A Brief History of Time: Chapter 2 Summary & Analysis

What we understand today about forces and motion dates back to Galileo and Newton. Before them, people believed Aristotle, who said an object was naturally at rest and only moved if a force was acting on it. According to that logic, a heavier object ought to fall more quickly to the earth when dropped.

Aristotle’s teaching also said we could understand the whole universe just by applying logic, so no experiments were required. Galileo was the first to bother to check out the theory about weights falling at different rates. The story goes that Galileo tried it out by dropping things from the Tower of Pisa, but actually he rolled balls of different weights down a hill and measured their acceleration.

Galileo found that each ball increased its speed at the same rate, regardless of its weight. The acceleration of the balls was directly proportionate to the incline of the hill, not their different weights. If one dropped a lead ball and a feather, the ball would drop faster only because air resistance slows the feather. Removing air resistance as a factor would see both fall at the same rate, as shown by astronaut David R. Scott, who performed exactly that experiment on the moon, where there is no air.

Newton used Galileo's measurements as the foundation for his laws of motion. He deduced that the force (the balls' own weight) was constant, and  this force caused the object to accelerate, not just set it moving. This meant the absence of force would leave an object moving straight ahead at a constant speed.

Newton was the first to put this idea forward, in 1687. It is now called Newton's First Law, which states an object's speed will change proportionally to the force that affects it. The object's deceleration or acceleration is also affected by its own mass—the same force will be twice as strong if the object is half as big, and vice versa. For example, think of a car. A more powerful engine will produce higher acceleration, unless the car itself is heavier.

Newton also discovered the law of gravity, which is the idea that every object attracts every other object proportionally to its mass; the bigger the object, the stronger its gravitational attraction. The gravitational force between two objects doubles if just one object’s mass doubles. If the other object were to triple its original mass, the overall gravitational pull of the two objects combined would be six times stronger than before. This is why all objects fall (or accelerate) at the same rate; if one ball has twice the weight of another, that effect is canceled out by the fact is also has twice the mass to move.

This law of gravity also states that the force is proportionally smaller the further away the objects are from each other. A star's pull is one-fourth of that of a similar star at half the distance. Applying this law helps us to accurately predict the orbits of the planets and moons. If force were not proportional, and did not increase or decrease more rapidly as objects approached or drew away from each other, the planets would either spiral into the sun or escape its pull altogether.

The main difference between Aristotle's approach and that of Galileo and Newton is the former's idea of the preferred state of rest, meaning an object would remain still if no force were acting on it. But Newton's laws of motion tell us there is no one standard of rest. For example, if we ignore the fact the earth is orbiting the sun, we could say a train is traveling over a still earth at 90 miles an hour. But you could also equally say the earth is moving south at the same rate, if you say the train is at rest. If you carried out Galileo's moving objects experiments on the train, Newton's laws would still apply—for example if you played table tennis on the moving train. Really, you can't deduce which object, of the train or the earth, is moving at 90 miles an hour and which is at rest.

Because there is no state of absolute rest, it is hard to determine if two events that took place at different times took place in the same location. If a table tennis ball bounces twice on a table on a moving train, someone inside the train will say the ball moved a few feet, but someone beside the track will think the bounces took place 40 meters apart as the train continued to travel along between bounces.

This means we cannot give an event an exact location in space, contrary to Aristotle's teachings. The two people on the train would not be able to agree on the positions and distances of the event, and there is no reason to side with either one's version of events over the other's.

This idea worried Newton, as a lack of absolute space didn't agree with his idea of an absolute God. He refused to accept the idea, even though his own laws implied it. Bishop Berkeley was one among many who criticized him for this, as he believed anything material, such as time, matter and space, were all illusions. Dr. Samuel Johnson in turn disagreed with the Bishop, and kicked a rock to show his dissatisfaction with such ideas.

Aristotle and Newton both believed in absolute time, meaning the interval of time between two events could be definitely measured. This meant time was separate from space, which seems commonsense. These ideas have since changed, although the commonsense approach still works when dealing with everyday object like apples or slower-moving things like planets. When looking at things that move near or at the speed of light, however, this commonsense approach doesn't work at all.

Ole Christensen Roemer, a Danish astronomer, was the first to notice that light had a finite, albeit very fast, speed, in 1676. He noticed that Jupiter's moons didn't seem to appear from behind Jupiter at a constant rate. He noticed the eclipses of the moons were later the further the earth was from Jupiter, and deduced it must be because the light takes longer to travel to the earth. His measurements were not very accurate, but it was still a remarkable achievement, especially as it came 11 years ahead of Newton's Principia Mathematica.

James Clerk Maxwell provided a full theory of the transmission of light in 1865, when he unified the theories that has been used to understand electricity and magnetism. He said wavelike disturbances in the electromagnetic field would travel at constant speeds, just like ripples in ponds. The different wavelengths (the distance between each wave crest) were different types of light; there are, for example, meter-long radio waves, centimeter-long microwaves, and smaller infrared, ultraviolet, X-rays, and gamma rays.

