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94. Interview with Steve Nadis, Co-author of 'Gravity of Math'
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Summary
**Tensor Poster - If you are interested in the Breaking Math Tensor Poster on the mathematics of General Relativity, email us at [email protected]
In this episode, Gabriel Hesch and Autumn Phaneuf interview Steve Nadis, the author of the book 'The Gravity of Math.' They discuss the mathematics of gravity, including the work of Isaac Newton and Albert Einstein, gravitational waves, black holes, and recent developments in the field. Nadis shares his collaboration with Shing-Tung Yau and their journey in writing the book. They also talk about their shared experience at Hampshire College and the importance of independent thinking in education. In this conversation, Steve Nadis discusses the mathematical foundations of general relativity and the contributions of mathematicians to the theory. He explains how Einstein was introduced to the concept of gravity by Bernhard Riemann and learned about tensor calculus from Gregorio Ricci and Tullio Levi-Civita. Nadis also explores Einstein's discovery of the equivalence principle and his realization that a theory of gravity would require accelerated motion. He describes the development of the equations of general relativity and their significance in understanding the curvature of spacetime. Nadis highlights the ongoing research in general relativity, including the detection of gravitational waves and the exploration of higher dimensions and black holes. He also discusses the contributions of mathematician Emmy Noether to the conservation laws in physics. Finally, Nadis explains Einstein's cosmological constant and its connection to dark energy.
Chapters
00:00 Introduction and Book Overview
08:09 Collaboration and Writing Process
25:48 Interest in Black Holes and Recent Developments
35:30 The Mathematical Foundations of General Relativity
44:55 The Curvature of Spacetime and the Equations of General Relativity
56:06 Recent Discoveries in General Relativity
01:06:46 Emmy Noether's Contributions to Conservation Laws
01:13:48 Einstein's Cosmological Constant and Dark Energy
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Transcript
Interview with Steve Nadus
00:00:00
Speaker
Welcome to another episode of Breaking Math. I am one of your hosts, Gabrielle Hesh, and I am joined by my host, Autumn Fanoff. How are you, Autumn? I'm well. How are you? Outstanding. Thank you for joining us. And I'm thrilled to say that we have a new book author today. We are interviewing Steve Nadus, who is the author of this awesome book called The Gravity of Math. How are you doing today, Steve? I'm doing great. Thanks, Gabe.
00:00:18
Speaker
Absolutely. I couldn't be more happy to have you join us. It's very, very exciting. Do you think about this book, Gravity of Math? This book is about, of course, the mathematics of gravity going back to Isaac Newton, including Albert Einstein, and gravitational waves and black holes, as well as some recent developments, I say developments, some
00:00:34
Speaker
Recent ideas involving things like string theory and how the concept of mass isn't really all
Shared Background Discovery
00:00:39
Speaker
that well defined. We've got a whole list of questions here about this awesome book. I also want to tell you a little bit about the authors. If you guys want to interrupt me because you've got more to say on this, please do so. First, I'll talk about Steve Natiss. Steve Natiss is a contributing editor to Discover magazine. He's also a contributing writer to Quanta magazine. Quanta, by the way, I think is one of my new favorite magazines. Every episode we've done so far, Quanta has done multiple articles on.
00:01:02
Speaker
The last one that we did with Quanta was involving some of their articles in the Lean Theorem Prover. Very interesting concept. Essentially, it's our previous episode. Imagine something like the game Operation where it buzzes you if you go out of line and it's supposed to help you to write math theorems without contradicting yourself or your established definitions. It's pretty cool. It's pretty cool. Quanta has written quite a bit about that as well. So I like hearing from other authors of that magazine. Also, I'd like to talk about the other
00:01:25
Speaker
Xingqing Yao is a mathematical professor at Xinguai University and a professor emeritus at Harvard University. He's also the recipient of the Fields Medal, the National Medal of Science, and a MacArthur Fellowship, and he currently lives in Beijing. And he has written, he is co-author, I should say, Steve Natus has co-authored five books with Xingqing Yao, and we have the latest book with us here today. So again, I already said thank you for joining us. And yeah, there's just so much to talk about here. During
00:01:52
Speaker
our pre-interview, we discovered something pretty cool. And that is that both my co-host, Autumn and Steve Natus went to the same university.
Podcast Origins and Diversity
00:01:59
Speaker
Can you tell us a little bit about that story? So we were chit-chatting about our backgrounds before the show and geographically it was like, where are you located? And he's, can you say Cambridge? And I'm like, okay, I'm in Massachusetts and I'm
00:02:13
Speaker
kind of by UMass. Well, kind of by UMass, the other side of the state, we proceed to discover I went to UMass and he's like, oh, my undergrad was in the area. And I'm going, OK. Right in that area, there's five colleges.
00:02:28
Speaker
taking classes on four out of five campuses during my time for my undergraduate career. And I was like, wait, did you go to Amherst? I'm thinking Houston physics. I'm like, did you go to UMass? No, no, no. Our favorite school Hampshire College. Think of the party hardy
Hampshire College Culture
00:02:44
Speaker
hipster college that has no grades is really awesome and teaches people how to think
00:02:51
Speaker
work creatively and very freely. On top of that, we share a couple of advisors and committee members who were Hampshire legends. I'd like to continue that. So what's kind of cool is with the Breaking Math podcast, initially our, how do I say our heritage, I suppose, was strictly out of New Mexico. We were strictly out of UNM, University of New Mexico, as well as a little bit of New Mexico Tech, which was myself and my original co-host, the late great Sofia Baca.
00:03:16
Speaker
And of course now it's awesome. So I've got a co-host who's got this entire cultural heritage from Hampshire. I'm sorry, I'm not saying it right. Hampshire. Hampshire. Yeah. In Massachusetts. And not only this, but also our guest author. And you guys even shared some of the same advisors. Yeah, please. We certainly had one physics professor in common. It's also the case that we were there in quite different eras. So it's only because that physics professor is over 80.
00:03:45
Speaker
Well, we know the same professor because I was there in the 70s and much more recently. I don't want a nitpick, but we probably wouldn't apply the term University to Hampshire, small college of not probably not much more than a thousand students at its peak. I think it got above a thousand, but yeah.
