ASTR 160: Frontiers and Controversies in Astrophysics

Lecture 12

 - Stellar Mass Black Holes


One last key concept in Special Relativity is introduced before discussion turns again to black celestial bodies (black holes in particular) that manifest the relativistic effects students have learned about in the previous lectures. The new concept deals with describing events in a coordinate system of space and time. A mathematical explanation is given for how space and time reverse inside the Schwarzschild radius through sign changes in the metric. Evidence for General Relativity is offered from astronomical objects. The predicted presence and subsequent discovery of Neptune as proof of General Relativity are discussed, and stellar mass black holes are introduced.

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Frontiers and Controversies in Astrophysics

ASTR 160 - Lecture 12 - Stellar Mass Black Holes

Chapter 1. Invariance in Special Relativity [00:00:00]

Professor Charles Bailyn: Okay, here’s the plan for today. I want to do one last foray into relativity theory. And this is going to be a tricky one, so I hope you’re all feeling mentally strong this morning. If not, we–gosh, we should have ordered coffee for everyone. And, in so doing, I want to introduce one key concept, and also answer at least three of the questions that you guys have asked before in a more–in more depth, and also relate the whole thing back to black holes. And then, having done that, we’ll have some more questions. And then, having done that, I want to get back to astronomy; that is to say, to things in the sky that actually manifest these relativistic effects. So, that’s where we’re going today. And along the way, as I said, we’ll deal with some of the questions you’ve been asking in a deeper kind of way. In particular–so, questions.

Watch out for the answers to these questions. Somebody asked, “What’s special about special relativity, and what’s general about general relativity?” How do they relate? So, we’ll come back to that one. Somebody also asked, “Why use the speed of light to convert time into space and vice versa, to get them in the same coordinate system?” So, why use cto convert time to space and vice versa? And then, also, there was the question of, you know, “What is the mathematical formulation of general relativity?” So, how to express general relativity in some kind of equation. And we’ll get to the key equation, which is something called a metric, for general relativity, and then we’re going to stop. Because to go on from there is fairly heavy calculus and we’re just not going to do that. But I want to get at least that far.

Okay, so let’s go back to special relativity for a minute. So, special relativity. Flat space-time, no gravity. And you’ll recall what happens. As you get close to the speed of light, all sorts of things that you thought were kind of constant and properties of objects, like mass and length and duration, and duration of time, and things like that, all start to get weird and change. So, length, time, mass, all these things, vary with the velocity of the person doing the measuring.

And so, you could ask the question, is there anything that doesn’t vary? Is there anything that’s an invariant? And the answer is, yes. There are some things that don’t vary. So, some things are invariant. And Einstein actually said later in his career that it’s actually the invariants that are important, not the things that change. And so, he should have called his theory invariant theory instead of relativity theory. Think of what that would have done to pop philosophy. Instead of saying, “everything is relative,” all this stuff, you would have had the exact same theory. You would have called it invariance theory. And the pop philosophy interpretation of this would be, “some things never change.” And it would have been a whole different concept in three in the morning dorm room conversations.

Okay, so some things are invariant, what things? Now, let me first give you a little bit of a metaphor and then come back to how this really works in space-time. Supposing you’re just looking at an xy-coordinate system and you have two points in a two-dimensional space. So, here’s a point and here’s a point. Now, if you arrange for some kind of coordinate system–here’s a coordinate system. This is x, this is y–and you ask how far apart these points are. Well, you can do that–let’s see. They’re separated in x by this amount here, which we’ll call delta [Δ] x. And they’re separated in y by this amount here, and that’s Δ y. But of course, those quantities depend on the orientation of your coordinate system. If I now take this coordinate system and I shift it like this, now it’s going to be totally different. Now I’m going to have x look like this and I’m going to have y, Δ y look like that. So Δ y has gotten a whole lot smaller. Δ x has gotten a whole lot bigger. And all I did was twist the coordinate system. Yeah?

Student: [Inaudible.]

