 Before we get started on this lecture, on the first glimpses into the general theory of relativity, I want to kind of put a little cautionary warning label at the beginning of this video. In any textbook at the level of a course like this, and certainly in this lecture video, I don't want you to walk away with a feeling of full confidence that you have completely understood the generalization of the theory of space and time that Einstein set in motion in 1905 with what we call the special theory of relativity. Relativity is an extremely rich subject. You can quite literally fill volumes on this particular bit of material, and in fact I'm holding one of them in my hands now that I'll show you later. The general theory of relativity is fundamentally built on a rich and complex set of mathematics that students at the level of a course like this simply have never seen and cannot be expected to master in a week or a month or even three months without really first having had the full breadth of undergraduate mathematics. Now that said, there are nuggets of ideas and mathematics that one can draw out of the general theory of relativity and use to motivate in the context of special relativity the implications of the grander theory of space and time. Since 1916, when Einstein first established the calculational framework, the reliable calculational framework that set the stage for the general theory of relativity and all the work that would be done with it, students have struggled with this material because it challenges many preconceived notions, many concepts that we walk into any standard science class cherishing. So I want you to be a little bit forewarned, first of all, that the nuggets that we will draw out of the general theory of relativity and analyze in the context of special relativity can have some stunning implications that either will challenge things you already believe to be true or which open your eyes to the grander scale of the cosmos that we inhabit. Now I mentioned this book earlier. This book is one of the seminal works in the field of physics on the whole of the general theory of space and time. It's entitled Gravitation. It's three co-authors are Charles Misner, Kip Thorne, and John Archibald Wheeler. Now all three of these individuals, each in their own way, are considered some of the brightest lights of 20th and 21st century theoretical physics. And this book is expansive in its treatment of the subject. Look first of all at how thick this book is, and if you flip through this you will quickly see that most of us would be out of our depth in the level of mathematical rigor and notation and variety and subject matter that is minimally expected in order to follow along with a text like this, certainly to its bitter end. All of this is to simply point out that the general theory of relativity is complex and rich and mathematically far beyond the scope of a course at this level. Now that said, we can draw nuggets of ideas out of the general theory of relativity and we can put them in context in our own course experience for a course at a level like this one in modern physics. Now some of the names on here may seem familiar to you. Kip Thorne, for instance, has become recently famous not only for winning the Nobel Prize in physics for one of his key bits of work on space and time, energy and matter and the theory that links them together in general relativity, but also because he has served as an advisor to film and TV including things like the movie Interstellar from 2014 which had some of the most advanced visualizations of physics based on the general theory of relativity in any movie that came before it or since then. John Wheeler is another bright light in the field of theoretical physics. He will feature briefly later on in this lecture in the context of an individual who could not only deal with the mathematics of this subject, but elegantly communicate to an audience even at the level of our course the grandest sweeping summary of the general theory of relativity and its implications for energy, matter, space and time. So with all of those caveats in mind let's pluck some interesting nuggets of ideas out of the general theory of relativity, place them in a local context and special relativity where we feel more comfortable with the mathematics, albeit with the caveat that the mathematics required to really do this treatment is far beyond our grasp at this stage in a physics course at the university level without the full breadth of undergraduate mathematics behind us quite yet. Let's see what those nuggets tell us should be revealed about energy, matter, space and time and then let's look at how those ideas have implications for the whole structure of the cosmos in which you and I live. So with all of those things in mind let's start digging into some of the basic ideas that motivated the general theory of relativity and take a look at some of those nuggets of ideas that we can couch in the picture of special relativity that we're a bit more comfortable with at this stage of our engagement in physics. In this lecture we will learn the following things. We will learn about the transition in thinking from the special to the general theory of relativity. This will by no means be comprehensive but merely a taste of some of the basic ideas that led Albert Einstein ultimately to construct this theory of space and time. We'll look at some implications of the general theory of relativity on specific physical phenomena and we'll look at some of the large scale implications for space and time. Let's talk a little bit about the transition from special to general relativity. Based on the speed of light in the first and second decades of the 1900s continued to yield no disconfirming evidence for the postulates of special relativity. Now Einstein's physics work in 1904 and 1905 and beyond earned him the faculty position that he had so richly sought after his PhD and he was able to escape the job as a patent clerk in Bern, Switzerland and finally take on the mantle of the academic position that he had hoped for after earning his PhD. He was sure that the special theory itself could be generalized to a complete theory of space and time, including, he hoped, an explanation of the nature of gravity itself, the very prize that had eluded Isaac Newton. All that Isaac Newton could establish was that his law of gravitation correctly described the behavior of gravity on all scales that could be observed at the time. Einstein ambitiously pursued the elusive prize that Newton could not grasp and that was to finally unmask the nature of gravity itself. Now this work would take another decade of struggle, Einstein would fail many times and in fact if you look at the record of Einstein's work in the decade that followed his miracle year there were some serious missteps in papers that he had published based on his work beyond the special theory of relativity and had anyone been able to mount an experimental test of the claims of the general theory of relativity prior to 1916, Einstein might have