PSW 2478 Einstein's Real Equation | Sean Carroll

foreign 2478th meeting in what is now the 152nd year since its founding on March 13 1871. good evening everyone my name is Larry milstein I am the President of DSW one of the oldest scientific societies of Washington DC committed to providing a forum to further scientific understanding and inquiry Welcome to our members guests and Friends including everyone here in the Palo auditorium at the Cosmos Club and everyone tuning in on YouTube to tonight's lecture by Sean Carroll the society is grateful to PSW full year sponsors PSW member Mike Helton and Hilton Associates LLC and the policy studies organization in cooperation with the American Public University we're also grateful to the PSW spring lecture series sponsor PSW member Tim Thomas for his generous support and the sponsors of tonight's lecture the intellectual property Law Firm melon white Solano and Branigan PC speaking of sponsors it mentioned at previous meetings the society is undergoing a fundraising campaign to ensure that it will be able to meet expenses in the coming years especially the much higher prices for facilities we use at the club if you want PSW to continue its activities and remain a vibrant organization please contribute to its financial support let us know of any individuals or organizations who might be interested to support the society I can be reached most easily by email at president at PSW science.org I am also pleased to announce the following new members Scott Carson a computer scientist currently a contractor to the U.S Office of Management and budget interested in technology in general and space programs in particular who learned a PSW through friends and colleagues Arthur house an arbitration and mediation attorney interested in ancient civilizations life sciences and cosmology who end up PSW from PSW president Larry milstein and tonight's speaker Sean Carroll who learned a PSW from the invitation to speak here tonight and whose interests will be clear in a small part from tonight's lecture we welcome them all to membership [Applause] now if you're not a PSW member please join it's very easy to do go to the webpage the homepage www.pswscience.org and click on the join blue button at the upper right hand corner and then follow the prompts all members are entitled to where the PSW science rose that whose symbolism is explained on the BSW home page the rosettes are 15 plus 90 cents DC sales tax and they can be purchased online or at the rosette table in the back please note that rosettes must be picked up at a lecture here at the Cosmos Club recording secretary Cameo Lance will now present the minutes of the 2477th meeting and the 92nd Joseph Henry lectures by Paul Kessler Eleanor Morgan Daniel innocente and Richard Clinton on Space architecture Cameo stage of yours thank you Larry on May 19 2023 from the Powell Auditorium of the Cosmos Club in Washington DC and by Zoom webinar broadcast on the PSW science YouTube channel president Larry milstein called together the 92nd Joseph Henry lecture and the 2477th lecture meeting of the society to order at 803 pm he welcomed new members and the minutes of the previous meeting were read on behalf of the recording secretary president milstein then introduced the four speakers for the evening Paul D Kessler an aerospace technical manager at Nasa Eleanor e Morgan a space architect and program manager at Lockheed Martin Daniel inocente senior space architect at Blue origin and raymie Raymond Corky Clinton senior technical advisor and Science and Technology office at Nasa Marshall space flight center president milstein noted that the opinions expressed in these lectures are not of the official opinions of NASA or Lockheed or any other organization that may seem to be seen somehow associated with the speakers the lecture for the evening was titled architecture for the new space age Kessler began by thanking the society for the invitation to speak Kessler first discussed what the vision of NASA is with respect to the exploration of space by humans he contextualized this reviewing of NASA's overall strategy to pave the way to human exploration next he reviewed the variety of habitation considerations and challenges the space environment he explains presents a variety for a variety of difficulties for long-term habitation in space including dust contamination radiation and the capability to maintain and repair systems Kessler concluded his portion of the lecture by emphasizing the Reliance of collaboration and noted that it is a key to success next Morgan took the stage and began by thanking the audience in PSW science Morgan reviewed technology developments at Lockheed Martin is working on to set the foundation for going to Mars these Technologies she explained are have very long lead times and Lockheed Martin intends to use the moon as a testing ground for the Technologies which are intended to bring humans to the red planet she went on to review their ongoing efforts in low earth orbit that are on the development path to assist with in-space transportation of crew Morgan commented that Artemis 3 mission will be the first mission to bring humans back to the lunar surface Morgan went on to discuss Transportation Technologies and difficulties and solutions for habitats that Lockheed Martin is working on she showed a video interestingly of a stress test on their current inflatable habitat module which she emphasized is more rugged than a metallic structure and very much not like a bouncy house Morgan went on to discuss habitats and her experience as a simulated astronaut and other efforts that Lockheed Martin is pursuing she concluded by reviewing concepts for potential architectures for a Mars Transit vehicle innocente an architect and space architect began by discussing the concept of space architecture which involves designing and building structures for human habitation and space he mentioned the historical development of space architecture the importance of considering factors like the environment and habitility habitability and health in space and the integration of various systems and Technologies this speaker went on to highlight specific projects and Technologies being explored such as the orbital reef and the carbon fiber structures and concluded by discussing the challenges and opportunities of space architecture on celestial bodies like the Moon the fourth speaker for the evening Clinton began with a Monty Python joke at which the audience laughed he then briefly reviewed the history of in-space manufacturing and extraterrestrial construction at Marshall space flight center the speaker then reviewed the objectives of the Moon to Mars exploration strategy emphasizing the need for advanced manufacturing autonomous construction capabilities a continuous human lunar presence and a robust lunar economy next the speaker outlined the phased approach to lunar construction starting with the development of planar surfaces and structures like landing pads and Roads followed by a more complex vertical structures and the eventual establishment of a lunar settlement with a Full Construction capability Clinton concluded by emphasizing the important role Partners play in this ambitious endeavor the question and answer period began one member asked whether there was consideration of placing a habitat into the lava tubes on the lunar surface to protect from radiation Kessler first responded that there is indeed consideration for this and placing habitats under a meter of regolith is being considered as well however for the initial development stages the habitats will likely be lunar surface bound Clinton next highlighted the pockets of availability for deposits of lunar regolith to protect from radiation and some of the structures being considered next a Swedish viewer streaming the lecture asked about the in-situ fuel resources Clinton responded that liquid oxygen and liquid hydrogen could be produced from water found on the lunar surface for fuel after the question and answer period president milstein thanked the speaker made the usual housekeeping announcements and invited guests to join the society president milstein adjourned the meeting at 10 20 pm the temperature in Washington DC 5 degrees Celsius the weather was partly cloudy and the attendance in the Powell Auditorium was 90 attending via live YouTube was 34 for a total in-person attendance of 124 viewers the number of online viewers in the first two weeks of streaming was 395. respectfully submitted recording tax secretary thank you Cameo are there any comments Corrections suggestions regarding the minutes hearing none all entertain a motion from a member to accept the minutes is read do I have a second all members in favor all members opposed the minutes are accepted as read and will be posted to the website in due course we now turn to tonight's lecture by Sean Carroll on Einstein's real equation mass energy and the curvature of space-time Sean is Homewood professor of natural philosophy in the department of philosophy and professor in the department of physics and astronomy at Johns Hopkins University and he is a member of the fractal faculty at the Santa Fe Institute I'll leave it to him to explain what that means Sean is well known for his work in theoretical physics and the philosophy of science his theoretical work focuses on the foundations of physics and his research Publications include work on Lawrence and variants closed time-like Curves in general relativity topological defects in field Theory extra space-time dimensions dark matter and its interactions with ordinary matter and with dark matter sorry dark energy and its interactions with ordinary matter and dark matter modifications of general relativity and aspects of quantum mechanics he is an author of many Technical and general Publications and books including the well-received textbook space-time and geometry and introduction to general relativity and his most recent book the biggest ideas in the universe volume 1 space time and motion Sean also has made two lecture series on physics on wondering the Great Courses and they're very good and he hosts a weekly mindscape podcast among many other honors and awards Sean has been awarded prizes and fellowships by NSF and NASA the Sloan Foundation the Packard Foundation APS the American Institute of physics through US side of London the Guggenheim Foundation and aaas Sean earned a BS in astronomy and astrophysics with a minor in philosophy at Villanova and a PhD in astronomy at Harvard University all questions will be fielded after the lecture in the Q a session Sean the stage is yours thank you thank you very much Larry thanks to the uh philosophical Society of Washington which I guess I'm a member of now all right go me that's wonderful I didn't even know um and some of you have uh been chatting to me and know that I just recently moved back to the east coast I lived in Los Angeles for 17 years I just want to say you know I'm living in Baltimore now it's good to be back in a region of the country where the idea of an exciting thing to do on a Friday night is to go to a physics lecture I think that's great and here we are and it will hopefully be exciting this is the book that uh inspired this lecture but it's not really a book talk I'm not going to try to go through what's in the book in the book what I try to do is to go through the parts of modern physics at least the first third there's going to be two more volumes coming but the parts of classical physics that are not speculative and Cutting Edge but they're going to last forever are going to be around with us hundreds of years from now and the trick to that this is not the first book to do that but the