It is indeed a pleasure to be here. As you know, Arun is fungible. He can do almost anything. I saw him at ARPA-E and I had the pleasure of serving on an advisory committee with him at ARPA-E, but I want to comment to the students in this class how lucky you are in this classroom, how lucky you are to be here.  He got early acceptance into Harvard. He was a world-class rower. He made the U.S. national team rolling, and he applied at Stanford. He was going to come to Stanford, but he didn't get in. So he went on and got a scholarship to go to Oxford, academic scholarship. But he's a row scholar. He really went so he could row another year. So he rode for Christchurch and worked here in San Francisco two years, did two startups, applied for an MBA program here at Stanford, and didn't get in. He applied at Oxford and Harvard, and was able to get in without any problems. So congratulations to the students. That is saying something for how tough it is to get into this school. But I want to talk about the power grid, and I title it, the future ain't what it used to be, because we're seeing probably the most rapid change to how things are operating across the grid. Arun has covered this. PJM, my former sponsor that I worked for the last eight years. The most important number on this screen is 21% of the GDP North America is in that one grid. So you can imagine the economic impact, the importance that the grid operation, the efficiency of grid and electricity is one where efficiency really matters in terms of the cost of energy and how the grid operates. But bottom line is already, Arun has already alluded to what is the single most important engineering achievement of the last century? The National Academy of Engineering, which Arun is a proud member of as well, said electricity and electrification has done more to improve our standard of living. It's a lifeline to our homes and the lifeblood of our economy. Nothing has improved our productivity or our standard of living more than electricity. And I might add of the top 10, six of the remaining engineering achievements are fuel by electricity. Fun things like computers and TVs. There's some fairly important things like refrigeration for our food, no botulism. Like water treatment, no cholera. The ability to have as many people working productively in North America. If we're smart, and I think we are, and Elon Musk is working with me on this, electrification of automobiles. I'm going to talk about it a little bit and how important it is. Now you hear a lot about distribution system in microgrids. I know you've those of you that are in the engineering department. How long do you think it will take to build a microgrid? I know the answer to this. It's about five days. This is Superstorm Sandy. This is Wall Street. Cables laid in the street. It took about five days to build out that microgrid. And we had Wall Street back up and running. There's something about as-built drawings that we're missing. I might add, and there's a few things you don't like, but the importance of when Superstorm Sandy came through, 8.3 million people were in the dark. But I want to emphasize efficiency matters in this business, and I want to zoom in on a micro microgrid in the middle here. This guy's found a market, an H market. He's charging iPhones. I looked at him. I said, hmm, pretty lean, mean. He could be a rower. He could row for Stanford, I bet. But you look at the efficiency. It's obviously internal combustion process that he's using here. I gave him about 20% efficiency on the conversion of calories to energy. I gave the bicycle 95%. It looked pretty good in terms of the chain drive, but that's an 18,000 BTU machine per kilowatt hour. A combined cycle gas plant runs about 6,000 BTUs per kilowatt hour, three times as efficient. But that's not the problem on this. The problem is in the background. 99 cent pizza. Now that's pretty cheap. I figure 300 calories is the best you can do with 99 cent pizza. He can charge about three or four iPhone 6 with one slice of pizza. But if you used him in your home, in your home's about 3,000 square feet, say, your power bill would be $35,000 per month. And that's mostly pizza cost, not to mention the labor. So how efficient the grid is makes a lot of difference. And how secure it is makes a lot of difference. And by the way, that is a pizza place in Washington. The fuel is what they call the pizza place. Things that keep me awake at night as being a CEO of the largest grid operator are the big events that happen and impact the grid. Obviously, Fukushima, I'll talk a little bit about that. DeRachos, which I didn't know what was 15 years ago. I know quite well what they are now. Hurricane Sandy. And we just went through a solar storm, so earth and solar weather. Cyber and physical security, including electromagnetic pulses are the big black sky vents that we worry about across the system. Now, I was trying to think, which substations do you think we need to protect? Metcalfe is right here in your backyard. There was an attack on Metcalfe. FBI re-enacted the attack. It may have been one person, maximum two people that did that attack. There's 45,000 substations, all of which you've got cyber and physical security concerns. So I tried to visualize what you're up against. If you're up against a cyber attack, it's not easy. Here's the way I visualized it. Anybody recognize? This is from Star Wars. I was questioned whether I had copyright rights to use this picture, but I can tell you it was so long ago. But this is the internet as I saw it last Friday. It's not a very happy place to be doing business. A lot of the controls and markets and whole business economy. So there is trillions of dollars a day of energy and market trading going under the Atlantic Ocean on fiber optic cable each and every day. So being able to protect the system for physical and cyber security is becoming a very high priority as we go forward. We've met with some of your computer science security folks today. My vision is that you move to a cloud computing, but it's a private cloud. It's under an iron dome and I show Titan computer. This is an 18 megawatt computer. I measure computers differently than the computer science department. But this is a supercomputer at Oak Ridge National Labs. I think the fastest computer in the world right now, it was last year. But how can we do the computational needs with parallel processing in a very secure environment that uses dark fiber that we already own to a large extent to be in a secure high performance computing environment for the analytics on the grid. Private cloud computing is very efficient in my mind. I prefer it being under an iron dome. I know Google will and Amazon and others will tell you that you can secure a cloud computing. But if I don't know where my data is and which computers it's running on, it makes me very nervous when I think about it. But it's a partnership between the universities to write the parallel processing code that we need and the national labs is the way I see that going. I want to hit on what are the major uncertainties we're faced with today? What is the demand for electricity going to be driven in part by Elon Musk and what's happening here in the development of electric vehicles? Extreme earth and space weather, I'm going to hit that pretty hard. The largest fuel switch as we move from coal being the backbone of the system to natural gas and renewables being the backbone of the system and the intermittency of that. I'll talk a little bit about the polar vortex and what happened with natural gas in 2014 and some solutions in terms of integration of demand side management. I started my career writing software to send control signals to generating plants for economic dispatch and control, EDC. As the load changed we would send signals to the generator. Now we're sending signals to the load as the intermittent generators change. Same kind of process, it's just a little harder when you have 10 million different devices that you have to send to as opposed. And then kind of the grid of the future, I see HVDC as a big solution and every one of these challenges involve not just IT but IT and communication. How do you secure the communication system as well? How hot was it? Now let's talk about weather weirding. The summer of 2011 I was actually in charge of the power system. It went to 108 degrees in New York, 108 degrees in New Jersey. It was 106 in my hometown Philadelphia, Washington DC in Richmond. It was so hot in Hershey Pennsylvania where they make the milk chocolate. The cows were giving evaporated milk and by the time we could get it over to the shore down the shore in Jersey it looked a little like this. This is not Phoenix. There's a Gulf Stream. There's an ocean. Welcome. I'm so glad you're here. Here's one of my partners from PNNL. So it was the hottest day ever measured in the history of the Mid-Atlantic. So on weather, strange conditions, here's another storm. I was actually in Paris and before I could fly from the Seagrey meeting in Paris, this storm had moved from Chicago to Washington DC. 200 miles wide, 600 miles long, 100 mile an hour peak winds, straight line winds, derecho means straight line winds. The largest storm, 90 large transmission lines on the ground, 4.2 million people in the dark and yes we committed a billion dollars to harden the grid in Washington DC following this storm. The biggest one that I've ever worked bigger than Katrina. I worked Katrina in the southeast. As a matter of fact was Hurricane Sandy. Anybody see PJM on this map? It's kind of covered up. Bottom line, this storm was not a hurricane. It came in at 75 mile an hour winds. We didn't have to shut down any nuclear plants along the coast which we thought we would. High voltage was our biggest problem and then no voltage became a very serious problem during the storm as the system came through. Right after the storm we committed 1.2 billion dollars in New Jersey and 1 billion dollars in New York to harden the system, raise substations, sea walls around substations. This month, Matthew, I don't know if y'all were following the news. I was closely mostly water. The right front quadrant where the heavy winds are stayed out at