Professor Charles Bailyn: You still get the same distance, thank you very much, that’s exactly right. The distance is the invariant. The x-coordinate and the y-coordinate, those vary with the coordinate system, but the distance is the same. That’s exactly the point. And so, the quantity–let me summarize this.

For points on a 2-D space, Δ x varies. Δ y varies. But there is a quantity that is invariant and that is–well, let’s call it (Δ x2) + (Δ y2), which is the distance squared, (Δ D2). And this is invariant. It doesn’t matter which way you twist things around, that will–that quantity will remain the same.

So, now, imagine that you’ve got events in space-time. So an event in space-time has three spatial coordinates and one time coordinate. So, it’s basically a point in a four-dimensional space. And as your velocity changes, the distance and time also change. That’s the equivalent of rotating the coordinate system.

But there is something that doesn’t change, and let me write that down. This is usually given the Greek letter Tau [T] squared. And this is equal to (Δ x2) + (Δ y2) + (Δ Z2) - c2 (ΔT)2. And this is invariant. This is an invariant interval, sometimes called proper distance. And as you change your velocity–as the space, as the mass, as the time all change–this quantity, describing the separation of two events–so, this describes the separation of two events, that quantity changes–doesn’t change. That quantity is invariant.

Okay, so now, this answers one of the questions that was asked before. Why does one use c2 or c to transform the space coordinate into the time coordinate and back? It’s because you need the c2 out here in order to make this invariant. If you’re calculating the distance, if you use x2 plus ½ y2 or some other constant times y2, you’re not going to get something that’s invariant. And it’s only when you use the c, here, that you end up with something that’s invariant.

And so, if you think about these as representing the four coordinates of the system, it’s clear this one coordinate is x, one is y, one is z, just as you would expect. And then, there’s this other coordinate, which is c times T, but it’s negative so it has to be times the square root of -1. So, the four coordinates in space-time can be thought of as x, y, Z and i cT, if you want to think of it that way. And the time coordinate is imaginary, because when you square it, you have to end up with a negative number. Don’t worry about the details of that. But the presence of the c2 here is why you have to use c, in particular, to get from time to space and back. And that’s necessary because this is the thing that doesn’t vary with velocity.

Chapter 2. Invariant Intervals and the Schwarzschild Metric [00:10:10]

All right. So, this is actually kind of a weird expression. Because unlike the distance between two points–and you’ll notice, these three terms put together, that’s just the distance squared, ordinarily, but that is not invariant anymore, because there’s this other term here, which can vary. Unlike the distance, this doesn’t have to be positive. You’ve got three different cases here. This interval can be zero for different points. In ordinary distance it can only be zero if the two points are the same, but this can be zero for different points, for different events. It can be zero, it can be negative, and it can be positive. So, what does that mean? What happens when it’s zero?

So, if the interval is zero, that means that the distance between the events in light-years, for example, is equal to the time separation in years, because that’s–because this term has to be exactly equal to that term there, in order for them to subtract out and get zero. And the c2 converts from light-years to years and back again.

And so, what does that mean? That means if you emit a photon at one event, that same photon can, if it’s going in the right direction, be present at the second event. So, if you ride along with light you’ll see both–you’ll participate in both these events. So, you sit at event one. You flash a light. You ride along with the expanding light waves from that event and you get to something one light-year away, exactly a year later. And so, if the second event is one light-year away in distance and a year later in time, that same photon will be present at the second event, as at the first event. So, things that have one of these intervals of zero are separated by an appropriate amount so that the same ray of light can participate in both of them. So, if the interval–let’s keep that up there for a minute.

If the interval is negative, what does that mean? The distance is less than the light travel time. So, the photon is already past the second event. So, if you were to emit a ray of light at event number one, it would have passed the second event by the time it occurred. The photon has already gone by. And similarly, if the interval is positive, then the light photon hasn’t reached event two–hasn’t yet reached event two. Now, this is important, because this means that you can’t communicate from event one to event two. So, if you’re at event two, you don’t know what happened at event one. Because even if you’d sent out a signal, a radio signal or whatever, it would not have reached you by the time event two takes place. So you can’t communicate from event one to event two. And similarly, you can’t travel from event one and reach event two, because you’d have to go faster than the speed of light to do it. These kinds of intervals, these negative intervals, these are called time-like, because the time term is larger than the distance turn. And these kinds of intervals are called space-like intervals. And you can only travel or communicate over time-like intervals. Yes?