00:04:02
Speaker
It's had a lot of different folks that have come in and they're also very well known for mathematics and mathematicians. So there's been a lot of interesting folks that have gone in there, even people who have done the aerodynamics of a frisbee as their thesis.
Advice for Hampshire Students
00:04:19
Speaker
So yeah, let's have an article to that in the show notes. I'm now curious about that myself.
00:04:24
Speaker
Yeah, so like I think we mentioned it in a prior episode, some of the inventions I got to find out that Stonyfield Farm yogurts, seventh generation cleaning products, there's medical devices where they have a backpack
00:04:40
Speaker
that was meant for a gynecological table that you could take to remote countries. They have a saying from Hampshire to Harvard or Hampshire to Hollywood. And it has so far proven true with the number of doctors, lawyers, and attorneys that come out of there. It's a really cool, interesting deal. So we're getting to Dr. one today. Awesome.
00:05:01
Speaker
I can't say that Frisbee was big at Hampshire. I started playing ultimate Frisbee in the early days. And I can't say that many of those people at Hampshire who were playing started the ultimate Frisbee Hall of Fame. Yes. When I was in Hampshire, I just never thought it was a serious sport. It was only when I came up during my graduation. I actually, I watched the game and I thought that was pretty fun. And maybe I'd missed out on that experience, but I had other experiences.
00:05:27
Speaker
I think it was studying about like the aerodynamics of the frisbee. I guess I'll ask you this before we move on to the book. And this is mostly for Steve. Do you guys have any advice for anybody who's considering going to school there for what ways to make the most of their education while they're there?
00:05:45
Speaker
Well, I would say that it's really a great place for people who are kind of self-motivated and don't need to have someone tell them what to do and give them all kinds of written assignments and tell them what to read and what to write and what to think about. If you are a little more independent and have ideas, then you can get the guidance and the resources to pursue them.
00:06:08
Speaker
That's a place for, I think, someone who is very independent, or at least would like to be and have the environment in order to kind of grow in that direction.
Unexpected Collaboration
00:06:16
Speaker
Fantastic. Well, thank you for sharing that bit of background. I think we'll go ahead and dive into the book here. So, out of curiosity, how did your collaboration with Yao start in 2006? It's been quite a long time to work with someone with a lot of books.
00:06:34
Speaker
Well, that's a great question. And as is often the case in situations like this, I never expected that, I guess it's 18 years later, we're still collaborating.
Genesis of the Gravity Book
00:06:44
Speaker
You can't see the future, but in any case,
00:06:49
Speaker
It basically came about as follows. Yao was contacted by a literary agent, I think, and then he talked with a friend of his, a physicist who was also from China, and as to who might work with him on it. And this physicist named Henry Tai at Cornell University. I'm not sure if he still has ties with Cornell. He's at a university in Hong Kong now. He put me in touch with
00:07:10
Speaker
Yeah, I sent you an email, you know, he wrote back very quickly. He said, show me some things you've written. I sent him a few articles and then he wrote back again. He said, well, when do you want to meet today? And so we met that afternoon. I thought we just have a general chat, but he wanted to get started working on the book right then.
00:07:29
Speaker
We didn't know what the book was going to be about, though, so that was a bit of a problem or not a problem, but it was so we started thinking about different options. He'd had a lot of accomplishments in math and was a question of figuring out what would be the best one. And it took a while, actually, to figure that out. In fact, we had a conversation with that literary agent and, you know,
00:07:49
Speaker
mentioned some ideas and what he had in mind when he had contacted Yao and the agent was a little vague. He said, well, you know, it's up to you to decide, but I'm sure it'll be
Mathematics and the Universe
00:07:59
Speaker
great. So then we went back to the drawing board and wrote the first book called The Shape of Inner Space. It's about the mathematics of string theory, which was an area that Yao had made really kind of fundamental contributions to. And that was all I thought we were going to do. But then at the end of that, he suggested something else.
00:08:17
Speaker
So when you're talking with Yao here, how did you decide to write a book about gravity specifically? Yeah, that actually did. It was a follow up to the first book, even though this turns out to be the fifth book. So there's a little gap there. But when our book on string theory came out, an editor from Cambridge University Press contacted Yao. I guess he thought he liked that book and wondered if he actually asked Yao whether we had an idea for a second book and
00:08:43
Speaker
Yao said, he said, how about a book on gravity? And he passed that on to me. And I kind of thought, you know, it's just I filed that idea away. But before we got to that one, Yao had become the chair of Harvard Math Department. And I guess the chair of the MIT Math Department had just sent him a book about some famous mathematicians at MIT. And now that Yao was Harvard chair, he said, well, we've had some pretty famous mathematicians over the years and thought maybe a book on the history of mathematics at Harvard would be a
00:09:12
Speaker
subject that became our second book called A History and Some and it was about what become chair of the math department and thought a book on the history of math at Harvard would be an order and that led to our second book together A History and Some which covered the years of 18th to 19th and 18th was picked just because that was when the first Harvard mathematician actually published a paper that a new mathematical research
00:09:36
Speaker
Up until that time, professors were supposed to focus on writing textbooks and teaching. And the idea of breaking new ground in math had been discouraged. And so that's when math officially started at Harvard in our view. And we took it up as close to the present as we could. But we didn't want to get right up to the very day because things were changing. You can't always tell how important the development will be unless you have to give it some time to see whether a result is really going to go somewhere or not.
00:10:03
Speaker
But going back to your earlier question, oh, I forgot about gravity. So we ended up writing four books. And then when that was done, I remembered what Yau had said about book on gravity. And that's what really led to this current book. And what basically what we thought was to
Tensor Poster Discussion
00:10:21
Speaker
focus on the mathematics of gravity, the mathematical underpinnings of our current theory of gravity and the mathematics
00:10:28
Speaker
has been used to kind of explore the implications of this theory called general relativity, which was invented in 19th by Einstein.
00:10:37
Speaker
Um, so then I should also say another reason that I thought this would be a good book is that, or a good book for us to do rather is that Yao is one of the world's experts in the mathematics of general relativity. And he's made some big contributions in that area. So we would be well poised to take on that subject. That's, that's sort of the, the genesis of it. What happened is we started to work on it in early.
