Frontiers and Controversies in Astrophysics

ASTR 160 - Lecture 14 - Pulsars

Chapter 1. Review of Post-newtonian Effects of General Relativity [00:00:00]

Professor Charles Bailyn: I’ve been talking about general relativity, and in particular, the post-Newtonian, relativistic effects. These are the first things that happen as the gravitational field becomes strong enough that Newton’s laws don’t perfectly apply. And let me just summarize those, because we’re going to see them all in action in just a minute.

So, here are the post-Newtonian effects of general relativity. And the first of them is the precession of the perihelion. I’m just going to list these. And this was seen in the nineteenth century already, and correctly interpreted by Einstein. It had to do with Mercury’s orbit.

I should say, this word, perihelion, that means closest approach to the Sun. And so, that only applies for planets if you have the same–you can have the same kind of effect in binary stars or in other things, in which case, you might–they call it a periastron. That’s the closest approach to a star that isn’t the Sun. Or, here’s a favorite word of mine: perigalacticon, I like that one. That’s the closest approach to a galaxy. You can have–the other side of the orbit that is 180 degrees away from the perigalacticon, that’s called the anti-perigalacticon. And you are challenged to use that in a sentence at any time over break. Anyone who successfully does that, send me an email recounting the circumstances, and I’ll give you an extra point or something.

All right, so, precession of the perihelion, that’s one post-Newtonian effect. Another post-Newtonian effect is the deflection of light by mass, just because mass curves space-time, and so, light will follow a curved path also. This was first demonstrated in 1919 by the famous eclipse observations of Eddington. And it also has been manifested recently in the discovery of so-called gravitational lenses, of various kinds, in which the appearance of an object is distorted by the fact that the light from that object has to pass by some other mass on its way to your eye.

And so, I have a couple examples, here, which I’d like to show you of gravitational lenses. This is the famous Einstein cross, so-called. And what this is, this is just a picture, I think, from the Hubble Space Telescope. And these four objects, here, are the same object. That’s a quasar. It’s just, for this purpose, a point of light in the sky. But it’s located–its true location is behind this thing. This is much closer to you than any of these guys. This is a galaxy about partway between us and the quasar. And what has happened is, if there were no deflection of light, you’d see this quasar, kind of, right behind the galaxy. And in fact, it would be hard to distinguish between the two. But what has happened is, because there’s this galaxy in the way, the light from the quasar starts coming this way and then bends around the galaxy and comes back toward us. And the same is true up here and down here, and down here.

And so, you get four different images of this quasar–of the same quasar. And they know it’s the same quasar, because they’ve taken spectra of the thing, and it’s the exact same spectrum on all four of these things. And so, these are just four different images of the same object. And the reason they’re not symmetrically distributed, here, is because the mass in this galaxy is not symmetrically distributed. And so, depending on what path you take, you get different amounts of deflections. So, that’s a classic example of gravitational lens. Yes?

Student: Why can’t we also see this quasar in the middle?

Professor Charles Bailyn: In principle, you could see it, right through here, but there are two things about the path that comes right at us. There are two problems with that. One is that for various–these guys are all magnified, and the one that comes through the middle is demagnified. Second of all, it’s right in back of the galaxy. But, in principle, there’s a fifth one down in the middle, that’s true. Yes?

Student: [Inaudible]

Professor Charles Bailyn: Sorry?

Student: [Inaudible]

Professor Charles Bailyn: Okay, so what–let me, let me pause, here, and tell you what a quasar is. A quasar is a quasi-stellar object. What it is, is, it’s a very bright source right in the middle of a galaxy. And so, it’s actually a point source of light. It looks like a star. These things turns out to be–what they really are, it turns out, are massive black holes in the center of galaxies, accreting material from the galaxy itself. But they appear to be very bright nuclei of galaxies and if you could look really hard, you could see the galaxy–the background galaxy around this thing. But in this particular case, it’s way too faint to see. The galaxy has a much larger extent, because it’s all the stars and all the stuff in the galaxies. It’s not coming right from the middle. And this one’s much closer than those, and so, you can actually see that it’s extended, in this case. Yes?

Student: [Inaudible]

Professor Charles Bailyn: Oh, it’s not the black hole that’s emitting the light. It’s the material falling into the black hole that gets heated up. I’ll actually talk about this on Thursday a little bit.

