model how we take a situation that could have just like exploded and gotten really nasty and to try to revert it into something that I hope could model a little bit of being productive.  So I like to make a joke because I think laughter is also really important. So the joke is, is that I'm five things that this president doesn't like. I'm Mexican. I'm an immigrant. I'm a journalist. I'm a woman. And I'm flat chested. So, you know, so when people are like talking about feeling fear right now, okay, you know, I'm like, hey, I wasn't born in this country. I wasn't born here. I became a citizen. And so I understand when people are talking about fear and how that fear looks. And I, every single day have to just be like, it's good. I think a lot about Martin Luther King and he rest in peace. And that it's just, it's good. It's good. This is all good. Nothing is going to happen. It's going to be positive. But at the same time, the peace that's coming up on Latino USA I think next week or the week after, we went to Milwaukee because a Peruvian-American citizen went to get a taco and some guy threw acid on his face, battery acid on his face. And great crimes against Latinos are the fastest growing. Have surpassed now hate crimes against Muslims. So this is real. The hate is real. Our last question. Hello. First of all, great panel. Thank you so much. My name is Kyra Kiles and I represent a nonprofit media organization called Wire Media, formerly Youth Radio. And my question is, you know, I've spent 20 years in journalism. I think one of the fixes that mainstream media has tried to offer up is fellowships, fellowships, fellowships. I know. And so what I'd like to hear from the panel is a meaningful way that we can make change within these mainstream newsrooms that are not hospitable so that we can end the, you know, fellowship apocalypse, as I would call it. I think that's so well said. You know, fellowships are an interesting thing, right? Because they seem like tremendous opportunity, especially the good paying ones that actually have benefits now, right? But what ends up happening with folks is that they're often the least senior and the tethered to often grant funding that can be fleeting because foundations have obviously a lot of priorities and there's a ton of need. And they are, and so all of a sudden they create this mirage of diversity because you have people that come in. They're doing great journalism, but they aren't in the C-suite. They're not making strategic decisions about the organization. They're not tethering things to revenue initiatives that are key, right? It's not connected to money. It's connected to content, but nevertheless they're still incredibly important and they provide great opportunities. I think what needs to really happen here is that there has to be a collective understanding that pipeline is key and that news organizations actually need to think of themselves as their own, dare I say, farm system. That you need to grow your own from within with a plan to bring people from here to here. That they don't just come in. They're not transient, right? They are a part of a strategic, a positive part of the strategic goals of the organization because fellowships while they're wonderful, they are not a fix for the diversity problem that is facing these organizations. So I think one thing philanthropy could do is maybe creating a collective pool that funds early on a fellowship program that could be sort of almost for the entire sector, right? But I think for organizations that are larger, have revenue, that they need to actually think of a strategy around diversity. How does this align? They do this all the time with revenue initiatives. So how does the people, how does making people a part of your, the hierarchy in the organization to go from junior to senior to leadership to heading it, how is that tethered to your most sustainable future? So it can't just be a program. It has to be about what the business tethered to the success of the operation. And on that, thank you, Martin. I think when you put it also back to newsrooms to ensure that journalists of color don't have to be poor to continue on fellowship after fellowship to thrive and get a full-time paying salary job. Also, I think, Maria, to your point on addressing online hate speech, we also have to hold newsrooms accountable. What can newsrooms do to protect journalists who are targeted and create conditions of safety? It shouldn't just be on the journalist to handle it. We need new systems and protocols to support journalists of color and ensure their safety and that they can thrive and do their best work. So I want to thank my well-esteemed panelists, Jeff Faroda, Martin Reynolds, and Maria Inosa, and thank you all for sharing in this conversation with us.


Polymer dielectrics have been used in many applications, but they often suffer from poor thermal stability and low energy storage capacity at higher temperatures. To address this issue, researchers developed a new strategy to tailor structural units for the design of high-temperature polymer dielectrics. They then synthesized 12 representative polymers based on their predictions and tested them experimentally. The results showed that the high-temperature insulation performance decreases when the band gap increases beyond a certain point, which is related to the dihedral angle between neighboring planes of conjugation in the polymers. By optimizing and predicting the structures, the researchers were able to increase the energy storage capacity at temperatures up to 250 degrees Celsius. This strategy could potentially be applied to other polymer dielectrics to improve their performance. This article was authored by Rui Wang, Yu Jishu, Jing Fu, and others. We are article.tv, links in the description below.


A Catholic priest has declared his intention to contest the governorship seat in Benwa State. In the 2023 general elections, Rev. Father Heisenth Alia visited the State Secretary of the All Progressive Congress, APC, to make his intention known. Alia said he has what it takes to run the race and win, citing his long years of service to humanity, in the church, in communities, as relevant qualifications. My presence before you and the State Expo again is to seek permission from you that are in this race to win it. From the backdrop of the church, in the church, our clergy are trained to be everything. We pitch pennies, but we build church structures. We pitch for the little we get, we place it where it is supposed to go. And behold, much on accountability, responsibility, and transparency. That must remain my foothold. As I seek your blessing to step further with consultations for the first, the number one seat in the state, I'm also before you to pray your energy blessings on me. Respondent, the Deputy State Chairman of the APC, Stephen Sav, extolled the leadership qualities of Father Alia, but charged all aspirants to work for their victories. May the request, you may the request, and you say we should allow you go out and contest selection. On behalf of the Party Secretary in Faki Esco, who are seated here, we say yes. We have allowed you to go all out for the nooks and crannies of Benway State, consult widely. What we will not do as an Esco is to adopt an individual candidate who will not do that. So you get notified about fresh news updates.