trick is I assume that you're smart I don't assume that you're educated so I don't assume that you're a physicist or mathematician or anything like that but I assume that the math that physicists use to discuss some of these things is not beyond the average person it's just that the average person doesn't want to do problem sets and homework so I teach you the ideas and the concepts Behind These equations and you actually see the equations and therefore hopefully you get a little bit more of an insight into what physicists are really thinking about when they talk about their theories and of course we've all heard of Einstein and we've all heard his equation here it is the famous E equals MC squared right and look it's not that hard energy is mass times the speed of light squared there's some footnotes there there's some details like what energy is it why is the speed of light there Etc what I'm here to tell you though is that if you talk to a professional physicist and they mention Einstein's equation they're not talking about this they're talking about this which if I were to read it out loud is arm you new minus one half r g mu Nu equals eight Pi G team you knew that's Einstein's equation now we're talking right I mean now we're being serious so for probably for most of you you are not familiar with this equation Uh there's Big letters and Tiny letters some of the letters are in Greek it's very strange we don't know what's going on an hour from now you will know what is going on I'm going to teach you what this equation means why we think that it is true how Einstein came up with it and what it tells us about the universe but to start we need to go back to where we remember physics back in the days of classical mechanics Allah Isaac Newton from the 1600s right before Newton came along Aristotle had given us the reigning Paradigm of physics and Aristotle had a very simple idea that if you want something to move you have to keep pushing it things had a nature to be in a certain place and they would just sit still and if you push something you could move look and then if you stop if I stop pushing it it stops moving sadly we're on a tilted surface here so this is not the perfect way to illustrate it but Aristotle's not wrong here's a picture of someone trying to move the car it's not going to move by itself there needs to be fuel in the car there's a little puppy that's trying to steer the car I don't know if you can see that but that's not very helpful some motive Force has to act on it Newton comes along and says something very different and of course it wasn't out of uh just the genius of Isaac Newton he was building on the shoulders of giants very famously he says that the acceleration not the velocity but the acceleration of an object is proportional to the force that you put on it and the proportionality is the mass so what that means is that if you have zero force on something if you're not interacting with it at all if you just let it go then it's not accelerating not that it's not moving it's not changing its motion so if you have an object that is moving at a constant velocity and no forces are acting on it then Newton says it will continue to move at that velocity forever it took a long time to get there obviously from Aristotle to Newton there were great contributions made in the Islamic Golden Age avocena had a you know a lot of insight here and nowadays we build spacecraft that actually move through the space and demonstrate this because the reconciliation is of course that the real world has friction and dissipation and air resistance and all those things so what I want to emphasize though is not this change from Aristotle to Newton but the existence of an equation like this this equation like equals mc squared this is not that intimidating f equals m a the your first year physics teacher will joke that this is the only equation you need to solve the exam you could in principle derive everything from this slight exaggeration but only slight this is at the center of classical physics the only slightly intimidating part is that there are little arrows over the F and the a indicating that those are vectors so they had not only a size but a Direction but you know that if you push in something you push it in a direction and that's the direction which it accelerates this is not really too scary what I want to emphasize about it is these two things that I've written here it is precise and it is universal so once you've written down the equation one Sir Isaac Newton writes down the equation it's not his equation anymore it's everybody's equation anyone can solve it anyone can use it anyone can understand it in principle just as well as Isaac Newton did and it's Universal it's not just this car it's not just this particular incident it's throughout the Universe literally this equation is supposed to hold true and that's why it is worth putting in a little bit of effort to understand what the equations are trying to tell us because the equations of physics have this character they are precise they tell you exactly the amount of force you need to exert if you want to get a rocket to a moon you better not just have a tendency to accelerate you better know exactly how much you accelerate buying and they're Universal but this equation is not that much use if you don't know what the force is so Newton also gave us his inverse Square law for Gravity there's a story that you get told of Newton sitting under the apple tree the reason you get told this is because Newton told this story he was later in life trying to burnish his own reputation I mean if Newton doesn't think his reputation is good enough I don't know what help hope there is for the rest of us but he was trying to tell stories to increase the impression of his genius he didn't discover gravity right like Newton didn't realize that apples fall from trees for the first time people knew that already the thing that he got was that the same force that makes apples fall from trees explains the Motions of the planets and the moon in the solar system that's the universal nature of Newton's law of gravity and what it says is once again it is a equation between vectors F on the left hand side so let's imagine we have a heavy object with mass capital M like the sun and a smaller object with mass Little M like the Earth and we draw a little Vector e pointing from the Earth to the Sun and all it does is point it has a unit length it doesn't get bigger or smaller it just points in the right direction so it's just telling us what is the direction in which gravity is pulling and the answer is toward the sun all the time then the next question is well how strong is that force of gravity and that's where the inverse Square comes in capital G which we now call Newton's constant is a constant of nature telling us how strong the force of gravity is and then there's the mass of one object the mass of the other object divided by their distance squared so the inverse Square law so this is just a way of making rigorous and precise the idea that gravity is strong when objects are heavy and nearby that gets weaker as they get further away of gravity we already had on the previous slide that Force equals mass times acceleration you can guess what's going to happen I'm going to set those two things equal to each other and a miracle occurs when we set these two things equal to each other so we have f equals m a force is mass times acceleration and f equals g m m over r squared so the Little M appears the same way on both sides of this equation forget about the F this turns out to be something where we can cancel the Little M that's a little math I'm trusting you to do that we can divide by Little M on both sides and what we get is an equation just for the acceleration so this is an equation for the acceleration of anything in the gravitational field of an object of mass capital M at distance r squared the illustration here this is a painting because there are photographs of this but they're crappy quality this is a real experiment that was done on the moon I think it was Apollo 15. I'm not completely sure is anyone 15 yeah good so uh Newton had predicted in the 1600s that if you drop a hammer and a feather they will fall at the same rate if it weren't for air resistance but he was surrounded by air resistance so it took us a while to get up to the moon and actually do the experiment and indeed the hammer and the feather fall at the same rate when you're on the moon why because there's nothing about the hammer or the feather that appear in this final equation anything accelerates at exactly the same rate okay so two things have just happened one is I'm just warming you up by showing you an equation and manipulating a little bit okay and that's fun and you you realize Ah that's not so scary anybody can do this I'm going to apply to graduate school tomorrow and but the other thing is that we've learned something deep and important about gravity which is that it is universal gravity doesn't care what color you've been painted it doesn't care what day of the week it is it doesn't care how you're spinning it doesn't care what your mass is you are going to be accelerated under the action of Gravity by the same amount as anything else at the same location as you in space-time so I'm going to drop that for a minute but keep it in mind gravity is a universal Force unlike let's say the force of electromagnetism where I can have a positively charged particle or a negatively charged particle and they act in an electric field but everything acts the same in a gravitational field so years pass here comes uh Albert Einstein you know when you you've all seen pictures of Albert Einstein they've probably been pictures from his Twilight years when the hair had sort of gone crazy and he was wearing the sweaters and everything and you forget that back in the early 20th century he was a sharply dressed young man someone was combing his hair and uh he was changing our Notions of space and time so Isaac Newton like we said he did these two he did many things but two of the things he did was classical mechanics and in particular the law of gravity Einstein in 1905 revised classical mechanics in a dramatic way with what we call the theory of special relativity and it wasn't just Einstein he was very much following up and basically putting the finishing touches on some ideas that have been growing since the mid-1800s when electromagnetism came along and we're not going to go into detail about special relativity usually there would be discussion of you know length contraction and twin paradoxes and so forth I don't really care about that we have other fish to fry what I want to tell you is that what he was trying to do was reconcile the new science of electromagnetism with the old rules of Newtonian mechanics couldn't do it realized we had to change our notion of what space and time are to make it work interestingly all the equations you need to do that had already been written down by other people what Einstein added was the conceptual Insight that there is no such thing as an absolute frame of rest as there would have been in Newtonian mechanics if you say how fast is this planet moving through space there is no answer to that relative to what is the only thing you can ask and that's where the idea of the theory of relativity comes from motion is relative the real kicker though was that there is a speed limit the relative motion of two objects can never exceed the speed of light that was what came out of electromagnetism as something special and to reconcile these two ideas he needed to really change how we think about space and time and that's where the whole story of length contraction