General Relativity

This book is the fourth volume of The Theoretical Minimum series. The first volume, The Theoretical Minimum: What You Need to Know to Start Doing Physics, covered classical mechanics, which is the core of any physics education. We will refer to it from time to time simply as volume 1. The second book, volume 2, explains quantum mechanics and its relationship to classical mechanics. Volume 3 covers special relativity and classical field theory. This fourth volume expands on that to explore general relativity.

Also by Leonard Susskind

Special Relativity and Classical Field Theory: The Theoretical Minimum

Quantum Mechanics: The Theoretical Minimum

The Theoretical Minimum: What You Need to Know to Start Doing Physics

The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics

The Cosmic Landscape: String Theory and the Illusion of Intelligent Design

An Introduction to Black Holes, Information and the String Theory Revolution: The Holographic Universe

Lecture 1: Equivalence Principle and Tensor Analysis

Andy: So if I am in an elevator and I feel really heavy, I can’t know whether the elevator is accelerating or you mischievously put me on Jupiter?

Lenny: That’s right, you can’t.

Andy: But, at least on Jupiter, if I keep still, light rays won’t bend.

Lenny: Oh yes they will.

Andy: Hmm, I see.

Lenny: And if you are falling into a black hole, beware, things will get really strange. But, don’t worry, I’ll shed some light on this.

Andy: Er, bent or straight?

Introduction

Equivalence principle

Accelerated reference frames

Curvilinear coordinate transformations

Effect of gravity on light

Tidal forces

Non-Euclidean geometry

Riemannian geometry

Metric tensor

Mathematical interlude: Dummy variables

Mathematical interlude: Einstein summation convention

First tensor rule: Contravariant components of vectors

Mathematical interlude: Vectors and tensors

Second tensor rule: Covariant components of vectors

Covariant and contravariant components of vectors and tensors

Introduction

General Relativity is the fourth volume in The Theoretical Mini-mum (TTM) series. The first three were devoted respectively to classical mechanics, quantum mechanics, and special relativity and classical field theory. The first volume laid out the Lagrangian and Hamiltonian description of physical phenomena and the principle of least action, which is one of the fundamental principles underlying all of physics (see volume 3, lecture 7 on fundamental principles and gauge invariance). They were used in the first three volumes and will continue in this and subsequent ones.

Physics extensively uses mathematics as its toolbox to construct formal, quantifiable, workable theories of natural phenomena. The main tools we used so far are trigonometry, vector spaces, and calculus, that is, differentiation and integration. They have been explained in volume 1 as well as in brief refresher sections in the other volumes. We assume that the reader is familiar with these mathematical tools and with the physical ideas presented in volumes 1 and 3. The present volume 4, like volumes 1 and 3 (but unlike volume 2), deals with classical physics in the sense that no quantum uncertainty is involved.

We also began to make light use of tensors in volume 3 on special relativity and classical field theory. Now with general relativity we are going to use them extensively. We shall study them in detail. As the reader remembers, tensors generalize vectors. Just as vectors have different representations, with different sets of numbers (components of the vector) depending on the basis used to chart the vector space they form, this is true of tensors as well. The same tensor will have different components in different coordinate systems. The rules to go from one set of components to another will play a fundamental role. Moreover, we will work mostly with tensor fields, which are sets of tensors, a different tensor attached to each point of a space. Tensors were invented by Ricci-Curbastro and Levi-Civita¹ to develop work of Gauss ² on curvature of surfaces and Riemann³ on non-Euclidean geometry. Einstein⁴ made extensive use of tensors to build his theory of general relativity. He also made important contributions to their usage: the standard notation for indices and the Einstein summation convention.

In Savants et écrivains (1910), Poincaré⁵ writes that “in mathematical sciences, a good notation has the same philosophical importance as a good classification in natural sciences.” In this book we will take care to always use the clearest and lightest notation possible.

Equivalence Principle

Einstein’s revolutionary papers of 1905 on special relativity deeply clarified and extended ideas that several other physicists and mathematicians – Lorentz,⁶ Poincaré, and others – had been working on for a few years. Einstein investigated the consequences of the fact that the laws of physics, in particular the behavior of light, are the same in different inertial reference frames. He deduced from that a new explanation of the Lorentz transformations, of the relativity of time, of the equivalence of mass and energy, etc.