sea as it came up the coast. It 143 mile an hour winds as it was working its way on to the coast at the seawall. That's nothing folks. This is Typhoon Hyatt. This is the fastest wind speeds ever measured when a storm came ashore. And over 5,000 people died. There wasn't a leaf left on the trees because the trees were gone in the Philippines when this storm hit. So those that say, hey, weather is not getting stranger, I haven't looked hard at where the data is these days. Polar vortex. You know, I've been working in Alberta on shutting down the coal fleet there. They called Polar Vortex in Alberta winter. In New Jersey, it was cold. I believe we got down to a minus 8 degrees in Philly and minus 20 degrees in the Midwest. It was so cold the cows in Hershey, Pennsylvania where they make milk chocolate. We're giving chocolate ice cream. This was a terrible day. The worst day I've ever had on the power system, by the way, 43% of the gas plants would not run. Almost half of them could not get fuel at any price. Natural gas on the gas pipeline. Home heating has priority over electric generation. Natural gas more seriously, the plants that were dual fuel had not seen this kind of weather in at least two decades and the fuel systems froze up on the trace heaters and we learned a lot about the importance of winterizing the plants before such an extreme event on the system. This event was forecasted. We had a pretty good forecast but the performance and the same thing would happen here in San Francisco if you had extraordinarily cold weather. It happened in Florida, Black Christmas 2009. It happened in Texas as well. Space weather, I'm not the kind of guy that says the sky is falling but it really is. Millions of tons of particles come our way. Every time there's a solar storm event we were very fortunate in this last solar cycle and where you never know some of the extremist solar events can happen off cycle but it's 11-year cycle. I've worked on four, 44 years I've worked on solar events. We work on it very hard every 11 years but at the end of the day a solar event like the Carrington event of 1859 was the last time we had a severe event that actually melted down the telegraph wires and caused all kinds of problems. There was no power outages then I might add because there was no power system. Renewables is the answer. This is the PJM build out. We have NRQ, enough wind to meet the RPS renewable portfolio standards for all the states in the 13 state region and I'll be frank with you I like wind a lot. I like the fuel cost but as I was telling Robert if you like wind you got to love storage and because the wind and solar are intermittent resources and we use electricity in a different way. Anybody see the problem here? This is the PJM average load factor for a year and that is the hourly average so if I gave you two second data it would be worse than that but you can see the variability the intermentancy of that resource and when you get to 50% of it here in California you're going to need storage. I really like pump storage. I wrote my graduate thesis on the optimization of raccoon mountain pump storage plant a 1700 megawatt plant. Robert here worked on the compressed air energy plant that's he was project manager almost on it down in LA lower Alabama back in Tosh Alabama is the location of this. We have a 20 megawatt flywheel. I love this battery. It's a one megawatt lithium ion titanate battery. We did 250,000 swings on it to do load frequency control on the PJM system. I love my job as a matter of fact I met with AES and I said I want to see this. I want to see it happen. I want to see it outside my window and I want to see it soon. Being CEO you know it was kind of a nice thought. Three days later there's alignment punching that hole in the ground for that pole and the the tractor trailer rolled in in less than a week. It took us longer to get Comcast in to give us broadband to the thing and it did to do a 34 kv connection to the system. Mobile batteries I'm going to talk about that's my Chevy Volt number 11 off the line and obviously when you when you think about water heaters and the fact that we have them connected the system. This is the first commercial fleet of electric vehicles anywhere on the grid anywhere. We have the University of Delaware and NRG and PJM have a program where we're paying cars to do load frequency control. So as the wind changes we actually change the charging on the car so that we can balance out the grid and pay them about $150 a month per car that won't pay for the car but will pay for the battery. And so we have a program using cars in load frequency control. If you look at carbon which I think we've got to take a look at all forms of carbon. 26 percent of the carbons comes from automobiles transportation. You don't hear a lot of talk about that in Washington DC. We love our cars. I'm serious you don't talk about it. If we electrify those cars which we can they'll be much more efficient and we can use wind, solar, shell gas which has 50 percent of the carbon as a coal plant to run those vehicles. This is the growth curve. My Chevy Volt is down here in 2010 and then I bought a Cadillac ELR in 2014. That's a Chevy Volt with leather seats. It's the exact same drive as the Chevy Volt but a little nicer. If I had 400,000 cars which already exist in North America I could have enough to do load frequency control across the 13 states of PJM. If we had those on load control with a broadband connection to them we could control the grid and the interment and see the wind. So as we look at cars it's more than just the economics of electricity being more efficient. It's also allowing the integration of the renewable resources. Nissan is considering EVs for emergency backup in essence having an uninterruptible power supply with your EV being part of that. We have 300 megawatts of battery storage. Lithium ion primarily hooked to PJM. That is enough to do most of the very rapid load frequency control. If you're in the battery business don't go to PJM. It's saturating. That market will saturate with time but it says in a very short period of time since we hooked that one megawatt trailer up we've gone to enough to do load frequency control for the whole system. This is my favorite of all storages because it already exists. It's in your basement it's in your attic it's in your garage. It's doing load frequency control with water heaters. In France where they're primarily nuclear 70 plus percent nuclear capacity 50 percent of the water heaters are under control so that you can charge at night you can vary the charge rates on those water heaters. That one is actually in my basement. They gave it to me as a gag gift at PJM as a retirement gift and they didn't know I was going to take it home but I moved it back to Tennessee. That water heater and this is a large tank in 105 gallon stores 26 kilowatt hours. My Chevy Volt source 16. My Cadillac store 16 and a half. There's 53 million of these hooked to the grid. If we controlled half of them which roughly would have some kind of broadband connection to we would have the equivalent of all the pump storage 23 gigawatts if I'm correct Robert could correct me in that range. A pump storage that we built out on the grid to enable the nuclear plant. We are converting very quickly to natural gas. We have reduced carbon more than Europe due to the availability of shell gas both of Marcellus shell and the Utica shell are coming into play in a big way as we look at it and I want to talk about CO2. I just came from Alberta Canada did a debrief of the premiere last week. They have decided to put a price on carbon of $30 per ton starting January 1 of this year and to shutter all the coal plants by 2030. Pretty ambitious goal if you look at it. In the U.S. we haven't had the courage to put a price on carbon but I want to tell you why we should. In the markets if you look at the green curve that's SO2 mostly SOX SO2 SO3 by having a price on it we reduced it by a factor of four in a decade. By putting a price on NOx nitric oxides we reduced it by a factor of three. We have reduced the carbon output average for the PGM system about 180 pounds per megawatt hour driven mostly by natural gas a little bit by the wind that you saw 5,000 megawatt wind but we can make major changes in the fuel and we are doing so as we speak and that's we talked about HVDC. HVDC adds controllability to the grid. If I have a HVDC terminal and I call this back to back it just goes across the Hudson River to Long Island from New Jersey but we can actually direct the power flow instead of it being an open network we can say I want to move a thousand megawatts today and dial that into the computer and move it and if there's a problem on the AC system we can change the DC very rapidly in order to take care of that contingency. So this is a voltage source converter South China presented a paper at my request where they're using the voltage source converter as a STATCOM device to control the AC voltage regardless of the throughput of the DC side. So back in my life I was in charge of electric system reliability in the 1980s. My biggest two challenges were lightning and squirrels. Now that sounds de minimus but we took on lightning with metal oxide varistors the West Point Mississippi Aperi device 1985 or six I think in there and we took on squirrels with squirrel guards on the transformers. Now the squirrel guards started tracking and they were blowing up we had to redesign but that was pretty simple world. In today's world I want to take you forward from where we were in the year 2000 and tell you where we are today just very quickly I'll walk through what we thought the future was and what it really turned out to be. In the year 2000 avoid gas at any price volatility is crazy the prices in the energy and gas market had gone off scale in the 1970s it was against the law to burn natural gas today natural gas is king due to technology created by DOE to a large part the horizontal drilling and boring the natural gas has half the carbon content and it may be public enemy number two here in California after Alisho Canyon but it's much better than coal as we look at nuclear we had the AP1000 I was involved in paying