been laughed off the scene of physics, but it took more than ten years to be able to make an observational test of one of the key predictions of the general theory of relativity and that ultimately saved Einstein, buying him the time he needed to fully develop the mathematics behind the general theory, put on firm ground theoretically speaking the predictions of the theory of general relativity, and to finally publish key papers in 1915 and 1916 that are considered the first accurate and fairly complete treatment of the subject from Einstein himself. Now interestingly in order to lay the groundwork for the general theory of relativity, Einstein required much of the advanced math that he had eschewed during his time in graduate school, one of the things that offended many of the faculty that had him in his classes. He was forced then to go and actually relearn subjects that he had actively avoided in some cases during his PhD education. He benefited from a close network of friends who were outstanding physicists and mathematicians in their own right, and through his network of friends he was able to build his own foundation strong enough to eventually lead to his key insights and firm mathematical grounding of the general theory of space and time. In this lecture we will explore some of the very basic ideas and tease the larger implications of the general theory. It's very difficult at this level as I have warned you before to give you the full treatment, but I will do my best with the aid of the textbook that we use in class to attempt to communicate some of the key nuggets and frame them in the language of special relativity which we've developed more carefully over the last few weeks. Let's begin with a tale of two masses. We take for granted in introductory physics that mass appears in a large number of equations. But if you really boil it down, mass as a concept appears in two distinct equations in introductory physics. The rest of the equations that we use that involve mass can be found to stem from these two laws of nature. Now what's interesting and what may not have been pointed out in introductory physics was that the two equations do not necessarily have anything specific to do with one another as regards mass itself. The two equations in question are Newton's second law of motion, F equals MA where mass appears as the multiplicative scalar factor in front of acceleration, the constant of proportionality between what an external force of a general nature exerts on a body and the responding change in the state of motion, the acceleration of the body. But mass also appears in an equation that describes the nature of a very specific fundamental force, gravity. The law of gravity states that the force of gravity between two bodies which we might label one and two is proportional to the product of their masses divided by the square of the distance between them. Now the M for mass that appears in Newton's second law has to do with inertia, the tendency of a body to resist changes in the state of motion. And so it's more honest to say that Newton's second law is concerned with a mass concept we might label inertial mass, the mass of a body that resists changes in state of motion. But the M that appears in Newton's law of gravitation has to do with the primary cause of the gravitational force between two bodies that have mass. This doesn't necessarily have anything to do with their tendency to resist changes in the state of motion, it has all to do with the degree of the gravitational attraction between the two bodies. This is more honestly referred to as gravitational mass, potentially distinguishing it from inertial mass. There is nothing in these two laws that says that these two quantities, these two kinds of mass, have to be fundamentally the same. And yet their equivalence, the equivalence of inertial mass and gravitational mass has been tested to a remarkable precision. Inertial mass and gravitational mass appear to be one in the same. Let's take a look at this by briefly stepping through some mathematics, now couched in the language of inertial mass as potentially distinct from gravitational mass, and revisit some conclusions we drew in introductory physics. So a consequence of their equivalence is often taken for granted. What if they weren't equivalent? Well if they weren't equivalent, then we would rightly state that two bodies that are acting on one another through the gravitational force can have their degree of acceleration explained by Newton's second law, but without necessarily equating inertial mass and gravitational mass. For example, if we consider the earth to be pulling on a body, say you, you jump off the surface of a table in an attempt to accelerate down to the floor and land on the ground. The earth is attracting you down toward its center. We can figure out the local degree of acceleration due to the gravitational force by taking the product of Newton's gravitational constant g, the mass of the earth, and dividing by the distance between you and the center of the earth squared. Now we multiply that acceleration, which we often denote little g for gravitational acceleration, times your gravitational mass, and we set that equal to your inertial mass times your total acceleration. Now if we then solve for the acceleration due to gravity, we find that this would be equal to this gravitational acceleration, little g, or big g times the mass of the earth divided by the distance between you and the center of the earth squared, times the ratio of the gravitational mass and the inertial mass. If we were to substitute for little g, 9.81 meters per second squared here, we would conclude that if gravitational mass and inertial mass are not the same, if for instance the gravitational mass were 10% of the inertial mass, that only 10% of your inertial mass has anything to do with causing the force of gravity, well then we might conclude that your acceleration might be very different than a body that has more mass. But we don't observe that when two objects of different masses fall in a uniform gravitational field, a field of uniform gravitational acceleration if you will, all bodies, even though they possess of different masses, appear to fall at the same rate. And so by eye you can already draw the inference that gravitational and inertial mass are, if not equal to each other, very close to one another. As far as the limits of our ability to test this have taken us, we've never seen a difference between gravitational and inertial mass, they really seem to be one and the same within the limits of experimental methodology. This leads us into one of the key insights that Albert Einstein had early on in the process of trying to generalize the ideas of space and time to include gravity. This is summarized by the phrase the equivalence principle. Now I've pointed out that observationally there seems to be an equivalence of gravitational and inertial mass, and this can lead you down the path, as it did along with some thought experiments for Albert Einstein, to a larger consequence, and