This theory gave fixed speeds to different types of light, but Newton's theory had overridden the idea of absolute rest, so that raised the questions as to what the speed of light was relative to. People suggested the idea of an ether that occupied all space, which light traveled through. The light would travel relative to the ether, but would vary according to different observers.

For example, light should travel faster measured in the same direction of the earth's movement around the sun (i.e. toward the source of light), rather than at right angles (away from it). But Albert Michelson (the first American to win the Nobel Prize for physics) and Edward Morley tested this in 1887, and found it not to be true according to observation—the speed of light was the same.

Many people tried to explain this result. It wasn't until 1905, when previously unknown Swiss patent office clerk Albert Einstein suggested there was no need for the idea of ether if you accepted time was not absolute. Henri Poincaré, a French mathematician, made a similar point soon after.

This new idea was called the theory of relativity, which meant that the laws of science were the same for all freely moving observers. This brought together Newton's laws of motion and Maxwell's theories on light. No matter how fast they are moving, all observers will measure the same speed of light.

Although a simple idea, it had huge ramifications. Mass and energy were equivalent, as summarized in Einstein's famous equation E=mc2, and the theory that nothing is faster than the speed of light. This means that an object's motion-related energy will increase its mass, which will make it harder for the object to increase its speed.

This is more significant for objects moving close to light speed. As an object gets closer to light speed, its mass rises exponentially, taking ever-increasing energy to speed up. Objects cannot reach the speed of light, as it would take an infinite amount of energy to do so. Normal objects are thus stuck within the limits of relativity and cannot reach light speed. Only light, or other things with no mass, i.e. waves, can get to light speed.

Relativity has changed the way we see space and time forever. Under Newton's theory, observers would agree on how long it took a beam of light to reach one place from another, but not the necessarily the distance between those points, because the idea of absolute space had been abandoned. If the time was constant, then the speed of light would have to differ between observers. But in relativity the observers must agree on the speed of light, so the time measured must differ. The time taken for light to travel equals the distance traveled (which the observers disagree on) divided by the speed of light (which the observers agree on). There is thus no absolute time. Observers all have their own measure, according to their own clock, and each observer's clock will not necessarily agree with others’.

If the observers used radar to record the place and time of an event, they would send a pulse to that event that would then be reflected back. The time of the event is thus halfway between the pulse going out and returning. The distance is worked out by multiplying half the distance of the round trip by the speed of light. A space-time diagram, can be used by different observers moving at different speeds, and no measurement is more correct than any other, though they are all related. So, one observer could work out the time and position another observer would calculate if only the former knew the latter's relative velocity.

Today, this method accurately measures distances, because we can measure time more accurately than length. A meter is defined as the distance light travels in a tiny fraction of a second. (The historical definition is a platinum bar that is kept in Paris.) We can also therefore use an accurate measurement called a light-second—the distance light travels in a second.

Under the theory of relativity, distance is determined by time and light speed, so every observer must agree on the speed of light. There is no need for a theory of ether, which we cannot detect anyway. We must also accept that time is not independent of space, but rather combined in an idea called space-time.

In everyday life we can locate a position according to three dimensions of space, or coordinates. For example, a point in a room is measured by its distance from two walls and the floor or ceiling; a point on the earth is defined by a specific longitude and latitude, as well as height above sea level. We can use any three suitable coordinates. But we could not use miles north and west from Piccadilly Circus and height above sea level to locate the moon. We could pick points from among the sun or planets, but these in turn could not locate our sun compared to the rest of the galaxy. Thus, the whole universe is a group of such overlapping layers of relevant reference points.

An event happens at a certain point in space and time and can be measured according to four coordinates: three in space, and the fourth in time. These can all be arbitrary. In space-time, there is no distinction between space and time coordinates, like there's no difference between space coordinates, if they are suitable. These place the event in four-dimensional space-time.

Drawing diagrams of two-dimensional space is easy, like maps of the surface of the earth, because any point can be determined by latitude and longitude. Space-time diagrams can show time increasing on one axis, and one dimension of space on the other, with the other space dimensions ignored or shown via perspective that implies a third dimension.

Hawking presents a figure showing time in years on the upward axis and distance in miles on the horizontal axis, as measured between the earth’s sun and Alpha Centauri, a nearby star. The path of each of the two stars is a vertical line, and the diagonal line that connects them is a ray of light, which takes four years to travel the distance between them.

Maxwell predicted the speed of light would be constant whatever the light’s source, which has since been proven true. As light is emitted, it spreads out like a sphere from a certain point similar to ripples that spread out from where a stone is thrown into a lake. Stacking up pictures of these ripples as they spread creates a cone, with the tip being the place and time the stone hit the surface of the lake. Light spreading from a source forms a similar cone, called the event's future light cone. We can also create a past light cone, which is the group of events that light can reach from a given event (Fig 2.4).

