It's a great pleasure to be here and I thank Professor, President Lenson for his very kind words.
I apologize that I come to Germany and speak to you in English, but in the long run it will work a lot better.
If I speak too fast, just tell me to slow down.
It is a great pleasure to be here.
I'm going to be giving a series of five talks over the next few days, all on the topic of science and philosophy meeting.
The first of these is going to deal with some work that Einstein did.
But before I talk about Einstein, I'd like to talk a little bit about Karl Friedrich von Weizsäcker,
in his honor I've come to speak today, because it is really his work that is inspiring the idea of a series of lectures
that talk about the connection of philosophy and of science.
I'm no expert in Weizsäcker's work, but one finds very quickly when you read his work that he had a very deep and enduring interest in philosophy.
These are remarks from the preface of one of his later very important works.
And he says here very clearly that philosophy was no distraction for him at all, it was indispensable.
And then he proceeds to read off a list of very significant philosophers that he sought to understand.
Now, it's interesting to me to learn that Weizsäcker was inspired to work on philosophy very early by Niels Bohr.
Niels Bohr, of course, was Einstein's great antagonist in the famous quantum debates that lasted over many decades between them.
So one might well wonder what his feelings towards Einstein were.
And we find in the same work admiration for Einstein.
He lists three names at the head of its book, Einstein, Bohr and Heisenberg.
And he indicates that he knew of Einstein from an earlier years and everyone was under the spell of his insights.
And he says that he unfortunately never met Einstein.
And I think that's rather a shame because what Weizsäcker and Einstein shared was a fascination in philosophy and the way it connects up with science.
So I sometimes think it was unfortunate that he never met Einstein and we could perhaps imagine what it would be like were he to meet Einstein.
And one of the joys of Photoshop is of course you can make things happen that you otherwise could only imagine.
So no, he never met Einstein but that's maybe what it would have looked like.
Okay, so let me move on and now turn to Einstein who is the topic of my present talk.
What is distinctive about Einstein is that he sought to integrate his thinking in science and his thinking in philosophy throughout his life.
And the interactions between Einstein's scientific work and his philosophical work are very many indeed.
Let me indicate some of the philosophical reading that Einstein did.
These are just a few of the names of the philosophers that we know he read and thought about and influenced his work very greatly.
And correspondingly and more so later in his life we find quite a few fundamental philosophical ideas being drawn out of Einstein's work and being brought over into philosophy.
Now there's a lifetime of study already in trying to tease out just the themes that I've put up on the screen here.
So what I thought I'd do for you today is pick out just two examples of where Einstein's science and philosophy interacted in very important ways.
I've chosen them with two thoughts in mind.
One, the episodes I'm going to tell you are not just abstract theorizing, they're part of Einstein working as a physicist.
The philosophy was integrated into his work as a physicist.
And then secondly I'm going to talk about them because I think they're less well understood.
I think they were very important to Einstein but in the larger arena I don't think people are so aware of just how important these particular ideas were.
Now the two episodes are going to be the following.
The first is going to be Einstein's discovery of the special theory of relativity.
There came a decisive critical moment in that discovery where Einstein needed philosophical analysis to complete the story.
He consciously was aware of that and he consciously used a particular piece of philosophy to finish up the theory.
And I'll give you the details of that in a moment.
And then the second thing I'll talk about is Einstein's work on his general theory of relativity.
And we're going to see as I work through that that his experience with the theory caused him to change his mind almost completely about the correct methods for doing science.
It's a very dramatic transformation in the way that Einstein thought about how science ought to be done.
Okay so let me start at the beginning with Einstein's discovery of the special theory of relativity.
The discovery itself was reported in this paper here in the Annaland der Physik in English translation on the electrodynamics of moving bodies.
The paper is presented as a complete and perfect whole and when you read it you don't get a sense of the many years of struggle that went behind it.
This paper was in fact the result of Einstein spending seven and more years, seven and more years on his own count, struggling with particular problems in electrodynamics.
The theory of electricity and magnetism that he'd learned from the 19th century physicists.
I don't want to go into the details of these problems. All I want to tell you though is that having worked for this long period on these problems, he reached a point of desperation.
And it's important to understand that it was a point of desperation. He did not know where to go next.
And I can characterize that point of desperation for you in very simple terms thanks to the way that Einstein set up his own theory.
The point of desperation looked something like this. He had to reconcile two apparently incompatible ideas.
So I've put a little geometrical figure up here which is meant to be the reconciling of apparently irreconcilable ideas.
On the one hand, he had the principle of relativity. He had been looking at experiments in the electromagnetic ether of the 19th century.
He'd been looking at the structure of electrodynamic theory. He'd been looking at classical mechanics.
And he'd become completely convinced from all of these areas that one has a complete equivalence of inertial motion.
Loosely speaking, physics is the same for every uniformly moving observer.
That was the first part of the two things to be reconciled.