for the first license 60 million dollars went into doing the analysis for the new standardization today a poise for a renaissance a Fukushima happened and today nuclear Diablo Canyon surprised even me that they would announce this early the shuttering of a nuclear plant here in California the nuclear renaissance pretty much stall we have plants now that are being priced out of the market in PJM because gas prices are cheaper than the operation of nuclear plants. In the year 2000 digital tech was forecast to speed up hard drives we were looking at digital hard drives we were working Intel was improving processor speed today digital tech is very much focused on distribution systems and energy storage you may have heard of a conference here it's called bits and watts that's where a lot of the bits and watts will be applied in the distribution so boring keep the squirrels out of the line keep the trees out of the line make it safe so people don't get hurt distribution was just plain old boring today distribution is the platform in New York for micro grids and other grid connected tens of millions of devices prices to devices in the market that we can do distribution has changed considerable carbon capture and storage worldwide emphasis on R&D we had projects going at US projects in China today there's two projects in North America one in Mississippi that's coming in at $11,000 per KW which is $3,000 per KW more than a nuclear plant on carbon capture and storage and there's a project up in Alberta right on the edge of Sasquatchian and in Alberta where I've been working that's demonstrating oil advanced oil removal with but not much moving on carbon capture and scores wind and solar too expensive not in anybody's budget we're not planning for it we're not building it nobody knows it's coming price of solar thanks in part to Arun he wrote a a goal based on JFK's moonshot he said sunshot for a dollar a watt now that's poetic but it also sends a signal that we want to get to a thousand dollars of KW that solar would cost about the same as a new gas plant in the system we have built more solar and wind than any other resource yet in the year 2000 we didn't have a budget or a plan to add that now that I've convinced you that we didn't know what the heck we were talking about in the year 2000 I'm going to take you forward in time and tell you what's really going to happen high future load growth low load growth Dan Riker's not here but something he said that I really like we built the fridge to the 21st century I said fridge not bridge each refrigerator is 50% more efficient each heat pump sear number is more than double what the heat pump that it replaces we've gotten much more efficient each light bulb led uses 20% of the energy we will have low low growth in electricity that means low revenue growth that also means an opportunity as we go to find ways to do that the only hope in seara I like the consulting firm Larry Malkovich thinks we'll have higher load forecast because of electrification transportation if that happens I think it's a very good thing carbon wise it's a very good thing efficiency wise my Chevy Volt I spend about four cents a mile my wife's car is 18 cents a mile in my Chevy pickup truck I don't know but the bottom line is I think the load growth will be very small less than 1% maybe negative concentration fuel portfolio high diverse fuel lots of mix gas is king right now and we will have a lot of natural gas coming in Alberta looks like Pac-Man after the year 2030 and the yellow being natural gas if you can imagine a pie chart with that shape so we will see natural gas we will see renewables but they're they're energy not capacity distribution self supply smaller units versus centralized supply of large units answer that yes you're going to have both the efficiency of a very large unit is 60 percent the smaller units is smaller so but because of reliability and the need for micro grids and backup supplies we'll see both of those develop and autonomous micro grids people going off the grid you hear that a lot not going to happen there will be a California ISO and a PJM long after I'm gone there will be larger now when there's a storm event there may be both so we're kind of the crossroads here between uncertainty and prosperity how are we going to go I want to say thanks to Eprey Robert I was on the Eprey board and this is something we worked on this is the grid as we built it as built drawing generation on the left load on the right nothing political about that just the way the drawing was made but the bottom line is one directional flow across the transmission network we're going to move very quickly to DER and and what we're working on we're going to say energy environment and efficiency are all one program that we work together and I might say that drives our economy which is fairly important to all of us pessimists complains about the wind the optimist says the wind will change a leader says I can't change the wind but I'll adjust my sales and I think if we look at the weather that we've been seeing the extreme we've got to change the way we're doing business the grid of the future will look more like a ring bus to those that are in the power department that has many opportunities to input and remove from the markets resources both demand side and supply side both small generators and large generators that has much more flexibility than the grid that we've built to date I want to close with a story on nuclear plants Fukushima how many people have heard of Fukushima everybody in the room how many people have heard about Fukushima Diana not die at you few great two different management approaches to two different disasters that happen from the same tsunami both of these Mach 1 GE nuclear reactors the earthquake did not damage either plant the tsunami caused a power outage to both plants in the case of Daiichi they got focused on how can I get pumps how can I get fire engines how can I get cooling into the reactor etc they were very much focused mechanical in the case of Diana they were very focused on where is the closest hot transformer that I can drop cable just like New York City and within the hours not days they dropped 5.5 miles of cable to a distribution transformer and started the pumps to cool the reactor and there was no there was no major hydrogen explosion no major damage to the fuel the fuel was damaged I didn't say that no maybe major damage to the reactor in essence so very different stories very close together in terms of how they happen but the more important thing is when things are written in stone especially by old geezers like me and Robert pay attention to to what they say this was actually a stone that was running and I had to go to a little bit of trouble of getting a real good picture of that stone from the New York Times but it was a tsunami stone and it said do not build your house below this elevation 1270 excuse me 127.6 feet and it was carved in stone and I might add we built two nuclear reactors and more below that elevation so we didn't learn from what happened in the past going forward this yeah 1896 not that far back when you think about it this is carved in stone as I wind down and I've got a minute and a half to go so I can tell this story 108 years ago this was carved in stone at Union Station Washington DC electricity care of light and power devour of time and space bearer of human speech over land and sea greatest servant of man nothing has improved our standard of living more nothing has improved our economy and productivity more than the electrification that we've done today and somebody way back in 1908 saw that coming and saw the value proposition of electricity and carved that last thought the leaders in this room especially the the graduate students in this room that are going to become the leaders in areas the best way to predict the future is to create it you you saw what we thought in the year 2000 what we actually have today the best way to get there and we're going to start on that tomorrow with a symposium that is set up here on campus and in bits and watts with that I've got 11 and a half minutes for questions yeah thank you so we have some time for questions let's have the students and the postdocs take the first shot and then we can go on to the faculty don't shoot please thanks for your talk yeah thank you wondering for energy storage what kind of time scales do you most want in the grid is it minutes hours date cycles I love 15 minutes I'm passionate about 10 hours for different reasons a battery doing load frequency control 15 to 30 minutes of storage doing this all day will help us manage the load frequency control I need about 400 megawatts of that in PJM 180,000 megawatt system if I'm controlling for the wind and the duck curve you've seen in California or I'm controlling for that hottest day of the year I need 10 to 12 hours and right now pump storage is the only thing that we have that's in that area Bob would argue that compressed air energy storage is but we only have one of those so at the bottom it's very it's proven technology I would say further than that compressed air energy storage firing natural gas with the compressed air that storage expands the gases and but the point is for different functions you need different you don't need a whole lot of the batteries and we're going a little bit crazy I can do with water heaters and mobile batteries what we can do with the large scale energy batteries we've hooked to the system to date I love flow batteries now that's the that's the other battery that has not discussed very much because we don't have a lot of them up and running but you can get to the 10-hour kind of storage yes sure some of your skepticism a little bit on that you know like the tower system the future being all everything with micro grids or entirely micro grids I'm curious though a lot of the concept about hierarchical controls do you see examples of that being deployed whether or not small parts of distribution system can island seamlessly do you see hierarchy control applied by the system operators to at least control DG whether or not it's an island you could argue Texas is an island it's a 50,000 megawatt island but