time dilation Etc comes from but rather than tell you that story I'm going to point out a wrinkle that happened a couple years later from this guy Hermann minkowski who was actually he had been one of Einstein's professors at University and minkowski was a mathematician so of course he was reading his star pupils papers right and Einstein was not a mathematician bless his heart sometimes people will tell you Einstein was bad at math that's an exaggeration Einstein was better than at math and URI are good at math okay but it wasn't his passion what he cared about was physics and he learned only enough math to get by with the physics that he needed to do so minkowsky who is a mathematician looks at Einstein's theory and says we can formulate this in a mathematically more elegant way if you don't just say I'm going to change our notion of space and our notion of time but instead you marry them together to form a single unified space-time and this famous quote where he says that space by itself and Time by itself are doomed to fade away into mere Shadows some people thought this was just the bee's knees this is the coolest thing space time this is awesome other people were sort of grumpy and skeptical and that includes Albert Einstein he was grumpy and skeptical about this he was like ah those mathematicians again you know they've got their handy their hands on my work and they've turned it into something you know recondite and hard to understand he was not a fan of the idea of space-time from the start his mind would change Einstein had many salutatory qualities maybe the best of which was he was very willing to change his mind when it became necessary to do so so if Einstein succeeded at uh uh overthrowing classical mechanics he was still left with the puzzle of fitting gravity into the game and if you go back to what minkowski said he was at heart a geometer he thought about the geometry of things and geometry had really witnessed a Renaissance in the 19th century with uh The Advocate the Advocate the Advent of non-euclidean geometry so minkowsky knew about this and he says Einstein you should think about space-time as a novel kind of geometry so what is that all about and this actually again is counter-intuitive but not at all too hard to understand if you just sit through and think about the steps think about literally the steps the steps you take when you're walking between two points okay we've all learned a little bit of euclidean geometry the geometry that came down to us from the ancient Greeks the geometry of the tabletop or the flat plane so if you have two points in the space and someone has put a set of coordinates on space so you have a field like a football field it's flat there are coordinates on the football field right there are yard lines so you know that between these two points there is a coordinate differential X and a chord differential y thagoras tells you what the distance is between these two points Pythagoras's Theorem D Squared is x squared plus y squared the shortest distance between these two points is a straight line it's not you walk down X and then you walk down y but this is the distance you would have to travel to get between those two points and you operationalize this this is what Einstein was really good at this is what the Spirit of special relativity is all about you say how would I measure it don't tell me what exists and what doesn't exist tell me what I'm going to measure so when I say that there's a distance between two points I could just measure it by my little odometer right you can have your smartphone or your smart watch do this for you count your steps or you could plug into the formula either way works and what minkowski is saying is it's the same thing for space-time but the formula is just a little different in space-time you can move between two events okay again not that spooky when you want to meet somebody for coffee you need to tell them where you're going to meet them and also when you're going to meet them if you only tell them one of those pieces of information that is not enough a point is specified in space-time by giving its location in space and its moment in time and you can travel in some slightly generalized notion of the word travel between two events in space-time and to travel in space-time in a straight line just means move at a constant velocity and how do you tell the interval that has passed when you move between two events in space-time you check your watch the implication of minkowsky and Einstein's work is that your elapsed time that you measure using your watch is a feature of space-time and the journey that you have taken through it and just like the distance that you get between two points in space follows a certain formula Pythagoras's Theorem the time interval that you experience on your watch obeys a certain formula it's the same thing as Pythagoras Theorem but with a minus sign in it so you have that the what we call the proper time this is our first Greek letter this is the Greek letter Tau the actual time that you experience is not the coordinate time on the universe when you meet someone for coffee and you say 8 pm that's an agreed upon time that the Universe sort of shares we put coordinates on space time we know where we're going to meet them when we're going to meet them Einstein says that you also have a personal time that you personally experience and measure on your wristwatch that personal time is Tau the universal time is T they are related by this simple formula Tau squared is t squared minus x squared so you see as an immediate implication of this that in Space the straight line is the shortest distance path right you can't go any shorter than just moving between the two points in a straight line in space time a straight line is the longest time path if you zoom out at the speed of light to Alpha Centauri and then you zoom back you've traversed a lot of X as well as a lot of T and they can almost cancel so you can travel between two points without experiencing much time at all the most time you can experience between two events in space-time is by going at the minimum speed that will get you there at a constant velocity so that's weird but it's just geometry with a minus sign it's actually fundamentally not that scary so can we somehow fit gravity into that picture minkowski comes along and says think about it geometrically Einstein says eh that's too hard I just want to think about my rods and my clocks but he wanted to fit gravity into the relativistic framework he failed at first it couldn't fit so maybe you just need an entirely new theory of something and he remembered what Newton had figured out that gravity is a universal Force but the way he thought about it was not by dropping Hammers and feathers on the moon but by saying that it gravity is such a universe Universal force that you can never be absolutely sure that you're in a gravitational field we think that right here we feel gravity I feel it pushing up on my shoes right I could test it by dropping the laser pointer and it seems to work but Einstein says all of those phenomena would happen exactly the same way if this room was in a rocket ship and a very quiet rocket engine accelerating Upward at one g of acceleration right there's no experiment you can do closed in a little rocket ship that would be able to distinguish between being accelerated and being in the force of gravity why because gravity is universal it acts on everything in exactly the same way now you and I if we had thought of that would say huh that's kind of cool and we've gone on to do something else but he was Einstein he kept going and he said so if gravity is different from all the other forces because of this special feature that it's Universal then in some sense maybe we shouldn't think of gravity as a force at all maybe we should think of it as a feature of space-time itself it's built into space-time and that's why everything in space-time responds to it in the same way what could the feature of space-time be that gives rise to gravity the geometry of space-time the curvature of space-time so minkowsky had said space-time has a geometry space and time are unified into space time and there's a geometry of it but that geometry was just souped up euclidean Geometry it was the equivalent of the Pythagorean theorem but the new hotness at that time was curved geometries and Einstein was smart enough to realize I need curved geometry for space-time to explain gravity he had no idea how to do that he did not know the math that you needed to do that but happily he was very plugged into the old boy Network really honestly Einstein had a friend from school Marcel grossmann who was once again a math whiz he knew the geometry he knew all about all this new work that had been done in the 1800s and he tutored Einstein this is what made Einstein Einstein it wasn't that he was just born knowing tensor calculus but when it realized I got to learn this stuff he just did it he just learned it and then he started using it as well as anybody so there's Marshall Grossman and Einstein uh Grossman's name is still remembered it's on one of the major general relativity conferences held every three years so imagine that I am Marshall Grossman and you are Albert Einstein I'm going to teach you the geometry of curved spaces so that you can understand how gravity is the curvature of space-time so we have to go back to geometry and now we're going to shift gears a little bit we've been like in second gear so far okay we're gonna get up to third gear now um Euclid you know Euclid Euclid proved a bunch of theorems about geometry he probably borrowed theorems from other people of course but one of the interesting things about euclidean geometry is it's based on a series of postulates one of which is called the parallel postulate which just seems a little bit weirder than the others the others are like there's there are things called points you know it's like very basic stuff but then the parallel postulate says if you start with a little line segment and you shoot off two lines perpendicular to that original line segment they will stay parallel forever that seemed very specific and people tried for centuries to prove it using the other axioms of euclidean geometry and they never could it wasn't until the 1800s they realized the reason they couldn't prove it is because it might not be true you could invent a system of geometry based on the parallel postulate but you could also say well I'm going to postulate that initially parallel lines eventually Converge on each other or you could say I'm going to postulate that initially parallel lines eventually diverge from each other and people realized that there were three different kinds of geometry all of which share the feature that euclidean geometry has that the geometry looks the same everywhere but it's different from the two other options there is flat geometry in which initially parallel lines remain parallel spherical geometry where initially parallel lines converge just like lines of longitude going up from the equator would Converge on a globe of the earth there's also what is called hyperbolic or negatively curved geometry where initially curved we're initially parallel lines diverge so these were all the options that were put forward back in the 1830s but there was still something missing there was still this was still there's two things going on here one is that when we draw these pictures we tend to think of them as being embedded in good old three-dimensional euclidean space right we live in three-dimensional flat space we put a sphere inside it we try to put the saddle shape inside it that's harder to do but we still kind of visualize