00:10:59
Speaker
2010, just as the pandemic was starting to rear its head. Yao had been thinking of going to China and he decided he better not wait too long because travel may be difficult. And so he moved on early March of 2010. So we basically worked on the book separately. And I actually haven't seen him since he hasn't been back to the US. Why did you think we needed another book about Einstein gravity and general relativity? There's a few out there already. So what makes this one different?
00:11:29
Speaker
Yes, to say there is a few out there is maybe an understatement, and I certainly have not counted, but I did read somewhere that there had been
00:11:38
Speaker
1700 books on Einstein and that doesn't necessarily include all the books on general relativity and gravity so yes there's a lot out there and our feeling was that a book that was really focused on mathematics would be new and could make a new contribution now not totally new in the sense that for college students and especially graduate students there are there are books on the mathematics of general relativity but those are not books that the general public would read they're quite dense so
00:12:08
Speaker
That was really what we were thinking, but also I think we realized that mathematics is an important way of understanding the universe and that might not really be widely appreciated. That is certainly a theme that is developed in our book.
00:12:23
Speaker
just because we learned about the universe through some of these expensive science missions like the Hubble Space Telescope and the LIGO gravitational wave observatories, Virgo gravitational wave observatories, this thing called the Invent Horizon Telescope, which captured images of two black holes or silhouettes of two black holes, very big projects. But before any of these projects were realized, there had been a lot of mathematical mathematics work
00:12:51
Speaker
that led to our understanding of these objects, understanding of things like black holes 100 years before we got a picture of them, and understanding of gravitational waves 100 years before the first ones were intercepted. So that's kind of, that's where we thought that, you know, with this book we talk about how we can really understand the universe through mathematics, not mathematics alone, but mathematics an important part of the
Black Hole Physics
00:13:15
Speaker
picture.
00:13:15
Speaker
Awesome. Awesome. Yeah. And I want to mention real quick a few things. First of all, you talk about the mathematics specifically of general relativity. My former co-host, the late Sofia Baca, created a poster all about the mathematics of general relativity. We just call her a tensor poster. I'm going to put that on the website very, very soon, but we sell it. And I think the last price that we were selling it for was e to the pi dollars.
00:13:37
Speaker
I don't remember what that was. I'm going to have to check that out. But essentially, it was just the cost of production. She wanted to do it for free, not for free, but for a very low cost just to maximize the benefit. It's a gorgeous poster. Full disclosure, I have no idea what it says. I shouldn't say that. That's not entirely true. So I have a master's degree in electrical engineering, and we did linear algebra. And we did calculus, up through calculus three, and we did a lot of physics. So it's got that, but it's an extension. So a lot of it that I don't even know what it says, but it looks so pretty.
00:14:04
Speaker
So for those who have an appreciation for the aesthetic and want to see some pretty math posters and want to try to figure out and reinforce your understanding of general relativity, shoot us an email. If I don't, if I don't have the shop set up on the website yet, shoot us an email at breakingmathpodcastatgmail.com. I'll send you a picture of the poster and see if you want a copy. And Mr. Natus, if you'd like a free copy, I like to give one to all of our guests. I'd love to send you one if you'd like.
00:14:27
Speaker
It would be great. I'd like to see it for sure. Fantastic. $8.05 is $23.14. Yes. There you go. There you go. Yeah. I believe that that was the last part. And we just, that was Sophia's idea and I thought it was terrific. So I'll send one for you and I'd like to send one to Xingqing Yao. I don't have to get an appropriate shipping address. We can talk about that after the episode. So yeah. And I think aside from this, the only other thing I'll say is we have other episodes too. We did an entire three episodes on evolving models of black holes.
00:14:54
Speaker
How they've changed has been just so fascinating. I'm not going to spoil all that here because I want this to be your episode, but yeah, the models of black holes has changed in ways where I can't even make intuitive sense of the current models that involve things like a firewall and an ice wall and things like that. So, so yeah, do you have much interest in black holes?
00:15:15
Speaker
yourself? Well, yes, quite a bit. We have a chapter in the book devoted to black holes. And I've written, I couldn't count the number of articles, but I know many dozens of articles on black holes. So it is, it's one of my favorite subjects. And there's a lot of mysteries still to be unearthed
Information Paradox
00:15:32
Speaker
there. So we haven't exhausted that subject yet.
00:15:34
Speaker
Awesome. You know what? And I may go off script here. I have, I have two questions. One of my favorite dinnertime conversations is trying to explain to somebody that doesn't, that's never heard of this, the difference between a neutron star and a black hole. I'll just add that because I love that conversation. I should make that script available for anybody who wants to download it, who's nerdy like I am. But my question is in the last, let's say 12 years, I have not kept up.
00:15:58
Speaker
with developments in ideas and models of black holes. Do you know of any major changes in thinking paradigms over these last 12 years per se?
00:16:15
Speaker
you know major advances in the last I guess probably eight or eight or nine years because in 2003 the first gravitational that was you know what the physicists believed product of two large black holes colliding and producing
00:16:30
Speaker
gravitational waves which were detected. And so that's some of the earliest solid evidence for black holes. There had been other evidence that it might be somewhat more indirect. But then a few years after that in 2009 was when the first image of a black hole was taken by this network of I think a dozen, at least a dozen radio telescopes around the globe, including Greenland and Antarctica and other continents, which
00:16:56
Speaker
got a picture. It's called the first picture, first image of a black hole. You can't actually see the black hole part of it, but you can see right up to the edge of it. And so that goes very far in terms of confirmation of the ideas. But I think you're referring more to some of the conceptual breakthroughs.
Einstein's Mathematical Collaboration
00:17:13
Speaker
And there are just many things to talk about, almost too many to go into here.
00:17:18
Speaker
But I can say that this did not get in our book because it's so new, but I met with two mathematicians just last month who have proven that type of black hole called an extremal black hole, which is a black hole which has the maximum possible charge or the maximum possible spin.