and I wouldn't call it a micro grid obviously but what I see is the micro grids will happen for reliability purposes and people are willing to pay more for reliability than they are for their normal efficient use of energy in the case of Princeton University in Hurricane Sandy they had a micro grid fired by very inefficient high sulfur diesel engines but they saved all the research and development that was being done in DNA during that storm so a lot of the movement to micro grids is a backup for the grid when it fails in the case of rooftop solar and if we get to a dollar watt it's in the money and you want it but it will be synchronous with the system until bad things happen and then a combination of solar and storage comes into play in my mind that becomes very attractive in a micro grid environment but you've got to be efficient and even the small modular nuclear reactors you've got to be efficient in terms of how you would build out something in a system to make it work but I actually think micro grids are going to happen I could argue that my future ex-wife's home because I left her so many times for storm duty in the dark she has a generator in her home that kicks in and can actually carry part of the grid part of the load in the home not the whole neighborhood I might add but micro grids are going to happen it's just because of the value of electricity and the value of reliability of electricity. So demand response has taken kind of a beating in the recent PJM capacity auctions and due to some rule changes what do you see as being the future of DR in PJM and other markets? I actually have a chart on that for tomorrow I still see it good but you have to aggregate it so that it's not just a summer product. Right now we have a summer product we got caught with our plants down in the winter during the 2014 so it's taking a beating because you can't just control air conditioning and meet the needs so the aggregator has to put the right mix of fuel I think it will continue to drop some because we have performance penalties if you don't perform but at the same time I see it as a huge saving as a matter of fact in the recommendation to that we're working on for Canada we're saying hey having more DR in the grid and Jerry Brown called me to California four or five years ago to say what can we do in California to increase the amount of demand response in the market I see it as an integral part but you have to perform you can't it's not a free ride to just get a demand charge for the summer period only but it will it will come back because it's much cheaper who would spend billions of dollars on something that's only going to be used three percent of the time football stadium and a peaking power plant that's it so you have to be able to to respond when the system needs you to respond in terms of demand response any other students opposed ox yes so in terms of the water heater storage with any storage system you're going to have losses from the conversion back and forth but I would think with the water heater the temperature is not being all that high are the losses much worse than with batteries or other systems and how the losses the water here you saw on the screen the losses there would be the best of all better than batteries because you're lowering the temperature in the tank not raising it you're not keeping you know 122 degrees all the time and more importantly you super insulate the larger tanks now I had the BTU police came after me at DOE on the larger storage tank saying that hey instantaneous water heater is what you need because it only heats it when you need it and you don't have any losses that is the worst thing from a system perspective that an operator ever had you know hitting you with 15kW for for every time somebody turns a faucet on is not what you're looking for on the system so bottom line is the overall efficiency is pretty good and something that I don't understand exactly the physics of but actually the element in a resistant water heater not a not a heat pump water heater lasts longer if you're lower if you're raising and lowering it rather than being full load on and off so that surprised me a little bit but pretty efficient in terms of turnaround efficiency in terms of the water heaters a better better much better in pump storage which is rocking mountains 76% turnaround efficiency for example another student question so when you look into using ease to help balance some of the fluctuations how do you balance the needs of the grid with the needs of that person who might want to drive in a suit of time excellent question because I'm my dual fuel sort of guy whether it's a combined cycle plant having some number two fuel oil on site or a car that has nine gallons both of my electric vehicles have nine gallons of stored fuel called gasoline on board in terms of an all-electric fleet which Tesla is pushing pretty hard you worry a little bit about not meeting the charge but any good graduate student doing optimization theory if you plugged into your computer when to be charged you could actually send a signal to optimize between now and then and I'm not talking about the deep discharge where I empty your battery when the grid needs it I'm talking about varying the charge rate as you charge the vehicle at night mostly cars are on the road two hours a day average a little less so 22 hours a day they would be available to do load frequency control midday you would have to have more plug ends and you'd need more parking that you got here for one day but you would need more plugs in terms of and PJM was a great place to work they had electric plugs and free electricity so you're in the day but the point I was going to get to is the cars are not in use a large part of the time so you can actually use those batteries and you avoid the warranty if you do the deep if you do vehicle to grid that you're trying to pull the grid and the first time we tried to control that one megawatt battery that I put in the parking lot I was talking about we emptied the battery and then we filled the battery and we emptied so we came up with an algorithm that says you do this while the slow what's through baller does the ramping for the system so okay let's open it up for other people yeah yep you have a point in the cost of per kilowatt hour for batteries to take over and this you have a if we get to this point then that'll we'll be able to go all battery I don't uh the best I've seen is eos talking 160 dollars per kilowatt hour in that range that's a very good cost I hope it has an inverter that's just for the battery that's just the battery and there's not an inverter as Bob is saying but back to my point the batteries for them to make it big they can do it in the load frequency control market uh raccoon mountain pump storage was 10 dollars per kilowatt hour and of course I had hair on top and it was a long time ago uh the bottom line is the pump storage has not been built lately because of the there's not a large diurnal swing in price now gas prices are so low you don't have the big differential peak to off peak but for the load frequency control and the wind integration I think you the externality that storage has to be priced outside the energy market it has to be ancillary services which in pjm is the case with demand side management we were talking about earlier it's paid through the capacity market well I think we've we've come to an end for all the questions we are out of time but let's just thank terry again it's just wonderful


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So let me begin with an overview of what I hope to cover. Unfortunately, I will probably have to erase this. But it can still be on the board for just a little while longer, maybe. So the plan for today is to give an overview of the cosmic censorship conjectures in general relativity. Can everybody see this, or should I write larger? It's okay? By the way, how many of you have heard of the cosmic censorship conjectures in general? Okay, so quite a few. Okay, very good. I will not assume that you will have, but in any case it's... ...repetition is the mother of learning. So after this overview, the next lecture, let's say, will be a discussion of spherically symmetric dynamical space times. So this is sort of a world which is very dear to my heart. And it's actually a world in which you can understand a lot of what's new and interesting in general relativity. And in particular, it's a world in which you can really understand what's going on concerning these conjectures. So in some sense my favorite way to really understand what the issues at stake are for these conjectures is really to understand this world. And another nice thing about this world is it's in a world where you can really make formal sense out of the notion of a Penrose diagram. So I'm going to add this here, and Penrose diagrams. So this is something that I'll use informally in today's lecture, and I know many of you have nodding familiarity with them. But in this context, we'll actually be able to sort of discuss these well-defined mathematical objects. And you can even state theorems just with pictures. So it's nice. Okay, so now I guess I'll actually have to rub out this nice outline starting. So the third, again, let's say, lecture, in some sense a continuation of this second topic. So in the world of spherical symmetry, in the simplest case, that's to say one of a self-gravitating scalar field, there's actually a complete proof of both cosmic censorship conjectures, the so-called weak and strong cosmic censorship conjectures, due to Sothulu. So I'm going to try to devote a whole lecture to at least explaining those statements and a little bit what goes into the proofs. So we'll see these acronyms a lot later, so I might as well initiate them already here. This is weak cosmic censorship and strong cosmic censorship for spherical symmetric self-gravitating. And this work really is sort of the place to start for anyone wanting to understand more generally these conjectures. This is what you have to understand. And well, the final lecture will be concerning sort of a result which is beyond spherical symmetry, in fact beyond any symmetry assumptions, and it actually