it that way what you really want is an intrinsic geometry a geometry that does not depend on what things look like from the outside and the other thing you want is to allow the geometry to change from place to place it's nice when you have a perfect sphere or saddle or plane but many real things are kind of lumpy we would like to be able to describe lumpy things in our geometry as well so the task of doing that fell upon this man Bernard Riemann he didn't want to do it by the way uh in the German system academic system you have to take many exams and you have to take after your PhD exam there's a whole nother exam you have to do to be allowed to teach in universities and riemann's the sis advisor was Gauss Gauss is one of the most accomplished mathematicians of all time so Riemann gives his advisor a list of possible topics for his exam and Gauss came back with the one that Riemann thought was the most boring namely the foundations of geometry but he was very smart he was already very accomplished mathematician there's a lot of things named after Riemann even though he died pretty young and it was his job to figure out the foundations of geometry and it's actually quite worth reading his paper because he's constantly fetching throughout the paper he's saying like ah this is not my style I don't really like this but I guess I have to do it and you know it very much humanizes one of the Giants of the field so what Riemann realized what he put forward was the idea the question is how do you capture the information that tells you the geometry of a space or a curve or a three-dimensional or even a 20 dimensional kind of thing there might be many different ways that you could quantify that this geometry that you care about and what Riemann hit on was the lengths of Curves basically if you know the geometry of a space entirely and I draw a curve in it then it's kind of intuitive that I could figure out the length of that curve right what Riemann postulated was you can go the other way if you know the length of every possible curve you can draw then I can figure out what the geometry is on a plane the length of every possible curve is just given by Pythagoras's Theorem but on a sphere or a pringle it will be a little bit different so he wants a general formula so this the strategy is actually not that hard if you have an arbitrary space and an arbitrary curve inside it you zoom in this was the old trick from calculus as invented by Newton and leibniz you always zoom in and when you zoom in close enough the curviest thing in the world begins to look straight so this is why what rebound invented is now called differential geometry because it's the calculus trick of zooming in on curves to make them look straight talking about what happens there and then adding them all together to form the whole thing so if you have a curve and you zoom in it looks straight we know how to calculate the length that it's whatever generalization you need of Pythagoras's Theorem if it's three dimensions rather than two dimensions for this three-dimensional line segment here you might guess and you would be right that D squared equals x squared plus y squared plus Z squared okay the three-dimensional version of Pythagoras's Theorem but what we want is to let that information change from point to point in our geometry so just uh we're going to warm up to this now we're shifting gears again fourth gear now okay so in euclidean space three-dimensional euclidean space it's just Pythagoras theorem with an extra little Factor there D Squared is x squared plus y squared plus Z squared minkowski's space time I told you Tau squared is t squared minus x squared but that's like a little toy two-dimensional version the real formula is t squared minus all of the spatial coordinates x squared minus y squared minus Z squared pretty straightforward but what we do is we stare at these two equations and we try to figure out what is the pattern what is the general principle that is being expressed in these two very very simple special cases and that might be hard but again Riemann he's very smart so he said okay what's going on here is on the left hand side I have some interval in space or space time squared on the right hand side I have all my coordinates times each other so I have x times itself y times itself but he because he's Riemann he's very smart he says I could have x times y I could have x times Z Etc so basically I'm adding up some number because sometimes it's plus sometimes it's minus maybe it could be other things some number I'm adding up times two coordinates and the coordinate might be two copies of itself so x squared x y x Z all the different combinations they all go into this sum to make it look a little bit mathier this is the mathy version of this okay but we're doing this in space time so we'll put T in there as a coordinate the proposal is from Riemann that you can think of the interval of a little tiny line segment in space time as some number times t squared plus some number times T times X plus some number times T times y Etc through all the different possibilities and here capital a capital B etc those are numbers that might be able to change from point to point the way that we generalize Pythagoras's Theorem might not be the same everywhere that's what's going to let us have different amounts of spatial space-time curvature at different points in our space so now we're going to make it more slick this is the part that mathematicians like the best we've already figured out what we're doing but we're going to invent some notation to make it look Compact and beautiful right so rather than calling the coordinates on space time t x y z we will call them x 0 x 1 x 2 x 3. very important if you know a little bit of math this is not x to the zeroth power x to the first power x to the second power x cubed it's not that we run out of things to do with our numbers so if you want to square something you got to like put parentheses around it or something like that these are just labels indices x0 is just t X1 is just X X2 is just y Etc okay you might say well why are we calling T equals x zero why not call it X4 well someday you're going to want to care about five dimensional space or six dimensional space time is always special so we make it x0 and then we allow you to have as many spatial coordinates as you want okay and then this formula that we already had the interval squared this is going to come from Riemann the interval squared is some number times t squared some number times TX Etc we're going to change out those numbers capital a capital b capital c for lowercase G Sub 0 0 for the t squared term because t squared is X 0 squared we're going to call its coefficient G zero zero the coefficient of x0x1 is going to be called g01 Etc and this is called in the lingo the metric tensor the metric tensor is what it's I didn't do anything between these two equations it's just new notation the metric tensor is carrying the information that we use to calculate the lengths of Curves in an arbitrary geometry in any number of dimensions that's what it's doing it's basically the completely souped up generalization of Pythagoras's Theorem so in principle what we've written here is g00 g01 ETC these could depend on where you are they could be functions of T and X and Y and Z and and you could use spherical coordinates or whatever coordinates you want this is perfectly General okay and in general we would call the components of the metric tensor G mu Nu when we have just spatial indices one two three we use Latin alphabet indices i j k Etc but when we have space time indices going 0 1 2 3 we use Greek letters that's where the Greek letters come in for the rest of the talk you saw those Greek letters on the first slide that's because we're talking about space time and we've put coordinates on space time that are indexed with Greek letters that's where the Greek letters come from so just to drive it home a little bit here is the formula we used for the proper time the time you measure on your wristwatch when you travel through space time okay it's t squared minus x squared minus y squared minus Z squared we can put that in the language of the metric tensor the metric tensor G mu Nu is really A four by four array of numbers or a matrix in the lingo there's g00 which we could call gtt g01 which we could call GTX Etc and in this particular case almost all of those numbers are zero but gtt is minus one sorry gtt is plus one t x x g y y and gzz are minus one so hopefully you can see the resemblance between the formula on the top and the Matrix on the bottom G mu Nu times x mu X Nu is what gives us the formula on the top and we can even interpret what's going on here all of the components of G the metric tensor with purely spatial indices X Y and Z that just helps us calculate distances in space like we're running down a path and we wanted to do that g00 tells us if you want to hold on your hats a little bit the rate at which time flows that's always a tricky one what do you mean the rate at which time flows if time Flows at one second per second right yes it does slow at one second per second but the time coordinate may go wacky and in general relativity it often does so g00 is telling us how quickly time flows with respect to whatever time coordinate T you've chosen to use and then you have these off diagonal things GTX gty g t z which represent time n space twisting into each other those were all zero when I showed you the oops yeah they're all zero here for the minkowski metric right but in principle they could be there you might ask under what circumstances would time and space twist into each other how would I ever come across such a crazy idea in my everyday life and the answer is at the multiplex this is an image famous image from the movie Interstellar directed by Christopher Nolan based on ideas from Kip Thorne my ex-colleague at Caltech Nobel prize winning General relativist and this is an image of a spinning black hole you've heard of black holes we'll beat them again in a second black holes spin usually and when they spin they twist space and time into each other you could not derive this picture this picture was actually generated in a computer by solving for the motion of photons in a metric tensor where space and time are curving into each other quite a lot you couldn't get this picture without thinking about those components of the metric and it was so detailed to do it that they ended up writing a technical paper in the journal classical and quantum gravity where the authors were Kip Thorne and three special effects Wizards at the special effects computer Graphics house in the UK this picture has a lot of Science in it this black hole picture okay I just got to tell you some anecdotes to keep you like slightly amused in the middle of the tensors and everything right I know what I'm doing here this is not my first rodeo back to the tensors Riemann tells us that in principle if you know the metric tensor everywhere so you know in principle the distance between any two little points that are not too far apart from each other he claims that you can calculate anything areas lengths of Curves whatever geometrical thing that you care about sometimes all you want to know is at one point how curvy are things how much curvature is there at that one location forget about the whole shebang just focus in on a little bit of space or space time tell me how curved it is and the answer is kind of complicated because there's no one number that says here's how curvy it is even at one point things can be curvy in all sorts of different ways as Riemann figured out so the answer is going to be something called the Riemann tensor which has