that is the principle of the equivalence of a system accelerated by a constant force, or alternatively experiencing a constant gravitational field. The principle of equivalence, in the language of relativity and space and time, is about the equivalence of two different situations. One in which a system is experiencing an external force of some kind that causes it to change its state of motion, like a rocket or something pushing on something else, the equivalence of that system, and a system that's experiencing a gravitational acceleration that pins objects to a low point in the system, and I will illustrate this. Einstein observed early on in his thought process about all of this, that due to the equivalence of these two kinds of mass, inertial mass and gravitational mass, there's really then no difference between being under the influence of a uniform and constant gravitational field, or source of gravitational acceleration, or instead being placed in a non-inertial reference frame, one where there is an observed net force acting on all the parts of the system, by the action of some other kind of external force. This picture illustrates the idea in a rather cartoonish but elegant way. The scenario I like to have that goes along with these set of pictures is the following. Imagine you wake up, and you find yourself in a room with no doors and no windows. There's no way to see past the walls of the room at all. As I like to joke with people, this is like the premise of the opening scene of some kind of cheap horror movie. You push yourself up from the ground, you feel gravity pulling you down, and you have to work against gravity to raise yourself up, that's what it feels like to you. Now on the floor next to you was a red ball. You lean down and you lift up the red ball, and you hold it out roughly at arms length and level with your shoulder, and you let the ball go. And indeed, you observe that the ball falls down to the floor of the room. You then check your watch. Think about the average height of a human being, measure roughly how long it takes that ball to make that drop from shoulder height to the ground, and you're relieved to find out that you seem to still at least be on earth, albeit you have no other external information to tell you where you are, because the ball appears to fall at a rate of acceleration consistent with G at the surface of the earth, 9.81 meters per second squared. But in reality, in this opening scene of this cheap horror movie, the camera zooms out and gets a view from outside of the enclosure in which you have woken up, and reveals that you're not on earth, but rather far from all planets and stars in empty outer space, being accelerated upward from your perspective by a rocket that you can't hear through the soundproof and vibration proof walls of your little prison, and that rocket is accelerating you upward at 9.81 meters per second squared. So from your perspective in the sealed room, you think that objects are falling down in a gravitational field, or that you have to work against a gravitational field to lift yourself off the floor, but in reality what's going on is the entire system is being pushed by an external force, experiencing an acceleration in one direction of 9.81 meters per second squared, which gives you the illusion inside the room that you're in a gravitational field, even though you're not. How would you be able to tell the difference between these two situations, a soundproof vibration proof, windowless, doorless room, with no external reference information to tell you that you're moving or not in outer space, and a gravitational field on earth under the same conditions, where yes you're on the surface of a planet, but you have no external information that tells you that. A ball dropped in either of those two environments will look and behave the same way. And it was this insight, or a variation on it, that led Einstein to realize that a constant acceleration due to gravity is no different from taking a reference frame and accelerating it at a constant rate. As a visual test of the equivalence principle, let's see if you can tell the difference between the following two situations, a zero gravity environment, and an environment that's in free fall in a gravitational field. Take a look at the video on the left, and the video on the right. Which one do you think is shot in a zero gravity environment? Which one do you think is shot in a free fall environment where a gravitational field is present? The answer is that neither of these is in a zero gravitational field environment. This may surprise you, maybe you recognized somebody in one of the videos and said aha that person's an astronaut, therefore this video must have been shot in zero gravity. But in fact, both of these videos are depicting life in a locally inertial reference frame in free fall in a gravitational field. The video on the right is shot in something known as a reduced gravity flight, an airplane that makes a parabolic arc through the sky and briefly enters free fall in the earth's gravitational field, close to the ground. The video on the left is shot in the International Space Station. The International Space Station may be far above the surface of the earth, but the acceleration due to gravity is actually quite strong in its orbital position. However, the International Space Station is orbiting the earth every 90 minutes, and as a result of this circular motion, it's actually in free fall constantly, it's just missing the earth, because it's moving to the side every time it falls down a little bit. It's almost impossible to the human eye to tell the difference between life in a free fall frame of reference in a gravitational field and life in a zero gravity environment. That's no accident, that's the equivalence principle in action. Einstein then defined the concept of a locally inertial frame by imagining not this situation I've described here, but a system in which a person for instance is in free fall in an external gravitational field. The concept of a locally inertial frame of reference is one in which all parts of the system are experiencing a constant acceleration due to gravity, but because all parts are accelerated the same way, it's as if the system is entirely free of any external forces. It's as if everything is in an inertial reference frame with no external forces because all of you are accelerating at the same rate at the same time. This is an incredible insight. It may not seem that impressive, but it frees you very suddenly from thinking of gravity and acceleration of an entire reference frame of things as different things. And it was this insight that freed Einstein to think about gravity in a completely new way as another aspect of space and time. The key idea here is that without external information, in any of the situations I've just described, being in free fall above the surface of the earth or being inside a sealed room with no windows and no doors that's vibration resistant and sound resistant, without external information there's