00:17:35
Speaker
can actually exist, whereas a law that had been so-called law, a physical law, had been proposed in 1960 years ago by John Bardeen, a Nobel Laureate, Stephen Hawking, and Brandon Carter, who said that an object like that could not exist. And other mathematicians have taken issue with that, and this is kind of brand new. They say that objects like this can exist and could be observable in the sky.
00:18:00
Speaker
I should say, though, that the fact that something is proven to be mathematically plausible does not guarantee that such things will exist in nature. But what it does show is that the arguments that had been presented before saying that they couldn't exist were faulty. And so that's a brand new possibility, and it has certain implications, which I could go into now.
00:18:23
Speaker
Basically, the idea was that if you had an extremal black hole and you were to add a little bit more spin to it, why don't I just say that there's been a lot of thinking on black holes and
00:18:35
Speaker
It's, you know, the subject is, uh, in some sense, uh, it's an old one, but they're, they're new papers on it. Maybe every day. I think, I think it would be surprising if there isn't a new paper on black holes every day. I'll have to see. Um, one of the things that we tried to do rather when I was trying to dive into black holes, I read the book, the black hole war. That's part of the title by Leonard Suskind. Are you familiar with that work?
00:18:59
Speaker
I'm not familiar with the book, but I think I am probably familiar with some of the ideas in it, but I can't comment on the book specifically. Yeah, that's an interesting book. I cannot say it is my favorite book, but it was still an absolute delight. I think that, not to go into great detail here, I understood about two thirds of the book, and then the last third was completely lost on me. This is Leonard Susskind.
00:19:19
Speaker
I think it has made himself quite a name. He tried to argue with Stephen- I mean, you probably know this. He argued with Stephen Hawking for over 30 years, arguing that the information paradox is incorrect, but he couldn't quite figure out why. And I think it was a documentary originally made in Britain, I believe, and I think it's just called the information paradox. Oh gosh.
00:19:38
Speaker
I'm going to have to Google search it, but essentially it was an entire documentary that talked about the staggering implications of the information paradox. And is that something that you feel comfortable talking to somebody about? And if not perfectly fine, are you comfortable talking about what, what is meant by the information paradox?
00:19:55
Speaker
Well, I could comment on that. It wasn't a subject covered in our book just because I don't know of sort of the mathematical aspect
Developing General Relativity
00:20:03
Speaker
of it. We focused on issues where mathematics has advanced our understanding, and I don't know that it has in this case. But what the basic issue here is the idea of Hawking radiation, the idea that
00:20:17
Speaker
Black holes, you know, you say are, you know, basically nothing can escape a black hole was the initial idea. But Stephen Hawking actually proposed this mechanism called Hawking radiation, where black holes would actually would leak and radiate away basically matter over time.
00:20:33
Speaker
process that can be compared to like evaporation, so that in an incredibly long time, and I can't even give you a number right now, that black holes, a large black hole would disappear. The question is whether information and quantum theory holds that information shouldn't be destroyed. And is that process of Hawking radiation and the evaporation's black hole, is that in conflict with this quantum theory of information conservation, basically?
00:20:59
Speaker
So that, that is the issue as to how I don't know the details of what Hawking and Wager and what Leonard Susskind said. But that's sort of what it's about. It's not something that we delved into in the book. Cause I hadn't heard of really important mathematics that, that I don't cast that subject in a new light and maybe notified us on that point. Okay.
00:21:19
Speaker
Awesome, awesome. Yeah, I'd be curious at some point. And I don't mean to spend so much time on that one. I'm very curious about your work as well. How did Einstein formulate the equations of general relativity in 1915? And what mathematical help did he get along the way? Yes, well, I guess one point that we made in our book, and before getting to the actual equations, which is quite a lengthy story, is that what Einstein did in formulating these equations was tremendous feat. And maybe one of the greatest theoretical advances, you know, in the
00:21:48
Speaker
history of science but it is often said that this was the greatest achievement of the single human mind and the achievement was amazing and no credit should be taken away from Einstein for what he did but it wasn't quite he didn't do it
00:22:02
Speaker
entirely on his own. He did get some help along the way. Once he realized that the problem was going to involve geometry and a kind of geometry called non-Euclidean geometry, basically geometry of curved spaces, something a subject he really did not know about. Then he realized that he was going to need help. And in 19, he went to a friend of his, a former classmate from college, Marcel Grossman, who was then at the same university in Zurich that he had gone as an undergraduate ETH and asked him
00:22:32
Speaker
for help because he was trying to he was wrestling with these ideas these ideas from physics for this new way of a new conception of gravity and it somehow involved a form of geometry that he was unfamiliar with and Grossman very quickly maybe within the course of a day told him that what he needed was ideas of a form of gravity that had been invented in 18 sexually 18 by Bernhard Riemann and he told him that that's what you need but that wasn't all you needed as a
00:22:59
Speaker
mentioned tensors and tensor calculus. This was something else that Einstein wasn't familiar with, and there was work that had been done, a kind of very important work by Gregorio Ricci and Tullio Levicivita, two Italian mathematicians who worked out tensor calculus, and Grossmann introduced Einstein to this as well. And that started to give some of the tools
00:23:21
Speaker
that one could use to kind of formulate the equations. But it was a long road. And even before he got there, I guess this is this is a big question because there are many steps in the process. It actually Einstein in 19 of published four major papers, one of which was on special relativity. And then the next step after that special relativity paper was to find a general theory.
00:23:43
Speaker
Special relativity describes physics of physical systems that are in constant motion. And to do a theory of gravity will require accelerated motion of something Einstein realized. And therefore, special relativity was a special case, a case when velocities are constant and objects move, you know, objects from relative to each other. Let me back up there.
00:24:04
Speaker
Special relativity describes a system, describes systems that are at constant velocity. Einstein recognized that he was going to need to generalize that theory to include accelerated motions in order to describe a theory of gravity. Now, how he came to that stemmed from a discovery he made, sort of a thought experiment in 19, something called the equivalence principle. This is something that Einstein thought said was one of the greatest things, these realizations he ever had. Basically that person falling from a roof during the
00:24:35
Speaker
at least the interval in flight, wouldn't feel any gravity. And what he basically discovered here, why it's called the equivalence principle, is he established an equivalence between gravitation and acceleration. And one way to think about that is that suppose you're in an elevator or something that seems like an elevator and you suddenly feel, you know, you feel yourself pressing against the floor. Is that because the force of gravity is pushing you down or is it because the elevator is accelerating upwards?