Student: What are these so-called events?

Professor Charles Bailyn: So, they’re events–you can think of them as points in space-time. So, they have a particular position in space and a particular point in time. So, they can be described by four numbers, three spatial coordinates and a time coordinate. They can be anything. You know, turning on a light, doing anything you want to do. Receiving a photon, whatever it is. But they are points in a four-dimensional space-time and therefore require four numbers to describe them. And you can only get from one to another if they’re separated from a time-like, that is to say, a negative, interval.

Okay, so this expression, which I’ll write down again, this is called a metric. And the particular metric that I’ve written down here is the metric for flat space. Because remember, this is special relativity. There’s no masses, no curvature of space, none of that stuff, yet. This is the metric for flat space with no mass present.

And there are many other metrics possible. Any time you add mass or do other things, you get different kinds of metrics more complicated than this. So, what’s special about special relativity is that you use the metric appropriate for flat space as opposed to the many other different kinds of metrics that you can use in general relativity, which has a much more general form for the metric.

I should say, you can also write this down. You can write down the spatial terms here in polar coordinates. Remember polar coordinates? Polar coordinates, you describe the position in space, instead of with x, y, Z, you describe it with a radius, a distance from zero, and some angles. And it turns out, that’s convenient to do so. Let me write this down in polar coordinates, or in polar. That’s an r. Let me write that explicitly. This is an Omega, that’s some angle. And then the Tthing remains the same. And I’ve pulled a little bit of a notational fast one on you here. I’ve gone away from the deltas and I’ve written these down as d. This is the differential d. Those of you who have taken some calculus will remember this. If this were a calculus-based course I would explain why I did that, but I’m not going to. So, just allow me this slight of hand, here. For technical reasons, these have to be differential. Yes?

Student: But you do need a second angle term for the [inaudible]

Professor Charles Bailyn: I do need a second angle term. I should say–good point. You need two angles and a distance in three space. This capital Omega here is actually–Omega squared is actually sin θ, d ?, d ?, which is the correct form. And so you could write out both terms here, but in fact, this one isn’t going to change. But you’re absolutely right. In principle, you need two angles.

Okay, why have I done this? Excellent question. I ask myself a question at this point. Where am I going? What I want to do now is write down a different metric. A metric that actually involves curved space and the presence of a mass. And this is something called the Schwarzschild metric. Remember Schwarzschild? He had a radius. And this is the appropriate metric for the presence of a single point mass at the center of the coordinate system, at R = 0. That’s why I put it into polar coordinates, because the presence of the mass is going to change the space-time as a function of distance–from radial distance from where the mass is. And so, it’s much more convenient for the Schwarzschild metric to use this in polar coordinates.

So, here’s the Schwarzschild metric, (d T2), that’s the–this is the interval, is equal to (dR) / (1 - Rs / R). So, that’s just like the flat term, except with something in the denominator there. Plus R2d Omega squared, that’s just like the flat metric. And then the–whoops this had better be - c2 (1 - RS / R) (cT2. Where RS is the Schwarzschild radius, which we’ve had before, which is 2GM / c2.

Chapter 3. Schwarzschild Sign Changes and Space-Time Reversals [00:21:01]

Okay, so this is just like the flat metric with two exceptions. It’s got a term in the radial part of this 1 - RS / R. And it’s got that same term, but this time in the numerator, in the time term here. Now, what do you do with this–with such an equation? Well, we’ve done–in special relativity, we’ve dealt with these kinds of things. What you do is you start taking the limiting cases. You say, okay, what happens when it’s getting really close to flat on the one hand, and what’s happening when it’s getting very seriously relativistic on the other hand? So let’s do that.