After 1905, Einstein began to think about extending the principle of relativity to any kind of reference frames, frames that may be accelerating with respect to one another, not just inertial frames. An inertial frame is one where Newton’s laws, relating forces and motions, have simple expressions. Or, if you prefer a more vivid image, and you know how to juggle, it is a frame of reference in which you can juggle with no problem – for instance in a railway car moving uniformly, without jerks or accelerations of any sort. After ten years of efforts to build a theory extending the principle of relativity to frames with acceleration and taking into account gravitation in a novel way, Einstein published his work in November 1915. Unlike special relativity, which topped off the work of many, general relativity is essentially the work of one man.

We shall start our study of general relativity pretty much where Einstein started. It was a pattern in Einstein’s thinking to start with a really simple elementary fact, which almost a child could understand, and deduce these incredibly far-reaching consequences. We think that it is also the best way to teach it, to start with the simplest things and deduce the consequences.

So we shall begin with the equivalence principle. What is the equivalence principle? It is the principle that says that gravity is in some sense the same thing as acceleration. We shall explain precisely what is meant by that, and give examples of how Einstein used it. From there, we shall ask ourselves: what kind of mathematical structure must a theory have for the equivalence principle to be true? What kind of mathematics must we use to describe it?

Most readers have probably heard that general relativity is a theory not only about gravity but also about geometry. So it is interesting to start at the beginning and ask what is it that led Einstein to say that gravity has something to do with geometry. What does it mean to say that “gravity equals acceleration”? You all know that if you are in an accelerated frame of reference, say, an elevator accelerating upward or downward, you feel an effective gravitational field. Children know this because they feel it.

What follows may be overkill, but making some mathematics out of the motion of an elevator is useful to see in a very simple example how physicists transform a natural phenomenon into mathematics, and then to see how the mathematics is used to make predictions about the phenomenon.

Before proceeding, let’s stress that the following study on an elevator, and the laws of physics as perceived inside it, is simple. Yet it is a first presentation of very important concepts. It is fundamental to understand it very well. Indeed, we will often refer to it. In lectures 4 to 9, it will strongly help us understand acceleration, gravitation, and how gravitation “warps” space-time.

So let’s imagine the Einstein thought experiment where somebody is in an elevator; see figure 1. In later textbooks, it got promoted to a rocket ship. But I have never been in a rocket ship, whereas I have been in an elevator. So I know what it feels like when it accelerates or decelerates. Let’s say that the elevator is moving upward with a velocity v.

Figure 1: Elevator and two reference frames.

So far the problem is one-dimensional. We are only interested in the vertical direction. There are two reference frames: one is fixed with respect to Earth. It uses the coordinate z. The other is fixed with respect to the elevator. It uses the coordinate z′. A point P anywhere along the vertical axis has two coordinates: coordinate z in the stationary frame, and coordinate z′ in the elevator frame. For instance, the floor of the elevator has coordinate z′ = 0. Its z-coordinate is the distance L, which is obviously a function of time. So we can write for any point P

We are going to be interested in the following question: if we know the laws of physics in the frame z, what are they in the frame z′?

One warning about this lecture: at least at the start, we are going to ignore special relativity. This is tantamount to saying that we are pretending that the speed of light is infinite, or that we are talking about motions so slow that the speed of light can be regarded as infinitely fast. You might wonder: if general relativity is the generalization of special relativity, how did Einstein manage to start thinking about general relativity without including special relativity?

The answer is that special relativity has to do with very high velocities, while gravity has to do with heavy masses. There is a range of situations where gravity is important but high velocities are not. So Einstein started out thinking about gravity for slow velocities, and only later combined it with special relativity to think about the combination of fast velocities and gravity. And that became the general theory.

Let’s see what we know for slow velocities. Suppose that z′ and z are both inertial reference frames. That means, among other things, that they are related by uniform velocity:

We have chosen the coordinates such that when t = 0, they line up. At t = 0, for any point, z and z′ are equal. For instance, at t = 0 the elevator’s floor has coordinate 0 in both frames. Then the floor starts rising, its height z equaling vt. So for any point we can write equation (1). In view of equation (2), it becomes

Notice that this is a coordinate transformation involving space and time. For readers who are familiar with volume 3 of TTM on special relativity, this naturally raises the question: what about time in the reference frame of the elevator? If we are going to forget special relativity, then we can just say that t′ and t are the same thing. We don’t have to think about Lorentz transformations and their consequences. So the other half of the coordinate transformation would be t′ = t.

We could also add to the stationary frame a coordinate x going horizontally and a coordinate y jutting out of the page. Correspondingly, coordinates x′ and y′ could be attached to the elevator; see figure 2. The x-coordinate will play a role in a moment with a light beam. As long as the elevator is not sliding horizontally, x′ and x can be taken to be equal. Same for y′ and y.