00:25:03
Speaker
there's no way for you to tell the difference. So in that sense, that was one of the principles that started him on this path, the equivalence between gravitation and acceleration. There are several other key elements that he needed in order to put this theory together, and this is where he got help, at least indirectly from someone else. His former math teacher invented the concept of space-time.
00:25:25
Speaker
He said that we can't no longer, for physics questions like this, can no longer think of space, three-dimensional space, and time is separate, but we have to think of them together as a four-dimensional space-time. Now, out of curiosity, didn't Emmy Neuter also have some sort of contribution in this effort?
00:25:44
Speaker
as well contributions come later. Yeah, maybe because I think I should get to describe the equations of general relativity before we get to that because she really was looking analyzing the equations after they were sort of formulated in 1950.
00:26:00
Speaker
So maybe I was going on too long. Yeah. There's, there's a lot to say here. What should we do now? If you want to talk a little bit about the equations and just go from there, what, like, where do they take place into it? And how does that add right into that little bit of a story? Okay. Yeah. If you were talking about, I'm sorry, it was, oh, you said his teacher, it wasn't Minkowski. Okay. Yeah. I was talking about Minkowski kind of, he invented the notion of four dimensional space time.
00:26:29
Speaker
That turned out. So should I pick off from there and maybe? Sure. Perfect. Perfect. Yeah. Okay. So, so let's just back up to 19. Einstein's former college math professor, Herman Minkowski gave a speech in which he talked about how we can no longer in doing physics or at least physics of this sword.
00:26:49
Speaker
Think of space and time as separate, that they had to be put together. We know of three dimensions of space that we can see, and there's time, and he said we have to think of this concept of spacetime, a four-dimensional spacetime, rather than three dimensions of space and one of time.
00:27:05
Speaker
Now, Einstein did not think much of that idea when he originally heard it. It took four years before, until 19th, Einstein recognized that it was the setting, four dimensional space time, was the setting in which he needed to write his theory of this new theory of gravity. And so he put his ideas about the equivalence principle, the equivalence of gravitation and acceleration together with this four dimensional space time. And then there was also Riemannian geometry that had been developed in the 18th. That was another kind of key ingredient.
00:27:35
Speaker
Those were some of the ingredients he needed, along with tensor calculus. So now he has to try to put together these theories of gravity. Initially he worked with his friend, the geometer, Marcel Grossman. And in 19, they came up with the first draft of the equations of general relativity. That paper that Einstein wrote the physics part, Grossman wrote the mathematics part.
00:27:56
Speaker
and it came very close to getting things right but it didn't quite work and there was one big defect in it in that one of the guiding principles from the start in this endeavor for Einstein and for Einstein and Grossman when they're working together was that this equations that they write should not depend on what coordinate systems we choose to label things and
00:28:17
Speaker
That concept is called general covariance. But they got into problems writing these equations and they had to give up on that to some extent. And that hurt the theory a lot. They decided then that they could not make a generally covariant theory and that it was impossible. But their arguments for that turned out to be wrong.
00:28:38
Speaker
It took Einstein a while to recognize that, and it took him actually two more years to establish a fully covariant theory. And what that means is a theory that means that physics is physics, and the coordinate systems are chosen arbitrarily by, say, human observers. How we choose to put the coordinates where we put, if we think of a graph, say, where we choose to put the origin, should not change how physics works.
00:29:01
Speaker
And that ultimately, you got a theory that did that, but it was two very hard years of work. I can describe the equations now. Should I say a little bit about them? Sure. Yeah. Yeah. Yeah, for sure. Right. So in November of 19, Feinstein published November 25th, 19, published these equations of general relativity, which are still with us today. And let's just say that, you know,
00:29:27
Speaker
You can't actually write this up. You could say it. I'll try to say it in words. Capital G with subscripts IJ equals capital T IJ. And in some ways, that's the whole thing. It doesn't seem so hard. Just two letters. It doesn't even fill hardly any lines on it. You know, maybe a single line on a page.
00:29:45
Speaker
But as these turn out to be, it's not just nearly as simple as that because the G term and the T term are tensors, which are basically kind of like matrices, four by four arrays. But each one of them is not necessarily a single number, but it can be a function. And what you end up with
00:30:05
Speaker
It's not just this simple equation, simple looking equation, gij equals tij. What you have is 16 equations, and they're 16 nonlinear partial differential equations, which turn out to be very hard to solve individually. Always are.
00:30:20
Speaker
always are. And you have to solve not just one, but 16 of them simultaneously, which makes it very difficult. But let me just say, I mentioned these letters, but what does it mean? What does it
Gravity as Spacetime Curvature
00:30:31
Speaker
have to do with? The left side of the equation has to do with the curvature of spacetime. And the right hand side has to do with sort of the matter and energy in spacetime. And so what the equation basically says is that matter and energy curves spacetime. And the curvature part of the equation includes different kinds
00:30:50
Speaker
Einstein needed to incorporate different kinds of, not just one kind of curvature, there are different kinds of curvatures and different dimensions. So it was a very detailed geometric description of how, say, concentrations of mass and energy, what they do to the time, the space and time around them, and how they curve things. Now, it sounds a little abstract, but I can try to make it a little clearer if you
00:31:14
Speaker
Think of gravity. Basically what Einstein said, the previous picture of gravity that we'd have for several hundred years that Isaac Newton developed and is valid in most cases, you know, in everyday situations. It's not valid in extreme cases like the environment around a black hole.
00:31:30
Speaker
But that notion invoked an visible force, an attractive force between massive objects. Einstein kind of changed that picture entirely. He said that massive objects curve space and time around it. You can think of, say, putting a bowling ball on a rubber sheet that stretched out, and you have little smaller balls that are on that sheet that will now be drawn towards the bowling ball because it's going to create a bit of a cavity around it.