And then the second was the light postulate. Maxwell's electrodynamics tells you that light propagates at a certain definite speed.
300,000 kilometers per second, 186,000 miles per second, and it's one definite number.
Now, there seemed no way for Einstein to get around that. He'd played around with Maxwell's electrodynamics at some length in the preceding years.
And that one result was immovable. He could not change it.
The difficulty now came when you put the two results together.
All uniformly moving observers are equivalent when it comes to the laws of physics.
The speed of light is always going to be the same number, 300,000 kilometers per second.
So you'll see the difficulty. If I measure a light beam to be moving at 300,000 kilometers per second,
and you chase after it, surely you would see the light beam or you would judge the light beam to be slowed down.
But no, if you take both of these principles seriously, you chasing after the light beam must still report back that no, the light is moving at the same speed.
That's the problem that Einstein was facing.
How is it possible that even chasing after light at very high speed, you do not find the light to slow down at all?
This produced the great impasse for Einstein, and here's a later recollection of how the impasse was resolved.
He talks here. I'll read this full quote later for you, but let me just point at the early remarks.
He talks about this paradox of chasing a light beam, the one that I've just described for you now,
and he says that the type of critical reasoning required for the discovery of the central point that solved the problem
was decisively furthered, in my case, especially by the reading of David Hume and Ernst Mach's philosophical writings.
So it's at this point when he reaches this impasse and he manages to break through, he later tells us it was reading Mach and Hume
that enabled him to make the breakthrough, decisively furthered by those readings.
Now, this particular passage in Einstein's writings is widely noted, but I think it's widely misunderstood.
It's normally presumed that when Einstein mentions Hume and Mach, he was thinking of the work that they did in philosophy of space and time.
Mach in particular is very well known for his critique of Newton's ideas of space and time.
However, when you start to read a little more carefully in the way that Einstein reports these episodes,
it starts to become clear that Einstein was not at all talking about anything in philosophy of space and time.
What Einstein was talking about was a very general piece of philosophy that was actually independent of the philosophy of space and time.
And in order to explain that to you, let me move on by telling you first what the breakthrough was for Einstein.
The key thing that Einstein needed was to develop a new view of the nature of simultaneity.
And to explain that, I'm going to use one of Einstein's celebrated thought experiments that explains the result concerning simultaneity.
I've simplified it slightly following a later account that Einstein gives in 1915, but this is pretty much exactly the way things were in his original 1905 paper.
Let's imagine that we have two clocks here, an A clock and a B clock, and they're set up to emit a flash of light.
Each of them is set up to emit a flash of light.
The clocks are put at either end of a very long platform, and an observer stands at the midpoint of the platform and waits for these flashes of light to arrive.
So the clocks flash, the light signals flash, the light signals leave, and approach the observer standing at the midpoint.
The observer then finds that the two light signals arrive at the midpoint at exactly the same moment.
The observer on the platform then makes the following judgment.
It follows, the observer says, that the two flashes happened at exactly the same moment. They were simultaneous events.
And correspondingly, if the two clocks read the same number when the flashes happened, then the two clocks are synchronized correctly.
And you'll see there that the clocks do indicate the same number. They both say zero when the flashes happen.
So this observer here is going to judge that the two clocks are indeed properly synchronized, and the two events, the two flashes, are happening simultaneously.
So far, so good. Nothing very puzzling happening.
But now let's re-describe the same process from the perspective of this observer who is moving relative to the first.
And let's recall that we are in the context of the two ideas that Einstein developed earlier on.
Each observer is equally good when it comes to laying out procedures and treating various laws of nature.
And each observer is going to judge the speed of light to be the same.
So here's how things look from the perspective of this second observer.
Here's the observer now at rest, who will then judge the platform to be moving past.
This observer will still agree that the two light signals arrive at the midpoint of the rod at the same moment,
but will give a very different account of why it is that they arrive at the same moment.
For you can see what's happening according to this observer.
The midpoint is rushing away from the A clock, from the A event, and is rushing towards the B clock, the B event.
As a result, the midpoint's motion is making it harder for the light signal from the A clock to arrive at the midpoint,
and making it easier for the signal from the B clock to arrive at the midpoint.
So if the two events were simultaneous, you would then expect the observer at the midpoint to notice the signal from the B clock first,
and then there'd be a delay, and then the signal from the A clock would come.
But it doesn't go that way. The two signals arrive at the same moment.
So the only judgment that this observer can come to is that the event of the flashing at the A clock happened earlier than the event of the flashing at the B clock.
Okay, that was a little tangled, but I'll give you the basic conclusion that you come to.
If you simply follow through the principles that Einstein has become committed to, you end up with this conclusion.
Whether two spatially separated events are simultaneous or not has no single answer.
There is no single fact as to whether two events, when they're spatially separated, are simultaneous.
You will get different answers to the question according to the state of motion of the person making the judgment.
Now this is the core result of the relativity of simultaneity around which Einstein then built up the special theory of relativity.