For the sake of clarity of the drawing in figure 2, we offset a bit the elevator to the right of the z-axis. But think of the two vertical axes as actually sliding on each other, and at t = 0 the two origins O and O′ coincide. Once again, the elevator moves only vertically.

Figure 2: Elevator and two reference frames, three axes in each case.

Finally our complete coordinate transformation is

It is a coordinate transformation of space-time coordinates. For any point P in space-time, it expresses its coordinates in the moving reference frame of the elevator as functions of its coordinates in the stationary frame. It is rather trivial. Only one coordinate, namely z, is involved in an interesting way.

Let us look at a law of physics expressed in the stationary frame. Take Newton’s law of motion F = ma applied to an object or a particle. The acceleration a is , where z is the vertical coordinate of the particle. So we can write

As we know, is the second time derivative of z with respect to time – it is called the vertical acceleration – and F of course is the vertical component of force. The other components we will take to be zero. Whatever force is exerted, it is exerted vertically. What could this force be due to? It could be related to the elevator or not. There could be some charge in the elevator pushing on the particle. Or it could just be a force due to a rope attached to the ceiling and to the particle that pulls on it. There could be a field force along the vertical axis. Any kind of force could be acting on the particle. Whatever the causes, we know from Newton’s law that the equation of motion of the particle, expressed in the original frame of reference, is given by equation (5).

What is the equation of motion expressed in the primed frame? This is very easy. All we have to do is figure out what the original acceleration is in terms of the primed acceleration. What is the primed acceleration? It is the second derivative with respect to time of z′. Using the first equation in equations (4)

one differentiation gives

and a second one gives

The accelerations in the two frames of reference are the same.

All this should be familiar. But I want to formalize it to bring out some points. In particular, I want to stress that we are doing a coordinate transformation. We are asking how the laws of physics change in going from one frame to another. What can we now say about Newton’s law in the primed frame of reference? We substitute for in equation (5). As they are equal, we get

We found that Newton’s law in the primed frame is exactly the same as Newton’s law in the unprimed frame. That is not surprising. The two frames of reference are moving with uniform velocity relative to each other. If one of them is an inertial frame, the other is an inertial frame. Newton taught us that the laws of physics are the same in all inertial frames. It is sometimes called the Galilean principle of relativity. We just formalized it.

Let’s turn to an accelerated reference frame.

Accelerated Reference Frames

Suppose that L(t) from figure 1 is increasing in an accelerated way. The height of the elevator’s floor is now given by

We use the letter g for the acceleration because we will discover that the acceleration mimics a gravitational field – as we feel when we take an elevator and it accelerates. We know from volume 1 of TTM on classical mechanics or from high school, that this is a uniform acceleration. Indeed, if we differentiate L(t) with respect to time, after one differentiation we get

which means that the velocity of the elevator increases linearly with time. After a second differentiation with respect to time, we get

This means that the acceleration of the elevator is constant. The elevator is uniformly accelerated upward. The equations connecting the primed and unprimed coordinates are different from equations (4). The transformation for the vertical coordinates is now

The other equations in equations (4) don’t change:

These four equations are our new coordinate transformation to represent the relationship between coordinates that are accelerated relative to each other.

We will continue to assume that in the z, or unprimed, coordinate system, the laws of physics are exactly what Newton taught us. In other words, the stationary reference frame is inertial, and we have . But the primed frame is no longer inertial. It is in uniform acceleration relative to the unprimed frame. Let’s ask what the laws of physics are now in the primed frame of reference. We have to do the operation of differentiating twice over again on equation (8). We know the answer:

Ah ha! Now the primed acceleration and the unprimed acceleration differ by an amount g. To write Newton’s equations in the primed frame of reference, we multiply both sides of equation (9) by m, the particle mass, and we replace m by F. We get

We have arrived at what we wanted. Equation (10) looks like a Newton equation, that is, mass times acceleration is equal to some term. That term, F – mg, we call the force in the primed frame of reference. You notice, as expected, that the force in the primed frame of reference has an extra term: the mass of the particle times the acceleration of the elevator, with a minus sign.

