Chapter 1 From Newton to Einstein

  Gravity is by far the weakest of the four fundamental forces that exist in nature—the others being the electromagnetic force, and the strong and weak nuclear forces. Yet over large distances it is gravity that dominates. This is because gravity is only ever attractive and because it can never be screened. So while most large objects are electrically neutral, they can never be gravitationally neutral. The gravitational force between objects with mass always acts to pull those objects together, and always increases as they become more massive.

  It is thanks to geniuses like Newton and Einstein that we understand gravity at all, yet it is gravity, more than any other force, that continues to present scientists with the most enigmatic of puzzles. To understand why, let's start at the beginning, and consider the historical development of gravity.

  The pre-history of gravity

  It seems safe to assume that mankind has always known that when we drop an object it will fall downwards. In this sense, we've always been aware of the existence of gravity. It was the cause of this motion that appears to have been the focus of many early thinkers.

  Aristotle, who's Physics played a dominant role in European science until as late as the 17th century, based his explanation of gravity on the idea that objects should move towards their natural place in the Universe. Where this place was, he reasoned, should be determined by the composition of the object in question. More specifically, it should depend on how much it contained of each of the four elements: Earth, Water, Air, and Fire.

  Aristotle argued that objects composed predominantly from Earth and Water should move towards the centre of the Universe. To him, the centre of the Universe was the place beneath his feet. Objects made from Earth, and that are thrown into the air, must therefore move down towards the ground. Water, he reasoned, is lighter than Earth, as can be verified by dumping some soil into a half-full glass of it. All Water must therefore come to rest on top of the Earth. Similarly, Air is lighter than Water, as bubbles rise in water. The natural place for Air is therefore above Water, and the natural place for Fire is above the Air.

  This framework provides a logical sort of order to the world that was observed, in terms of what were thought to be the basic constituents of matter at the time. It even allows one to make some statements about the speed at which objects should fall. Aristotle argued that the velocity of a falling object should be in proportion to its weight, and in inverse proportion to the density of the medium through which it travels. That is, Aristotle thought that an object that weighs 2kg should fall twice as fast as one that weighs 1kg.

  Sadly, Aristotle's theory cannot be correct. We now know that there is no centre to the Universe for objects to move towards. We can also demonstrate, by direct experimentation, that the rate at which an object accelerates due to gravity is not in proportion to its mass. In fact, it can be shown that all objects fall at the same rate. This discovery was one of the milestones in the modern understanding of gravity, and so it bears some further explanation.

  The fact that all objects accelerate under gravity at the same rate is not an obvious one. In fact, if I were to drop a feather from my left hand, and a lump of iron from my right, I should not expect them to hit the floor at the same time. The lump of iron will get there first. So what is meant by the statement 『all objects accelerate under gravity at the same rate』？ To understand this phrase, we have to think about all of the forces that act upon these objects.

  When I let go of the feather it is acted on by the force of gravity, but it is also acted on by other forces. As it starts to drop, there is resistance to its motion from the air around it. This slows the feather more than it does the heavier lump of iron. Any slight gust of air will also have a large efect on the feather while perturbing the motion of the lump of metal by only a tiny amount. So, the meaning of the phrase 『all objects accelerate under gravity at the same rate』 is not a statement about the motion of objects in our immediate environment. Rather, it is a statement about what should happen to an object if it were to fall under the influence of gravity alone. That is, if all other interactions are suppressed, then all objects should fall at the same rate.

  

  Galileo is widely credited with having shown the truth of this proposition. In 1638, he supposedly dropped cannonballs with diferent masses of the top of the leaning tower of Pisa. The cannonballs were found to fall at the same rate, independent of their composition. More recently, and perhaps even more dramatically, the same result was demonstrated by the Apollo astronaut David Scott. Scott dropped a feather and a hammer while standing on the surface of the Moon. There is no air on the Moon to slow the motion of the feather, and so both objects landed at his feet at the same time (see Figure 1). Today we refer to this phenomenon as the Universality of Free Fall. It is a key ingredient in both Newton's and Einstein's theories of gravity, as we will see later.

  1. (a) Galileo dropping cannonballs of the leaning tower of Pisa; and (b) a painting of David Scott dropping a hammer and a feather on the surface of the moon.

  Newton’s theory of gravity

  Sir Isaac Newton’s theory of gravity and motion was first published in his Principia Mathematica in 1687, and it changed the world forever. This was the first genuinely scientific theory of how gravity worked. Unlike Aristotle, Newton didn’t try and seek an explanation for gravity. Instead he quantified its efects, and in the process deduced physical laws that accurately described the motion of not only objects on Earth, but also the motion of the Earth itself, as well as all other bodies in the Solar System.