If RS / R goes to zero, then the metric turns into the flat metric. Because if RS / R = 0, this term disappears because it’s 1 - 0, and it just cancels. This term disappears and you recover the flat metric. This happens in two cases. If the mass goes to zero, then RS goes to zero, and you recover the flat metric. Or if R gets really big, then RS / R goes to zero, and again, you recover the flat metric. So in–there are two situations where Schwarzschild metric blends smoothly into the ordinary flat space. One is if the mass is zero, that’s not surprising. If the mass is zero then space-time isn’t curved. Or alternatively, if you’re really, really far away from the mass. If R is much, much bigger than the Schwarzschild radius, you’re way out there. There’s no gravitational effect. Space-time remains flat. So, these the–this is the limiting case where you recover special relativity.

Now, the other case is when R gets close to the Schwarzschild radius and approaches it. So then 1 - RS / R approaches zero, because these two are going to get closer and closer together. 1 - 1 = 0. What happens then? So, this is now, first of all–in physical terms you’re getting really close to the Schwarzschild radius. So now, what happens to the metric if you do that? The dR term gets very big, because it’s got a zero in the denominator. The dT term gets really small, because it’s got that thing that’s going to zero in the numerator.

Fine. What does that mean? Well, remember, this is the negative term. This is the positive term. So the positive term is getting really, really big. The negative term is getting really, really small. And that means that all intervals are gradually becoming space-like. What do I mean by that? Well, the negative term is getting small. One of the positive terms is getting big. So the sum of those tends to be positive. It’s becoming more and more positive. Positive intervals are these space-like intervals, and you can’t communicate or travel across space-like intervals. When you get all the way to the Schwarzschild radius, this blows up completely, becomes infinite. This becomes zero, and there are no time-like intervals.

There are no time-like intervals that cross the event horizon. That’s why you can’t get out. This takes us back to the basic principle of black holes.

So, cannot communicate or travel over space-like intervals. And so, you can’t cross R equals the Schwarzschild radius. All right. Let’s see. Let me write the thing down again, here, for you. Okay, so that’s the metric we’re worrying about here.

And now, let’s think about what happens inside the Schwarzschild radius. R less than RSchwarzschild. That means the dRterm becomes negative, because 1 - RS / R. If RS is bigger than R then this term is–this term is greater than 1, and this whole thing is less than zero, and the signs change. And the dT term becomes positive. So, that means this is the time-like term, where this one is the space-like term, because it’s positive.

That’s what I meant three, four, five lectures ago, when I said that inside the Schwarzschild radius, when you’re inside the Schwarzschild radius, space and time reverse. It’s a sign change in the metric. That’s what it means. And you can only travel along negative intervals. That means you have to move in R the same way outside the Schwarzschild radius you have to move in T. But notice it’s only the radial term. This term hasn’t changed. You could go around in circles, but whatever you do, you still have to move, as it turns out, toward the center of the thing in radius, in order to have a time-like interval. And so, motion in R is required for inside the Schwarzschild radius, whereas motion in T is required outside. So space and time reverse.

All of which is very nice, but I’ve left out something–I’ve left something out, which is the fact–inside the event horizon, how do you know that this is still the metric? One could invent some function that looks just like this outside the Schwarzschild radius, but then looks like something else inside the Schwarzschild radius. And because no communication across the Schwarzschild radius is possible, you’d never be able to test it. And so, this is how one gets away with doing non-testable physics. You say, well, we’re just going to assume that the metric hasn’t changed. Why should it change? After all, it’s the same equation. But inside the Schwarzschild radius you can’t actually test this.

Outside the Schwarzschild radius you can test it, because you see whether objects behave as they ought to behave in a space that’s curved in this particular way–in a space-time that’s curved in this particular way. And so, this is what I meant by, five classes ago, by saying space and time reverse. These two quantities reverse their signs.