absolutely no experiment you could do in any of those situations that will tell you that the system is either away from a gravitational source or simply in free fall in a gravitational field. These all seem like inertial reference frames as a result of that. Now, since there is no difference between gravitational acceleration and the act of changing a whole reference frame into a non-inertial reference frame, you can analyze phenomena in a situation for instance where an inertial frame of reference is considered instantaneously inertial, that is, although it experiences overall some acceleration, like taking a whole room, dropping a rocket to the bottom of it and accelerating the entire room and all its contents up at 9.81 meters per second squared, you're doing that equally and fairly to all parts of the system and so at any moment in time all elements of the frame will have the same velocity. Now of course if you take an object off the wall of this frame of reference, hold it out and drop it, it will appear to fall down because once you let go of it it's no longer part of the frame of reference, it's not bound to it in any way and so it will appear to fall down as if under the influence of an external gravitational field, but of course what's really happening is that the floor of the reference frame is being accelerated up toward the object, now freed from the bonds of the reference frame at 9.81 meters per second squared. Perspective is everything. It's the relativity of whether you're falling in a gravitational field or whether the floor is rushing up toward you at the same rate. It's the ambiguity in those perspectives that lead to the key insights that blossom into the general theory of relativity. So in order to help us to picture this, let's consider reference frames in the same way that we've done this in special relativity before. Let's imagine a frame that we're going to always take to be exactly and absolutely at rest. We choose which frame that is and then we define it as the rest frame. It's our choice to make. It doesn't matter which one we pick. I'm going to choose this one with X and Y coordinates as the absolute rest frame. Now the frame over here with our friendly observer in it, it's labeled as having its own axes, X prime and Y prime for instance, and I've exaggerated the X axis here only because I'm going to need some room on this as I start to play around with it. But at first, at time zero in our little thought experiment here, the rest frame and this frame are in the same state of motion. The velocity of what will become the moving frame is instantaneously zero at time zero. And so it too, instantaneously at time zero is a rest frame. But this frame, which I've labeled with prime notation, is actually experiencing a net constant acceleration A. And in the next instant of time, its velocity changes from zero to something non-zero. In this case, it goes to being a teensy bit above zero, a little differential of velocity dv above zero in the i-hat or positive X prime direction. So it was at rest, same as the actual rest frame. And an instant later, it is no longer instantaneously at rest. Instantaneously now, it represents a moving frame, S prime, at velocity dv relative to what we consider the actual rest frame. But that acceleration continues to act. And so in the next instant of time, the velocity is increased again to twice dv. And so now, while at this instant of time, it's another inertial reference frame, albeit with a different velocity relative to the rest frame, it is the result of an acceleration that has been acting on the system the whole time, from time zero to time one to time two. At each instant in time, this frame is inertial because it has a well-defined velocity at that moment with respect to the rest frame. But overall, we can clearly see that this is a noninertial reference frame, one that is experiencing a net acceleration, and a person in that frame would conclude overall that there must be some external force acting on the system because they will see objects freed from their frame of reference to behave as if an external force is acting on them. Now, let's start to dig a little bit into some of the implications of the more general view of space and time now that we freed ourselves and allowed for the equivalence of a gravitational force to a frame that's noninertial, experiencing an external acceleration by any means necessary. Now, using this imagery of frames that are instantaneously inertial, but overall noninertial frames of reference, let's analyze an observation of light that has been emitted during this period of slight accelerations of the frame of reference S'. So let's consider a light source, this black dot here that's pegged to the y-axis of frame S'. It's fixed in that frame. It's bolted to the wall. We can consider the y-axis to be a wall in the frame of reference. The x-axis is like the floor of the frame of reference. The person is firmly rooted on the frame of reference and they're experiencing only a slight acceleration. It doesn't totally knock them off their feet to be accelerated. It's a very dental acceleration. The speeds that we will consider always in these examples will be very much less than the speed of light. This will help us to get at the implications of general relativity without having to dig into the full general mathematics of relativity, which is much harder. So this is our situation. At time zero we have our happy observer. They're looking at this light source on the wall that could emit a pulse of light at any time. And in fact, at time zero, we're going to allow the light source to send out a wave of light. So at that moment, t equals zero, it emits a wave front. But at the same time, the frame is accelerating. It's been accelerating and instantaneously at time zero, its velocity happened to be zero. So it was in our rest frame as we've defined it. But at that moment, time zero, the light source pulses, emits a wave front, and that wave front, of course, being light, is going to travel at a velocity of exactly C from the perspective of any observer in any frame of reference. And it will travel from the left to the right from the above the origin of the S prime coordinate system where the light source is pegged toward the observer over here at some other coordinate along the horizontal axis in frame S prime. So we have a light wave, a wave front, traveling at C, released from its prison in the light source at time zero. Now at some time later, the light wave will cross the gap between the light source and the light observer, and the light observer will see it. But what will the observer see? Because in that time that it took the light wave to cross the gap, the frame has changed its state of motion from zero to some velocity dv. Will the observer see the light wave as it was emitted from the source, or will they see something else? Let's take stock of the key elements of this question from the picture that I showed you on the previous slide. First, the light was emitted originally from a source that was