Given an event, as represented on the graph as P, all other events can be classified into one of three groups. Events that can be reached from event P by anything at or slower than the speed of light is the future of P. Only these events will be effected by P. Events in P's past are those from which P can be reached at or  under the speed of light. The elsewhere of P is everything else. These events are not affected by nor affect P.

For example, if the sun went out at this very moment it would not affect events on earth right now, as it takes 8 minutes for the sun's light to reach us. After those 8 minutes, the earth would be in the future light cone of the event of the sun going out. In the same way, we do not know what is happening right now in distant space. We are seeing the universe as it was in the distant past, on the far end of past events’ future light cones.

Ignoring gravitational effects, like Einstein and Poincaré did back in 1905, the resulting theory is called the special theory of relativity. All light cones would be identical and point in the same direction as light speed is the same at every event and in every direction. Any object's path is therefore represented as a line in every relevant light cone. This approach was successful at explaining why the speed of light seems the same to everyone, and what happens when traveling near the speed of light.

But this theory is inconsistent with Newton's laws on gravity, in which distance is a factor, meaning moving an object would affect the force applied to it instantly. This implies gravitational effects take effect instantly, which doesn't work with the special theory of relativity's idea that nothing moves at or above the speed of light.

In 1915, Einstein put forward the idea of the general theory of relativity. He suggested that gravity is not like other forces. Rather, it is the result of the fact space-time is not flat. It is curved according to the mass and energy distributed across it. The earth is not forced to move in a curved orbit by gravity. Instead, it takes what is closest to a straight path in curved space. This is called a geodesic, the shortest path between two points. A geodesic of the earth is called a great circle, and is used by airline navigators to determine the shortest distance between two airports.

In general relativity, objects take a straight route in curved, four-dimensional space-time, but seem to take curved routes in three-dimensional space. For example, an airplane flying straight will have a shadow that seems to take a curved path on the two-dimensional ground.

The sun's mass and resultant gravitational force curves space-time so that although the earth travels straight in four-dimensional space-time, it looks like it follows a circular orbit in three-dimensional space. Newton's law of gravity predicted the planets' movements fairly accurately. But the gravitational effects are so strong on Mercury, which is closest to the sun, its orbit looks very elongated. This extra-long axis causes Mercury’s orbit to rotate by one degree every ten thousand years. This fact was accounted for in the new theory, and helped to confirm Einstein's new proposition. Even smaller deviations have been found elsewhere and confirmed the theory's predictions since.

Light also seems to not take straight paths through three-dimensional space. Light should also be bent by gravity, according to general relativity. Light cones near the sun ought to bend slightly inward, because of the sun's mass. Light from a distant star that passes by the sun ought to bend, making the star appear to be where it's not. If the light from the star always passed near the sun, we would not be able to tell. But the stars move relative to each other as the earth orbits the sun, so different stars pass behind the sun from our perspective.

It is hard to see this effect of light bending around the sun because of the latter’s gravity, because the sun's light is brighter than that from more distant stars. But it is possible to observe the effect during a solar eclipse, because the sun’s light is blocked, allowing the bent light from the distant star to be measured.  This effect has since been observed and measured, confirming the theory.

According to general relativity, time should also run more slowly when closer to objects with large mass, like the earth. The higher light's energy, the higher its frequency (or waves per second). Light loses energy to escape the earth's gravitational field, making its the frequency slow and in turn making it look to an observer above the earth like everything below is happening more slowly than where the observer is. This idea was tested in 1962 with a pair of very accurate clocks on the top and base of a water tower. The clock at the base ran more slowly than that at the top, as predicted. This has great significance on navigation systems for satellites.

Newton's laws ended the idea of absolute space and relativity ended the idea of absolute time. If twins separated, with one living on top of a mountain and one living by the sea, the first would age more quickly, and would be older when they met again. In this example the difference is small, but if one twin took a ride in a spaceship at the speed of light, he would be much younger than his brother by the time he returned to earth. This is known as the twin paradox. Really, there is no absolute time, instead each person has their own measure of time.

Space and time were considered fixed and separate arenas before 1915, unaffected by what took place within them. People thought they both went on forever. But with general relativity that thinking has changed considerably. Space and time are affected by objects' movement and forces, and space-time in turn affects the movement of those forces and objects. Just as space and time affect everything in the universe, there is no meaning to space and time outside the universe.

The world was now dynamic, rather than unchanging, was expanding, and possibly finite, with a beginning and an end. This was the start for Stephen Hawking's own work in theoretical physics, and later he showed with Roger Penrose that Einstein's general theory of relativity suggested there was indeed a beginning and end to the universe.