four Greek letter indices R Lambda rho mu Nu and here's what I'm going to tell you what they mean kind of hand wave my way through what they mean here's the basic idea remember the existence of curvature is in some sense telling you the failure of the parallel postulate right if Euclid had been right about the parallel postulate space would be flat and the parallel postulate says start with a point draw a line segment through it draw two lines perpendicular to the initial line segment and they will remain parallel in a curved geometry they will fail to remain parallel so Riemann says start with a point draw a little line segment so you have to tell me what kind of line segment you're drawing that is information contained in the Riemann tensor then you have to shoot out two perpendicular lines you have to tell me in what direction you're shooting things out also containing the Riemann tensor and then what's going to happen is you plug that in and the Riemann tensor tells you how they curve how they move away or come together or you can twist around you can have a corkscrew pattern in a more than two-dimensional space lots of crazy things can happen they're all encoded in the Riemann tensor and ribon himself didn't actually get this far in his paper but you can calculate the Riemann tensor from the metric riemann's promise to us is that we only in principle need the metric the distance between two points that's all you need to calculate anything so there's a formula for calculating the Riemann tensor from the metric now I'm not telling you it's easy so momentum in space-time you know momentum is a vector so space time unlike space has four dimensions not just three so there are four vectors momentum is a four Vector in space time that means it has four components PT p x p y p z the metric is a tensor with two indices you saw what that is it's a four by four array the Riemann tensor has four indices which means it is four by four by four by four or if you like it is a four by four Matrix of 4x4 matrices this is why it's fun this is why general relativity earns its keep because there's a lot of different ways that space-time can be curved and the Riemann tensor tells you all of them Einstein had to learn all this you can imagine that he might have been tearing his hair out a little bit maybe that's how the hair got must over the ears I don't know but he did it he stuck through it he learned all this stuff about the Riemann tensor and so forth and I just want to boil it down what is the lesson here so we don't get lost in all the indices the upshot is that the geometry of space-time can be arbitrarily complicated it's not just a matter of positive or negative curvature it can be wildly different from place to place we know how to encode that information we encode it in the metric tensor and from that we can in principle derive the entire geometry and so that's what the metric tensor is good for and from that we can calculate the Riemann tensor and from that Einstein would like to invent reinvent gravity so Einstein had this physical idea gravity is the curvature of space-time right fifth gear now gravity is the curvature of space-time curvature of space-time is captured by the Riemann tensor how do we turn gravity into a statement about the Riemann tensor well we know what gravity looked like back in Newton's world right we canceled out the little m's on both sides acceleration is proportional to the one over the distance squared and also to the mass the heavier you are the more gravity and therefore the more acceleration you generate around you so somehow we need to replace acceleration with some measure of the curvature of space-time and we have to replace Mass with some expression for matter and energy and heat and momentum and all those things bundled up into one because at heart relativity is a story of unifying different ideas together so we need a tensorial four-dimensional geometric way of talking about something acceleration like and something Mass like for the right hand side for the mass making long story short here we replace it by something called the energy momentum tensor it is again a 4x4 tensor you don't need to know much about it but it's all of the energy like things in some collection of matter radiation whatever you want t 0 0 is literally the energy including the mass because E equals m c squared the diagonal spatial parts are what we know is the pressure like literally the pressure in a box of gas and the off diagonal Parts have to do with anything twisty once again right momentum heat flowing stress all of those things go into the off-diagonal parts you don't need to know any of that feel free to forget this it's just that there is there is a tensor T mu Nu and it tells you everything that there is to know about energy momentum Mass all of those things it will appear on the right hand side of whatever equation it is we're going to make that's good it's also bad why is it bad because mass and energy and all those things are in team you knew and that is a tensor with two indices the Riemann tensor is a tensor with four indices we can't set them equal to each other or even proportional to each other that I mean Einstein literally faced this problem we have a way that we thought was good of characterizing the curvature of space-time we wanted to set it equal to the amount of matter and energy and so forth but they're different kinds of tensors we can't do that happily there were people there were some Italians who figured out how to boil down the information in the Riemann tensor to other kinds of tensors you could make something called the Richie tensor this is a very strange story that I have a footnote in my book Richie was not the guy's last name his last name was Richie Dash carbastro and in every paper he ever wrote this Italian geometer he signed it Richie carbostro except for one paper where he invented the Richie tensor I think maybe he thought this would become famous and he wanted a catchier name to adhere to it so anyway we call it the Richie tensor you can boil down the Riemann tensor to a tensor with just two indices and immediately immediately you say well I bet I should set that proportional to the energy momentum tensor because it has two indices I'm I hope you thought that because that's what Einstein thought but it doesn't work so Einstein again he was he plugged away that you can boil down again the curvature uh tensor down to the curvature scaler and then you find a combination of them and the combination you find that works in the sense of preserving conservation of energy and so forth is our mu Nu minus one-half r g mu Nu and then there is some constant proportionality you work it out because you want the Earth to go around the Sun in one year you can do that and you find that rather than Newton's constant G it is eight times pi times G which is kind of nice so this is it this is Einstein's equation this is literally the reasoning that Einstein went through to derive this equation for Gravity this is the equation that tells us how the gravitational field in general relativity responds to matter and energy you see why all the symbols are there you see why there are Greek letters because they index different directions in space time you see why they're tensors because there's a lot going on sadly this equation is very complicated so remember I told you that the metric can be used to derive the Riemann tensor and from the Riemann tensor you can derive the Richie tensor in the curvature scalar easier said than done so here is one component of the Riemann tensor expressed in terms of the metric tensor so it's easy for me to tell you oh here's a metric tensor calculate the Riemann tensor it is it is hard for you to do it it was hard for me to do it these days the kids had computers and they just plug it onto a computer and they don't know what it's like to spend to pull an all-nighter trying to calculate the Riemann tensor and occasionally writing down a mu when you meant new and things like that but the relationship is very complicated it's so complicated that Professor Einstein himself when he presented his equations and he was very triumphant he's very proud of himself but he thought that no one would ever be able to actually solve them exactly he solved them approximately like he he came up with some clever approximation schemes so he predicted the procession of mercury the deflection of light but look at this no one is ever going to solve that exactly except two years later someone did Carl schwartzshield was a German astronomer and mathematician who was in the army look at the years it was World War One he was literally at the Eastern Front he was a professional mathematician astronomer so he wasn't you know out there with uh rightful he was calculating trajectories for uh missiles and things that you fired at the at the Russians but even when you're in the Army you get occasional um I don't want to call it Shore leave what is it called when you get a week off just go home when you're in the army not AWOL it was allowed he was absent with leave he gotta leave and r r yeah so his version of r r was to go to Berlin and sit in on lectures by Einstein on general relativity and so he picked this up he goes oh yes general relativity how hard can that be I'm gonna solve Einstein's equation and he did so this is the famous Schwartz Shield solution to general relativity and it comes about by making the right assumptions by simplifying your life by saying I'm not looking for anything that changes with time what I care about is the metric of space-time around the Sun I want to calculate things like mercury in the Earth orbiting around the Sun so I'm going to make everything perfectly spherically symmetric absolutely static make a lot of simplifying assumptions solve the equations and here's the answer and Einstein in the inside Einstein's like yes that is it he got it he loved this this was very good and we don't have time to go through all the implications here but I do want to point out one feature so this is literally the metric you know what a metric is now you didn't know 15 minutes ago what it was but now you're like oh yes that is how I that's a line element on space time that's how we calculate the distance between two points sure it's all inherent in here but unlike the minkowski metric which is plus one minus one minus one minus one there's little dependence in here which it should be because it's going to be the gravitational field it's going to fade away as you go further away which is fine but then you look at two left hand side terms and it looks like when R equals 2gm the first term is zero right I think I can do this watch me do this right there this this puppy is zero when R equals two GM and this one is one over zero which is infinity and that sounds bad Einstein knew it was bad short Shield knew it was bad they just didn't know what to do about it so they said don't go there don't visit R equals 2gm and this was always supposed to be only supposed to be the space outside a planet or a star and you can check that for the real sun outside the sun means that R is much much bigger than 2gm so it's not a big worry that you have but we know better what is going on at R equals 2gm it's a black hole is the answer and you can see that and you and literally I want to get this across to you you can see that now because you know that this thing oops that this term up here is the rate at which time passes with respect to T and so if it goes to zero then we've plotted this term G zero zero here it's the rate at which time flows and it goes to zero at this place we call the Event Horizon and what that means is for someone very very far away for someone far