00:31:56
Speaker
That is basically the essence of gravity. Gravity is a consequence of the curvature of space-time, and the curvature of space-time can be considered. It tells you the shape of space-time, or its geometry. So gravity is really a consequence of geometry, and therefore you have this equation on one side, you have mass and energy, the stuff that kind of creates the gravitation, and then on the left side you have the curvature of space-time that corresponds to it.
00:32:23
Speaker
And it's an entirely different picture of what we had before, but it's a picture that's held up for very well for more than a hundred years. You know, I think what you just said there, if I may be simple to interrupt here, when you talked about the rubber sheet analogy, when I was in high school, maybe I was eighth grade, I want to say eighth grade or so. That was the first time I ever saw that. And nothing impacted me in terms of my love of science and physics specifically, as much as seeing the analogy of the rubber sheet, CGI analogy of the rubber sheet example of space time.
00:32:49
Speaker
Now I know that that's not a completely accurate model. I've heard plenty of criticisms, but I wanted to point out a few examples of where the audience, the listeners can find it. I think it's available on the PBS or Nova websites. I will find it and put the link in the show notes. And then I also realized I have permission to even show a small clip of it. The first time I saw it, I think was in the
00:33:07
Speaker
elegant universe by Brian Greene where you again you've got a CGI example where he did this great three-dimensional grid with these white lines and he showed where the Sun is and he showed how the grid lines come together exactly as they would on a rubber sheet and then it just shows the movement of the planets that that follow the contours of the bending of space they are utterly spectacular
00:33:26
Speaker
He does it not only in the elegant universe, which I will hopefully have a clip of, but he also does it in a follow-up called The Fabric of the Cosmos, where I think he used a billiards board pool table instead. But again, just that visual, that visual alone, I think communicates the idea in such a glorious, spectacular fashion that, yeah, just the fact that, you know, in a simple visual, you can show this beautiful analogy. I think it's wonderful. I think it is absolutely wonderful. Gosh, have you, are you familiar with any criticisms of that analogy?
00:33:54
Speaker
I think that the basic analogy holds it. I'm sure that there will be some renderings of it that would be maybe some poetic licenses taken or may not be accurate. But that is the essence of general relativity. And as I said, it's not a new idea. It goes back to 19th and even to the earlier work of
00:34:14
Speaker
Einstein and Grossman didn't get things quite right. But going back even a couple of years before then, but it is not necessarily how people think of gravity, you know, everyday life. And it's not necessarily germane to our everyday experience. But it is it is actually a very mathematical and geometric interpretation, which is sort of kind of at the heart of what this book is about to show the importance of mathematics and geometry, not only in developing the theory
00:34:41
Speaker
and in creating these equations that are, you know, geometric, you know, really crazy about geometry, but also in understanding what is unpacking that theory because it's been a hundred years and it hasn't been fully unpacked even
Gravitational Wave Discoveries
00:34:54
Speaker
yet. And there's a lot of mathematics has paved the way for some scientific discoveries made, you know, through astronomical observations and experiments in laboratories.
00:35:03
Speaker
Actually, I do have one big question for you. So gravity has been around for 109 years. What else is there? What's new to discuss about it? Well, let's just say that gravity has been around. I mean, as you know, gravity has been around before we were here. This is certainly this way of thinking of gravity.
00:35:24
Speaker
has been around a hundred years, but I guess there are new things to say about general relativity, and there are papers on it pretty much coming out, probably more than one a day, and big discoveries. Some of the discoveries are actually just ever more stringent, precise experiments testing Einstein's ideas, and so far they've all held up, and others just kind of amplify on what we know before. For instance, in 2000, the first gravitational waves were discovered.
00:35:51
Speaker
Now gravitational waves Einstein predicted in a 19th paper came out one year after the equations of general relativity were published. So he produced equations that described what happens when a massive object moves through space-time could create ripples, so-called gravitational waves, kind of like
00:36:09
Speaker
If you have a boat, you send it on a powerboat and send it on a pond that's completely placid and it speeds along. It's going to create waves where there were none through that motion. And Einstein believed that a phenomena like that would happen with massive objects moving through space time.
00:36:25
Speaker
As I said, that was 19 in 2000 years later, scientists from the LIGO and Virgo collaborations, gravitational waves announced the discovery of the first gravitational waves. Now, just last year, for instance, something new. And I should say that in the intervening years, gravitational waves have been detected from maybe a hundred other sources. Sources typically include collisions between black holes, a collision between neutron stars, or even a collision between a black hole and a neutron star. You need truly massive objects.
00:36:54
Speaker
moving very fast, possibly at half the speed of light in order to create waves that can be detected here. As to what's new, I should say that, well, here is the silver evidence that there is not just we have detected isolated cases of gravitational waves. It appears to be a background.
00:37:11
Speaker
of gravitational wave everywhere you look in the sky. That product of not a single collision between two black holes, but rather sort of the combined residue of hundreds, maybe hundreds of thousands of black hole collisions or maybe even millions of collisions over, you know, eons. And that's kind of, it seems to be a permanent feature, a kind of a background feature in the sky. So that's something that wasn't known before.
00:37:33
Speaker
There are new ideas about black holes. Our book focuses on that's not on the math and what I just discovered. Our book focuses more on the mathematical things. But there was a paper that came out by two mathematicians last year, which looks at whether black holes could exist, not just in the three dimensions of space that we know about or the four dimensional space time in which general relativity is set. But could there be black holes in higher dimensions? Now, Stephen Hawkins proved
00:38:00
Speaker
that in three dimensions, spatial dimensions, or four dimensional space time, black holes have to be spherical in shape. And by saying the shape of a black hole, referring to the event horizon of a black hole has to be spherical. These mathematicians prove that in higher dimensions, you could have all kinds of different shapes.
00:38:17
Speaker
now, and black holes can exist in these higher dimensions mathematically, and they wouldn't be limited to having a spherical shapes. That does not mean that these things actually exist in nature, but they're possible now. It's something that one could look at for, and one may even be able to look for signs of them in accelerated physics experiments like the Large Hadron Collider in Geneva.