considered to be at rest. That light was emitted with frequency f at the source, f source, and wavelength lambda also defined at the rest frame of the source. Now the light source and observer remain a fixed distance apart the entire time in this question because they both accelerate together. The system is experiencing only gentle accelerations, not enough to knock the observer away from their spot on the x-axis, so they remain planted at their position. The light source is bolted to the wall. The whole thing is accelerating together at the same rate, so their state of motion is changing instantaneously in the same way at every moment of time. And so there is no change in the distance, call it capital L, between the source of the light and the observer of the light. Light travels at c 2.998 times 10 to the 8th meters per second, no matter the frame of reference in which it was observed. It was emitted in a frame that was at rest. It will be observed in a frame that is moving, but no matter the state of motion or the change of the state of motion of that frame of reference, the observer, if they were measuring the speed of this wave, is always going to say it moved at c. The light will take a finite non-zero time to travel to the observer from its source. It has to cross a gap. That's going to take some time. And by the time the light reaches the observer, they will have entered a state of non-zero velocity from the cartoon on the previous page. They went from zero to zero plus a little bit due to their acceleration, and I've called that little bit a differential of velocity, dv. Therefore, the light wave in the end will be observed in a frame of reference that is now moving with respect to the frame of the source, which had been a rest frame. What does this sound like? This sounds an awful lot like a Doppler shift problem. Light being viewed in a frame that's moving with respect to the original frame of ignition. This basic insight then guides the math within the framework of special relativity that we can do to calculate just with the observer we'll see. So let's do some very basic calculations with this, building on top of all of the stuff that we've been looking at over the previous lectures and time in our course. I have emphasized this before. I'm going to codify it now. I want us to assume that we are in an instantaneously inertial reference frame. That is, at any moment in time we have a definite velocity that's well defined, albeit changing from moment to moment to moment to moment. We want the velocity of that frame to be greatly less than c, not even close to the speed of light, less than 1% the speed of light, or even smaller. And that's so that we can have v over c, which we've previously defined as this nice number beta, to be much, much less than 1. This is going to come in handy very quickly in this problem. Let's assume, again, as I pointed out before, that the distance from the light source to the observer, which is fixed to this whole time, is some length l. Because the light source travels in the same frame as the observer, l remains constant the whole time. The light will take a time, which will denote delta t, t2 minus t1, t1 being the time of emission, t2 being the time of observation, and that's going to be given by l divided by c. Light has to travel across a gap of length l. It does so at a fixed speed c, the speed of light, the time that will take is l over c full stop. Now in that time, delta t, the frame of the observer will have accelerated by an amount a, up from rest to a velocity v. And we can actually then analyze this using the very same equations of motion from introductory physics, which are still valid here for the conditions that we're looking at. There we could relate initial velocity to final velocity by considering the acceleration of a system and the time over which the acceleration acts. This equation will do nicely. The final velocity v will be equal to the initial velocity v subscript zero, or v naught, plus a term that's the acceleration times the time that is passed over which the acceleration has acted. Now we can plug in some specifics here for us. v will be equal to zero, the initial velocity of our instantaneously inertial reference frame, plus the acceleration times l over c, which is delta t, the time it takes for the light wave to get from the source to the observer. This then leads us to the conclusion that v is equal to a times l divided by c, and if we transform this into an expression for beta, we find that beta, which is v divided by c, is given by the acceleration times the distance divided by the speed of light squared. Now let's take this information and let's put it into the context of the Doppler shift, the relativistic Doppler shift, specifically the special relativistic Doppler shift. So we're going to treat the case of small velocities relative to light. Beta is a small number. The Doppler shift of the light wave by the time the observer sees it will be given simply by what we did before. We take the frequency of the source, we multiply it because the observer is in a frame that becomes a frame that's moving away from where the source was. We have to multiply by the square root of the quantity one minus beta divided by one plus beta. This represents a lengthening ultimately of the wavelength of the light, a red shift. But we want to get acceleration of the frame, the distance between the light source and the observer, and the speed of light into this equation. We want to put these things from our picture into this equation, and the way we can do that is by doing some binomial expansions of the numerator, the square root of one minus beta, and the denominator one over the square root of one plus beta. Well, if you do those two things and multiply them, do the binomial expansion of the square root of one minus beta of one over the square root of one plus beta, multiply those together, you find you get expansion products that look like this, one minus a half beta plus terms that are higher order in beta, and one minus a half beta, the same thing again, plus again, higher order terms in beta, which I've just left out, but indicated that they're supposed to be there from these three dots. The product of these things is multiplied by the frequency at the source of emission. Now, because we're working in the case that beta is a number much, much, much, much smaller than one, because v is much, much, much, much smaller than c, we only have to keep the leading terms in all of this, and if we multiply out this product and then only keep the leading terms in beta, we wind up finding out that this product is approximately equal to just one minus beta, all times the frequency at the source of emission. Now, we have an expression for beta in terms of the acceleration of the frame, the distance between the source and the observer, and the speed of light, and if we plug that in, we get a final form of this approximate equation for the frequency that the observer should see. The observer, in the moment after the light has been transmitted, they