A Brief History of Time

A Brief History of Time

A Brief History of Time Chapter 2 Summary & Analysis

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 once again offers an example of how not to suggest new scientific laws. Of course, it's easier to see he was wrong in hindsight, as Hawking notes that everyone was happy to accept Aristotle's teaching, without enquiring any further themselves.
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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 was the first to properly challenge Aristotle's teachings about object's natural state of rest, and actually checked it out for himself. In this way, Hawking shows the importance of an inquisitive mindset, as well as the necessity of double checking and observation.
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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.
By setting up an experiment to check Aristotle's claim, Galileo showed that people had been wrong for centuries, simply because no one had checked. His finding changed how people viewed objects’ movement and how forces worked on them. Later, when humans had advanced enough to travel to the moon, they performed further experiments to confirm his findings. Experiments, Hawking shows, gain better results than guessing.
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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.
Just as Galileo had built on, and challenged Aristotle’s earlier claims, Newton built on Galileo’s findings, digging deeper into the science of acceleration. This not only confirmed Galileo’s claims, but also revealed new findings.
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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 massthe 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’s findings are foundational to people’s understanding of the movement of objects to this day. The fact that the effects of these forces are proportional to the object affected means that people can predict and therefore control such movement.  The idea seems simple now, but it took several round of further curiosity to unveil this fundamental law.
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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.
Just like his laws of motion, Newton’s law of gravity states that the force acts on a object proportionally to that object’s mass. This makes it a fairly simple rule to work with, something that suits Hawking’s definition of a useful scientific theory. Objects therefore fall to the earth at the same rate no matter their weight. Bigger balls will experience a higher gravitational force, but that force has to pull on a bigger object.
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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.
Distance is also a factor in gravity, but again, as the force acting on objects is proportional to their distance from each other, the model is fairly simple for scientists to work with. This also agrees with observation, as the planets remain in fairly stable orbits around the sun, instead of careering off in another direction when a slight disturbance interferes with the balance, such as a passing meteor.
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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, for example, a table seems to be perfectly still when you look at it, Aristotle never thought otherwise. But Galileo and Newton tested the theory out by analyzing the basics of how forces apply to objects. Newton’s discoveries led to the next logical step, that you would have to find a perfectly static object to compare that table to. A seemingly obvious answer might be the earth, but the earth is hurtling through space around the sun, taking the table along with it. Objects, then, are never truly at rest. 
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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.
The next step, then, involves relating two, or more, moving objects to each other. If neither object is truly at rest, then the question is which point of observation one should take. Hawking provides an accessible, everyday example to show this is not a cosmic quandary only, but that ordinary situations involve the same issues. This is a fundamental question of how to perceive and explain the world.
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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.
Each observer will have their own measurement of how the objects moved in relation to one another according to their own position and movements, and there is no reason to prefer one over the other because each viewpoint is valid.
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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.
Again, the reader sees the mighty fall victim to their own assumptions. In this case, Newton’s religious beliefs create an inner crisis for the scientist, as his own discoveries challenge his worldview. Interestingly, Hawking chooses a religious thinker as an example of an exasperated onlooker, showing there is not necessarily a religion vs. science divide, only stubborn people and their own demons.
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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.
Commonsense is not necessarily the most accurate of measurements. Here, Hawking does not blame Aristotle and Newton for their out of date ideas, as not everyone can discover everything all in one go, and he doesn’t suggest they obstructed any discoveries related to the function of time. Instead, Hawking explains that in daily life, time moves at much the same rate for everyone, so time would seem absolute to the casual observer. But when it comes to much faster events, objects, or forces, the everyday approach cannot be trusted.
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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.
Roemer’s curiosity led him to inquire further into the irregular orbits of Jupiter’s moons. Instead of simply accepting what he saw, Roemer set out to understand why the moons appeared from behind the planet at different times. His curiosity led to the discovery that light has a fixed speed. He even had a try at calculating that speed, although given his work came before Newton’s crucial laws of motion and gravity, he wasn’t very close.