00:38:37
Speaker
and your ideas how you might see evidence of higher dimensions, which we don't even know that the higher dimensions exist. But that might be one way of finding out if you see an object that couldn't exist or remnants of an object that couldn't exist in four dimensions. Spacetime, that would be, yes. So those are just a couple of, there's just new findings coming out all the time. This theory was presented in 1915 and people haven't stopped analyzing it and making new discoveries.
Defining Mass in Relativity
00:39:04
Speaker
I have a couple of questions to myself. There's something interesting that I want to talk to you about in this book, and that's that Jaak talks about mass specifically not being a very well-defined concept in general relativity. Can you explain what does he mean by not being well-defined? Yes, that is a good question.
00:39:22
Speaker
It's kind of hard in a way because you would think that it's almost the simplest thing. And, you know, we've had mass, you know, if you go back to Isaac Newton's equations from the force equals mass times acceleration, mass is kind of just a basic idea and it doesn't seem that hard. And it is, I can't actually explain why it has been so elusive in general relativity. But let me say,
00:39:45
Speaker
that the way it has been defined, and it's the most rigorous way of defining mass in general relativity, I think, well, OK, is to think of the following. You can think of the mass of an object that's isolated. Say you have a black hole, and you go very far away. In theory, you would say almost infinitely far away. And if your theory halts, that is, you get far and farther away from massive objects, dense objects like black holes that curve spacetime,
00:40:12
Speaker
The farther and farther away you go, the flatter and flatter space gets. From that point, you can actually determine the mass of that physical system. But we don't have great tools in general relativity to work out the mass of things from close up. For instance, it's considered impossible to define the mass at a single point.
00:40:33
Speaker
What work that my co-author Yao has been involved in and made very big contributions in is trying to get a more, let's say, a sharper picture of mass than having to go out very far away, if not to infinity, just far enough away from, say, a black hole that
00:40:52
Speaker
things will be calm enough that you can do it. This is called quasi-local mass. You could look at a system that's not just a single object, but you could have a system that it might include two black holes and you could try to get a sense of the mass of each of them rather than looking at the whole system far away and determining its mass. Now, I wish I could answer your question better as to why this is so difficult.
00:41:19
Speaker
No, no, it's all good. So I didn't want to say how people have attacked this problem a little bit. And the first, the first kind of definition of quasi-local mass, which means was advanced in the 19th by Stephen Hawking. And he thought, well, you can take, take like a black hole and you have to assume that it's a perfectly spherical object. You look at how light rays come in and how they would be deflected by that mass. And then you can get an estimate of what the mass is.
00:41:47
Speaker
That's one approach that people have taken. The only thing is that spherical symmetry is kind of a special thing to insist on and not really found necessarily found in nature. Another mathematician, Robert Bartnick, came up with an idea that you could take this physical system with some mass and put a surface around it and then
00:42:06
Speaker
that surface out towards infinity or to a great distance from which you can then determine the mass inside. But he said you can expand that surface like you could think of it as like a flexible thing that could be expanded in different ways. And depending on how you expand it, the mass that is smallest is the best definition of that mass inside there.
00:42:26
Speaker
Now, Yao and his co-author Mutao Wang at Columbia University have what's now considered to be the best definition. And again, it's very technical. It's a little frustrating that mass is so difficult. But what they basically say is that you can find a two-dimensional surface to surround a physical system. It could be their solar system, or it could be black hole or binary black hole. And you can look at basically the curvature of that surface. And the curvature of that surface
00:42:54
Speaker
will give you the measure of the mass. And so it's very hard. You can't just put things on a scale. And part of the fact that this theory is non-linear means everything is affecting everything else. And there's all kinds of strange feedback that makes things hard. But right now, I think that most people believe that what Yao and his co-author Mu Tao Wang have the best definition of quasi-local mass, mass that you don't have to actually go out as far as infinity to measure.
00:43:20
Speaker
But let me just say that, even now that's the best, you have to solve some very difficult differential equations, it's not easy to use.
Yao's Mathematical Contributions
00:43:27
Speaker
The Bartlett solution is considered very mathematically elegant, but no one knows if it can actually be solved. So it's not, I should say that even today this problem isn't solved, but there have been advances
00:43:38
Speaker
and their recent advances by Yao and his colleagues, once they have this definition of quasi-local mass, then they can get definitions of angular momentum. You have a mass that is moving and maybe orbiting in a circle, and now they have a new definition of quasi-local angular momentum.
00:43:53
Speaker
Now, can you tell us a little bit about Yao's contributions? Yes. Well, his contributions, I think I've been alluding to some of them and those have. So he's helping, you know, and made some big contributions to helping us think about mass in general relativity and think of other basic concepts like angular momentum.
00:44:14
Speaker
But I think he's also contributed in several other areas, kind of important contributions. I believe I referred to this earlier in our conversation. 19th was the first of Roger Penrose papers on singularities theorems. He basically talked about how certain kinds of
00:44:30
Speaker
situations in which singularities would arise. Something called a closed trap surface. It proves if you have a closed trap surface that will lead to a singularity. Now what is a closed trap surface? It's basically a surface that is collapsing in a way such that if you have say light, a source of light in the middle, you will shoot it out towards the edges of this surface and it will wrap around and come back into the middle. And so
00:44:52
Speaker
In any case, so, Roger Penrose wrote a series of singularity theorems starting and then including some work with Stephen Hawking. Penrose won a Nobel Prize in Physics in 2003 for that work, but his work didn't show mathematically how you get a closed-trapped surface. And this is work that Yao and
00:45:12
Speaker
Yao and his co-author, Rick Shane, did what's called a black hole existence proof. So basically, Penrose showed that if you have a closed trap surface, you'll get a singularity. But in order to say how you get black holes, you have to show how you get that closed trap surface. And so this is work that Yao did with Shane called Black Hole Existence
00:45:31
Speaker
theorem. So basically proved mathematically how you can get a black hole. And this was done in the late 70s before there was actually proof that black holes exist, even though there was some, you know, evidence from astrophysicists that there were black holes out there. And so having this black hole existence proof, it's mathematics. It doesn't mean that there are black holes out in the sky or in the center of our galaxy, which we now believe to be the case in the center of every large galaxy, basically. But
00:45:57
Speaker
That's kind of what you might call a precondition. Mathematical existence proofs show that these objects make mathematical sense. They're mathematically consistent and soon thereafter we started accumulating more and more evidence that black holes really do exist and are quite abundant in fact.