Chapter 2. Gravitational Lensing [00:06:15]

Other questions about the Einstein cross, here? Let me show you another one. So, this is a Hubble Space Telescope picture of a cluster of galaxies. See all these different things, all these extended sources, those are all individual galaxies. Galaxies, it turns out, come in clusters. And so, here is a big bunch of galaxies.

And some of the galaxies you see here–most of the galaxies, all these big ones, are part of this cluster. This is a cluster that’s relatively nearby to Earth. But some of the galaxies are actually not in the cluster. They’re much, much further away. They’re in back of the cluster. And these things–these, sort of, streaky things you see all over the place here–what these are, are galaxies that are in back of the cluster of galaxies, whose light has been lensed when it passes through the cluster of galaxies. And it’s been stretched.

It’s because, what happened is, this thing here, for example, ought to be a tiny little galaxy that you can barely see here, except the light from the top end of that galaxy got bent up to here. And the light from the bottom end of that galaxy got bent down to here. And that’s why they’re all sort of in this kind of–all these things are kind of circular around the center of this thing, because this is a very, very massive galaxy. And that’s the thing that’s doing the lensing. And so, it effectively distorts the shape of the background galaxy. And you can see a couple more of them around this massive galaxy, here.

These are pretty, but they also can be used in order to understand the distribution of mass inside the galaxy cluster. Because the particular way in which these galaxies are distorted depends on how much mass there is in this galaxy cluster, and how that mass is distributed. And so, these are used to study the mass distributions of galaxy clusters. And so, this is a different kind of gravitational lensing.

And finally, let me show you a joke. This is a picture of The Smithsonian Institute in Washington, D.C., where there are a group–well, actually, The Smithsonian Astrophysical Observatory is actually in Cambridge, Massachusetts. And there, there are a group of people who study gravitational lensing. And they have taken this picture. And imagine what it would look like if, about halfway between where you’re looking and this building, there were a black hole the mass of Saturn. And here’s what it looks like. And all these distortions are simply due to the change in direction caused by the gravitational force of this imaginary black hole. And so, you could see what happens. The black hole is, sort of, right down the middle, here. And so, off at the edges, there’s only a little bit of distortion. And then, in the middle, everything goes haywire.

Let me take this back a notch. Take a look at these two central towers, this one, here, and that one. And so, they’re kind of–what’s happened, here, is that you’re seeing light that has gone out and then been bent back towards you, so, it looks like the tower’s over here. But look, that same tower is down here. And so, that’s light from the tower that went down and underneath the black hole and then got pulled back up to reach your eye. So, there are two paths from that tower to your eye, and you can see both of them.

Look at this thing here. What’s that? Let me go back to the original picture. That’s this statue. And what’s happened is, light from the top of the statue has gone up and over the black hole, and come down towards you. Whoops–and so, here it is. And similarly, the clouds that were up here have been bent down–have gone underneath the black hole and been bent back up, so you can see them here.

And so, they worked out all the distortions of this, and so, they like to show this kind of amusing picture. They actually had a movie, which I couldn’t find on the Net, last night, which is too bad, where they take the black hole, and they kind of move it across the front of this thing. You can see all the distortions coming out. And so, you can have enormous fun with this kind of thing. Needless to say, this is not–this particular picture has not ever actually been observed. That’s calculated, not actually observed. But, you can see the kinds of bizarre effects that gravitational lensing can create. And, as I mentioned, in some cases, these are scientifically useful things, because you can determine the distribution of the mass that’s causing the lensing. Questions? Yes?

Student: So in that particular case it’s always [Inaudible] that all the light from the other side get lensed?

Professor Charles Bailyn: Well, what you have to do–it depends a little bit on the exact mass distribution of the lens. In this case, I think they’re assuming a point source. And so, it is, in general, true that if there’s a gravitational lens, you end up seeing more light than you otherwise would, because it’s focused in towards you. So, the total amount of light that you see, if you’re seeing a lensed object, is typically larger than the amount you would see in the absence of the lens. So, there are more paths of light that are curved towards you than are curved away from you. If you’re standing over on the side, you’ll lose a little bit of the light, because it will be curved in some other direction. And so, there are more different way–you can see the thing–you know, it’s kind of an early twentieth century art thing. You can see the same object from two different sides at once, and you get these curious distortions. Yes?

