Three Roads to Quantum Gravity (9 page)

The future and past of Olga’s second return. Note that Sam being confused is in neither set of events.

Is our universe such a causal universe? General relativity tells us that it is. The description of the universe given by general relativity is exactly that of a causal universe, because of the basic lesson of relativity theory: that nothing can travel faster than light. In particular, no causal effect and no information can travel faster than light. Keep this in mind, and consider two events in the history of our universe, pictured in
Figure 9
. Let the first be the invention of rock and roll, which took place perhaps somewhere in Nashville in the 1950s. Let the second be the fall of the Berlin Wall, in 1989. Did the first causally influence the second? One may argue about the political and cultural influence of rock and roll, but what is important is only that the invention of rock and roll certainly had some effect on the events leading to the fall of the Berlin Wall. The people who first climbed the wall in triumph had rock and roll songs in their heads, and so did the functionaries who made the decisions that led to the reunification of Germany. So there was certainly a transfer of information from Nashville in the 1950s to Berlin in 1989.

The invention of rock and roll was in the causal past of the fall of the Berlin Wall because information was able to travel from the first event to the second.

So in our universe we define the causal future of some event to consist of all the events that it could send information to, using light or any other medium. Since nothing can travel faster than light, the paths of light rays leaving the event define the outer limits of the causal future of an event. They
form what we call the future light cone of an event (
Figure 10
). We call it a cone because, if we draw the picture so that space has only two dimensions, as in
Figure 10
, it looks like a cone. The causal past of an event consists of all the events that could have influenced it. The influence must travel from some event in the past at the speed of light or less. So the light rays arriving at an event form the outer boundary of the past of an event, and make up what we call the past light cone of an event. One is pictured in
Figure 10
. We can see that the structure of the causal relations around any event can be pictured in terms of both the past and future light cones. We
see from
Figure 10
also that there are many other events which lie outside both the past and future light cones of our particular event. These are events that took place so far from our event that light could not have reached it. For example, the birth of the worst poet in the universe, on a planet in a galaxy thirty billion light years from us is, fortunately, outside both our future and past light cones. So in our universe, specifying the paths of all the light rays or, equivalently, drawing the light cones around every event, is a way to describe the structure of all possible causal relations. Together, these relations comprise what we call the causal structure of a universe.

Many popular accounts of general relativity contain a lot of talk about ‘the geometry of spacetime’. But actually most of that has to do with the causal structure. Almost all of the information needed to construct the geometry of spacetime consists of the story of the causal structure. So not only do we live in a causal universe, but most of the story of our universe is the story of the causal relations among its events. The metaphor in which space and time together have a geometry, called the spacetime geometry, is not actually very helpful in understanding the physical meaning of general relativity. That metaphor is based on a mathematical coincidence that is helpful only to those who know enough mathematics to make use of it. The fundamental idea in general relativity is that the causal structure of events can itself by influenced by those events. The causal structure is not fixed for all time. It is dynamical: it evolves, subject to laws. The laws that determine how the causal structure of the universe grows in time are called the Einstein equations. They are very complicated, but when there are big, slow moving klutzes of matter around, like stars and planets, they become much simpler. Basically, what happens then is that the light cones tilt towards the matter, as shown in
Figure 11
. (This is what is often described as the curvature, or distortion of the geometry of space and time.) As a result, matter tends to fall towards massive objects. This is, of course, another way of talking about the gravitational force. If matter moves around, then waves travel through the causal structure and the light cones
oscillate back and forth, as shown in
Figure 12
. These are the gravitational waves.

A massive object such as a star causes the light cones in its vicinity to tip towards it. This has the effect of causing freely falling particles to appear to accelerate towards the object.