All right, that’s as far as we can go, because the next thing that one would want to do is, you find out what the equation is for finding out how things move in these curved space-times. Basically, you remember, things go from one event to another in the shortest possible path, that’s the equivalent of a straight line. That means if you integrate over dT, it has to be minimized. So, you minimize this integral. That tells you how things move. We’re not going to do that. Sighs of relief? And because–for obvious reasons. So, this is as far as we can go, just to write down the metric here. So, let me know pause for questions, and then we’re going to go back and talk about astronomy–about things in the sky that actually exhibit these relativistic behaviors. Yes?

Student: You were talking before about intervals, and how all the intervals are negative. What exactly is one interval? [Inaudible]

Professor Charles Bailyn: Sorry?

Student: What exactly is one interval? [Inaudible]

Professor Charles Bailyn: Oh, an interval. So, what I’m doing is I’m taking two events, each of which is one of these points in space-time, and I’m asking, “What is the interval between them?” What is–you could measure the distance between them, you could measure the time from one event to another. But as it turns out, those aren’t invariants. And so, there’s this other thing, the metric, which is invariant. And so, that’s a measure–an invariant measure of how separated these two events are. So, you take two events and you ask yourself, “Are they separated by zero, a positive quantity or a negative quantity?” Where by separation, I mean, this curious combination of space and time.

Student: So the intervals are before the metric, before the interval?

Professor Charles Bailyn: Yeah, it’s an interval–think of–let me go back to the analogy I started with. Here’s–in two spatial dimensions, x and x, here are two points. And depending on how I set the coordinate system up, the x-distance–the x difference between them and the y difference between them can change, but the distance is always the same. So now, I’ve got two points with–each with and x, y, Z and a T. And depending on how I change my velocity or my coordinates the particular values of x, yZ, and T can change, but this (Δ T2) defined by–I’m in flat space now, right? This separation, this interval between those two points, this is the invariant–in the same way that the distance between two points doesn’t change if you change the coordinate system, even though the x and y separations do.

Student: [Inaudible]

Professor Charles Bailyn: It gives you–no. Well, it combines these four things into one thing that doesn’t change. That’s the point. Yes?

Student: Is there a way that you can–like as an interval from zero basically describes two events that appear simultaneous?

Professor Charles Bailyn: As the–yeah exactly.

Student: So, is that then collapsed into a Newtonian theory, things with two–things appear simultaneously if they happen at the same time –

Professor Charles Bailyn: Well okay. So, there’s two different ways things can appear, quote, simultaneously. One is if they are two different events in space-time and light travels from one, and they’re exactly separated by–the amount of time between them is the same as the distance between them if you multiply by c2. They can also appear simultaneously if they are the same point as each other. And then everything goes to zero. And it’s only in that second case that it–that the Newtonian concept of simultaneous kicks in. Simultaneous is usually taken to mean that the time separation is zero. Two things happen simultaneously when they happen at the same time.

Student: [Inaudible]

Professor Charles Bailyn: On Earth the–the distances and the velocities and the gravitational fields are never so strong that you have any trouble–that the Δ T changes significantly depending on what your point of view is. So, in our everyday life, we have a strong concept of simultaneity. It’s two things that happen at the same time. Turns out, though, that if you move at close to the speed–if you observe two events to be at the same time and I’m moving at close to the speed of light, I don’t observe those two events at the same time, even though you do. And so, at that point, you have to abandon the Newtonian concept that Δ T = 0 tells you that two events are simultaneous. And the whole concept of simultaneity takes on a different task. Other questions, yes?

Student: [Inaudible]

Professor Charles Bailyn: Okay, so these units can be any units of length you like provided that–any units of length you like, provided the time units are related to it by c. That is to say, if your distance units are light-years, your time units have to be years. If your distance units are meters, then your time units are some kind of meter, light-meter-second thing. And so, the only restriction on the units you use is that the time and the space units have to be convertible into each other through c2.

Or, alternatively, another way of saying it is, you use any units you like, and as long as you express the speed of light in those units. If you have a time unit and a space unit, if you’re in–if you’re measuring your space in furlongs and your time in fortnights, as long as your c is in furlongs per fortnight, it’s going to come out okay. So, as long as it’s convertible. Other questions?