  Newton’s achievement was truly a work of genius. He created new branches of mathematics, and for the first time demonstrated that the laws of physics that apply to us, on Earth, are the same as those that apply to the heavenly bodies. All the complexity of motion, which Aristotle had tried to explain, was boiled down to a handful of simple laws. Newton’s theory was glorious,and went unchallenged for over 200 years. In one book, Newton revolutionized the worlds of science, industry, and warfare, and provided the framework within which many of us still think today.

  The basic ingredients of Newton’s theory are the existence of absolute space and time as the stage on which all motion occurs, and the existence of a universal gravitational force that acts instantaneously between every pair of massive bodies in the Universe. And that is all.

  Space to Newton was, as it appears to most of us in our day-today experience, simply the eternal and unchanging arena within which all matter exists. I can position object X at one point in space, and work out the distance to any other object, Y, by simply using a measuring tape held out in a straight line. In Newton’s mechanics there is no ambiguity in this process. Objects X and Y may move in space, but the space itself is fixed and forever unchanged.

  Likewise, Newton’s conception of time followed the everyday intuition that most of us grow up with. Instants of time unfold in Newton’s theory, one after the other. Objects can change their position during an interval of time, but time itself is universal and the same for everyone. In Newton’s theory all true clocks measure time in the same way, just as all true measuring tapes measure the same distance between any two given objects.

  Now, according to Newton, all bodies move at a constant rate, unless acted on by an external force (this was a departure from Aristotle’s physics). If a force should act upon an object, then the efect of that force is to cause the object to accelerate. More force means more acceleration, and, if an object has more mass, then it needs more force to reach the same acceleration. Within this framework, gravity is simply an external force acting on all massive bodies, pulling them together.Newton deduced that his gravitational force must be proportional to the mass of each of the objects it acts between, and inversely proportional to the square of the distance between them. That is, the force of gravity between two massive bodies must be given by an equation of the form:

  where M and m are the masses of the two objects in question, and r is the distance between them. This simple equation, together with Newton's laws of motion, is suicient to get a good approximation to the motion of the vast majority of astrophysical bodies as well as all bodies that exist on Earth.

  That all bodies fall under Newton's gravity at the same rate, and hence obey Galileo's observation, can be seen from the fact that in Newton's mechanics a given force causes a heavy body to accelerate more slowly than a light one. Consider this together with the fact that, according to Newton, the gravitational force must increase with mass. In Newton's theory these two things happens at exactly the required rate to cancel each other out. A body being acted on by Newton's gravity, and obeying Newton's laws of motion, must therefore accelerate at a fixed rate, independent of its mass. That this happens is no accident: the Universality of Free Fall is built into Newton's theory from the start.

  The first great success of Newton's theory was that it could be used to derive the laws of motion for the planets. These laws had been deduced empirically by Johannes Kepler earlier in the 17th century, using cutting-edge astronomical data. Kepler's laws stated that:

  ·The orbit of a planet traces out an ellipse, and the Sun is at one of the foci of this ellipse.

  ·If a line is drawn between a planet and the Sun, then, as the planet orbits the Sun, that line will sweep out an equal area of space in any two intervals of equal time.

  2. An illustration of a planetary orbit. The shaded regions, marked A and B, are of equal area, if they are swept out by the orbiting planet in equal time.

  ·The square of the time taken for a planet to complete one orbit is proportional to the cube of the distance between the two furthest points on that orbit (the major axis of the ellipse).

  The quantities described in these three laws are all illustrated in Figure 2 (with the exception of the orbital period). The good thing about Kepler's laws was that they seemed to apply to all known planets, even though at the time they had no known basis in any physical theory. They were simply seen to be a good fit to the data.Newton knew about Kepler's laws, and included in his Principia Mathematica an explanation of how they could be derived from his laws of motion and from his universal theory of gravitation. This derivation was one of the great success stories of physics. Empirical laws, derived from cutting-edge observations of the planets, were explained for the first time in terms of simple mathematical equations. Newton had shown that the same laws that describe the motion of a cannonball fired on the surface of the Earth could also be used to describe the motions of the planets themselves. This was the beginning of much of physics as we know it today.

  Einstein’s theory of gravity

  More than 200 years after its publication, Newton's theory of gravity was supplanted by Albert Einstein's. If Newton's theory was simple and useful, then Einstein's was beautiful and truly universal. Einstein didn’t just change the equations involved in Newton's theory, he pulled up the very foundations upon which it rested. Einstein changed everything.

  As with much of the progress that happens in physics, Einstein's theory was motivated primarily by the inconsistency of existing ideas. Newton had given us a theory of how gravity and motion operate. But Newton's ideas were not compatible with the theory of light that James Clerk Maxwell had developed in the middle of the 19th century. Maxwell's new theory stated that everyone in the Universe should measure the speed of light to have the same value: just under 300 million metres per second. This might not sound terribly profound, until you take a few moments to consider what it means.