away R is a very big number one two GM over R is close to zero so G zero zero is one so time Flows at the rate of t compared to you far away someone who is falling in to this object you see them slow down and eventually stop when they hit the Event Horizon at R equals 2gm later it took literally decades to figure out that they wouldn't see that they would actually just fall in to the black hole but if you look at it really really far away you see people falling in and also you could predict that there is a red shift the light emitted from the in-falling object or the in-falling astronaut gets stretched and made redder and redder as if they're a little bit embarrassed to be doing something as silly as falling into a black hole and now we can take a picture of it this is a picture of a black hole taken by the Event Horizon telescope rather recently and my point of going into this is neither Einstein nor short Shield had any idea they didn't invent black holes the idea of a black hole wasn't made clear until after they had both passed away but the idea was implicit in the equation right the equation and this is the final slide the equations are much smarter than we are we are just smart enough to write down the equation but inside that equation there's a lot of knowledge that it takes us a long time to squeeze out because the laws of physics are precise and Universal just like Newton's law of gravity is so implicit in Einstein's equation is all these things the large-scale structure of the universe gravitational waves ways to see Dark Matter maybe even the existence of the Big Bang itself the equations remain by the way smarter than we are so we don't know what all their implications are stay tuned as we keep finding more and more of them thank you [Applause] I suspect there might be a few questions we have a procedure for questions okay first question would be Richard Talbot I believe uh Richard Talbot I'm a proud member of PSW that's a red microphone sounds good okay uh engineer so I want to apply your math here for a minute uh so I read an article that said for fission reaction of uranium that we get a 0.1 percent conversion into energy is a strong force going into energy and we get a big bang out of that second article said that when you have matter of atomic matter falling into a black hole then you have a 10 conversion so intuitively it makes sense that the black hole was ripping apart the nucleus and throwing out the strong force much more than the uh the fission process so then the question is when you have spaghettification as you're falling in and then you get the Hawking radiation so you don't even need Atomic matter apparently the black hole is ripping apart space itself okay and then you have the idea of uh particles the Hawking radiation particles going into the black hole and the black hole evaporating uh so I've always intuitively had a hard time understanding how it is that you can have matter falling into a black hole and then having the black hole shrink I was just wondering if you could deconflict any of what I just said sure um but there's there's a whole bunch of things going on so I have to deconflict a lot there's an interesting thing happening here that even though uh can I have the clicker back okay yes so you know what we wrote down the this is the metric this is the metric that you learn you got about one semester's worth of graduate general relativity in the last hour and you would learn to solve what is going on here in this metric and the what is happening at the Event Horizon is it's a point of no return once you go into the Event Horizon you can never come back out but in a very real sense the gravitational field does not feel that strong at the Event Horizon it is a global feature of the space time if you have a very big black hole like at the center of our galaxy or like the one we took a picture of here these are very large more than a light year-sized black hole when you fall in and when you pass the Event Horizon you don't even know you don't even notice it's I mean you've made a regrettable decision and you will notice soon enough but it's not until you fall into the singularity at R equals zero that you notice so there's a story you alluded to of spaghettification and tidal forces tearing you apart but that doesn't happen at the Event Horizon that happens deep inside a macroscopic black hole so when you get a 10 conversion rate that's not because the black hole is pulling apart Atomic matter near the Event Horizon that's just because there's good old-fashioned astrophysical processes heating up matter and doing things to them as they swirl around the black hole and eventually fall in the Hawking radiation is a whole other story because that's what happens when you consider Quantum field theory in the background of a black hole so you don't even need matter you have empty space with a black hole and Stephen Hawking says when you have quantum mechanics plus a black hole the situation is not stable it is going to leak out particles there one way of thinking about it which is very hand wavy and not exactly right but Hawking says it so I can say it too is that there's a pair of particles produced near the Horizon through a Quantum fluctuation one falls into the black hole but it has negative energy one escapes to the outside world with positive energy so the overall feature that the black hole shrinks but it's shrinking because of quantum mechanical effects in empty space not because of matter falling into the black hole thank you Kirby Runyan I am a member I'm a Staff scientist at planetary Science Institute so thank you and I'm just a dumb geologist so thank you for explaining all the scary map you mentioned that as as Things fall into the black hole to an outside Observer the time dilation makes it look as though they just stop as they approach The Event Horizon and I've read that before my question then is why don't we see a train wreck pile up of lots of crap accumulating at the Event Horizon when we look at black holes because I lied because when I say there's a reason why I lied because if you take literally this solution from Professor schwarzshield and you ask what happens when a tiny little particle Falls in well what do we mean by tiny little we mean we're gonna ignore its own gravitational field right it's so tiny that it's up falling to a big black hole like when I think about dropping the laser pointer and the gravitational field about of the earth I don't calculate the gravitational field of the laser pointer I just let it fall it's called the test particle approximation right in that approximation what you would see is the test particle slow down as it approaches the black hole but never cross the Event Horizon in the real world the things falling in add to the gravitational field because they have matter and energy in the event horizon expands a little bit to swallow them up so what we see at the end of the day is not things that are infinitely redshifted in moving infinitely slowly they move slower and slower and slower and then they are in fact swallowed up by the black hole hi lupicor the University of Maryland uh I recall that this is a very static picture of the universe and really Einstein was faced with the idea that space can expand but I'm just more historical in a way I think he didn't like that and put in a special term to cancel it I was wondering if you would expand on that and did he ever accept it eventually and and why yeah this is I'm very glad you asked that question because the more that I learned about the history of this believe it or not I'm going to come out with a dramatic claim that Einstein was underrated we actually there's this thing this story that I don't know maybe it makes us feel better but we show pictures of Einstein You know looking all rumpled and we say yeah when he got older he couldn't quite keep up with all the new developments and he was an old buddy buddy all that is a hundred percent wrong and this particular this is the story of the cosmological constant which is part of that discourse so Einstein very soon after in fact in 1917 very soon after the field equations that he wrote down in 1915 applied it to the universe he said I have a universe I scatter it full of matter completely uniformly and I say what is the solution to my equations and this was a crucial step because you could do the same thing in Newtonian gravity but there was no unique solution the the question was Ill posed in Newtonian mechanics whereas in relativity there's a perfectly clear answer and the answer was the universe is not static it is either expanding or Contracting okay and you're right that this Disturbed Einstein but not because he didn't like it but because the year was 1917 and he went to his astronomer friends and said is the universe expanding or Contracting and they said no it's static so he said uh oh my equations are in trouble how can I alter them and he altered them by adding a term which we now call the cosmological constant it was in the 1920s that Hubble and others showed the universe actually is expanding and Einstein hugely regretted adding this term because he could have just stuck by his guns he could have said nope I bet the universe is expanding or Contracting and he could have become famous who knows the irony is that we now know that there is a non-zero cosmological constant but it is not balancing anything it is not keeping the universe static it is pushing the universe apart at an ever-growing rate from the web we have two questions from the web both of them are relatively expert I'll ask the less expert one first Carl Miller Merrell a member asks could you expand on the effects on time if the universe as a whole were rotating as calculated by codel universe as a whole we're rotating well with the effects on time be and there's I'm going to give a slightly unsatisfying answer to this sorry about that but you know we have to try to be careful one is what do you mean the universe is rotating it's very hard to Define exactly what that would mean there's various solutions to Einstein's equations where at every point there is some kind of rotation and famously there was a solution of that form derived by Einstein's good friend Kurt girdle the mathematician and logician who worked at The Institute for advanced study girdle's theorem incompleteness upending The Rock Solid foundations of logic and so forth in his spare time he solved Einstein's equation for a spinning universe and he found that there were what we call closed time-like curves that is you can travel there's a rule in special relativity you can't go faster than the speed of light in general relativity which is this still true but what you mean by going faster than the speed of light is changing because space time is curved so we can imagine curving space time so much that you personally are just traveling in your spaceship or whatever going faster than the speed of light and you end up at the same place you left before you left you you end up in your own past No One Believes the real universe is like that it's the short answer to that so the ability to find solutions to Einstein's equations with dramatically weird non-physical looking properties is an interesting one but as far as we know we cannot realize them in the real world black holes no black holes are all over the place those are easy to make but there was a time when people thought it was a an honest you thought all sorts of things all right so uh one more from the web the other question from the web the more difficult one I think are Richie tensors eigenvalues of Riemann tensors no that was much easier than the first one their contractions of the Riemann tensor so they are sums