sort of leads to a reformulation of strong cosmic censorship, or if you want sort of a disproof of the original version of strong cosmic censorship. So this is a recent work of myself and Jonathan Luke from Stanford University. And well, I'll just write the statement here and later it will be clear what is its relevance for strong cosmic censorship. The story is a bit complicated and it doesn't fit in a tweet or in this part of the board. So let me just write for now the C0 stability of the... So this is my plan. I think I'm giving four lectures and these are four topics. Although I have to warn you that so I'm from Greece and I'm not so good in planning ahead. And so it could be that the plan sort of disintegrates before our eyes. But in any case, this way you know what my plan was. Okay, so let me begin this overview. And by the way, feel free to interrupt me at any time. But also if you don't quite understand everything yet, that's fine. You shouldn't because the whole point in some sense of topic two is topic two will in particular explain formally in a certain world everything I said today. But in any case, nonetheless, do feel free to stop. So let's begin our overview cosmic censorship conjectures. Okay, what are these conjectures about? So these conjectures, the very big picture, whatever they're about, they're about the nature of singularities that form dynamically in general. At the end of the day, both of these conjectures you should think of them as being about the nature of singularities as they form dynamically. So I want to focus for a second on that last word dynamically because until you formulate these questions in terms of dynamics, you will not be able to rationally study them. So before you say anything about these conjectures, you have to understand the notion of dynamics in general. So let me sort of review that very briefly. Possibly this has already been discussed a bit last week and actually I know that Harvey Rayell, his lectures will be about the initial value problem in, well, not just in classical GR, but beyond classical GR. So in any case, this is something very important. Without dynamics, we could not talk rationally about it. So let me fully deserving of a bullet point. So dynamics. So what is dynamics? Well, let me first discuss the vacuum. So for the vacuum equations, Ricci curvature equals zero. We learned that there is some notion of initial data and the initial data is a three manifold which I've drawn here. So it's some three manifold with a Riemannian metric and with the second fundamental form or what will be the second fundamental form. And maybe you heard last week that these actually are not completely free. They have to satisfy these constraints and that's a nuisance in the theory various people study. So the fundamental theorem of dynamics, which is due of course to Yvon Chocabra from 1952, is that given sufficiently regular such initial data, you can solve these equations. So what does that mean? So there exists a Lorentzian four manifold that satisfies these equations and admits this three manifold as a hyper surface which is space like. And such that this is the induced first and this is the induced second fundamental form. Moreover, and this is actually important for being able to talk about dynamics. So the way I said what it means for a space time to solve this initial value problem, then solutions are severely non-unique. Okay, because you could always then take sort of a type of subset of this and this would also solve the initial value problem. Or maybe you could take another space time which is a bit bigger in one direction that also solves. So actually in 1969 Bob Garrosh and Yvon Chocabra, they were able to show that there exists a maximal Cauchy evolution. So there exists a maximal solution. So there exists a maximal Cauchy development of, okay. So this object is globally hyperbolic. So globally hyperbolic means that all inextendable causal curves in the space time, they intersect the initial sigma three once and only once, okay. And it is the largest globally hyperbolic solution of this initial value problem in the sense that any other solution embeds isometrically into this. So anyway, I can't emphasize more how from the point of view of language, this object is absolutely fundamental in rationally discussing all problems and dynamics in general relativity. Because you should think that any problem essentially in dynamics is a problem of connecting properties of initial data, okay, to properties of this object. Any question in general relativity essentially can be phrased in that way. So I can't resist saying a certain side story that actually the proof of this theorem by Bob Garrosh and Yvon Chocabra in 1969 actually appealed to Zorn's lemma. So if you know what that is, that's one of the equivalences of the axiom of choice. So that sort of already hint that it's non-trivial to construct this object. And the reason is that if you have two globally hyperbolic developments, then they're not sitting, they're not living on the same manifold. And it's not clear how to compare them a priori, which is bigger. And that sort of is essentially why one had to appeal to Zorn's lemma. For those of you who like the philosophy of physics, you don't have to identify yourselves. But for those of you, you might think that that's actually quite problematic. If indeed, you know, talking about dynamics in general relativity relies on the axiom of choice, you might be sort of worried. So actually a bright young graduate student in Cambridge a few years ago, Jan Zbierski, he succeeded in disornifying this proof. So actually this proof is now completely constructive. So I don't remember the year of that, so let me guess that it was 2015. So actually his paper on this is I think also the best reference to discuss sort of, you know, this proof from a modern point of view. Okay, so that's great. I fade lip service to dynamics, and we will certainly see in use this object later on. And we will see the significance of global hyperbolicity later on. So if you don't think you have a sense for what it means, don't worry. Sort of everything will become much clearer in lecture two. Okay, so that's the first thing we have to pay lip service to say anything. So we have a notion of dynamics. So the second thing that we have to remember, and again, we can only say these words because of dynamics, is that bad things happen to good data. Okay. Bad things happen. So, okay, so what do I mean? So this is data. Okay. So the first thing that, okay, you can know this in this framework is that there are space times where the data is impeccable. There's nothing wrong with the data. The data is complete. It's asymptotically flat. There's no singularities in the data. In fact, the data can be, you know, very nearly flat in some sense, very close to Euclidean space. And yet bad things happen. So happen always means the maximal crochet development of the data. The maximal crochet development of the data has, in quote, properties. Okay. So this, if you want, you already see this from the explicit solutions that we know in law. So Schwarzschild, something which I'll talk about later, Oppenheimer-Sneider, in which it's even more clear, in some sense, the, you might quibble that the data in Schwarzschild or Kerl, let me also add Kerl. Okay. It's complete, asymptotically flat, but it has a funny topology. So maybe you don't like that. So here in Oppenheimer-Sneider, which I'll show you very soon, you can't quibble about anything. So bad things happen to good data. And again, ah, means it's the biggest. It's the biggest. So it's the biggest solution with this property. And biggest because, again, you can't compare a priori. So biggest means any other one isometrically embeds to this one. So any other one. So any m tilde. Okay. G tilde, which is globally hyperbolic. Okay. And admits sigma t bar k. Okay. As sort of hyper-surface with induced dot as a Cauchy surface. Any other one isometrically embeds into m comma g. Okay. So that this diagram commutes. All right. So to get back here, so bad things happen to good data. And again, keep in mind, especially with the discussion of, let's say, Schwarzschild. Okay. You see, in the anti-Diluvian discussion, and I mean before the relevance of dynamics, sort of, was clear, then people saw Schwarzschild. They saw that it was singular. Actually, there was a load of confusion of what was singular, et cetera. They even lost over that. They saw that it was singular. And many people's reactions, including Einstein, was that, okay, that just means we should throw this solution out. And, okay, it's sort of a reasonable thing to do in theories which are not governed by dynamics. But the point is that you're not allowed to throw, you cannot sort of throw out the maximal Koshy development just because you don't like one of its properties. You can only throw out the initial data. That's to say, you know, judgment calls can only be made on the level of initial data. Okay. And the initial data is flawless. Okay. And it's not singular. So somehow, the understanding that dynamics is the important problem, it fundamentally sort of changed the point of view of these solutions. Okay. Anyway, all this will become clear later, but it's worth saying this sort of philosophy at the beginning because, you know, anyway. Okay. Bad things happen to good data, but the story gets worse. And the reason that the story gets worse is that, again, these here, so Kerr, of course, was discovered later than Schwarzschild and this Oppenheimer-Sneider that I'll talk about. So these solutions, of course, they were discovered rather early, this one already in actually in December 1915, this in 1939. We'll see in just a second. And okay, from the point of view of dynamics, they exhibit this principle. But of course, these are just some explicit solutions. You might hope that the reason that bad things happened was because these solutions are very symmetric. Okay. So even though, indeed, bad things happen to some very particular good data, at this point, you could still be hoping that for generic data, bad things don't happen. Okay. Now this was very much the hope in the early 1960s and the hope was completely