  The point is that, according to Newton's mechanics, if I fire a bullet forwards at 1,000 miles per hour while seated on a train moving at 100 miles per hour, then an observer on the side of the track will see that bullet moving at 1,100 miles per hour. In mathematical terms we say that the velocity of the bullet and the velocity of the train add linearly. Now consider that I turn on a torch, while still seated facing forwards on the same train. From my seated position I will see the light from the torch propagate forwards through the train carriage at the speed of light (i.e. at about 300 million metres per second). Consider again the observer watching me from the side of the train track. If you listened to Newton, you might expect this person to see the light travel at the 300 million metres per second, plus 100 miles per hour (the speed of the train). But according to Maxwell this is not what happens. Maxwell says that the person at the side of the track sees the light propagate at the same speed as the person on the train. That is, Maxwell’s equations imply that velocities do not add linearly.

  The contradiction just described is a profound one. If we are unable to agree on how to add velocities, then we are unable to use physics to calculate the motion of objects at all. It was impossible for both Newton and Maxwell to be correct. At least one of them must be wrong. A lesser scientist might have tried to re-write either Newton’s or Maxwell’s theories, but this was not what

  Einstein did. Einstein treated both Newton’s and Maxwell’s work with the utmost respect. He recognized their great strengths, and worked to solve the contradiction in a truly ingenious way.Einstein hypothesized that if the speed of light was the same for everyone, then time and space cannot be universal concepts. Instead, he reasoned, each observer must have their own personal concept of time, and their own personal concept of space. According to Einstein’s new theory, a clock carried by a person on the train is seen, by a person standing beside the track, to tick slower than a clock they carry themselves. Likewise, the person on the train sees the clock of the person standing beside the track to tick slower than his or her own.

  This result initially sounds odd, but that’s only because we』ve been programmed from a young age to think of time as universal. What Einstein showed us is that our childhood understanding of time is mistaken. Time is not a universal concept, unfolding at the same rate for everyone. Time is a personal thing, and depends on our relative motion, with respect to others. Likewise, space is not the fixed backdrop that we think it is. What we think of as distances, and the lengths of objects, are actually dependent on how we are moving.

  These are startling ideas. At first they can seem unsettling, as if the crutches we've used to understand the world have suddenly been kicked away. But we need not despair. There is a concept involving space and time that survives in Einstein’s theory, and that maintains an observer-independent reality. This is what is known as space-time. Instead of Newton’s concepts of universal time and universal space, what we are left with is a larger structure that encompasses them both. A person or object, like you or me, follows a line through this structure, known as our world-line. Our personal time is measured along our world-line, and while my world-line might be diferent to yours, they both exist in the same space-time (see Figure 3).

  So, it is the promotion of space and time to space-time that allows us to make Newton’s mechanics consistent with Maxwell’s. This realization was one of Einstein’s early contributions to science, and it is the backbone of what is now known as the Special Theory of Relativity. It has a wide variety of profound consequences, many of which have been confirmed experimentally. The most famous of these is probably the equation:

  3. An example of two world-lines, for two observers, moving in the same space-time. The observers meet where the lines cross.

  which tells us that mass and energy are intricately connected (a fact that became devastatingly apparent with the advent of nuclear weapons). Other consequences are the longer lifetime that unstable particles appear to have when they move quickly, and the fact that nothing can ever move faster than light.

  It is this last result, together with the new concept of space-time, which led Einstein to his theory of gravity. Again, the impetus for this development was an apparent contradiction. And again, it was Newton’s theory that was causing problems. This time, however, it seemed that there was a contradiction with Einstein’s own work. This was because Newton’s gravity acted between objects instantaneously. That is, if the Sun were to suddenly explode, according to Newton we should feel the gravitational efects of this at the very same moment it happens. But Einstein now knew that this was impossible. First, he had found out that nothing could move faster than light. Second, he had shown that there was no such thing as universal time, so the idea of two things happening simultaneously in two diferent places made no sense at all (if they happened simultaneously for one observer, they would not be simultaneous for any other who was in a diferent state of motion). So, once again, something was wrong and needed fixing.

  Einstein’s solution to this problem was even more amazing. He hypothesized that gravity, instead of being a force that simply pulled things through space, was the result of the curvature of space-time. The fact that massive objects were drawn towards each other was then, according to Einstein, just a result of those objects following the shortest paths they could in the curved space-time in which they existed. The idea was that mass and energy caused space-time to curve, and that this curvature caused the paths of the objects that move through space to appear to bend towards each other. The beauty of this idea is that we now no longer need to include gravity as an extra force that exists in the Universe. In this new picture, the only thing responsible for the attraction of massive bodies is space-time itself (which has to be there anyway). This is the fundamental idea behind the General Theory of Relativity.