Once you understand the way that relativity is put together, you see this is actually the central effect,
and everything else can be derived out of it through various manipulations.
In fact, this is the content of the first section of Einstein's paper.
I'll indicate briefly for you how it goes about reconciling the problem that we saw earlier.
If you're an observer, say on Earth, who wants to measure the speed of light, all you need is a ruler, two clocks,
and then have the light signal propagate along the ruler.
Now, in order to have a good measurement of the speed of light, a very important condition has to be satisfied,
and that is that the clocks that you place at either end of the rod have to be properly synchronized.
If they're not synchronized, the measurement results that you get will not be reporting correctly what the motion of the light signal is.
So you can understand the situation.
The observer on Earth measures the speed of light to be some number, let's call it C, 300,000 kilometers per second.
What now happens if there's a second observer who is moving very rapidly with respect to the first observer?
We know from Einstein's postulate that the second observer must still get the same number C.
So how is this observer then going to explain the result that the Earth-based observer gets?
And the answer is very simple.
This observer can then watch this process and say, oh, wait a minute.
I understand what you're doing here, but your clocks aren't properly synchronized.
I don't find your results to be reliable.
You should have measured a smaller speed.
And the reason you didn't measure a smaller speed or a different speed is because you're using clocks that aren't properly synchronized.
So that's the basic idea of the relativity of simultaneity.
I don't want to labor it any further.
The only thing I want to say about it is the following.
It takes a considerable amount of courage to take that step.
In fact, Einstein, in later correspondence, in later discussions, called it the step.
Der Schrit, if I remember in German correctly.
It was a step that Einstein took in desperation because it involved overturning the foundations of the known theories of space and time,
in particular, Newton's ideas about space and time.
Somewhere, somehow, Einstein had to get the courage to take that step.
Now, the mere fact that it solved his problem, that it brought together the principle of relativity and the light postulate,
is not enough.
Saying something crazy, right, because it conveniently solves a problem for you, is only a part of what you have to do.
You have to be secure. You have to be warranted in taking the step that you're about to take.
And that's where the philosophy came in.
The philosophy gave Einstein the courage to take the step.
So, the central insight was a philosophical view of the nature of concepts.
Einstein explicitly adopted the view about concepts and how they can be used in scientific theorizing.
And this was the thing that gave him the courage to move forward.
Now, I'm going to summarize that view of concepts in two ideas.
The first is this very simple idea, and that is that concepts have to be properly grounded in experience.
Now, I'll just mention here that I've put up a fair number of quotes here,
and I'm not going to try to read them to you because it starts to get burdensome if I stand here and read long passages off the screen.
However, I've set up a website that you can get through the lecture series,
and if you go to the website, I've put up these transparencies so that you can, if you're interested enough,
you're free to download the PowerPoints and see all the quotes.
But since many of my philosophical colleagues are very text-based,
they like to know that there's a full quote there that supports the claim that's being made.
So, here's Einstein saying it in his words,
our concepts and laws of space and time can only claim validity insofar as they stand in a clear relation to experiences.
And so, he gives an example in an interview with Max Wertheimer, the psychologist in 1916.
It's not an example I think we would use now, so forgive me, but it is Einstein's example, so I will use it.
He asks the question, you know, if someone wants to use the word hunchback, how can we know when to use that concept?
Well, he says, if that concept is to have any clear meaning, there has to be some way of finding out whether or not a man has a hunchback.
And this is Charles Lawton you might recognize in the hunchback of Notre Dame, which a movie I think was made a little later.
Okay, so that's the first point. The concepts have to be properly grounded in experience.
And then the second point is this. If a concept does not have a proper grounding in experience, then we are free to discard it.
Okay, or we are free to simply accept it as a conventional element in the theory.
And he says there's a very great danger if we don't do that.
If we allow the concept to have an authority over us when it has no grounding in experience, then we risk importing presumptions into our science that have no place there.
Now I hope you can see this maps exactly onto Einstein's work on simultaneity.
Newton's views of space and time presumed that there is a fact as to whether two events are simultaneous when they're spatially separated.
This is the concept of absolute simultaneity.
What Einstein recognized in 1905 was that there was no basis in experience for that.
There was no way to give an unequivocal experiential basis to two separated events being simultaneous.
And when you recognize that, then you were free to start introducing alternate conceptions of simultaneity.
So this for Einstein was the great breakthrough. And so there are again two parts to it.
Concepts have to be anchored in experience. And when they're not anchored in experience, watch out.
You might be importing presumptions into your theory that are going to mess things up.
And that's exactly what happened. He says with earlier ideas about simultaneity, we were unable to see the compatibility of the principle of relativity and the light postulate
simply because we made a bad assumption about a concept.
And so that's now the full quote here. And let me read it to you. I'll read this one because this summarizes the whole story.