accelerate, they get up to a velocity v relative to where the source had been, and then they observe the light, they will see the frequency shifted by an amount of one minus the quantity Al over c squared. This represents a shortening of, this represents a decrease in the frequency relative to the source, or an increase in the wavelength of the light. You can play around with this yourself and convince yourself that that's the case, but we basically conclude that the observer, who now, at the moment of observation, has been put into a new frame, that's not that in which the light source was at rest, when that emission had originally occurred, will now observe the light to appear shifted from its source frequency, and in this case, it's a red shift. If the observer, when our frame accelerating in the opposite direction, in the direction from the observer toward the source, rather than from the direction of the source toward the observer, so a becomes minus a, then the light would instead appear blue shifted, shifted to smaller frequencies, or shorter wavelengths. So let's think about light as viewed in an accelerating frame of reference. We've found by making this approximation that we have a frame of reference that's all accelerated at once, so that the light that was emitted at time zero is observed by an observer in a frame that's no longer at rest with respect to where the source of emission had been, that the observer will see a frequency as they accelerate to the right in the direction from the source toward the observer, they'll observe a reduced frequency of the light, a lengthening of the wavelength. But let's dig back to the equivalence principle. The equivalence principle states that there is no difference between an entire frame of reference that's all experiencing an acceleration due to some external force, and a frame of reference that is merely experiencing an external gravitational acceleration. As a result of the equivalence of these two things, one is forced to conclude that the shifting of light must also occur in a gravitational field of acceleration. In other words, if the source of that acceleration is gravity, you know, for instance a equals g times the mass of the earth divided by the radius of the earth's square, because you're standing on the surface of the earth, that's just 9.81 meters per second square, the old little g from introductory physics. And imagine instead we're viewing light from a source above us, we are downstream in the gravitational field and there is a light source down on the ground below us, sort of upstream in the gravitational field. We would conclude that the light, as we observe it, emitted up from the ground toward our eye, must be shifted, in this case a red shift, in frequency. This phenomenon is real. It has been confirmed repeatedly by experiments over and over and over again, and we'll look at some of those through problem solving in the class. It's a real phenomenon. It must be taken into account when you are dealing with electromagnetic radiation and gravitational fields, and it's known as the gravitational red shift, or depending on the problem is set up, gravitational blue shift of light. In this example, if we were laying on the ground looking up at a source that's above us and looking at light that's emitted down toward us, because we are further upstream in the gravitational field of acceleration, we would see the light in that case as blue shifted. It's equivalent to switching around the acceleration sign. Now, this very same phenomenon, the red shifting or blue shifting of light merely because of its transmission in a field of gravitational acceleration, has other implications, including for the very nature of the passage of time in different parts of a gravitational field of acceleration. So by implication from this previous example, the Doppler shifting of light by a gravitational field, one can also predict that time itself will pass at different rates, at different heights, different locations, in a uniform gravitational field. We saw that the frequency of light in a gravitational field is altered depending on the degree of acceleration. If I increase the acceleration of a frame of reference, or equivalently, increase the amount of gravitational acceleration a system experiences, I will increase the Doppler shifting effect. Frequency, of course, looking back at the discussion of waves and the Doppler shift and other things related to waves, frequency is a measure of the rate at which events happen, the time between events, effectively. So consider observing time at a height zero above the surface of the earth. We'll call that person the lower observer, somebody right at ground level looking at time passing, say by looking at pulses of light, or ticks of a clock, or something like that. And instead, a person who's way higher up, more upstream in a gravitational field of acceleration, a higher observer, also looking at their clock or their light pulsing or ticking away. From our exploration of frequency and period, we know that the frequency of a wave is given simply by one over the period of the wave. You can think of that as the passage of time between regularly spaced events. So the period is just a difference in time, it's a delta t. And so really frequency is another way of saying that we're looking at one over a time difference between regularly spaced events. In other words, frequency is really probing time structure. Now if we were to be looking at the time between regular events at our higher altitude in the gravitational field, this would be related to one over the frequency of events at that higher altitude. And we already know how to relate those through the Doppler shift to the frequency of events at the lower altitude. We just have to take this Doppler shifting equation again, do the binomial expansion, and we find out that we are just multiplying the time duration at the lower altitude by a quantity one minus beta. And because we are experiencing a gravitational acceleration here, a height h above the surface of the earth, this is equivalent to one minus gh over c squared in this approximation. One minus say 9.81 meters per second squared times your height above the surface of the earth divided by the speed of light squared. You take that quantity and you multiply it by the duration of time between regularly spaced events at lower altitude, and you get the time at higher altitude. And so as a consequence of this, we expect time to pass more slowly for observers who are lower down in a gravitational field. If you were to take this to some extreme, imagine a person deep down in a gravitational field, they might experience an hour, but a person higher up in the gravitational field might observe that days, weeks, or months pass depending on the degree of difference of location in the gravitational field. Time that passes higher up in a gravitational field is always multiplied by a number whose value is less than or equal to one, meaning that less time passes lower down in the field. This is a real effect, and