  Even more impressive is that Einstein’s idea explained Galileo’s result that all objects fall at the same rate. Recall that in Newton’s theory this result wasn’t really explained at all. It was simply taken as a fact, and a law of gravity was devised that was compatible with it. Einstein went one better. Now, in Einstein’s theory, there is no external force called gravity; the motion of every object is just a result of the curvature of space-time. But all objects are moving in the same space-time, so all objects must follow the same paths. In other words, all objects must fall at the same rate, just as Galileo had observed.

  These ideas can sometimes be confusing, so let’s think about an example. Imagine the paths of two objects that have no forces acting on them. In a flat space the paths of these objects are straight lines, as shown in Figure 4.

  If the space is curved, however, then this is no longer true. Consider the simplest curved space: the surface of a sphere. The shortest path between any two points on the surface of a sphere is called a great circle (the equator is an example of a great circle-on the globe). If two objects follow two diferent great circles, on the same sphere, then they will initially move away from each other, before finally coming together again, as shown in Figure 5.This is how Einstein envisaged gravity working. He imagined that it was curvature that was responsible for the paths of objects meeting, and not anything external that pulls them left or right as they move. The curvature of space-time is usually more irregular than the surface of a sphere, but the basic idea is the same. As far as the consequences for gravity go, the largest efects of Einstein's new theory were found to look very similar to the law that Newton had prescribed 200 years earlier. The diference is that this law now comes with a new understanding of space and time. It also comes with the prediction of a variety of smaller, more subtle, efects (to be discussed in the following chapters).

  4. Two straight lines, indicating the paths that two particles might follow if they travelled through a flat space, without any external forces acting upon them.

  5. Two great circles, indicating the paths that particles might travel in a spherically curved space, if no external forces act upon them. The lines are no longer parallel forever, but meet at a point.

  Think now about what all of this means in a practical situation. Consider a skydiver jumping out of an airplane. The skydiver falls freely, up to the efects of air resistance. According to Einstein, the skydiver’s path is the straightest line possible through the curved space-time around the Earth. From the skydiver’s perspective this seems quite natural. Except for the air rushing past her, the skydiver feels no perturbing forces at all. In fact, if it weren’t for the air resistance, she would experience weightlessness in the same way that an astronaut does in orbit. The only reason we think the skydiver is accelerating is because we are used to using the surface of the Earth as our frame of reference. If we free ourselves from this convention, then we have no reason to think the skydiver is accelerating at all.

  Now consider yourself on the ground, looking up at the falling daredevil. Normally, your intuitive description of your own motion would be that you are stationary. But again this is only because of our slavish regard to the Earth as the arbiter of what is at rest and what is moving. Free yourself from this prison, and you realize that you are, in fact, accelerating. You feel a force on the soles of your feet that pushes you upwards, in the same way that you would if you were in a lift that accelerated upwards very quickly. In Einstein’s picture there is no diference between your experience standing on Earth and your experience in the lift. In both situations you are accelerating upwards. In the latter situation it is the lift that is responsible for your acceleration. In the former, it is the fact that the Earth is solid that pushes you upwards through space-time, knocking you of your free-fall trajectory. That the surface of the Earth can accelerate upwards at every point on its surface, and remain a solid object, is because it exists in a curved space-time and not in a flat space.

  With this change in perspective the true nature of gravity becomes apparent. The freely falling skydiver is brought to Earth because the space-time through which she falls is curved. It is not an external force that tugs her downwards, but her own natural motion through a curved space. On the other hand, as a person standing on the ground, the pressure you feel on the soles of your feet is due to the rigidity of the Earth pushing you upwards. Again, there is no external force pulling you to Earth. It is only the electrostatic forces in the rocks below your feet that keep the ground rigid, and that prevents you from taking what would otherwise be your natural motion (which would also be free fall).

  So, if we free ourselves from defining our motion with respect to the surface of the Earth we realize that the skydiver is not accelerating, while the person who stands on the surface of the Earth is accelerating. Just the opposite of what we usually think. Going back to Galileo’s experiment on the leaning tower of Pisa, we can now see why he observed all of his cannonballs to fall at the same rate. It wasn’t really the cannonballs that were accelerating away from Galileo at all, it was Galileo that was accelerating away from the cannonballs! If I leave a number of objects at rest at some position in space and accelerate away from them, I shouldn’t be at all surprised that the distance between me and each of these objects increases at the same rate. So it is with Galileo and his cannonballs.

  For some, the beauty in this description is obvious. For others it is the fact that it is possible to probe Einstein’s theory experimentally that is most compelling. These experiments range from looking for small perturbations to the orbits of the planets, to the bending of light around the Sun, and many, many more besides. We will explore these exciting phenomena in the chapters that follow. But bear in mind throughout: it is the curvature of space-time that is responsible for it all.