of certain sets of terms within the rebound tensor hi Sean my name is Brett Feldman I'm a member thank you very much my question is related from I guess from special relativity to general relativity and I guess my understanding is that uh special relativity deals with an inertial reference frame and Einstein sought to generalize that for an accelerating reference frame so the question is is that is that true in finding finding that what gravity actually was was a consequence of that or did he set out to find gravity and use the tools of such relativity uh to get there it's just completely false sorry there's you will often read exactly what you said including in textbooks devoted to relativity but special relativity is a hundred percent compatible with acceleration and accelerated frame of references and general relativity has nothing to do with generalizing to accelerated frame of references the reason why people make that mistake is think about just space forget about space time right and you say in the two-dimensional plane draw a Cartesian coordinate system which means X and Y coordinates at perpendicular angles everywhere right you say okay I can do that no problem so someone says here's a sphere draw Cartesian coordinates on the sphere well you can't you can't have coordinates that are perpendicular everywhere on a sphere because it's curved a similar thing happens in general relativity and special relativity and special relativity because space time is flat you can write minkowski coordinates t x y z which are the ones that would naturally be constructed by an inertial observer in general relativity where space time is curved you in general can't draw the space-time version of Cartesian coordinates but couldn't have done any other coordinate system back in special relativity so it really does it doesn't have anything to do with acceleration thank you red microphone [Music] Bob Cherry I'm a member and I'm curious about this problem of rotation is there oh an experiment to do that would falsify the concept of rotation for the universe is there an experiment to do that would identify what kind of special rotations it might show how do you gain access to this through experimentation because obviously someone came up with a solution that satisfied these equations with all this rotation in it there's nothing I priority to to say it can't happen so how do you how do you test or falsify that that concept yeah that's a very good question the answer is you make astronomical observations so in order to falsify it you can't falsify the statement a universe with rotation because that doesn't it's not specific enough to say anything you have to tell me what exactly kind of universe you're talking about which means writing down a metric in general relativity as Kurt girdle did once you've done that I can easily test it I can look at gravitational lensing at the cosmic microwave background at statistics of clusters of galaxies and things like that I can look at our uh here's a good story do I have time to tell a story this is a good one yeah absolutely so one thing you might ask is okay there are galaxies out there in the universe they're rotating is there a sense of rotation is every Galaxy rotating in the same direction in some sense so before we had Advanced machine learning and anything but you know just a few years before when we had pictures of galaxies being taken by large surveys they would take pictures of galaxies and they would say we want to classify these are they spirals twisting clockwise or spirals twisting counterclockwise and bless their hearts they gave them to volunteers and said tell us whether this is clockwise or counterclockwise and it was like 75 percent clockwise and they're like wow that's an incredibly important Nobel prize winning result and then they started randomly flipping the images of the galaxies and it was still 70 clockwise because the human brain just sees things going in some ways rather than other ways so the current best data is no there is no sense of uniform rotation of galaxies in the universe no galaxies are rotating but that doesn't mean the universe is right as far as we can tell right yeah okay okay so but my is when you see singularities like this in most of the theories could you speak up a little bit yeah sorry so when you see singularities like this in most of the series uh you find ways to pad them out you find ways to renormalize the system and presumably those ways of trying to renormalize this Singularity would tell you something how this how these materials couple to other fundamental forces I mean G up there is a friend from you know Newton's time I mean suppose how does how does this tensor couple to electromagnetics or does it can it ever uh and and what what phenomenal would you see if it did yeah gravity bends light that's the coupling between the light isn't it bending light because the space just Twisted yes because there's no Mass to light there's energy yeah right gravity is universal right okay everything feels the force of gravity right so it's the energy feeling the force of gravity is okay first one I think is straight up your ballpark how does mass exactly affect space-time how does it make it curve there it is that's the answer you solve that equation you know at the at some point it's not Turtles all the way down there's a bottom Turtle there is an end to the why how types of questions as far as we currently know why does gravity act that way because it does now there might be someday a deeper theory of gravity there probably is because this crap this theory is classical at the end of the day does not take into account quantum mechanics but at the end of the day it's going to be laws of physics and the laws of physics work in some particular way and this is the best idea we have about how the curvature of space-time works right now and the second question is what would be the effect of antimatter parentheses negative M question mark on event Horizons and inside a black hole the good news is gravity is universal and the sine of M doesn't sorry I shouldn't say it that way anti-matter has exactly the same mass that matter has the mass of a proton is exactly equal to the mass of an anti-proton we can't have particles that have negative masses that would be bad for all sorts of reasons that in the end of the day amount to it is ruled out by the data uh yes uh Charles Clark I'm a member of the philosophical Society of Washington uh veteran of the National Institute of Standards and Technology where all all equations are expected to start by being dimensionally correct so if I look at the one in the upper uh my upper left corner it looks to me that that should be 2gm over r c squared where C is the speed of light right and I know that you're using a convention where C is equal to one okay I assume that yes but if also say if you put the c in there then that it seems that gives you relatively directly the fact that the escape Velocity is the speed of light or to say it another way if the speed of light were to go to Infinity then that that number would just reduce to you to one and I guess space would be flat throughout is that is that a correct interpretation nope you were doing so well but uh this is why you don't want to put the sea in there because for one thing you can't change the value of C I can always use units where time is measured in years and and distance is measured in light years no matter what the speed of light is you only it's relativity there's only speeds relative to other things and the thing is not the relevant fact about the speed of light is not that it's the speed that light goes at but is it the speed it is the speed that defines the relationship between space and time and whatever other things are happening in the universe that's always something we can set equal to one and also in the short Shield solution there's no such thing as the escape Velocity that is a good old Newtonian notion that has to be updated in this context okay but I think if you do put the seat you don't disagree that putting the C in in the formula makes it dimensionally correct and it's dimensionally correct now and calculable you know in based on you know the values of things like a gravitational constant the mass of the star and the rate the mass of the object and the radius you're welcome to do that thank you yes my name is Scott Matthews and I'm a member in this Matrix where did that come from and can you say something about the physical significance of it there's not a lot to say it's just the angle between the Equator and the North Pole we're working here in spherical coordinates not Cartesian coordinates so we have R is the distance data is I forget which is azimuthal and whatever but you know Theta goes from the equator to the North Pole actually goes from the South Pole to the North Pole technically but you know what I mean and then Phi would be around the equator it doesn't matter because we're spherically symmetric here so that's why I did not dwell on it hey Sean Liz Landau I work at Nasa gotta renew my my membership so my question is you know scientists are very excited right now about um constraining things like dark energy modified gravity do you think that this equation is actually going to hold up as we get more observations from telescopes like Nancy Grace Roman bear Reuben will this hold or do you think you know 20 30 years from now we might change it that's an excellent question of course I would love it if this equation were wrong right every physicist wants to you know you don't become a famous physicist by proving your predecessors right right that's just not the way um anyone you know anyone who gets a PhD in physics should have at least once proposed a alternative to Einstein's equation I know that I certainly have proposed several the data sadly seemed to be favoring Einstein over and over again and it's more than just that when you dig into it when you think about that the world is quantum mechanical at the end of the day that uh the the parts of the universe that are described in astrophysics and cosmology seem to obey the rules of quantum field Theory we we not only are able to guess at equations but we're able to say that certain guesses seem reasonable and certain ones seem unreasonable and the fact is we've made the reasonable guesses for what could be the alternative to Einstein's equation if they were an easy natural way to alter Einstein's equation we would have seen it already so on the one hand we're open-minded we're always hoping to get something interesting and different um I would say that Lisa the gravitational wave Observatory is probably the best bet for finding deviations maybe the Event Horizon telescope will see something but the smart money says that it's just going to be Einstein cosmologically and once we're going to get deviations from general relativity if at all on Ultra small scales not Ultra large scales my name is Eric I'm a rising Junior at Hopkins I study Physics so I just took your last Master with Professor bury and when I was taught the Einstein's field question the duration came from the variational approach of writing down the Einstein Hilbert action and there's two parts that the matterfield part and then there's other part my questions that like you can actually modify the action terms and you'll end up with the different equations so is grd only uh correct uh field metric theory for a classical metric theory for Gravity good I mean this is related to um Liz's question it's certainly not the only Theory we can write down there is a famous Theory written down in the 60s called bronze sticky Theory uh by Carl's Bronson and Robert Dickey where they throw in a scalar field