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# ASTR 160: Frontiers and Controversies in Astrophysics

## Lecture 12

## - Stellar Mass Black Holes

### Overview

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.

Step by Step into a Black Hole

Courtesy of Ute Kraus, Max-Planck-Institut für Gravitationsphysik, Golm,

and Theoretische Astrophysik, Universität Tübingen

http://www.spacetimetravel.org/expeditionsl/expeditionsl.html

Black Holes and Neutron Stars

“Written Description of Visible Distortion Effects”

Courtesy of Robert Nemiroff, Michigan Technological University

http://antwrp.gsfc.nasa.gov/htmltest/rjn_bht.html

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html## Frontiers and Controversies in Astrophysics## ASTR 160 - Lecture 12 - Stellar Mass Black Holes## Chapter 1. Invariance in Special Relativity [00:00:00]
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 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
For points on a 2-D space, Δ 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 Okay, so now, this answers one of the questions that was asked before. Why does one use c to transform the space coordinate into the time coordinate and back? It’s because you need the c out here in order to make this invariant. If you’re calculating the distance, if you use ^{2}x^{2} plus ½ y^{2} or some other constant times y^{2}, 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 ## 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 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?
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
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 So, here’s the Schwarzschild metric, ( dR) / (1 - R_{s} / R). So, that’s just like the flat term, except with something in the denominator there. Plus R^{2}, d Omega squared, that’s just like the flat metric. And then the–whoops this had better be - c^{2} (1 - R_{S} / R) (cT) ^{2}. Where R_{S} is the Schwarzschild radius, which we’ve had before, which is 2GM / c^{2}.## 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 - If Now, the other case is when 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 And now, let’s think about what happens inside the Schwarzschild radius. 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 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
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 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, 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, [end of transcript] Back to Top |
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