Emmy Noether's Conservation Laws
00:46:13
Speaker
So that is one thing. Yao also worked with the same co-author Rick Shane on work. It's called the positive energy theorem, which in some ways, it's a little oversimplification to say this, but shows that proves that an isolated physical system has energy must be positive or at least none. Let's just say energy is none.
00:46:34
Speaker
Negative energy is either zero or positive. And that proof in some ways could be applied to our universe. In some ways, a physical system, an isolated physical system, the energy has to be non-negative. And it's not quite fair to say that applies to our universe, but it could apply to our universe, tell us that we have positive energy.
00:46:53
Speaker
That is a bit of a leap going to the universe. There are some caveats when we're attached to that. But those are some of the areas that he's made big contributions. And he's made huge contributions to string theory, which is a theory that combines our theory of gravity with the theory of quantum physics. So he has been quite a big player in this area. And so I was lucky to have him, certainly very fortunate to have him as the co-author of this book on the mathematics of gravity.
Cosmological Constant and Dark Energy
00:47:18
Speaker
Yeah. So I guess the one thing I want to ask you are things you most would like to talk about in these last five. It could be more of the same book. It could be your other books, things that you'd like to, uh, gone in the future or anything that you would like the floor is yours. Well, I guess I would like to take up Autumn's question one and maybe another one that was on the list. Sure. Absolutely. Do I need to just, I don't need to, so I'll just go right in.
00:47:40
Speaker
Yeah, we can cut it out if need be. Yes, Adam asked about Emmy Notter's contributions to this and I think that's kind of an interesting story. I didn't pronounce the name so well. It's a German name, Emmy Notter. A mathematician from Germany who has made huge contributions in the area of algebra.
00:47:57
Speaker
A great pioneer had a difficult, interesting story in that she could not go to women were not admitted to universities when she was of college age. She had to audit classes. Eventually she was able to get permission to study and earned a PhD. And then she did great work, but she couldn't actually get a job.
00:48:16
Speaker
because of being a woman in Germany. And she was also Jewish, and I don't know the years, but I believe sometime in the 30s emigrated to the US. And she finally did get a job teaching at Brynmore College, although she became ill and did not live long after she had finally gotten a professor job, even though she was quite an illustrious mathematician. But let's go back up to 19. Einstein come out with these theories of general relativity, the equations of
00:48:41
Speaker
described gravity, and he was then doing his work in Got, Germany, and there's a famous mathematician there, David Hilbert, at the University of Got, which was the world's leading center of mathematics, along with Felix Klein, another major mathematician there.
00:48:56
Speaker
invited Emmy Notter to come there to look into the issue of energy conservation and the theories of the equations of general relativity and make sure energy was conserved, which is kind of a key physics principle. And it's interesting that they picked her to do that, and she was someone Einstein was eager to have work on it, even though she was not a working mathematician in that she couldn't get a job in the field.
00:49:19
Speaker
But she did in fact, she did in fact prove that energy was conserved in general, the equations of general relativity, so long as you
00:49:28
Speaker
So long as you took a so-called global view, you had to be far enough away from things. And if you look from a far enough vantage point, energy was conserved. And she proved that in a theorem. And the basic idea might be if you had, say, a bowl of water or something, and you're in a desert and the water's in the sun, it's going to evaporate. Will that water be conserved? You say it would change forms from liquid to gas. And it would be if you looked at a large enough volume
00:49:55
Speaker
of space to capture all the water molecules that had evaporated. And so that's the basic idea. She proved that, but she also proved something else at the same time, somewhat related, related to conservation laws in general. She said every so-called continuous symmetry is conservation law.
00:50:11
Speaker
That's a little bit abstract, but I can say that the discovery of hers, which was made maybe, I think, published around 1980, has been a guiding principle in physics ever since. And when people, particle physicists, tried to have discovered new particles, they've often thought about symmetries first, and then how these symmetries may be embodied in certain particles. And so Emmy and I, there's principles, and her theorem that she proved
00:50:35
Speaker
about the link between symmetry and conservation laws is just a basic pillar in physics today. And it may have somewhat sprung from her, you know, being, it came to some extent from her being asked to come to Gottingen and to look into these equations of general relativity. Okay.
00:50:51
Speaker
Well, let me just say one thing very quickly as Einstein talked about his greatest blunder was adding this term called the cosmological constant to his equations of basically the equations of gravity. And he thought that in order to describe the universe as a whole, that the equations needed an extra term. Now he inserted this term because he believed that the universe should not
00:51:13
Speaker
should have a constant radius, a constant size, never expand or shrink. And he put that term in there, this cosmological constant, into these equations. For that reason, yet his instinct that the equations needed something was good. It turns out that he was very wrong in his reasoning. The universe was soon discovered to be expanding, and that was known in the 20s by astronomers like Edwin Hubble. And for the past, I think, 40 years, we now know that expansion is accelerating.
00:51:42
Speaker
Now what we see is the term, the cosmological constant, may be what's called dark energy. Dark energy, turns out physicists believe, points out to 70% of our universe, 70% of the universe is matter and energy. So the thing that Einstein considered his greatest wonder is actually the biggest thing in the universe there
Conclusion and Appreciation
00:52:00
Speaker
is. It's dominating the universe. So that's kind of an interesting story as to how that unfolded.
00:52:05
Speaker
Yes, I have heard that. That's actually quite fascinating when they had a Oh gosh, what was it on the science channel in 2012? I want to say they had a special all about that. Oh, no, I'm looking at the name. I'll have to Google search the name, but they had a really cool CGI animation where they showed real matter clusters, and then the dark matter clusters that are identified only through gravitational lensing. So
00:52:23
Speaker
Fascinating stuff. And we are out of time. Thank you. This has been a phenomenal interview. I really enjoy it. I enjoyed your time here. I appreciate all the work that you and Xingtun Yao put into this awesome effort. And I look forward to reading your other books as well. I wish I had more time for this, but it's been a pleasure and thank you for joining us. Thank you.