General relativity is the theory of space and time and gravity. The essence of the theory is simple: gravity is geometry. The effects that we attribute to the force of gravity are due to the bending and warping of spacetime, from falling cats, to orbiting spinning planets, to the motion of the cosmos on the grandest scale. The purpose of these lectures is to explain this.
Before we jump into a description of curved spacetime, we should first explain why Newton’s theory of gravity, a theory which served us well for 250 years, needs replacing. The problems arise when we think about disturbances in the gravitational field. Suppose, for example, that the Sun was to explode. What would we see? Well, for 8 glorious minutes – the time that it takes light to reach us from the Sun – we would continue to bathe in the Sun’s light, completely oblivious to the fate that awaits us. But what about the motion of the Earth? If the Sun’s mass distribution changed dramatically, one might think that the Earth would start to deviate from its elliptic orbit. But when does this happen? Does it occur immediately, or does the Earth continue in its orbit for 8 minutes before it notices the change?
Of course, the theory of special relativity tells us the answer. Since no signal can propagate faster than the speed of light, the Earth must continue on its orbit for 8 minutes. But how is the information that the Sun has exploded then transmitted? Does the information also travel at the speed of light? What is the medium that carries this information? As we will see throughout these lectures, the answers to these questions forces us to revisit some of our most basic notions about the meaning of space and time and opens the to door to some of the greatest ideas in modern physics such as cosmology and black holes.
There is a well trodden path in physics when trying to understand how objects can influence other objects far away. We introduce the concept of a field. This is a physical quantity which exists everywhere in space and time; the most familiar examples are the electric and magnetic fields. When a charge moves, it creates a disturbance in the electromagnetic field, ripples of which propagate through space until they reach other charges. To develop a causal theory of gravity, we must introduce a gravitational field that responds to mass in some way.
It’s a simple matter to cast Newtonian gravity in terms of a field theory. A particle of mass $m$ experiences a force that can be written as
$\mathbf{F}=-m\nabla \mathrm{\Phi}$ |
The gravitational field $\mathrm{\Phi}(\mathbf{r},t)$ is determined by the surrounding matter distribution which is described by the mass density $\rho (\mathbf{r},t)$. If the matter density is static, so that $\rho (\mathbf{r})$ is independent of time, then the gravitational field obeys
${\nabla}^{2}\mathrm{\Phi}=4\pi G\rho $ | (0.1) |
with Newton’s constant $G$ given by
$G\approx 6.67\times {10}^{-11}{\mathrm{m}}^{3}{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$ |
This equation is simply a rewriting of the usual inverse square law of Newton. For example, if a mass $M$ is concentrated at a single point we have
$\rho (\mathbf{r})=M{\delta}^{3}(\mathbf{r})\mathit{\hspace{1em}\hspace{1em}\u2006}\Rightarrow \mathit{\hspace{1em}\hspace{1em}\u2006}\mathrm{\Phi}=-{\displaystyle \frac{GM}{r}}$ |
which is the familiar gravitational field for a point mass.
The question that we would like to answer is: how should we modify (0.1) when the mass distribution $\rho (\mathbf{r},t)$ changes with time? Of course, we could simply postulate that (0.1) continues to hold even in this case. A change in $\rho $ would then immediately result in a change of $\mathrm{\Phi}$ throughout all of space. Such a theory clearly isn’t consistent with the requirement that no signal can travel faster than light. Our goal is to figure out how to generalise (0.1) in a manner that is compatible with the postulates of special relativity. The end result of this goal will be a theory of gravity that is compatible with special relativity: this is the general theory of relativity.
The goal that we’ve set ourselves above looks very similar to the problem of finding a relativistic generalization of electrostatics. After all, we learn very early in our physics lives that when objects are stationary, the force due to gravity takes exactly the same inverse-square form as the force due to electric charge. It’s worth pausing to see why this analogy does not continue when objects move and the resulting Einstein equations of general relativity are considerably more complicated than the Maxwell equations of electromagnetism.
Let’s start by considering the situation of electrostatics. A particle of charge $q$ experiences a force
$\mathbf{F}=-q\nabla \varphi $ |
where the electric potential $\varphi $ is determined by the surrounding charge distribution. Let’s call the charge density ${\rho}_{e}(\mathbf{r})$, with the subscript $e$ to distinguish it from the matter distribution. Then the electric potential is given by
${\nabla}^{2}{\varphi}_{e}=-{\displaystyle \frac{{\rho}_{e}}{{\u03f5}_{0}}}$ |
Apart from a minus sign and a relabelling of the coupling constant ($G\to 1/4\pi {\u03f5}_{0}$), this formulation looks identical to the Newtonian gravitational potential (0.1). Yet there is a crucial difference that is all important when it comes to making these equations consistent with special relativity. This difference lies in the objects which source the potential.
For electromagnetism, the source is the charge density ${\rho}_{e}$. By definition, this is the electric charge per spatial volume, ${\rho}_{e}\sim Q/\mathrm{Vol}$. The electric charge $Q$ is something all observers can agree on. But observers moving at different speeds will measure different spatial volumes due to Lorentz contraction. This means that ${\rho}_{e}$ is not itself a Lorentz invariant object. Indeed, in the full Maxwell equations ${\rho}_{e}$ appears as the component in a 4-vector, accompanied by the charge density current ${\mathbf{j}}_{e}$,
${J}^{\mu}=\left(\begin{array}{c}\hfill {\rho}_{e}c\hfill \\ \hfill {\mathbf{j}}_{e}\hfill \end{array}\right)$ |
If you want a heuristic argument for why the charge density ${\rho}_{e}$ is the temporal component of the 4-vector, you could think of spatial volume as a four-dimensional volume divided by time: ${\mathrm{Vol}}_{3}\sim {\mathrm{Vol}}_{4}/\mathrm{Time}$. The four-dimensional volume is a Lorentz invariant which means that under a Lorentz transformation, ${\rho}_{e}$ should change in the same way as time.