Here we go. All these efforts, that is to solve this problem, were condemned to failure as long as the axiom of the absolute character of time or simultaneity
was rooted unrecognized in the unconscious. To recognize clearly this axiom and its arbitrary character already implies the essentials of the solution to the problem.
And then he goes on to say that this is what he learned from reading human Mach.
Okay. Once you realize that what Einstein is saying that he's drawing from human Mach is a theory of concepts, an empiricist theory of concepts,
it's now very easy to go back and look at both the writing of Mach and Hume and see exactly what Einstein said in their theorizing.
Now, in the case of Mach, Einstein did it for us. When Mach died in 1916, Einstein wrote an obituary for Mach in which he described what he, Einstein,
regarded to be the essential ideas of Mach's philosophical system. I won't read you these quotes, but I hope you'll take my word for it.
The Mach that Einstein describes here espouses pretty much exactly the conceptual system, the theory of concepts that I just described for you.
And it concludes with a wonderful assertion that he thinks that had Mach been of young and fresh spirit, he might even have been able to hit upon the special theory of relativity himself had he thought about simultaneity.
Now, let's move on to Hume. What's interesting about Hume is this. The presumption you might have is that the work of Mach being closer to Einstein,
being in time, and also being more concerned with space and time, that that would surely have been the stronger influence.
But no, Einstein was very clear. It's the other way around. He says it was still much more Hume who had an influence on him. And here this comes from a letter Einstein wrote to Schlich in December of 1914.
Little confusion over exactly which of Hume's works were read, but I'll leave that for another time.
And we can do the same exercise of going through Hume's writings, and you find exactly this idea of the complete dependence of concepts on experience and the inapplicability of a concept if there's not a corresponding experience.
And finally, I want to mention something for philosophers here because those of you with philosophical backgrounds are now wondering, well, why Hume more than Mach? Why Hume more than Mach?
My conjecture is the following. Mach was more rigid than Hume. Mach really wanted to purge any concept out of science that did not have a proper grounding in experience.
Whereas Hume was willing to tolerate concepts that did not have a proper grounding in experience. If you want the obvious and clear example of that, for Hume, the idea of cause and effect or causation as an unseen hand that necessitates, that is not grounded in experience.
But he's willing to allow us to use that as a kind of habit of thought. I think Einstein found that more congenial than the rigidity of Mach.
And he has some fairly stern things to say about Mach and some fairly comfortable things to say about Hume. In particular, the other place that Hume is mentioned in Einstein's later recollections is with regard to the theory of causality.
Okay, so here's a summary of how Einstein used his theory of concepts in the development of special relativity. The basic claim is that concepts have to be grounded in experience, else they fail to represent the physically real and affictional.
He's getting that from Hume and Mach. But in addition, he then says that concepts without proper grounding, they don't need to be purged out of your theory entirely. You can leave them in as conventional elements.
Einstein introduces simultaneity by definition, as he says. It's the title of the first section of his paper. And no harm is done as long as you keep in mind that these are purely conventional elements of your theorizing that should have no prior authority over you.
And then finally, I make the point, of course, that this enabled Einstein's breakthrough. Okay.
Now let me move on to the second part of the talk. I want to say a little about Einstein's discovery of the general theory of relativity. Again, it was a process that took very many years, roughly from 1907 to 1915.
And in this case, we have a lot of information about the various stages that Einstein passed through, and that's going to prove very helpful to us.
For those of you who don't work in the physics, the general theory of relativity is Einstein's greatest discovery without any question. It's the one that towers over everything else that he did.
The basic idea is the following beautiful luminous idea, and that is, gravitation is to be understood as a curvature of the geometry of space time. So it's a pulling together of gravity, the falling of apples, and the geometrical curvature of the structure of space time.
It's a beautiful and wonderful idea, and I won't be explaining it to you at all. So let's move on. What I am interested in, though, is something that is, I think, easier to grasp at the first pass.
Einstein thought a lot about the methods that he used, and from looking at his papers and correspondence in the period that he was working on general relativity, you can see that Einstein himself conceived of there being two ways that you can advance to a physical theory.
And he described these, and we will describe them, as a physical approach and a formal approach. And it's the tension between those two approaches that I'm going to be tracing through over the next few minutes of speaking here.
Let me say a little bit about the physical approach first. The physical approach is the one that we think of in conjunction with the earlier Einstein.
The idea is that you advance to new physical theories by finding physical principles that have an evident empirical support. For example, the principle of relativity or the conservation of energy. All of these things are closely grounded in experiment and experience.
And the next thing that you do is you pick out certain limiting cases where you are very secure of how the physics goes. So Einstein knows what's going to happen in space and time if there's no gravity at all.
That's a clear case. Or if the gravitational field is very weak, then things must look Newtonian.
And then the extreme example of this would be the idea of physical naturalness, and that gets expressed most purely in Einstein's beautiful thought experiments.
These are thought experiments in which certain physical ideas are laid out for you in beautiful pictures. You see the thought experiment and everything clicks and you just get the idea.