Okay, if you don’t get all the details and nuance of what I’ve said this period, don’t worry too much about it. I just wanted to get the concept of the metric out there and show you how, if you look at that equation these concepts of space and time reversing, and so forth, have a kind of mathematical consequence, as well as just spouting words. And if you get, sort of, the basic outline of the argument, that’s fine.

Chapter 4. Evidence for General Relativity in Astronomy [00:36:27]

Okay, back to actual things in actual–that actually exist. So, what I want to talk about now is evidence for general relativity from astronomical objects–real black holes, stuff like that. Now, one of the curious things about this is that when Einstein thought all this stuff up, he thought it up from basically these philosophical concerns about mass ‒ that the inertial mass turned out to always be equal to the gravitational mass. Why would that be? And there wasn’t, when he thought it up, a great body of evidence for his theory in the real world.

This is in contrast to special relativity. Special relativity, there were all these experiments that needed to be explained. General relativity–very, very little. In fact, when Einstein first put forward the theory in 1917, there was only one thing that had ever been observed that actually showed an effect of general relativity, and that was the orbit of Mercury, which you’re reading about for this week’s problem set.

So, just going back a little bit, in the nineteenth century, people had observed the orbits of planets in great detail. And they found out that two of the planets were moving in ways they couldn’t quite explain. There were very small deviations from the predictions orbit. In particular, the orbit of Uranus was a little weird. And that was quickly explained by the presence of an unknown–hitherto unknown planet, which was also exerting a gravitational force on Uranus and pulling it out of the orbit that it should have been, by a very small amount, because the gravitational force of another planet is very small compared to that of the Sun. But by the middle of the nineteenth century they could measure such things. And they therefore predicted the presence of this other planet, of Neptune, and they calculated where it should be. And some guy went off and observed in that spot and found it–predicted presence of Neptune and discovered it in the predicted place. Big triumph! Everybody–if they had had Nobel Prizes back then, they would have won it for this, for sure. And then, there was a whole big kerfuffle because they couldn’t decide whether the French guy had done it before the English guy, or vice versa. And they argued with each other for decades about who gets the credit. But in scientific terms, there was a prediction, and the prediction was verified. Excellent news.

Now, there was also a problem with the orbit of Mercury–also perturbed, from what you would expect. And having had this big triumph in the Outer Solar System, they figured, well, we know how to deal with this. There’s got to be another planet in there. So, they predict the presence of a planet called Vulcan, which then disappears from the scientific literature until it’s resurrected by Star Trek. But Vulcan–the concept of Vulcan was, this was going to be a planet that’s closer to the Sun than Mercury. That’s why they haven’t been able to find it, because it’s too near the Sun to be easily observed. And it’s going to pull on Mercury in such a way that it’s going to explain the problems with the orbit of Mercury. And so, they then look for Vulcan in the predicted place, and they find it. And then somebody else finds it. And they find it many times and each time it’s different–all different.

And it gradually becomes clear that everybody’s fooling themselves. That there’s no–this is a really hard observation to make, right? Because the thing is right near to the Sun. And so, it turns out that all of this is wrong. None of these observations are really any good. It’s not repeatable–so, not really.

And so, after some attempts to find Vulcan–and then, they rule out the presence of Vulcan in various places. So, then the people calculating the orbits have to go back and say, well, if Vulcan isn’t there, maybe there are two or three planets combining together to do the thing that we originally wanted Vulcan to do. This gets sort out of control after a while. And at a certain point, people just kind of give up, and they say, well, it’s a great big mystery about Mercury. And after a while, after that, people kind of even stopped caring. Because, you know, we know Newton’s laws worked. This is just some weirdness about Mercury that we don’t understand.

And then, when Einstein creates his new theory of gravity, he then computes in the new theory of Mercury’s orbit. And he now gets something that agrees with the observations, without the need for a new planet. And so, what happened was, Mercury’s orbit is a little different from the Newtonian prediction. The general relativity prediction is a little bit different in just the same way to explain this problem that people had been trying to solve for fifty years unsuccessfully. And so, this was the first verification, empirical verification, of general relativity. And if you think about it, you would expect that Mercury would be the place you would find this out.