The fact that the source ${J}^{\mu}$ is a 4-vector is directly related to the fact that the fundamental field in electromagnetism is also a 4-vector
${A}_{\mu}=\left(\begin{array}{c}\hfill \varphi /c\hfill \\ \hfill \mathbf{A}\hfill \end{array}\right)$ |
where $\mathbf{A}$ is the 3-vector potential. From this we can go on to construct the familiar electric and magnetic fields. More details can be found in the lectures on Electromagnetism.
Now let’s see what’s different in the case of gravity. The gravitational field is sourced by the mass density $\rho $. But we know that in special relativity mass is just a form of energy. This suggests, correctly, that the gravitational field should be sourced by energy density. However, in contrast to electric charge, energy is not something that all observers can agree on. Instead, energy is itself the temporal component of a 4-vector which also includes momentum. This means that if energy sources the gravitational field, then momentum must too.
Yet now we have to also take into account that it is the energy density and momentum density which are important. So each of these four components must itself be the temporal component of a four-vector! The energy density $\rho $ is accompanied by an energy density current that we’ll call $\mathbf{j}$. Meanwhile, the momentum density in the ${i}^{\mathrm{th}}$ direction – let’s call it ${p}^{i}$ – has an associated current ${\mathbf{T}}^{i}$. These $i=1,2,3$ vectors ${\mathbf{T}}^{i}$ can also be written as a $3\times 3$ matrix ${T}^{ij}$. The end result is that if we want a theory of gravity consistent with special relativity, then the object that sources the gravitational field must be a $4\times 4$ matrix,
${T}^{\mu \nu}\sim \left(\begin{array}{cc}\hfill \rho c\hfill & \hfill \mathbf{p}c\hfill \\ \hfill \mathbf{j}\hfill & \hfill T\hfill \end{array}\right)$ |
Happily, a matrix of this form is something that arises naturally in classical physics. It has different names depending on how lazy people are feeling. It is sometimes known as the energy-momentum tensor, sometimes as the energy-momentum-stress tensor or sometimes just the stress tensor. We will describe some properties of this tensor in Section 4.5.
In some sense, all the beautiful complications that arise in general relativity can be traced back to the fact that the source for gravity is a matrix ${T}^{\mu \nu}$. In analogy with electromagnetism, we may expect that the associated gravitational field is also a matrix, ${h}_{\mu \nu}$, and this is indeed the case. The Newtonian gravitational field $\mathrm{\Phi}$ is merely the upper-left component of this matrix, ${h}_{00}\sim \mathrm{\Phi}$.
However, not all of general relativity follows from such simple considerations. The wonderful surprise awaiting us is that the matrix ${h}_{\mu \nu}$ is, at heart, a geometrical object: it describes the curvature of spacetime.
We can simply estimate the size of relativistic effects in gravity. What follows is really nothing more than dimensional analysis, with a small story attached to make it sound more compelling. Consider a planet in orbit around a star of mass $M$. If we assume a circular orbit, the speed of the planet is easily computed by equating the gravitational force with the centripetal force,
$\frac{{v}^{2}}{r}}={\displaystyle \frac{GM}{{r}^{2}}$ |
Relativistic effects become important when ${v}^{2}/{c}^{2}$ gets close to one. This tells us that the relevant, dimensionless parameter that governs relativistic corrections to Newton’s law of gravity is $\sim GM/r{c}^{2}$.
A slightly better way of saying this is as follows: the fundamental constants $G$ and ${c}^{2}$ allow us to take any mass $M$ and convert it into a distance scale. As we will see later, it is convenient to define this to be
${R}_{s}={\displaystyle \frac{2GM}{{c}^{2}}}$ |
This is known as the Schwarzschild radius. Relativistic corrections to gravity are then governed by ${R}_{s}/r$.
In most situations, relativistic corrections to the gravitational force are very small. We can, for example, look at how big we expect relativistic effects to be for the Earth or for the Sun:
For our planet Earth, ${R}_{s}\approx {10}^{-2}\mathrm{m}$. The radius of the Earth is around $6000\mathrm{km}$, which means that relativistic effects give corrections to Newtonian gravity on the surface of Earth of order ${10}^{-8}$. Satellites orbit at ${R}_{s}/r\approx {10}^{-9}$. These are small numbers.
For the Sun, ${R}_{s}\approx 3\mathrm{km}$. At the surface of the Sun, $r\approx 7\times {10}^{5}\mathrm{km}$, and ${R}_{s}/r\approx {10}^{-6}$. Meanwhile, the typical distance of the inner planets is $\sim {10}^{8}\mathrm{km}$, giving ${R}_{s}/r\approx {10}^{-8}$. Again, these are small numbers.
Nonetheless, in both cases there are beautiful experiments that confirm the relativistic theory of gravity. We shall meet some of these as we proceed.
There are, however, places in Nature where large relativistic effects are important. One of the most striking is the phenomenon of black holes and, as observational techniques improve, we are gaining increasingly more information about these most extreme of environments.