Student: If you’re looking into a galaxy cluster and there are a lot of streaks and a lot of galaxies and you don’t know the mass of the galaxy to try to find the mass [Inaudible]

Professor Charles Bailyn: Yeah.

Student: How do you know what streaks belong to which galaxies?

Professor Charles Bailyn: What streaks belong to which galaxies? Okay. You know that the streaks must come from galaxies that are located behind the galaxy cluster, that are not themselves part of the galaxy cluster. And, in order to understand what that streak is, in principle, you’d need to know the exact shape and size of that galaxy. You can go and take a spectrum of the thing and figure out how far away it is and what the relative distances are. But, in principle, you would need to know the exact shape of that galaxy before it was distorted, which you don’t know. And so, what you have to do, to interpret these things, is to take many streaks and assume that the galaxies that caused them are kind of normal–a normal distribution of galaxy shapes. And so, it’s more of a statistical process than using them one at a time, in that case.

The quasars are a little easier to deal with, because you know that they’re points. And this–and there are various kinds of limits to how much information you can get out of this. And it’s quite tricky to do.

What’s easy to do is the thing that they did in that picture. You imagine that there is a mass in the way, and you figure out what happens to the light from the background. What’s hard to do is to observe the light in the background, not know what it ought to look like, really, and infer what the mass distribution is. That’s a harder problem. And there’s a lot of theoretical effort, at the moment, going into trying to solve that problem of, here’s what you see, infer what the mass distribution has to be in order for you to see that. Yes?

Student: Have we ever observed where it was the perfect lensing when you have a ring around [Inaudible]

Professor Charles Bailyn: Yes, there are some Einstein rings. I don’t have them here. Hubble Space Telescope has observed a bunch of these things. Typically, they’re not full rings, because the mass distribution is not perfectly symmetric. But there are, as you can see, you can see, those were short, kind of, little arcs. And if you get it set up just the right way, the arcs can go a large fraction of the way around the circle. So, these are, kind of, fun things to observe, and also give you useful information. So, that’s the second of the post-Newtonian effects. The first was the precession of the perihelion. Now, the deflection of light.

Moving on, number three, here, is what we talked about last time, which is the gravitational redshift. And here, we have delta lambda over lambda [Δ λ/ λ], or alternatively, in the case of the problem set you’re working on, the pulse period, in either case–these are usually given the letter Z, that’s redshift. That’s a definition of this quantity, Z, which is often used. And this is equal to this quantity. And this is the Schwarzschild radius of some mass. And this R is the distance of the light source from that mass.

And what this equation signifies is how much redshift is there, if you observe this thing when you are sitting at infinity–do you observe from infinite distance. And what do I mean by infinite distance? What I mean is that R_S / R goes to zero. That’s what infinite R means, in this case. So, you’re sitting out there a long, long, long way away from any gravitational effect. You’re looking at a light that is emitted inside a gravitational field, relatively close to some mass. That light is redshifted by this amount. And pulse periods are also lengthened by the same amount, and I’ll come back to that in a minute.

One thing I want to say right now is, what happens if you are an observer and you’re not infinitely far away? What happens if you’re just a little bit further away than the light source is, or perhaps even if you’re closer to the object than the light source is? And, in that case, if the observer is not at infinity, then the observed redshift is equal to the redshift from the source to infinity, which you could calculate by this formula, minus the redshift from the observer to infinity, which you can also calculate with that same formula. So, you have to subtract two things.

Basically, what happens is, imagine the light going from–it starts out at the source, it goes to the observer, and then it keeps going all the way out to infinity. And you want to ask, what does the observer see? The observer sees that portion of the redshift that happens between the source and the observer. So, to figure out what that is, you figure out the redshift from the source to infinity. You figure out the redshift from the observer to infinity. And you subtract those things. And that gives you the redshift from the source to the observer. Okay, yes?

Student: [Inaudible]

Professor Charles Bailyn: No, no, no, but look what happens. When R_S / R is equal to zero that means there is no redshift. That means there is no redshift. It doesn’t mean that the observed wavelength is zero. It means the change in the wavelength is zero. Yeah, that’s important.

And, just to comment on the problem set here, the idea on the second problem of the problem set is, you do have an observer at infinity. So, here’s the observer. Here’s some neutron star. And you have a light source that can either be on the surface of the neutron star, or it can be raised off the surface of the neutron star. And basically, what you’re observing is the change in the light source. So, this has some redshift, some pretty large redshift, because it’s sitting on the surface of a neutron star. This, because it’s further away from the neutron star, has slightly less redshift. And what you are interested in is the change in the redshift.