That's one way of proceeding. That's the physical approach. But then there's a formal or mathematical approach. In the formal or the mathematical approach, you look at other things in the physical theories.
So one of the things that became very important for Einstein is how quantities change when you start changing the coordinate systems that are being used to evaluate them, or when you have group properties that I won't go into.
It then becomes an important tool of theory construction to start looking at the mathematical theorems that govern the transformation properties of these particular quantities.
And then finally, the correlate of this idea of physical naturalness is formal naturalness, and then the very strong technique, the analog, if you like, of the thought experiment is to pick out the equations that are somehow mathematically most simple.
And down the most simple equations is a way of getting to a physical theory. Now you can see these are two very different approaches. This one is very much based in the physical understanding of the world and physical instincts.
This one's very much based in a presumption that the mathematics is leading you in the right direction, always leading you in the right direction.
I recognize there's some overlap between these, and you can find that one or the other of these ideas is in some sense an expression in different words of similar ideas, but I'll pass over that.
Now in having laid that out, let me just mention that the idea of looking at these two different approaches is something that I worked on in collaboration with some colleagues of mine at the Max Planck Institute for the History of Science in Berlin.
And we produced this four-volume work, The Genesis of General Relativity. It's not cheap, but you get four very big volumes that will impress your friends.
But still, if you want to see vastly more that I'm going to say here, in vastly more detail than any reasonable person could ever want, then you'll want to get hold of these books.
And I don't think I get royalties. So let's now get back to Einstein. The story that we're going to see is a complete transition in the way that Einstein thought the methods of physics should run.
And I want to remind those of you who know a bit about Einstein, those of you who don't, I'll be telling you for the first time.
Early in his life, Einstein had a very clear disdain for mathematics. There are numerous very short anecdotes about how Einstein was disparaging of mathematicians. I'll just mention one of them.
In 1907 and 1908, Herman Minkowski took Einstein's special theory of relativity and cast it in a mathematically beautiful form of the four-dimensional formalism.
Einstein regarded this, as he said to, well, I can't remember who he wrote that to now. This was in correspondence fairly late on, 1911 or 1912.
He regarded that as superfluous learnedness. He thought it was simply unnecessary. All of his early work was done with very simple mathematical methods, just the calculus of many variables, nothing terribly sophisticated.
Now, it's important for us to nail down that this attitude was clearly in place quite late in the story. Remember, the discovery of general relativity spanned a period of 1907 to 1915.
Well, as late as 1912, something very interesting happened. Max Abraham produced an alternative theory of gravity, and he produced this alternative theory simply by writing down which were the mathematically most natural equations to write down if you wanted a relativistic theory of gravity.
If there are any physicists here, you'll instantly recognize that that's about as the simplest four-dimensional generalization that you can possibly get of Newton's theory of gravity.
The details of the theory don't matter to us. What matters to us is that Einstein really hated this theory. He started writing to all of his friends, and I just started collecting words from the various letters that he wrote to his friends,
telling them that the theory, this is to Ehrenfest, it's totally untenable, to Wien, totally unacceptable, to Lawrence, incorrect in every respect, and to his friend Sangha, totally intelligible. It's just this barrage of complaint.
And what's most interesting is the basis upon which he complains about the theory. He complains about the theory, as I give you in these two quotes, because it's based on the idea of writing down the mathematically simplest equations.
So he says, it, this theory, has been created out of thin air, i.e. out of nothing but considerations of mathematical beauty and is completely untenable.
Okay, I want you to just keep that in your mind. Einstein is damning something. He's saying, look, you've put a theory together out of nothing but considerations of mathematical beauty. Don't you know better than that, right?
This is the complaint of 1912. Remember that, because you're going to see things change a little. Well, the first moments of change began sometime late in the summer of 1912. Einstein moved to Zurich and began collaborating with Marcel Grossman.
Marcel Grossman was a mathematician and Einstein's university friend from Zurich as well. And Grossman started helping Einstein plow his way through a rather difficult and large mathematical literature.
And in particular, it was Grossman who pointed Einstein towards the calculus of Ricci and Levicivita, which has the modern mathematical methods that are used in general relativity.
And in it, you find the mathematical structures out of which the general theory of relativity is built. In particular, at some point, we're sure of this, Marcel Grossman would have said to Einstein, this is the thing you need to work with.
This is the Riemann curvature tensor. It's this enormous four index quantity, second order derivatives, nonlinear. It's just a nightmare for a physicist to want to even look at.
But we know there was a moment when Marcel Grossman must have done this. We can only imagine Einstein's reaction. Now, it's in the middle of this that Einstein now starts to make concessions.
He talks about all the difficulties he's working with and he's saying, one thing is certain, never before in my life have I toiled anywhere near as much and I have gained enormous respect for mathematicians whose more subtle parts I considered until now in my ignorance as pure luxury.