this effect has been confirmed experimentally over and over and over again. And it plays a major role in the operation of key systems to modern existence, such as the global positioning satellite or GPS system. All modern navigation typically relies on a system of about 24 satellites. Each satellite orbits the earth twice per day, so it's moving very fast around the earth as a result of that. These are not so-called geosynchronous or geostationary satellites that always sit above the same point on the surface of the earth rather the GPS satellites orbit and they make about two rotations around the earth per day. Three satellites at any time are required to make a triangulation measurement on the surface of the earth and they do this using very precise clocks that they carry along with them that have been synchronized to clocks on the ground. And this system allows you to make position measurements on the surface of the earth but the problem is, first of all, that those satellites are traveling actually very fast relative to the surface of the earth, so they experience a special relativistic time dilation. Observers on the ground would claim that their clocks are running a bit more slowly than an equivalent clock on the ground because they're moving and people on the ground argue that they're at rest. So there's a special relativistic time dilation effect. But in addition, because humans who are down on the ground making these observations are lower in a gravitational field, an observer on the GPS satellite would argue that, well, okay that's true, there's a special relativistic effect, but there's also a gravitational effect, a general relativistic effect, because the clocks on the earth that we're supposed to be synchronized to are lower down in a gravitational field than the clocks in orbit around the earth, and so for those clocks there's a general gravitational slowing of time and these two factors must be taken into account in the modern GPS system, and in fact any guideline document that you look at for engineering systems for the GPS network will warn you about these corrections, spell them out for you, and tell you how to do them so that you can properly synchronize clocks taking into account all of these time effects between the ground and in orbit around the earth. These are real effects with real consequences on things like basic day-to-day navigation and without the general and special theory of relativity we would never have understood these had we launched a GPS system before understanding space and time at this level we would have failed to construct a working GPS system. Now one other implication of general relativity, and this can be relatively kind of quickly looked at in a cartoonish way by referring to our accelerated frame of reference, our sealed vessel. This other effect that we'll take a look at here is the deflection of light by a gravitational field. Now this might seem novel to you, but in reality the deflection of light by a gravitational field, the falling of light near the surface of the earth, was not a new idea in the time of Albert Einstein. It was actually quickly realized within certainly decades or a century after the work of Isaac Newton had established the laws of mechanics and gravitation that since all objects regardless of their mass fall at the same rate in a uniform gravitational field. Think of dropping a wadded up ball of paper and a bowling ball at the same time from a few feet above the ground. If you drop them so that their bottoms are starting at the same height they'll hit the ground at the same time. The mass of the paper and the mass of the bowling ball seem to play no role in the rate at which their velocity changes as they head toward the surface of the earth. Well if mass doesn't matter for gravitational acceleration then even one might argue a massless phenomenon like light should fall in a gravitational field. Now the specific reason why this would happen was put on much firmer footing thanks to the equivalence principle and I'll walk through an example of that argument here. So consider the cartoon at the right. We have our sealed vessel it's sound and vibration proof no windows no doors no way of knowing whether you're on earth or far out in space away from all planets and stars. Now in reality this system is being accelerated upward by a rocket you can neither hear nor feel nor see and it's doing so at 9.81 meters per second squared constantly. So you're in this sealed room there's a light source on one wall and you can push a button and fire away front a pulse of light across the room so that it strikes the wall on the other side. Now at the moment that the pulse is emitted and that's illustrated here on the left the line connecting its location of emission points straight across the room to a point on the other side of the wall but by the time that the wave reaches the other wall and that's illustrated here on the right the wave freed from its connection to this frame of reference that in the meantime has changed its state of motion the light wave will travel on that absolute straight line but from the perspective of a person inside the vessel looking at where the light wave strikes the wall if they had very precise equipment or if the speed of light were much slower than it actually is then they would actually observe that the light wave strikes the wall at a point that's lower than where it was emitted from so in an external frame of reference that light traveled on a real straight line but the frame moved up in the time during which it crossed the room from a perspective of an observer inside the frame who doesn't know that any of this is going on they see the light wave strike the wall at a lower point some vertical displacement below where it was expected to strike that is at the level of the emission source so the light wave reaches the wall but it does so in this case at a lower point now by the equivalence principle there is absolutely no difference between this frame of reference being accelerated by a rocket or a similar sealed room that's sitting on a planet experiencing a gravitational acceleration downward of 9.81 meters per second and so because of the equivalence of an accelerated frame of reference and a frame that's merely experiencing a gravitational acceleration light must also fall in a gravitational field because there's no distinction between these two cases it turns out that this is actually generalizable to anybody with mass bending the path of light and this is actually the key insight that Albert Einstein's general theory of space and time the general theory of relativity had that helped to distinguish it from Isaac Newton's original theory of acceleration and gravitation in Newton's theory the deflection of say starlight around a massive body like another star is smaller than the deflection predicted in general relativity which is supposed to be the more correct description of space and time and the way that energy and matter respond to space and time so in the general theory of relativity the