And the idea behind the problem is, you want to set this up so that that change in the redshift is the same as the change in the redshift due to the Doppler shift, if this whole thing was in orbit around something. That’s the plan of that particular problem.

Okay, so that’s the third of the post-Newtonian relativistic effects, and there’s a fourth one. The fourth is gravitational waves, which I may have time to come back and talk about in detail. At the moment, though, I just want to say what the effect of these things is, rather than talk in detail about what they are. The effect is that the orbital period decreases. Now, it turns out, this is a relatively small effect, and it is not observed in the Solar System, even in Mercury, which has the strongest relativistic effect.

Chapter 3. Jocelyn Bell, Binary Pulsars, and General Relativity [00:21:05]

Okay. So, those are four post-Newtonian effects of general relativity. And what I want to talk about now is what has become probably the best laboratory for general relativity that we know of. The problem with general relativity is you can’t actually study it all that much in the lab, because, in order to study it, you need strong gravitational forces. And it’s very difficult to create strong gravitational forces in a laboratory. What would you need to do? You’d need to take some massive object, crush it down to the size of a black hole or something, and have it sit there in your lab. And this is not something that one can do. And so, you have to go and look at astronomical objects in order to find suitable situation in which to study these relativistic effects.

So, the best lab for general relativity–lab for G.R., here, is a set of objects. And one object in particular, which is referred to as the Binary Pulsar. This is discussed in the readings for this week’s problem set. The discovery of this particular object led to a Nobel Prize for its discoverers, and that’s what’s discussed in that article.

So, the first thing to talk about, here, is what is a pulsar? So, pulsars were discovered in the late 1960s by a woman named Jocelyn Bell. And she wired up a whole field in Cambridge, England, to make it behave like a radio telescope, and discovered that there were objects in the sky, celestial objects, that seemed to be emitting pulses of radio waves. So, the observed signature of these things–what was observed is pulses of radio waves. So, you get kind of a blip every few seconds. It goes blip, blip, blip, blip, and so forth. And these pulses come by–so, the pulsations happen once–it’s sort of with a period of somewhere between milliseconds and ten seconds. So, they come at you pretty rapidly. Most of them are a few seconds or maybe a fraction of a second. So, these things are blipping away.

And they had no idea what these were. And the first four of them they gave the nickname LGM1, LGM2, LGM3, LGM4, for “little green men,” because the thought was that these might actually–they thought, briefly, that these might be signals from some extraterrestrial intelligence, because, you know, it’s a very regular signal. It’s hard to imagine how those could have been created. But, fortunately, before they published their paper, somebody suggested a better solution, which turns out to be correct. That what these things are, are rotating magnetized neutron stars. And this was the first direct evidence that neutron stars existed.

And the reason they thought they were neutron stars is because the speed with which these things pulsed is supposed to be reflected in the rotation of these neutron stars. And something that rotates ten times a second had better be pretty small. And so, there was strong size limits on the neutron stars. It’s not going to matter, too much, exactly what these things are, but I’m going to explain it to you anyway. But don’t worry too much about it. For the purposes of relativity, the key point, here, is that it’s an extremely accurate clock–that you get these blips coming out of the thing, and that creates a very accurate clock. And then, you can see very small changes in the pulse rate of these things.

Chapter 4. Measurement Errors and Testing Strong Field Relativity [00:25:16]

So, now let me just briefly tell you what these are. Oh, I should mention–so, the discovery of neutron stars, this was a very big deal. This was the first strongly relativistic object that had ever been discovered and they awarded a Nobel Prize to Jocelyn Bell’s thesis supervisor. And there is some feeling among some folks that this was an injustice. And so, there is–people discuss the story of Jocelyn Bell and the discovery of pulsars. To her credit, Bell has never complained. Bell and the discovery of pulsar–

Student: Did the thesis advisor do anything?

Professor Charles Bailyn: Oh yes, well that’s actually an interesting question. The question is, “Did the thesis advisor do anything?” As a thesis advisor myself, I take that question seriously. Well, this is the whole debate about who should have gotten the Nobel Prize. The thesis supervisor thought up the experiment, and Bell did it. So, who ought to get the credit for that? It’s actually a subtle question, and it bears on problem four, or whatever it is, of this week’s homework, so it’s worth thinking about.