Compared with this problem, the original theory of relativity is child's play. So we see Einstein's weakening. He's now starting to develop respect.
It all leads up to this paper here. It's a two-part paper written with Grossman in 1913. It's an outline or a sketch of a generalized theory of relativity and a theory of gravitation.
It, in fact, has the complete theory of general relativity in there. The entire theory is there, the complete structure. It's all there except one piece.
And the one piece that's missing is the most important part. And these are the so-called Einstein gravitational field equations. If you work in physics, you know what that is and you know that not having that is bad.
If you don't work in the physics, the analog of what just happened is this. It's imagine Newton creating the complete Principia with all the theorems and everything intact but not having an inverse square law of gravity.
It's trouble. Whatever else he's going to put in there, it's just not going to work. That was the situation that Einstein was in.
Now, it was interesting. It's not that Einstein didn't think about the Einstein equations. It's not that he didn't consider something pretty close to them.
In Grossman's mathematical part of the paper, the standard set-up was approached. I mentioned to you the Riemann-Christoffel-Kervich attention.
This is the basis out of which general relativity is built. And there it is. And then they take the first steps that are needed. They do a first order contraction.
They do just the first contraction down to a two index symbol. All the steps are being taken as you would expect. And then there's a sudden disaster.
Then Einstein just announces, or Einstein and Grossman just announce that it doesn't work. You don't get Newtonian theory out in the limit.
Now, if you've worked on general relativity, this has to be a jaw-dropping moment because here's Einstein with the complete structure of general relativity then telling you that the Einstein equations are the wrong equations.
I'm surprised that this could happen. Then it gets worse. Then Einstein and Grossman set them aside and cook up field equations for the theory that have very bad formal properties and then publish the paper.
It's not a good thing that's happened. And of course, one of the big puzzles for us in history of relativity is to try and figure out what on earth happened.
How could this happen? That Einstein could take such a devastating wrong turn at the very last moment.
Well, the good news is that we can solve the problem almost completely because it turns out that the notebook of calculations that Einstein kept during this period has survived.
And it's possible for us to go through this notebook of calculations step by step and understand what happened.
I flipped it over because this is one side of it and it says notes for lectures on relativity. It's mislabeled. They aren't notes for lectures. This was the way it was kept in the archive.
Here I think you can see faintly the words relativity. So Einstein worked from both ends in the book.
And I'll tell you in brief what the story will be. No need to keep you in suspense.
When you go through the book, what you discover is Einstein using the two methods that I described.
This physical approach, the mathematical approach and his hope, his expectation was the two approaches would meet and give him the same result at the end.
But unfortunately, when he started working through the material, it didn't go that way.
The physical approach led one way and the mathematical approach led another.
And he had to make a choice. Which way do you side? Do you side with the physical thinking or do you side with the mathematical thinking?
He had to choose and he chose for the physics. And it turned out to be the wrong choice.
And he visited some two to three years of misery upon himself while he tried to figure out what went wrong.
So that's the story that's coming. I'll show you one or two pages of the notebook, but I don't think there's time for me to linger here because it's getting on.
This is an amusing page. This is the inside cover. I don't know if anyone can recognize what this is.
I'll give you a hint. This is a puzzle in recreational mathematics to do with rearranging geometrical figures.
What you can do is you can cut up a square and reassemble the pieces and discover it has a different area than if you've ever seen these.
I'm expecting these are puzzles for children. This one here is a standard puzzle from the time about shunting railroad cars around on railroad tracks.
And we've figured out these two. I have no idea what this one is. And if anyone has any idea, think recreational mathematics as to exactly what this thing could be.
The rest of it's more serious. Very briefly, this is Einstein writing down the metric of space time for the very first time.
We know that because he's using a non-standard notation that he never used again.
This is very early. He's getting this notation out of Lauer's textbook. And then now he starts working on stuff. I'll pass over this.
I've put a lot of this up on my website, by the way, in what I call a goodies page where you can get a lot of details on the notebook and unpack exactly what Einstein's doing there.
I want to move on to this page because on this particular page, once you know how to read it, you can see Einstein using the two approaches and getting exactly the result that he wants to get out.
So how does this work? The heading here is gravitation. And Einstein is interested in recovering the law of conservation of energy.
How do you get the law of conservation of energy that will live within the theory?
The first part of the page is to go through the physical calculation.
What he does is he takes something he knows well, and that is what a speck of dust will do if left to go in freefall in a gravitational field.
And then he says, okay, I'm going to now imagine that I have many specks of dust forming a cloud, and I then know how that entire cloud will move.
And then out of that, I can read off the general law of the conservation of energy.
And now he goes through that through various steps. I won't give you the details of that.
All I'll report for you is this particular result. This is an equation that describes the law of conservation of energy for a dust cloud moving freely in gravity.
If anyone does any physics here and you know some of the older notation, this is simply the vanishing of the covariant divergence of the stress-energy tensor of matter.