in addition to these fields you could also do funny things with higher powers of the Riemann tensor r squared and things like that I wrote a paper which had one over R in there because why not and they just they're both more complicated and they don't fit the data as well so it's not that you can't it's just the motivation for taking those seriously is just diminishing over time okay name's George not yet a member hope to be next year uh question hopefully it's an easy solution if you go to the slide that has the two rocket ships two rocket ships I'll get there yeah always wanted a better explanation and go back keep go from there uh you said that you there isn't an experiment that you could differentiate whether you are feeling the force of gravity or filling the force of acceleration um in that situation doesn't the accelerating rocket ship if you're continuing accelerating at 1G you're ultimately going to approach the speed of light It ultimately breaks down what's a better explanation that takes in that full sort of dynamic that yes there is experiment do it and you you will ultimately not be able to accelerate at 1G so that turns out not to be true you can accelerate at 1G forever and ever and ever and you get closer and closer to the speed of light without ever getting there so inside the rocket ship remember relativity there's no such thing as going close to the speed of light it's only relative to what I can go I can send something out which to me is moving at 0.99999 the speed of light and to it it's stationary so there's no limit to how much you can accelerate yes I agree with that but the point would be that in order to actually do that experiment the amount of energy you'd have to expend from the rocket would approach Infinity as you approach the speed of light so it's not a doable experiment well that's true also it would be noisy I mean if we're listing real world obstacles there are many of them the technical uh wiggle room here is that if you look up in my book for example or elsewhere the statement of the principle of equivalence it is limited to small regions of space and time so you can't just let it go on forever there's a much simpler example of what you're thinking of just in these two rocket ships hang two pendulums in the rocket ship they will be exactly perpendicular exactly parallel to each other on the earth they will Point slightly toward each other because they're pointing toward the center of the Earth but the principal equivalence applies in the limit as your region of space and time becomes very small I'm Carrie list I work at the Johns hop University Applied Physics laboratory and I'm asking my own question not from the web one comment the very last slide showing what the astronomers have very lightly proven that Einstein was right he was in the ligo chirp when two black holes merge together if my understanding is that classical gr did a fine job of explaining the the frequency and the structure of the if you will the gravity wave that we saw one back and then the question I was going to ask you is what do you think are the prospects for emerging quantum physics quantum gravity how can we put that together and where do you think we're going to go with that to have some experimental data that was relevant to the reconciliation of quantum mechanics and gravity the simple but very important problem is gravity is a super duper weak Force quantum gravity is not expected to become important when gravity becomes strong it's expected to become important when you reach the Planck length the distance of whatever it is 10 to the minus 35 centimeters Etc so in astrophysical black holes you are super duper far away from the regime which we expect any Quantum effects to become relevant now maybe we're wrong that's why it's important to do experiments but I wouldn't again bet that we're going to see some quantum gravity there Ed pack and uh I'm a member retired from Naval research laboratory the idea of gauge endurance seems to be uh the foundation of the other world of physics uh has anybody done anything to try to merge that with these uh general relativity ideas how does that work and what's what happens sure the question is that there's this idea called gauge invariance which is basically in various theories of particle physics and Quantum field Theory there are Fields like the Quark which are not just a single field but live in a little three-dimensional space right red green blue different kinds of quarks you're told if you hang out with the wrong street corners that there are three types of quarks red green and blue that's not exactly right what is exactly right is the Quark field is a vector in a three-dimensional Vector space there's a certain amount of red a certain amount of green a certain amount of blue now it's not really red green and blue those are just colorful as it were labels and also it doesn't matter how you Orient your axes right you can Orient them any way you want and what is more you can Orient them one way here and another way over there and another way over there so there is a huge amount of symmetry you can change where your axes of Quark color space are pointing at every point in space time this is called gauge invariance and making it work out mathematically leads to a force of nature leads to a field that does the job of telling you how to relate the different little vectors the different points in space it's exactly the same story and general relativity except the vectors we're talking about are real honest to goodness vectors pointing in space-time so you can completely think of all of general relativity as a theory of connections on a certain mathematical space with a certain kind of gauge invariance it's different because space time is different than an internal as we say Vector space like color space is so there are important mathematical differences also but the underlying Spirit of it is exactly the same read my book which one you need to read both of them you read volume one and volume two to get that I should mention uh that these two previous PSW lectures one on hello what's going on on the plank length and that was by Nema our community Ahmed called the Doom of space-time his main contention being that all physics breaks down the black point then we don't really know what's going on and the other one is on uh on the Event Horizon results where we had three guys who talked about those results in physics behind it so for those of you who are interested in these things you might want to look at those David brosen lifetime member a question I know gravity is universal but the other forces are not so uh in in Einstein's equation you were you that you wrote up there uh where are the mechanical forces and proper acceleration and all that stuff with the mechanical as I was said you could buy at a grab grab a graphometer and you know on eBay very inexpensed three thousand dollars cheap and measure the proper acceleration but I don't see that in that equations well this equation you know as good as it is all it's doing is telling you what the curvature of space-time is of course in a complete Theory you have other equations as well this is not by itself a complete Theory even the right hand side of the equation where we've just written team you knew needs to be abetted by Maxwell's equations and the equations of particle physics and all that stuff so there's a whole bunch more equations you need to describe the entire universe there are other things in the universe Fields particles what have you they have a feature called their energy and momentum that is team you knew but that is not enough to tell you how they behave for to do that to tell you how the other particles and Fields behave under the influence of other forces you need other equations it's going to be an easy one I think uh I'm intrigued by your title professor of natural philosophy because natural philosophy I think medieval I think of Albert the Great and I'm wondering how how you came about using that title thank you I didn't pay her anything to ask this question but it's a great question I'm glad you are asking it you know when I got the job at Johns Hopkins I was allowed to invent my title that was part of the deal and so I named myself the professor of natural philosophy which was what Newton or Galileo would have called themselves right that was a time before physics and science more generally had broken off from philosophy the joke is that as long as we don't know what we're doing it's philosophy as soon as we start knowing what we're doing it splits off from philosophy and becomes science psychology whatever but it turns out I think that there are very important questions that are about science at the end of the day because they're about nature you know what is the quantum mechanical wave function why is the past different from the future how do we calculate probabilities in the universe and things like this for which the techniques and skills and tools of philosophy are really very necessary so it's not that there is something called philosophy of science but that is largely studying how science works natural philosophy is studying how nature works but using the style of a philosopher as well as of a scientist interestingly PSW stands for philosophical Society of Washington and it was named philosophical Society because the people at the time thought of themselves as natural philosophers which was their word for science so we're very much in keeping with it and with that uh thank you for your talk which was fabulous foreign token of thanks we have we have three things for you which you can see there and they're also over here it's just a frame copy of the announcement of your talk signed by all the members of the general committee on behalf of the membership a rosette which I'm sure you're going to wear at all important functions and a copy of volume one of the philosophical Society of Washington now known as PSW science and in that volume you will find why they called it the philosophical Society what they intended to do at a very very cogent statement of how science should be conducted so with that thank you very much thank you very much before everybody goes we have a very few housekeeping announcements the 2479th meeting will complete the spring lecture series on June 16 the speakers will be Bill Merrill and Sam Brody of Texas A M University at Galveston and they will be speaking about the 30 billion dollar that's current estimate multiply by three dollar Ike Dyke project being planned to control Rising ties and were surges that threatened to engulf Houston and flooding and storm damage you can see in the picture the results of one devastating hurricane on the Texas coasts this is one of the largest civil engineering projects being contemplated in the United States please check the PSW website often for the up-to-date information on meetings before you go let's thank our crew Rosemary Collins and John sorath for room Manet and rosettes Cameo Lance for reading the minutes and preparing them Robin Taylor Who's hidden behind the monitor back there for doing the video sound live stream carry lists for handling livestream chat Robert Thompson Lloyd Mitchell and Andrew Gunn for running cameras Bill Mitchell who will do the video editing and me thank you for coming here I hope you enjoyed it and there will be Refreshments in the back and I'll entertain a motion to adjourn the meeting second all members in favor all members opposed meaning is adjourned

Channel: PSW Science

Views: 109,868

Keywords: PSW Science, PSW, gravity, spacetime, physics, Einstein Equation, gravitation, Sean Carroll

Id: 3NQ7xeLcTwI

Length: 108min 31sec (6511 seconds)

Published: Mon Jun 05 2023