Jocelyn Bell and the discovery of pulsars–he also got the money. That’s not trivial sometimes. And the moral of the story–it depends on who you ask, for exactly the reason that you bring up. One moral of the story is that grad students, particularly female grad students, never get any credit and that this is a great injustice. As you get older, your attitude towards this problem changes.

And the alternative moral is that thinking and doing are not the same, and that thinking is better than doing. Now, one could raise various objections, and people have, to this line of thought. And interestingly enough, the next time they had to give a Nobel Prize for something to do with pulsars, which was the discovery of the Binary Pulsar, they did give the prize also to the graduate student. And it’s the graduate student’s description of what he did that you’re actually reading for this problem set. So, you can opine on this topic yourselves on the problem set. Anyway, a matter of continuing controversy in scientific circles.

But let me just briefly show you what’s supposed to be happening, here. Here is a neutron star–very dense thing, yeah? And it’s rotating, so, it’s got some axis of rotation, here–a different color. So, here’s its axis of rotation. And it’s rotating around its axis.

And it’s also got a magnetic field. But the magnetic field, the pole of the magnetic field, is not the same as the rotation pole. That’s true on the Earth as well. Magnetic north is not exactly in the same place as true North, the center of the Earth’s rotation. And if you’re in northern Canada and you try and follow a compass signal north, you’re going to get yourself into big trouble. Because, you know, the actual magnetic pole is somewhere in northern Canada, and if you’re north of that, you’ll be going south when you think you’re going north.

Anyway, imagine that here’s the magnetic field. Magnetic fields have these sort of field lines that look like this. And what happens in these things is that there’s radio emission coming out of the magnetic poles. So, there’s radio emission coming out this way. And so, if you’re observing this from here, say, you see radio emission when the pole is pointing towards you. But then, if the thing is rotating, then half a rotation later–so, later, but not too much later, because this thing rotates really quickly. Here’s the neutron star, still, and now the pole’s over here, because it’s rotated around by a factor of two, by half a rotation. So, now the radio emission’s coming this way. And if you’re still sitting observing the thing down here, you don’t see anything. So this is on, and this is off. So, you watch this pole.

It’s sort of like a lighthouse, you know. A lighthouse has a beam of light. And when it’s pointing towards you, you see light. And then, as it’s going around in a circle, you don’t see light. Then it points towards you–you see light again. And so, the consequence of this is that you get something that looks–if you plot radio emission versus time, you get these very regularly spaced blips. And it is the time between one blip and the next that’s the pulse period.

And so, for our purposes here–I mean, pulsars are very interesting objects, but we’re not going to go into them in any depth. As objects, the key thing is that it’s got this really accurate clock on it. And if you imagine a pulsar in orbit around something, when it comes towards you, each one of these blips happens a little closer to you. So, the light travel time from the blip to you is less. And so, it seems like the blips happen closer together, because each one of them has emitted a little closer towards you. Similarly, when it’s going away, each blip is emitted a little further away. It takes extra time to get to you, and the pulse period, the length of time between the pulses, appears to be longer. By how much? By exactly the amount that the Doppler shift formula would suggest. And so, the pulse period obeys the same rules in terms of its length as wavelengths of light do. Okay.

So, for example, it is true that Δ pulse period over pulse period is equal to the usual Doppler shift formula, which is approximately equal to V_R / c, in the Newtonian approximation. Oh, and I should say, in problem one on the problem set, you can use the Newtonian approximation, because, as you’ll discover, the velocity of the object as it goes around in its orbit is significantly less than the speed of light.

Okay. So, they discover this Binary Pulsar with a short orbital period. The orbital period–this is now the orbital period–is around eight hours, which is pretty short, but it’s still not very relativistic. But you can measure these things to incredible accuracy. So, you cannot only see the velocity curve that we’re used to. You can also see tiny deviations from the Newtonian velocity curve. And from those tiny deviations from the Newtonian velocity curve, you can see these post-Newtonian effects.

In particular, it turns out–so this in an elliptical orbit. It’s not a circular orbit. It’s a highly elliptical orbit. And you can see the precession of the periastron as the ellipticity–as the direction of the orbit, as the position of the orbit, changes, you can see the precession of the periastron. And the precession of the periastron happens at the rate of four degrees per year. Now, compare that to Mercury, in which the same effect can be observed at 43 arc seconds per century. So, this is a much stronger effect.