destroyed in 1965 by Roger Penrose. Okay. So Penrose, what he showed in 1965 is the following. So these three solutions, they have various bad properties. They have, in this case, they have singularities, whatever. They have various types of bad properties. Okay. Or what were in any case considered to be bad properties. Okay. But one thing that they have in common is that they are geodesically incomplete. Okay. And what Penrose showed in 1965 is that geodesic incompleteness is stable to preservation. Okay. If something is true, and this something is actually true in all these examples, then geodesic incompleteness is stable. So to perturbation of what? To perturbation of initial data. Because again, to make sense of all these things, you have to always refer to dynamics. It's stable to perturbation of initial data. So again, to go back to this picture, what is the claim? So all these, okay, you can think of them as solutions of the problem of dynamics with some special initial data. Okay. So the result of Penrose says that if you wiggle ever so slightly the initial data, okay, and you solve the Einstein vacuum equations, or while in this case, more complicated system of Einstein-Matter equations that we'll see in just a second. So you consider the maximal Cauchy development of wiggled initial data. Okay. Then the wiggled initial data will still be geodesically incomplete. So this theorem is traditionally in the literature known as Penrose's singularity theorem. But it is time that that terminology be retired, and that it be called what it is, namely Penrose's incompleteness theorem. Okay. Because that's actually what it says. So, and it will be clear later on why, okay, this is just semantics, but why this is a much more useful semantics. Okay. So to recap where we are so far after these three bullet points, the fundamental problem is one of dynamics. Bad things happen in dynamics to good data. Those bad things, or some bad thing, namely incompleteness, is stable to perturbation. So it wasn't because the data was symmetric. It wasn't because the data was fine-tuned. Okay. These bad things, they actually happen sort of generically, or they happen sort of with non-zero quote probability. So maybe I should say it like that. Okay. So where does that leave us? Well, this is now where enter the cosmic censorship conjectures. The cosmic censorship conjectures, they enter to make the best out of a bad situation. Okay. So cosmic censorship, what is it all about at the end of the day? It's about making the best of a bad situation. And if you want, so let me make a quote to carry on from here, and we'll understand in stages this quote as we go along. So the statement basically is given the above situation. Okay. Classical GR is as good as it gets. So let me say it like that. But there's one little caveat, and that caveat I'll write it here, generically. Okay. So with probability one, let's say it like that for now. So that's what cosmic censorship is supposed to be about.  Okay. This is not even a formulation. This is what is... This is, okay, a quote connecting to both cosmic censorship conjectures. Because as we will see, there are two sort of cosmic censorship conjectures. And then as time goes on, we will put meaning to these quotes. So, okay. So maybe I'll move back here now. Say again. Yeah. So there was not... So that's a good point. But yeah. So the question was that if you take 1965 and you subtract 1969, you get the number which is negative. And so did he use close time like curves? I think this was the question. So the theorem was not originally formulated with respect to the maximal Cauchy development. Because of course that notion didn't exist. But Penrose's theorem, one of its great strengths is that it's actually a purely geometric fact. So it's not even about a solution to a PD. In fact, it's about a solution to a partial differential inequality. But somehow in order to interpret the theorem, it turns out that the natural object to interpret it too is the maximal Cauchy development. Because if you are not interpreting it with the maximal Cauchy development, then all this is saying is that, well, maybe you should extend further. So somehow it is really the sort of, indeed, you should really think of this as a corollary of Penrose's incompleteness theorem. But this is actually the corollary that has content for classical Gauss. It's a good point. Okay, so this is sort of the quote. This is what the cosmic censorship conjectures are supposed to be about. So let me try to make this slightly more specific, referring to both these sort of separate statements. So the first cosmic censorship conjecture, so this was actually the original cosmic censorship, and it's now called weak cosmic censorship to differentiate it from something else, which is called strong cosmic censorship. But if you look back at the literature, this is what used to be just called cosmic censorship. So what does this conjecture try to say? So again, I'll give you a quote, quotations for the quote. So this conjecture says the following, despite the fact that the maximal Cauchy development may be incomplete, as Penrose's theorem tells us, far away observers, whatever that means, and we'll make that precise later, far away observers live forever. So far away observers, which you should think of as sort of geodesics that are far away, some sense of that word, they are complete. So in particular, they never see anything that deserves to be called a singularity in finite time. So this is, if you want, the original form of the conjecture or the original informal form of the conjecture. And one thing that actually really somehow first came out from Stolulu's work in spherical symmetry, because this really is an impeccable model problem, is that this, one doesn't expect this is true without adding the caveat generically. So, yes, you have incompleteness, okay? But generically, there's this class of observers, the so-called far away observers, and they live forever, okay? And they don't, in particular, they don't see any singularities in finite time. So as we'll see, this is a property which is shared by all of these examples, okay? So if you want, it's motivated by what we see in the black hole spacetimes that we know and love. So that's weak cosmic censorship. And what is strong cosmic censorship about? So this is sort of harder to explain in a quote, and it will hopefully become much, much clearer when we start looking at examples and drawing Penrose diagrams. But nonetheless, I guess I should write something. So one comment I should immediately make is that these names are actually quite unfortunate, because this is not a weaker statement than what I'm going to write here. In fact, what I'm going to write here, the way I'll write it, it will look as if it has nothing to do with this. Of course, there are relations between these statements and there are reasons why, the historic reasons why they have these two names. But if it were possible to retire these names, then maybe we would. And I could imagine better names for both of these constructions. On the other hand, these are very cool names that have stuck, so I wouldn't, I'm not supporting retiring this. Here it's different, the reason is that in fact, there's nothing wrong with an incompleteness theorem, after all, Godot proved an incompleteness theorem. He's a great guy, and I don't think anyone thinks less of him, because it was an incompleteness theorem and not a singularity theorem. So I don't, I mean, I really think this, this amount should stick. Here we can, okay, I will keep these words, but we should, maybe it is good to sort of remember that they're not maybe so great. So what is this conjecture? What does it say? So it says the following, so generically, again, and I'll tell you a little bit about history in this context of the word generically. Generically the, well, maybe I'll say it like this. All incompleteness is due to singularity. Let me say it like this. So this actually goes right to the heart of this distinction, okay? So Penrose's theorem, commonly thought of as a singularity theorem, is actually just telling you that spacetime is geodesically incomplete. And there can be many reasons why geodesically, a spacetime is geodesically incomplete. One of them being that it sort of has an edge which is everywhere singularity. But that's not the only way that spacetimes can be geodesically incomplete. And somehow what strong cosmic censorship is saying is that generically, that is why spacetime is geodesically incomplete. It's incomplete because of singularity, because its edge is singular. So you might think that it'd be very, very strange at this point that this be thought of as something good. But I claim to you, and this will only become more clear for those of you who don't know the story, when we really understand the ramifications of this not being true. So this is good, okay? And in fact, secretly, this is the statement that general relativity uniquely predicts the classical fate of classical observes. So let me write it like this. So this secretly is somehow the statement that the classical general relativity in its regime is deterministic, okay? Anyway, why that is the case might not be clear yet, but it's hopefully will be as time goes on. So secretly if you want in a more, for those of you who are sort of more, ah, there's a question, yes. Well, so actually the, yeah, so the question was, why all the fuss about geodesic completes? So the answer is actually, you sort of, so as we shall see in some sense, the, okay, it is not, so geodesics are convenient because, okay, these are sort of unquestionably good objects. So, okay, they're a convenient test of various things. But certainly, we can talk about classical physics fields without ever mentioning geodesics, okay? And if you want the, sort of, the rigorous formulations of these statements, they don't sort of refer to geodesics manifest. So they actually find their proper formulation just as in the world of the Einstein equations coupled to a billion sort of fields. So everything is in the solution. To pick it, to say it another way, in classical physics, you don't, you know, you don't need things from outside the, you know, the universe in order to measure the universe. Everything is in the universe, okay? So in particular, in, in, in classical general relativity, okay? We study the vacuum equations, and more generally the vacuum equations coupled to a billion fields, okay? And everything is there. So the observers, they're also there. They're, you know, they are one of the fields, let's say, okay? And so, so actually the proper way to say all these questions are as properties of that system of PDE. So you, you fundamentally, you don't have this issue that, okay, maybe you have in quantum physics that, okay, you have to talk about what, what is an, an observation. In classical physics, it's all, it's all in the system of PDEs. So all the, the questions fundamentally are statements about the, the system of PDE. So when we appeal to observers in time like Geodesics, it's just some limit of that system. So it's just something to help us, okay? But the, the, the primal formulation of everything is in a pure physics of fields. Because that's, because this is classical physics. Well, I, I, by which theory? By, by, well, first of all, these, these are not theorems. These, these are conjectures. Well, again, if it can, but, you know, Minkowski space can also have sort of, you know, such curves.   