So it's a familiar result written in very unfamiliar ways.
Okay, that's been the physical pathway. He's got one half of the story.
Question, is this now a reasonable mathematical result?
And the way that Einstein tests that is interesting. He says, if this is a reasonable mathematical result, I should be able to break up that equation into two pieces.
One of them is a mathematical structure here that represents energy, and another is a differential operator that eats it up and acts on it in an interesting way.
So Einstein then needs to check whether the equation so broken up has given him a general operator that has nice mathematical properties.
And that's what he proceeds to do in the next step. He takes that operator and he throws it at the only other quantity he has readily at hand, which is the metric tensor.
He expects this thing to either be a zero or a four vector. He then does the calculation and out comes the zero that he expects, a familiar result for anyone who does relativity.
What I like very much about this is that there can be no doubt that Einstein's satisfied with the result, because having finished the calculation all the way down here, you could imagine him saying to himself, yes, it works.
But of course he's German and so this is what he says.
If you can read this here, this is written right down at the bottom of the page here. So that's the way things should go.
You use simple physical ideas to build up a law, you then have the idea of mathematical naturalness, you bring that in, everything should match up beautifully.
So all that's left is to do the same exercise for the thing that will be the Einstein field equations, the analog of the inverse square law of gravity.
And here it starts. I mentioned to you earlier on that there must have been a moment when Grossman said to Einstein, here, you need this, the Riemann-Christoffel curvature tensor.
Well, here's that moment. It's right here. Einstein writes down for the first time the Riemann-Christoffel curvature tensor, the first time in the notebook.
Do we know that he's getting this from Grossman and that Grossman's telling him what it is?
Well, there are very few words in this notebook, but those are the words right there.
That says Grossman tensor, fierter, mannishfalischkeit, Grossman tensor of the fourth rank.
So this is the moment when Einstein now starts to work on the Riemann curvature tensor.
This is the moment when it all should have finished, should have gone very, very fast.
Einstein then goes ahead, contracts this to produce what we call the Ritchie tensor, and starts calculating, and then runs into a problem.
There are three terms here that Einstein has to make go away if he wants Newton's theory to come out at the end, if he wants the appropriate simple physical limit to be realized.
And you can see here, he says that they should vanish, but they don't vanish.
He has Zolder for Schwinden, but they don't.
Well, I don't want to go through what happens next, but over many more pages Einstein keeps struggling and struggling.
He keeps throwing things around and playing around with things.
He simply can't get this Riemann Christoffel curvature tensor to spit out the equations that he needs.
He starts doing very long, very tortured calculations, and then it all comes to an end here on this particular page,
where he writes rather, rather sadly, Spezialfar Varsheinlich und Ritchie.
The special case that is of his earlier theory, the expected to get back from this, is apparently incorrect.
So there was now a direct collision between following the natural mathematical pathway and following the pathway mapped out by the physical reasoning.
And there's more in the notebook, but at this point it's really all over.
It's not going to work anymore. And then eventually Einstein gives the, this is a summary of the derivation of the equations that he gives in the 1913 paper.
It's quite easy to map this back onto what you find in that particular paper.
Okay, so it's all over. Einstein's deflated and exhausted.
This is Einstein writing to his friend Besso in 1914.
The reason I've put this quote up here is you can see clearly Einstein reflecting on what just happened at the level of methodology.
He says, the general theory of invariance was only an impediment.
In other words, a mathematical theory was only an impediment.
He wasted a huge amount of time calculating things he then thought he didn't need to calculate.
The direct route, in other words, plain old-fashioned physical thinking proved to be the only feasible one.
This was 1914. Now, what you're looking at is something like the equivalent of the Titanic after it's hit the iceberg.
It's going to sink, right? It isn't obvious that it's going to sink.
It is listing a little, but it is going to sink.
And this is what's happening to Einstein's theory.
And over the next year or two, Einstein gets more and more dissatisfied, more and more unhappy with what's in his theory.
And then finally, it all comes to a head in the fall of 1915.
I won't go through the details, through the clues that Einstein found that revealed to him that his theory was just wrong.
But what did happen was the following.
At the beginning of 1914, he reported a new set of equations to the Prussian Academy.
He says very clearly that he gave up on the old approach.
And he went back to the Riemann-Kirvacher tensor and started using it to write down his equations.
And then in rapid succession, one communication each week, he gradually fixed up the mistakes that he'd made earlier on in November of 1916,
communicating the Einstein equations as we know them.
It was in the middle of all of this that Einstein had the greatest scientific moment of his life.
In the middle of this, even before he'd finished getting to the final Einstein equations,
he discovered that his new theory could account for one of the outstanding anomalies of astronomy.
And this is the odd motion of Mercury.
Mercury had this recalcitrant advance of its perihelion,
a very slight discrepancy that was simply not yielding to any of the normal methods of astronomy.
There ought to have been another planet perturbing it, but when you looked at where the planet should be, it wasn't there.