For Mercury, RS / R, this is the Schwarzschild radius of the Sun because that’s what’s doing the gravitating, is the biggest in the Solar System. Because the R, the distance from Mercury to the Sun, is the smallest of any of the planets in the Solar System. And so, the relativistic effects, the general relativity effects, are relatively large. But, you know, this is still a really small number. This is 3 kilometers, the Schwarzschild radius of the Sun. Mercury is way out there somewhere.

So, even though this is the most–this is the biggest relativistic effect in the Solar System, it still isn’t that huge. Let me just remind you what this effect is. Here’s the Sun. Mercury’s going around the Sun. And it’s going around in a slightly elliptical orbit. I’m going to draw a very elliptical orbit, here, but it’s really not that big. And there’s a point in the orbit where it is closest to the Sun. That point is called the perihelion. “Peri” for close, “helios” for Sun–of Mercury. And in the Newtonian theory, you should have exactly the same orbit every time. You should come back and the perihelion should be in the same place in each successive orbit. The orbit doesn’t move or doesn’t change.

But, in general relativity, the perihelion moves. So, after a while the perihelion will be here. The whole orbit will kind of tip this way, and it’ll look like this. So, this is the perihelion later. And it looks like that. And the angle which the perihelion makes with the Sun has changed. This angle is called the angle of the perihelion. And this precesses. So this is called the precession of the perihelion. And it’s measured in some angle per time. Because the question is, “How long does it take for the perihelion to precess across some angle?” And the key number for Mercury is 43 arc seconds. Remember arc seconds? Those are small angles. Per century ‒ a really small movement, but something that can be measured, and had been measured. And it’s not surprising that this is small, because the relativistic effects are going to be small, because the Schwarzschild radius of the Sun is really small compared to the size of the orbit of Mercury.

But this was observed before Einstein made his theory. Nobody understood it. Einstein came up with his theory. It turned out it predicted a precession of the perihelion in a way that Newton didn’t, and it turned out to work out precisely. So, that was good. And at the time Einstein published the theory, this was the only piece of evidence that it was correct. Pretty small empirical verification.

And so, let’s just write down the fable, here. This is Einstein and the precession of the perihelion. And there are two versions of the moral. Sometimes in textbooks, you know, they make a big deal out of this. They say, oh, there was this terrible problem with Mercury, and then Einstein came along with this great new theory, solved that problem. In the same way that they say, there was this terrible problem with the speed of light being constant from all frames, and Einstein came along with special relativity and solved that problem. That’s a misreading of history.

This was a by-product of Einstein. It wasn’t that there was a problem with the data and he went out to try and fix the theory to conform with the data. There was very little data. So, the moral here is aesthetic considerations, aesthetic–perhaps you want to call this philosophical, considerations can lead to a good new theory because he didn’t really do it to explain the data.

This is, however, the only time I can think of where this actually happened this way. Every other major advance in science came about because the observers or the experimenters had a problem–but not G.R. Only for general relativity.

Now, subsequent to that, between 1917, when this theory was promulgated, and now, there have been a variety of tests of general relativity using astronomical objects. You always have to use astronomical–or almost always have to use astronomical objects to test this, because you need really strong gravitational fields, and it’s hard to produce a really strong gravitational field in the laboratory. You’re kind of limited to what the Earth provides you with, and that isn’t such a strong gravitational field. We computed, at some point, the Schwarzschild radius of the Earth, R / RSchwarzschild is the relevant quantity, and it’s really small from the Earth’s gravitational field. So, it’s hard to see these things. So, all these tests tend to be astronomical in nature. Since then, a variety of tests of G.R., and the punch line is that it passes all of them. G.R. is still a good theory. There’s no contradictory data. But the tests still aren’t as strong as you might have hoped they would be. And we’ll talk about some of those on Thursday.

[end of transcript]

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