I mean, that's not sort of, that, okay, that's not inconsistent or in contradiction with anything. Anyway, we'll, we'll, we'll see sort of a lot of these properties and a lot of these subtleties actually in, when I talk about actually, well, hopefully very soon. So, okay, let me, let me try to put a little more meat into this by giving you what really you should think about as the sort of main model for gravitational collapse, at least historically, that sort of, if you want to shed light on, you know, what, what, what these statements are really saying. And this is Oppenheimer's night, okay?  If I get something wrong, there's at least one member of the audience who can help me out. So, this was first written down in a, in a paper from 1939. It was not presented in any sense in the way I'm going to present it. Actually, this solution in some sense is already implicit in the work of Lemaitre from, from a few years earlier and was from that work that it sort of eventually, yes? Ooh, sorry. Okay, okay. Yeah, I mean, in some sense this, it is already in, in, in Lemaitre, the, so Lemaitre was the first person to understand that the, the event horizon of, of Schwarzschild was not singular. And in fact, the reason he was thinking about that was sort of, because he was thinking about dynamical space times, dynamical space times in cosmology, but there is this connection with, with gravitational collapse and he sort of, so he already describes sort of this solution. He doesn't sort of write, write it down explicitly. So somehow there is, that's somehow where, where it's actually originating from. Anyway, so, so what is this a solution of first of all? So this is not a vacuum. This is a solution of the Einstein equations coupled to, well you can think of it as a perfect fluid where there's no pressure, however. So we often call this a dust, okay? So that's, I guess the energy momentum tensor is equal to this, where, where rho is pressure. And moreover, it is a, it is a solution of the system which evolves from very special initial data, initial data that looks as follows. So, so the initial data is, is, is spherically symmetric first of all. So if you want, I can think that I have a, a coordinate R, okay? That goes from zero. This is what I think about as a radial coordinate R. Okay? Which goes from zero to infinity. So this is time equals zero. Okay? And so the initial data is spherically symmetric but moreover it's actually homogeneous in some finite ball. So I have some R equals capital R, okay? And in this ball, I have rho, the density, constant and outside this ball I have rho equals zero. So this is the initial data. So if you want this, I can think of this as the initial state of a star. So this is the data, okay? So how does it evolve, okay? So to draw the space time that evolves on what I'm actually drawing is a Penrose diagram. Okay, of course these type of drawings did not exist at the time, but in any case. So, so I'll just use this now as a, as a, as a picture. And those of you who already, you know, have nodding acquaintance can nod. And while those of you who don't, well, this will be motivation for understanding tomorrow's lecture where in the context of spherical symmetry, I'll tell you exactly what these objects mean. So, so the evolution, it turns out that it looks like this. So the, if you want the surface of the star, okay? Follows a path like this in, in, in space time, okay? And eventually reaches a singularity where r equals zero. And of course outside of the support of the star by Birkhoff's theorem, okay? See, the outside of the support of the star, the solution is vacuum, but spherical symmetric. So it has to be Schwarzschild. And many of you have already seen the Penrose diagram of Schwarzschild, okay? So this is the boundary at infinity known as null infinity. Already you should be thinking this is related to far away observers in the radiation zone, okay? The past of future null infinity is bounded by a horizon known as the event horizon. So the, the compliment, right? Of the past of future null infinity is this region here. This is exactly the black hole region. And, and so we have, we have this picture, okay? So, so you should think that by the way maybe I'll label this that r equals infinity in some sense on this boundary, okay? So we'll understand exactly how and why whatever next lecture. All right, so this, this is the, the, the, the Penrose representation of, of Oppenheimer-Snoijder. And let me try to interpret now this picture, okay? In the terms that I talked about here, okay? So you'll understand sort of, so the claim is that this picture exhibits both this property here, okay? And this property here, okay? So Oppenheimer-Snoijder if you want is the model par excellence for sort of what we want to be true generically in evolution in general activity, okay? So let's see why, okay? So, so of course, again, just to make a sort of historic point. So very often when one talks about weak cosmic censorship, the way one would think about it in, again in slightly anti-Diluvian days was as follows. So this boundary here is a singularity, okay? We'll get back to that in a second. And this singularity is cloaked by a horizon. This is horizon, okay? Remember the horizon is just the past of, the boundary of the past of future null infinity. This is future null infinity. This is its past. The horizon is the boundary of the past, okay? And the singularity is sort of cloaked by the horizon in the sense that observers who are outside the horizon, okay? If they remain outside the horizon, they don't encounter the singular. But actually, it's sort of, and this insight in some sense goes back to Bob Garrosh and Gary Horowitz and Gary Horowitz's thesis is that you should, you know, to capture the property that we're interested in, you should actually focus on future null infinity itself. Because actually, you know, the statement after all that observers who remain outside the event horizon don't encounter a singularity is meaningless if all observers eventually cross the event horizon. So what you really want to say is that there is a class of observers, okay, who do not encounter the singularity. That's really what you want to say, okay? So, all right, one thing that you might want to, one way you might want to say it is talk about find certain classes of time like geodesics, et cetera, but then you run into this problem which is partly related to your question of, you know, which geodesics and why, whatever. So actually, it turned out that the cleanest way of thinking about this is thinking of the completeness as a statement that lives at null infinity, okay? So, I claim that you can make sense to the following words that future null infinity is complete, okay? So, this is a funny statement because, okay, whatever future null infinity is, it looks like it is asymptotically a null hyper surface. And of course you can't talk about the length of null curves because the length is zero no matter how long they are, okay? But nonetheless, I claim and we'll discuss it next time, that you can actually make sense of these words, okay? And, well, a very pedestrian way to understand it for the experts is that if you normalize in a geometric way a retarded time coordinate, okay, along null infinity, then the range of this coordinate will be from minus infinity to plus infinity, okay? But you have to normalize it geometrically, that's the point, okay? So, you should think of it as meaning that, you know, very far away observers, they live forever in both positive time here and negative time there. Another way of thinking about it, sort of related to the meaning of null infinity, so you should think that null infinity is actually where gravitational wave experiments take place. So, another way of thinking of this is that those experiments can go on for as long as they're funded, okay? All right, so this, so if you want, this is what weak cosmic censorship is telling you that for now generic initial data for the Einstein vacuum equations and for a good class of Einstein mother systems and which class we'll have to discuss later, future null infinity is something that one should be able to define and it should be complete, okay? So that's this statement, okay? So what about strong cosmic censorship? So, so first of all, although this is complete, this day's time is geodesically incomplete. In fact, it turns out that there is, ah, there is colored chalk. Turns out that all, let's say time like geodesics which cross the event horizon, okay? They will be incomplete, okay? And in fact, in finite time, they are going to reach this boundary where r equals zero. So this is geodesically incomplete and in fact, one way to show that this is geodesically incomplete is of course to do this computation, which is not so difficult, it's probably the easiest, but another way is to notice that any sphere here, so if you think these are r equals, you know, these are sort of spheres of some r value, okay, so this is a sphere in Schwarzschild actually, because this is outside the support of the mother. And well, if you know your Schwarzschild metric, then you'll realize that this sphere is something called a trapped surface and that's exactly what you need to apply panerosis in completeness theorem, okay?  