So, Einstein’s theory of gravity is a theory of causal structure. It tells us that the essence of spacetime is causal structure and that the motion of matter is a consequence of alterations in the network of causal relations. What is left out from the notion of causal structure is any measure of quantity or scale. How many events are contained in the passage of a signal from you to me, when we talk on the telephone? How many events have there been in the whole history of the universe in the past of this particular moment, as you finish reading this sentence? If we knew the answers to these questions, and we also knew the structure of causal relations among the events in the history of the universe, then we
would know all of what there is to know about the history of the universe.

A gravitational wave is an oscillation in the directions in which the light cones point in spacetime. Gravitational waves travel at the speed of light.

There are two kinds of answer we could give to the question of how many events there are in a particular process. One kind of answer assumes that space and time are continuous. In this case time can be divided arbitrarily finely, and there is no smallest possible unit of time. No matter what we think of, say the passage of an electron across an atom, we can think of things that happen a hundred times faster. Newtonian physics assumes that space and time are continuous. But the world is not necessarily like that. The other possibility is that time comes in discrete bits, which can be counted. The answer to the question of how many events are required to transfer a bit of information over a telephone line will then be a finite number. It may be a very large number, but it still will be a finite number. But if space and time consist of events, and the events are discrete entities that can be counted, then space and time themselves are not continuous. If this is true, one cannot divide time indefinitely. Eventually we shall come to the elementary events, ones which cannot be further divided and are thus the simplest possible things that can happen.
Just as matter is composed of atoms, which can be counted, the history of the universe is constructed from a huge number of elementary events.

What we already know about quantum gravity suggests that the second possibility is right. The apparent smoothness of space and time are illusions; behind them is a world composed of discrete sets of events, which can be counted. Different approaches give us different pieces of evidence for this conclusion, but they all agree that if we looks finely enough at our world the continuity of space and time will dissolve as surely as the smoothness of material gives way to the discrete world of molecules and atoms.

The different approaches also agree about how far down we have to probe the world before we come to the elementary events. The scales of time and distance on which the discrete structure of the world becomes manifest is called the Planck scale. It is defined as the scale at which the effects of gravity and quantum phenomena will be equally important. For larger things, we can happily forget about quantum theory and relativity. But when we get down to the Planck scale we have no choice but to take it all into account. To describe the universe at this scale we need the quantum theory of gravity.

The Planck scale can be established in terms of known fundamental principles. It is calculated by putting together in appropriate combinations the constants that come into the fundamental laws. These are Planck’s constant, from quantum theory; the speed of light, from special relativity; and the gravitational constant, from Newton’s law of gravitation. In terms of the Planck scale, we are absolutely huge. The Planck length is 10
^-33
centimetres, which is 20 orders of magnitude smaller than an atomic nucleus. On the scale of the fundamental time, everything we experience is incredibly slow. The Planck time, which must be roughly the time it takes for something truly fundamental to happen, is 10
^-43
of a second. That is, the quickest thing we can experience still takes more than 10
⁴⁰
fundamental moments. A blink of an eye has more fundamental moments than there are atoms in Mount Everest. Even the fastest collision ever observed between two elementary particles fills more elementary moments than there are
neurons in the brains of all the people now alive. It is hard to avoid the conclusion that everything we observe may still be incredibly complicated on the fundamental Planck scale.

We can go on like this. There is a fundamental Planck temperature, which is likely to be the hottest anything can get. Compared with it, everything in our experience, even the interiors of stars, is barely above absolute zero. This means that, in terms of fundamental things the universe we observe is frozen. We begin to get the feeling that we know as much about nature and its potential phenomena as a penguin knows of the effects of forest fire, or of nuclear fusion. This is not just an analogy - it is our real situation. We know that all materials melt when raised to a high enough temperature. If a region of the world were raised to the Planck temperature, the very structure of the geometry of space would melt. The only hope we have of experiencing such an event is by peering into our past, for what is usually called the big bang is, in fundamental terms, the big freeze. What caused our world to exist was probably not so much an explosion as an event that caused a region of the universe to cool drastically and freeze. To understand space and time in their natural terms, we have to imagine what was there before everything around us froze.