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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.
Further work on the nature of light uncovered its numerous varieties, as defined by its wavelength. This refers to the distance between each wave crest, or peak, and the next. Hawking makes a point of noting that Maxwell’s discovery came from unifying the theories of electricity and magnetism. He hints at the wonders that might be uncovered at the unification of physics that he seeks.
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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.
Having discovered the different types of light, scientists were bursting with more questions about how this all worked. The new focus of their curiosity was how light moved. It figures it must move through something. They proposed the idea of ether that gave a static position against which to measure light.
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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 observationthe speed of light was the same.
Because the earth should be moving through the ether too, while light would travel at a fixed speed through the ether, it ought to appear to move at different speeds to observers at different angles to the earth’s movement, for example moving toward the light source (the sun) as compared to at a right angle from it (looking away into space). But this was found not to be the case. More questions therefore arose.
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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.
With perhaps the hardest new idea to swallow so far, Albert Einstein earned his fame by suggesting time was not absolute, just as Newton had found centuries earlier that space was not absolute. By allowing different measures of time, the unfounded idea of an ether could be abandoned, as it had not agreed with observations.
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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.
While the laws of science apply to all observers in the same way, and the speed of light is fixed, Einstein’s theory of relativity stated that every observer will have a different measure of both space and time, relative to their own motion.
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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.
Einstein’s suggestion was a completely new way of looking at the universe. If mass and energy are equivalent, the faster an object moves the more its mass will be. This means objects need an exponential amount of energy to keep accelerating, as each step up in speed requires more energy than the last step to shift the object’s ever-increasing mass.
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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.
Normal objects simply cannot gain enough energy to make the speed of light because the amount of energy needed is infinite. Thus, nothing can travel at the speed of light. But if the accelerating objects has no mass, and is a wave, it could be possible.
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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’.
In the theory of relativity, time and distance traveled are variable quantities, while the speed of light provides a stable measurement to use in calculations. This means each observer will have their own measurement of time and distance teveled, but must agree on the speed of light. Following in his forebears’ footsteps, Einstein built on Newton’s earlier work. His new approach answered questions that had plagued scientists for centuries, but it was not to be a final answer.
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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.
Einstein’s ideas did not stay simply that—ideas. His discovery has been applied into real-life, everyday situations to make measurements more accurate. Here, the key to accuracy is understanding the differene perspectives of all related observers, each with a different and equally valid viewpoint. This allows each observer to know the different measurements another observer would have, an obvious advantage.
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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.
Being able to apply Einstein’s findings into real life situations leads to greater accuracy than previously available—a strong vote in favor of the advancements of science. Hawking provides the historical platinum bar as a static, inflexible contrast to Einstein’s more dynamic solution.
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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.
Out with ether and in with space-time—Einstein’s theory of relativity asked the scientific community for a total overhaul in approaching seemingly simple ideas like distances and time, yet Hawking does not hint that there was any reluctance to accept this. Perhaps Einstein’s finding was so accurate it withstood any challenges, or perhaps the time was ripe for increasingly rapid scientific discovery.
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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.
Human curiosity drives the desire to know where something is, specifically. Describing a location, however, is not always a simple task—the location is relative to other nearby locations, as not every point of reference will be directly applicable. Hawking’s example show that humans require specifics. Knowing the moon moves around the earth is not enough; humans have calculated just how far its orbits sits, etc.
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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.
As everything is always moving, an event describes what happens at a location in three-dimensional space with the added dimension of time. The moon was a certain number of miles from the sun two minutes ago, but it is no longer in the same exact place. Space-time is therefore four-dimensional.
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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.
Humans are not content to merely observe. They plot, measure, determine, and so on. With the creation of the idea of space-time came the creation of space-time diagrams, to accurately plot, measure, and determine events in four dimensions.
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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.
Humans have calculated that it takes light takes four years to travel to the earth’s sun from a nearby star called Alpha Centauri. The graph Hawking uses to depict this fact takes into account the four dimensions required by space-time.