It has to be because I told you that this was the maximal Cauchy development, so I can apply panerosis theorem, this must be incomplete. So somehow, panerosis theorem already told you that this was it and I remarked that if you want for a reason. Okay, so let me just, maybe, since I bothered to say it, this is a trapped surface, trapped surface, so panerosis already tells you, okay? So, okay, but anyway, so this is an incomplete geodesic, okay? And what happens, as I just told you, is that all incomplete geodesics, so I just drew a bunch of them, they end up at r equals zero. Now, what's written in all of the textbooks is that r equals zero is a curvature singularity, that's what's often said. So the curvature blows up. Now, for instance, just so that that's unambiguous, you can look at the Kretschman scalar, which is a quadratic expression in the Riemann curvature tensor, which is scalar, okay? And that blows up as r approaches zero, okay? So that's great, of course, but the fact that the curvature blows up is not too justice to the nature of this singularity, because in reality, the singularity in, so by the way, this I should mention is true, both of these geodesics, okay, who sort of approach r equals zero in the Schwarzschild region, and these geodesics who approach r equals zero, you know, always inside the star, okay? So the curvature blows up, but this does not begin to do justice to how singular this boundary is. So in reality, not only does the curvature blow up, but observers torn apart these observers by infinite tidal deformations. So what does this mean? This means that if I'm a, so let me say it first in this language of observer, so if I'm an observer, so I'm following one of these geodesics, but I also have hands and legs, okay? And if you want, you can think of the hands and legs as Yacoby fields, you know, your differential geometry, because the ends of my hands and of my legs, they also follow geodesics, okay, which are infinitesimally close to sort of where my heart is, okay? So if you look at the hands and the legs, then depending on their orientation, they are infinitely stretched or infinitely compressed as I approach r equals zero. So I'm killed as an observer. This is very different from just the statement that curvature blows up. That's an instantaneous force, okay? That's just an instantaneous force, which is infinite. So if I just know this, okay, then no one says that I am killed as a classical observer, okay? But this statement here tells me that I'm killed. So, but again, in the spirit of that question, can I say these two statements without referring manifestly to observers just in the language of the field, okay? So can I somehow capture these statements just in the language? Yes, question? Yes, of course, yeah, and we will see this later on. Okay, so actually this, it's sort of funny. So I think the first person to claim this statement was Roger Penrose, but he actually doesn't show it and the first calculations sort of show this in some sense are actually in Misner Thornin Wheeler. So there's a section of Misner Thornin and Wheeler about this. Okay, so how can you say this without referring to observers just in terms of the metric? Well, the statement that the curvature blows up as observers approach R equals zero, that's very easy to say in terms of the metric. This is just the statement that the metric, though the curvature blows up, so this is just saying that, let's say it like this, the Lorentzian manifold, M comma G, okay, is inextendable as a manifold with C2. So C2 means twice continuously differential, okay? So that's very easy to infer because of course, suppose you could extend, so here's a little bit of an extension, okay? Then very elementary geometry says there has to be some time like geodesic that passes into this extension, okay? But the curvature over here was blowing up as you approach there, okay? And that contradicts that this is a C2 metric, so locally nothing should go wrong. Okay, so that's what curvature blow up means. It means that the metric is inextendable as a C2 metric. So it turns out that this statement with observers being torn apart, well it's sort of difficult to directly translate because then you sort of get into this issue of which observers, maybe there are some special observers that are oriented in some way, such that. But the claim is that the correct purely field theoretic interpretation of this, okay, is the statement that the metric M4G is inextendable as a manifold with continuous metric, okay? So if you want, this is a, so from the point of view of differentiability, this statement lives at the same level as this one. I claim that these are not equivalent statements but in the spirit of what I told you, this is sort of the purely field theoretic description that the singularity R equals zero of Schwarzschild or Oppenheimer-Sneider is extremely strong, okay? Okay, so it's sort of of the strength that destroys observers, destroys. All right, so actually I should maybe even put an asterisk on this statement. So this statement is true about Schwarzschild but it was only very recently shown again by Zbierski. Okay, let me say, I don't know, 2018. And I don't know it to have been shown actually for the interior of the star. So any bright young student in the audience might want to read Zbierski's paper and see if you can actually extend this C zero, so continuous sometimes we C zero, C zero and extendability to the interior of the star. Okay, because this is really the statement that sort of, the singularity is strong. Okay, so in particular, let me try to now interpret sort of this statement here, okay? So that we see in the, let's say, the first part of the statement, okay, that indeed all incompleteness, all incomplete due to these six, okay, they go to a singularity. All right, that sort of this informal, version of the statement, okay? But the more correct version of the statement should be that there should be some obstruction to extending the spacetime. So spacetime may be incomplete, okay? But there should be some obstruction to extending it, okay? And what we see is that there is first of all the naive statement that indeed the spacetime is an extensible as a manifold with C two metric, okay? But there's actually a much, much stronger statement secretly there, okay? Which is that the spacetime is inextensible as a manifold with continuous metric. So let me now spell it out for you and try to interpret what I said here. So let me first say it in the language of observers and then I'll say it in the pure field theoretic language. So why is this telling us that GR uniquely predicts the classical fate of classical observers and why am I focusing so much on this destruction of observers as opposed to infinite curvature? So what's going on in the spacetime? Well, there are two classes of observers, right? They're the observers who don't cross the event horizon. Okay? And it turns out that these observers they all live forever. This is related of course to this completeness that I claim is the way we in the modern world the formula we cosmic censorship. So these observers live forever and of course the classical spacetime is telling me everything about this observer, okay? And then we have these yellow observers who only live for finite time and reach R equals zero. And what classical theory is telling me is that these observers as classical objects are actually torn apart, okay? So even if you believe, and certainly most people do that sort of maybe very, very close to the end we have to go beyond classical physics to understand sort of what happens in physics as far as this observer is concerned they have already been torn apart. They no longer exist as a classical object they're now some quantum soup and okay you can then use your favorite quantum theory of I don't know what to speculate what happens to that quantum soup. But as far as the classical physics is concerned classical physics has given up this classical observer to whatever is beyond classical physics as a non-classical object. So classical physics is in this sense self-contained. It tells you what happens to all classical objects, okay? So from the PDE point of view and let me just end with this comment, okay? So if I don't want to talk about observers, all right? Then I should go back to sort of just the initial value problem, okay? For my spacetime. So again, given initial data there is a unique object this maximal Cauchy development, M comma G. But the M comma G in what sense is it maximal? It is maximal as a globally hyperbolic Cauchy development. So no one says that that cannot be extended further just not as a globally hyperbolic spacetime. It is only maximal as a globally hyperbolic Cauchy development. We'll see exactly, I'm slightly behind so we'll see at the beginning of next time exactly in what way this can fail for instance, okay? So what we see here in this property, okay? Is that this object, the maximal Cauchy development is not just maximal as a Cauchy development it is maximal in any reasonable sense of the word. There is no possible way to extend this, okay? So in the sort of pure PD language again this is telling you that this unique object unique just by fiat is really unique in any sense it is the unique sort of classical object that we can associate to initial data, okay? So it is in this sense that this property represents determinism in general ultimately. Okay, so let me stop because I'm already two minutes over time and we'll sort of maybe in the beginning of the next lecture I'll explain so I'll maybe draw this picture again and I'll explain the many ways in which these nice properties could have failed to happen, okay? And then we'll start in earnest discussing spherically symmetric spacetimes where we'll be able to make formal sense of this picture and sort of really sort of show what you can prove, okay? Thanks.