Here you see a source that provides us with particles whose mass we want to investigate. Positively charged particles, negatively charged, or even neutral particles are shot out of this source. Nobody knows how heavy these particles are. And exactly therefore, you need a mass spectrometer to find out their mass. For a simple mass spectrometer, you first need a velocity selector. This consists of a plate capacitor whose plates are electrically charged. By charging the capacitor, an electric field is formed between the plates. Each time a particle enters this electric field, it experiences an electric force. This force deflects the particle either downwards or upwards, depending on how you have charged the plate capacitor. You can calculate the electric force acting on the particle with q times e. q is the charge of the particle and e is the electric field strength between the plates. You can control the field strength by changing the electric voltage u at the capacitor. In contrast to the electric field strength, you know the voltage because you set it yourself on the voltmeter. Therefore, you write e as u divided by a. a is the distance between the two plates. Next, you put the plate capacitor into an external magnetic field. So that the formula for mass does not become too complicated, we align it in such a way that its field lines are perpendicular to the direction of the particle's motion and perpendicular to the electric field lines of the plate capacitor. From your point of view, we allow the magnetic field lines to point out of the screen or into the screen. The strength of the magnetic field is described by the so-called magnetic flux density b. The larger b is, the stronger the magnetic field. If a charged particle enters the magnetic field, it experiences a magnetic force, in this case downward. It is also called Lorentz force and its direction can be determined with the so-called right hand rule. If the direction of motion of the particle and the direction of the magnetic field are perpendicular to each other, like here, then you can calculate the Lorentz force with q times v times b. Where v is the velocity of the particle. The knowledge of the velocity is important to determine the mass of the particle. The problem is, you don't know the velocity explicitly. But don't worry, if you now add an aperture to the right side of the capacitor, then you can determine the velocity yourself. With this aperture, you determine which particles are allowed to pass with a certain velocity and which are not. All other particles are stopped by the aperture. This experimental arrangement is called a Velocity Selector, or Wien Filter. A particle that has passed through the aperture must have traveled straight ahead through the electric and magnetic field present in the velocity selector. It was neither deflected by the electric force nor by the magnetic force, because otherwise the particles would not have made it through the aperture. The forces have acted on this particle with the same magnitude, but in opposite directions, which is why it didn't get on the wrong track. You express this balance of forces mathematically by equating the Lorentz force with the electric force. q times v times b is equal to q times u divided by a. To find out the velocity, you rearrange the equation for v. The charge q cancels.  So change the electric voltage and the strength of the magnetic field to set your desired velocity. Wonderful. You now have particles with known velocity landing behind the aperture. Now pay attention. This part is very crucial for a mass spectrometer. Behind the aperture there is also a magnetic field. To make the formula for the particle's mass easier at the end, we take the same magnetic field as inside the plate capacitor with the same strength and the same direction. So the particle flies out of the velocity selector into this magnetic field and experiences again a magnetic force, the Lorentz force, which deflects the particle into a circular path. After a short semicircular flight, it lands on a detector plate. On this plate, you can read how far the landing point of the particle is away from the aperture. In other words, you measure the diameter d of the circular path. The particle performs a circular motion because the Lorentz force is perpendicular to the direction of motion and thus acts as a centripetal force. So set the formulas for centripetal force and Lorentz force equal. q times v times b is equal to m times v squared divided by r.  It is part of the centripetal force formula, so rearrange the equation for mass. You can replace the velocity of the particles with the formula derived before. Replace it with u divided by a times b and the radius r is simply half of the diameter d. You can measure it at the detector plate. Now you have besides a built mass spectrometer also the corresponding formula to calculate the mass of a charged particle. m is equal to q times a times b squared times d divided by 2 times u. As you can see from the formula, to determine the mass of the particle, you must know the charge of the particle. Otherwise, you can only calculate the specific charge q over m. Furthermore, you obviously cannot determine the mass of uncharged particles, for example neutrons, because they do not experience Lorentz force in the magnetic field. Thus they do not perform a circular motion, but fly unaffected straight ahead. Well, then you cannot determine a circular diameter and the derived formula would make no sense. This formula also tells you some other useful facts about mass spectrometers. If you rearrange the formula for the diameter d, you will find that a heavy particle flies a larger semicircle than a light particle. A larger charge, on the other hand, will cause a smaller semicircle. So, that's it. Now you know how a mass spectrometer works. If you want to know more about it, feel free to visit my physics website when you visit denger.org. With this in mind, bye and see you next time.