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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).
Hawking describes a handy graph depicted on a two-dimensional sheet or screen but implying three dimensions with perspective (i.e. imagining the depth into the paper or screen). Scientists use such graphs to plot other events in relation to an original event at the tip of the cone, allowing for greater analysis in four dimensions.
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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.
The events analysed in relation to the original event, P, can then be classified, as the observer seeks ever greater specificity. Unsatisfied with analyzing events in isolation, people have devised these graphs to allow closer cross-analysis.
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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.
Just knowing the sun’s light reaches the earth was not enough. Scientists have precisely measured the time it takes for the light from the sun to reach earth. The next step is to realize that observers from the earth are always seeing the sun in its past, and the step beyond that, that the same view of the wider universe is even more outdated.
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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.
Despite the fact that nothing, including people, can travel at the speed of light (as Hawking explained earlier), scientists were still determined to devise a way to map out such movement. In such models, light speed becomes a fixed number, allowing greater analysis of fast-moving obejcts.
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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.
Einstein’s theory had caught a snag. Gravitational forces could theoretically move faster than the speed of light, which didn’t fit with his model. Einstein could have taken the same route as Ptolemy, when the latter ignored the fact the moon should have appeared twice as big sometimes if it followed the route his model suggested.
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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.
Einstein went back to the drawing board, and came up with a new and improved theory that better fit observations. By accepting the limitations of his previous theory, Einstein made yet another landmark suggestion that changed the way people see the universe, again. Time is not absolute, and space is not flat.
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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.
Einstein’s discovery had real-life applications in that it helps to understand how planes navigate around the globe and how large bodies pick out routes in the cosmos. His concept turned previous perceptions on their head, to offer a different perspective never before considered.
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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.
Einstein’s new theory built on his earlier theory that had built on Newton’s earlier theory, and so on. Einstein’s idea could predict the planets’ movements more accurately because he started from and expanded on his predecessors’ work. In this way, Hawking describes the ever-advancing progress that can be achieved by continuing to challenge and expand on the theories currently used. Hawking shows that, so far, there have always been more layers of knowledge to dig into, and humans’ curiosity has never exhausted.
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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.
The next question in the long line of science’s ongoing inquiries into the workings of the universe was how Einstein’s new version of his relativity theory would affect light’s movement through the cosmos. The phenomenon of the sun bending light might not affect day-to-day life on earth, as most things people need to see to survive are a lot closer than that. Still, scientists theorized that light ought to bend around the earth if the theory of general relativity was right, and set out to check.
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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.
Undeterred by the difficulties of measuring this effect, scientists waited until the opportune moment, during an eclipse, to see if their guess was right. Their suggestion was found to match with observation, fulfuilling Hawking’s requirements to be considered a solid theory.
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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.
Although it might not have seemed directly useful initially, the discovery that light bends according to gravitational effects has indeed had real-life applications. By extension of the theory, light should lose energy as it tries to escape the earth’s gravitational pull, causing time to slow. Now that humans have put satellites into orbit, such differences in time are critical to ensuring their safe operation. Hawking provides this example to show that greater scientific understanding does improve humanity’s ability to survive—as satellites acting erratically and crashing to earth would pose a very real risk.
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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.
Hawking now provides a more human example to emphasize the previous point and bring to life the reality and consequences of the theory of relativity. The more humanity advances, the greater the effects such concepts will have on daily life. It will no longer be satellites alone that have a totally different measure of time, but even family members.
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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.
Einstein’s theories asked for a considerable change of perspective among the scientific community. Throughout his explanation of the theory of relativity, Hawking does not comment on any opposition to it. It could be that no challenges to the theory are worth the space they would take up in this book. It could also be that Hawking intends to impress the significance of Einstein’s revolutionary model. Either way, Hawking shows that scientific progress will continue apace, as knowledge supports and spurs on ever more knowledge.
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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.
To wrap up the chapter, Hawking hints toward the next pages in the history of scientific progress to be discussed in the book. The lineage continues.
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