What is interesting about the “fictitious force” – mg, in equation (10), is that it looks exactly like the force exerted on the particle by gravity on the surface of the Earth or the surface of any kind of large massive body. That is why we called the acceleration g. The letter g stood for gravity. It looks like a uniform gravitational field. Let me spell out in what sense it looks like gravity. The special feature of gravity is that gravitational forces are proportional to mass – the same mass that appears in Newton’s equation of motion. We sometimes say that the gravitational mass is the same as the inertial mass. That has deep implications. If the equation of motion is

and the force itself is proportional to mass, then the mass cancels in equation (11). That is a characteristic of gravitational forces: for a small object moving in a gravitational force field, its motion doesn’t depend on its mass. An example is the motion of the Earth about the Sun. It is independent of the mass of the Earth. If you know where the Earth is at time t, and you know its velocity at that time, then you can predict its trajectory. You don’t need to know what the Earth’s mass is.

Equation (10) is an example of fictitious force – if you want to call it that – mimicking the effect of gravity. Most people before Einstein considered this largely an accident. They certainly knew that the effect of acceleration mimics the effect of gravity, but they didn’t pay much attention to it. It was Einstein who said: look, this is a deep principle of nature that gravitational forces cannot be distinguished from the effect of an accelerated reference frame.

If you are in an elevator without windows and you feel that your body has some weight, you cannot say whether the elevator, with you inside, is resting on the surface of a planet or, far away from any massive body in the universe, some impish devil is accelerating your elevator. That is the equivalence principle. It extends the relativity principle, which said you can juggle in the same way at rest or in a railway car in uniform motion. With a simple example, we have equated accelerated motion and gravity. We have begun to explain what is meant by the sentence: “gravity is in some sense the same thing as acceleration.”

We have to discuss this result a bit, though. Do we really believe it totally or does it have to be qualified? Before we do that, let’s draw some pictures of what these various coordinate transformations look like.

Curvilinear Coordinate Transformations

Let’s first consider the case where L(t) is proportional to t. That is when we have

In figure 3, every point – also called event – in space-time has a pair of coordinates z and t in the stationary frame and also a pair of coordinates z′ and t′ in the elevator frame. Of course, t′ = t and we left out the two other spatial coordinates x and y, which don’t change between the stationary frame and the elevator. We represented the time trajectories of fixed z with dotted lines and of fixed z′ with solid lines.

A fundamental idea to grasp is that events in space-time exist irrespective of their coordinates, just as points in space don’t depend on the map we use. Coordinates are just some sort of convenient tags. We can use whichever we like. We’ll stress it again after we have looked at figures 3 and 4.

Figure 3: Linear coordinate transformation. The coordinates (z′, t′) are represented in the basic coordinates (z, t). An event is a point on the page. It has one set of coordinates in the (z, t) frame and another set in the (z′, t′)frame. Here the transformation is simple and linear.

That is called a linear coordinate transformation between the two frames of reference. Straight lines go to straight lines, not surprisingly since Newton tells us that free particles move in straight lines in an inertial frame of reference. What is a straight line in one frame had therefore better be a straight line in the other frame. Not only do free particles move in straight lines in space, when we add x and y, but their trajectories are straight lines in space-time – straight in space and with uniform velocity.

Let’s do the same thing for the accelerated coordinate system. The transformation equation is now equation (8) linking z′ and z. The other coordinates don’t change. Again, in figure 4, every point in space-time has two pairs of coordinates (z, t) and (z′, t′). The time trajectories of fixed z, represented with dotted lines, don’t change. But now the time trajectories of fixed z′ are parabolas lying on their side. We can even represent negative times in the past. Think of the elevator that was initially moving downward with a negative velocity but a positive acceleration g (in other words, slowing down). Then the elevator bounces back upward with the same acceleration g. Each parabola is just shifted relative to the previous one by one unit to the right.

Figure 4: Curvilinear coordinate transformation.

What figure 4 illustrates is, not surprisingly, that straight lines in one frame are not straight lines in the other frame. They become curved lines. As regards the lines of fixed t or fixed t′, they are of course the same horizontal straight lines in both frames. We haven’t represented them.

We should view figure 4 as just two sets of coordinates to locate each point in space-time. One set of coordinates has straight axes, while the second – represented in the first frame – is curvilinear. Its lines z′ = constant are actually curves, while its lines t′ = constant are horizontal straight lines. So it is a curvilinear coordinate transformation.

Let’s insist on the way to interpret and use figure 4 because it is fundamental to understand it very well if we want to understand the theory of relativity – special relativity and even more importantly general relativity. The page represents space-time – here, one spatial dimension and one temporal dimension.

Points (= events) in space-time are points on the page. An event does not have two positions on the page, i.e., in space-time. It has only one position