degree of deflection of light around a massive object by falling in a gravitational field if you will is twice as big as predicted in Newton's original mechanical theory combined with his law of gravitation that's a key distinction between the two ideas the general theory of relativity and the old theory of mechanics married to the law of gravitation it was that prediction that was tested in the late nineteen teens and led to the confirmation that Einstein's work was probably the correct description of space and time and energy and matter and this catapulted Einstein into global fame it also led to a host of other predictions for other interesting phenomena because light can be deflected by large masses we could imagine being able to see objects that shouldn't be visible to us using larger rays of telescopes and looking out into the distant sky we can look for cases where we see a background object whose light has been bent around a foreground object allowing it to reach our telescope this so-called gravitational lensing allows astronomers not only to see objects that would otherwise be obscured behind other foreground objects things that sit between us and the thing we want to look at but because the general theory of relativity gives very specific relationships between the amount of mass and the degree of the deflection of light one can use the deflection of light itself to measure the mass of objects with which you can never hope to have physical contact gravitational lensing is one of the many tools that general relativity gives to us as human beings to better understand the universe even parts of the universe that are very old very distant or both so as you can see the general theory of relativity has some fairly impressive large-scale implications if you remember something back from your calculus the second derivative of something with respect to something else tells you about the curvature of the system that you're studying with the derivative now we've considered the fact that space and time are really part of a singular structure they really should be thought of as part of one four-dimensional framework which is called spacetime in special relativity we see that space we see that space measurements in one frame can turn into time measurements in another space and time are constantly getting traded for one another or tangled up in one another in calculations of motion from one inertial frame of reference to another there's a link between space and time and that link comes from the fact that they're really part of one interchangeable four-dimensional framework spacetime and it's in this framework that matter and energy can be described to move and change so general relativity is really a theory of spacetime a general broad theory of space and time and ultimately it concludes that what we call the force of gravity is really due to the fact that mass and energy cause space and time to curve or in more colloquial language bend or warp the second derivative is a sign of curvature and so it should have been a clue that since there's no distinction between accelerating a frame of reference or subjecting that same frame of reference to an external gravitational field there must be no difference between curvature and gravity and in fact that's one of the broad conclusions of the general theory of relativity energy and matter curve space and time and so other bits of matter or even light that travel past that object that's bending spacetime will follow the curvature of spacetime and the result of this is that from our perspective in three dimensions they appear to accelerate what is a ball doing when you hold it out at shoulder height and drop it it's not being pulled down by the mass of the earth rather it's following a path in spacetime that's curved due to the presence of the mass energy of the earth bending that space and time that is what gravity is that is what einstein was able to achieve the very thing that isaac newton could not grasp the nature of gravity curvature of spacetime space and time tell energy and matter how to move energy and matter tell space and time how to bend or curve or warp this elegant summary paraphrased from its author is a beautiful way of remembering the implications of the general theory of relativity writ large and it comes from the mind of luminary theoretical physicist john archibald wheeler the universe is observed to expand in all directions at once and the more distant an object you view in the universe the faster it appears to be moving away from us this tells us that overall spacetime is curved now on the grandest scales the largest distances that we can reasonably observe in the universe the universe's space itself appears to be very flat and smooth but just because space is flat and smooth overall doesn't mean that spacetime is and the expansion of the universe's evidence that spacetime itself is curved the curvature of spacetime leads us to conclusions about the origin and the fate of the entirety of the universe and it tells us that the universe as we know it now space and time and energy and matter was born 13.78 billion years ago in an event we have yet to fully understand but which is described by the phrase the big bang let's review what we have learned in this lecture we've looked at the transition in thinking from the special to the general theory of relativity we've looked at some implications of the general theory of relativity on physical phenomena specifically we've considered what it means for light to travel in a gravitational field from a higher to a lower vantage point in that field we've concluded that light should Doppler shift either red shift or blue shift depending on the direction in the field that you observe it we've also concluded that light should be bent in its path of travel in a gravitational field and we've drawn all of these conclusions by using the equivalence principle to map behavior in an accelerated frame of reference onto a frame that's experiencing an external gravitational acceleration we've then looked at some of the large-scale implications for space and time the bending of distant starlight around massive objects that intervene between us in the universe the use of the warping of space and time and the bending of light to infer the mass of objects that we can never hope to weigh by putting them on a scale and the overall implications for the nature of space time as a framework in which energy and matter play out the fact that energy and matter tells space and time how to curve and the curvature of space and time tells energy and matter how to move and how the overall curvature of space time indicates to us the origin and possible fate of the entire universe itself these grand themes all stem from the elegant thinking of a brilliant physicist who accepted observational evidence from experiment about the nature of light thought deeply about the world around him learned the math necessary to describe the universe and in that elegant language spoke a volume about the cosmos that we are still reeling from today