This was the moment that I think changed Einstein more than any other moment in his life.
There are numerous reports to friends and colleagues about that moment.
He says he had heart palpitations. He says he was so excited he couldn't sleep.
He says he felt as though something inside of him snapped when he saw that calculation come out right.
So this was the great moment for Einstein, and there was tremendous urgency here as well.
Because in Göttingen, the previous summer, he'd gone and given a talk on his theory.
The great mathematician David Hilbert had learned the theory from Einstein
and was starting to write down gravitational field equations and was starting to get very close as well.
So Einstein really had to move things along here.
Anyway, this was over. Einstein pulled it all together.
He has his great success, the greatest success of his scientific life.
And now he's starting to look back and think, you know, I could have had this three years ago.
I had already taken into consideration the only possible generally covariate equations
which now proved to be the right ones three years ago.
Why didn't he accept them? Only with a heavy heart did we detach ourselves from them
since the physical discussion had apparently shown the incompatibility with Newton's law.
It was the physical discussion had drawn him astray.
He could have gotten there three years ago
if only he had not taken the physics so seriously, if only he had taken the mathematics more seriously.
He says here, this time the most obvious was correct.
Where most obvious is now, of course, the mathematical pathway.
And then listen to this. This is now, now, you've got to remember
the Einstein of 1912 complaining about mathematical simplicity as a guide.
The Einstein who was disparaging to Max Abraham for his theory.
And now this is Einstein writing.
Hardly anyone who has understood it can resist the charm of this theory.
It signifies a real triumph of the method of the general differential calculus
founded by Gauss, Riemann, Christoffel, Ritchie and Levitjevita.
So now suddenly Einstein is on the other side and he's,
and he's touting the idea of mathematical simplicity.
Now this was a transformative moment for Einstein.
It began a complete transition in the way that he did physics.
This idea of tracking down the mathematically simplest equations
as the right equations flowed from this.
There's a very beautiful statement of all of this that comes later on.
And this is a lecture that Einstein gave in Oxford June 10th of 1933, the Herbert Spencer lecture.
And I'll just quote from you. It's a rather long lecture.
But I'll just quote from you the central part because it really gives you the idea in a nutshell.
Einstein starts out by asking the following question.
How can we know which is the right way to go?
In other words, which is the right theory to pick?
When you can get many theories that do justice to experience well enough.
You can get many theories and they all work near enough.
How do you know which is the right one? Is there a way?
And he answers, yes, without hesitation there is in my opinion a right way.
And then he starts out with this wonderful sentence.
Our experience hitherto.
Now when Einstein says our experience hitherto,
I don't think he's just meaning our communal experience.
I think he has in mind in particular that experience that he had in 1915.
It's real experience. It's the experience of a working physicist.
Our experience hitherto justifies us in believing that nature is the realization
of the simplest conceivable mathematical ideas.
I am convinced that we can discover by means of purely mathematical constructions
the concepts and laws connecting them with each other
which furnish the key to the understanding of natural phenomena.
Now those of you who know some philosophy will recognize this as out and out Platonism.
This is a straightforward Platonism.
You consult Plato's heavens to find the true laws.
Does Einstein realize the heritage he simply brought into?
I think he very clearly does. Here's the next paragraph.
I'll just read the last sentence.
First he's very clear. Experience has an important role.
But he says, the creative principle resides in mathematics.
And then comes the clincher.
In a certain sense therefore I hold it true that pure thought can grasp reality as the ancients dreamed.
I think this is an amazing reversal for Einstein.
The Einstein who was disparaging of mathematics
and now regards this as the royal road to the truths of nature.
This of course became the foundation of his work on the unified field theory
to find the mathematically simplest laws.
So let me wind up.
I could give you a more elaborate conclusion
but I think Einstein can give you a nice conclusion here.
Those of you who work in philosophy might see familiar echoes here.
The reciprocal relationship of epistemology in science is of a noteworthy kind.
They are dependent upon each other.
Epistemology without contact with science becomes an empty scheme.
Science without epistemology is in so far as it is thinkable at all primitive and muddled.
Beautiful, beautiful synthesis of ideas there that really describe the way that he did his science.
So if you want to read more about this, I keep a pretty rich and elaborate website.
You can go there and sort of click on things and have a bit of fun.
I've got a section on history of relativity and I've got some popular writing there I try.
I try and tease out things that aren't easily understood.
Further down, if you scroll down here, you come to a nice presentation on the Zurich notebook.
More importantly, I've put together a page specifically for this lecture series
and it has this particular talk in it.
You'll get the PowerPoint that I spoke to and you can download papers that explain more of what I was talking about.
So here's one paper you can get on the story of the relationship between Henmark and Einstein.
And then the story behind this wonderful quote,
Nature is the realization of the simplest conceivable mathematical ideas.
And again, you'll find that paper there.
And that's my story. Thank you very much.