Okay. You can also see the gravitational redshift. Now, let me show you how this comes about. But you have to be careful about exactly what you’re seeing. You’re not seeing the gravitational redshift, due to the fact that the clock is sitting on a neutron star. Because, okay, you’re seeing the effect of the companion object of this other object in the system, which happens to be also a neutron star, but not a pulsar.

And what happens is this. Here’s your clock, and it’s going around in some kind of elliptical orbit. Here’s the center of mass. Here’s the other; so, there’s a clock that’s going around like this. And then, there’s some other object that’s going around like this. They’re approximately the same mass. And so, sometimes these objects are close to each other. For example, here–when they’re here and here, this is–call this position 1. And then, later on, half an orbit later, they’re out here in position 2, and this one’s gone all the way out to here.

So, at position 1, the objects are close, and so, the clock is redshifted by the presence of the other object. And so, it has to–there’s a redshift due to the other object, which is nearby. Objects are close, so, the redshift caused by the other object is large. At position 2 the distance to the other object is much greater–distance is greater. And so, redshift is less.

Now, it’s also true that there’s a redshift caused by the fact that this clock is sitting on neutron star. But the thing about that redshift is it doesn’t change during the course of the orbit. So, you can’t actually tell that it’s there. Because that’s one of the differences between measuring pulse periods of pulsars and measuring wavelengths of light. With these spectral features in light, you know what it’s supposed to be if it isn’t redshifted.

In the case of the pulsars you don’t know, in advance, what the pulse period ought to be. So, you can’t tell what it should be if there was no gravitational field or was no Doppler shift. All that you can tell is the amount by which it changes during the course of an orbital period. And so, you’re comparing the redshift when the objects are in this position to the redshift of the objects in this position. The redshift caused by the pulsar itself doesn’t change, but the redshift caused by where its companion is, and the gravitational force exerted by the companion, does change.

The consequence, by the way, is that if these things are in a circular orbit, you can’t see the gravitational redshift, because they’re always the same distance away from each other, and the gravitational redshift doesn’t change. There is a gravitational redshift, but it doesn’t change during the course of the period, and it’s the change that you can observe. Yes?

Student: So, in this case, redshift refers not to the frequency of, like, the light in the pulses? It refers to the [Inaudible]

Professor Charles Bailyn: It’s the time between pulses. Yeah. You could imagine that the pulses might have a spectral feature and then you would see it in the spectral feature, too. In fact, that’s extremely hard to have that. That doesn’t happen. It’s a continuum of light. So it’s the time–what you observe is the time between pulses. And that can be observed extremely accurately. You get 20 decimal places of accuracy. It’s really kind of frightening. Yes?

Student: [Inaudible]

Professor Charles Bailyn: Well, if it’s a circular orbit, no. If there’s a circular orbit, they’re always the same distance around each other. They’re chasing each other around in a circle. That’s what it means to be circular orbit, if every point on the orbit is the same distance to the center for both objects. So, each object is the same distance from the center of mass at all times. That’s what makes it a circular orbit. And they’re always on the opposite side of the center of mass, so, the distance between them is always the same for a circular orbit.

I’ll draw it. Here we go. Here’s the center of mass, supposing these objects are the same mass as each other. Here’s one of them, it’s going around this way and here it is right now. And here’s the other object, it’s going around this way, and here it is right now. And they’re just going to march around this circle, each being on the opposite side of the center and the distance between them won’t change.

Student: [Inaudible]

Professor Charles Bailyn: Oh, they only overlap if the two objects are the same mass. So, if one object is more massive than the other, then you’ve got a little circle going this way and a big circle going this way. Let’s see, it would have to be there. And then, they’re still the same distance from each other at every point along the way. Because each one of them is always the same distance from the center of mass, in the middle, and they’re always on opposite sides.

Student: [Inaudible]

Professor Charles Bailyn: So, the way ellipses work is–so, here’s an ellipse. And the ellipse can be big or little–here’s an ellipse. They’re doing ellipses, and the center of mass is at one focus of the ellipse. It’s not at the center of the ellipse. That’s the key thing. And so, when you’re furthest away from the focus, you have a long distance from the center. When you’re closer, you’re closer. And it’s because, each one of these things, you don’t maintain a constant distance from the center of mass.

Okay, so, because of this effect in elliptical orbits, in Binary Pulsars, in elliptical orbits, you can observe the gravitational redshift. Also ‒ and this is what they got the Nobel Prize for, really ‒ you can see that the orbital period decreases. It’s getting shorter. The two objects are spiraling in together. Decreases due to gravitational waves.

So, you can observe three different post-Newtonian effects in these systems. So, this constitutes a serious test of general relativity in the following way. If you could do only Newtonian measurements, what would you get? Well, we did this before, for the planets. What you can get out of a Newtonian measurement of velocity–remember, what you’re doing is you’re measuring the Doppler shift; you’re measuring radial velocity. What you get is, you can determine M_total, the total mass of the system, if you know the inclination. That’s something we tended to leave out. We always claim these things are edge-on because it makes the calculations easier. But, in fact, you get a different answer depending on what the inclination is, and you don’t necessarily know the inclination.

So, if you now measure the precession of the periastron, then this also–the amount of that also depends on M₁, M₂, the two masses, and the inclination. So, you can solve for the inclination now. This isn’t actually precisely how you do it, but conceptually, this is how it works. You now solve for the inclination. You then know M_total, but not each individual mass.

Student: [Inaudible]

Professor Charles Bailyn: M₁, M₂ and the inclination, this is i. But you don’t know the individual masses. Now, if both objects were pulsars and you could see both orbits, then you would also know the individual masses. But you don’t. You only see one of these objects. Okay.

So, now, the next effect is gravitational redshift. And this, again, depends, in a different way, on M₁, M₂, and the inclination. And so, now you can solve for M₁. And since you know the total mass, you also get M₂. So, at this point you know M₁, M₂ and the inclination.

What’s happening here is that each time you measure an effect, you get an additional constraint. And you started out having three unknowns and one equation. And, of course, you can’t solve for three unknowns with only one equation. But then, you observe two more effects. So, now, you have three unknowns and three equations. So, now, you can solve for all of these things.

Now you know everything. At this point you can predict what the period change should be, because you know everything. Then you can measure that. And so, you have a specific prediction created from general relativity. You can measure this effect, and the prediction turns out to be correct, to within the measurement errors.

And the measurement errors are getting more precise all the time, because as time goes on the amount of change in the period, since they first observe it, increases. Because the period’s changing more, it’s slowing down and the period’s getting faster and faster, the objects are falling in. And so, over time that change mounts up. You can measure it better and better and better as time goes on. And so, the precision of this test of general relativity is going up all the time.

So, it is now clear that general relativity is correct, to some number of decimal places. I’m going to put correct in quotes, because it depends how accurately you can measure it, but the accuracy is going up all the time–in the post-Newtonian approximation. Remember, all these effects are post-Newtonian only. Actually, they’ve now observed a couple of post-post-Newtonian effects, which are the next term in the series expansion. And those are also correct, but they’re not measured as precisely yet. So, it’s clear that general relativity is correct as long as everything is still pretty close to the Newtonian thing. You can actually still have orbits.

But you could imagine a theory which is like Newton in the first term of the expansion, like general relativity in the second term, the post-Newtonian term, but different in very strong gravitational fields. So, you’ve got to get the first term to work, and it’s got to come out like Newton. You’ve got to get the second term to work. It’s got to come out like general relativity. But then, you can imagine all sorts of different functions that work with this, work with this, and are different in the higher order terms. And the higher order terms are the things that give rise to all these exciting effects, like event horizons, and things like that.

So, what would be nice–it would be good to have some way of testing strong field effects. And by that, I mean, situations in which the Schwarzschild radius over r becomes close to 1. Because, in that case, all the terms in this series expansion, contribute. And, if you can get it–if the theory is right there, then all the terms have to be the same, and you don’t have this nagging doubt about whether you could come up with some theory that looks like Newton, in the same way that general relativity looks like Newton, but is different once you get higher order terms. Looks like general relativity in the second term, but turns out to be a totally different theory.

Needless to say, the theorists have come up with, literally, an infinite number of such theories. Because there’s a whole infinite family of things which satisfy this, satisfy this, and look different at the upper level. So, what we’re going to talk about next time is how one might go about testing strong field relativity. This is a subject close to my heart because it’s what I do for a living, and we’ll talk about it on Thursday.

[end of transcript]

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