This article is a reproduction of parts of a chapter from the Ronald C. Pine book "Science and the Human Prospect". To return to the Astronomical Teaching Resource home page, click HERE.

"Understanding the Theory of Relativity"
by Ronald C. Pine

"In principle one could reach the year 2000 in a few hours."

- Paul Davies

"Modern scientific epistemology ... justifies discoveries of such far-reaching consequences as would, in former times, have been merely empty speculation, fantasies without empirical foundation."

- Hans Reichenbach

Slow Time and Fast Time

"If you do not ask me what is time, I know it; when you ask me, I cannot tell it."

- St. Augustine

When you're sitting in a room listening to a very boring speech or lecture, it seems to take forever, and time "drags"; when the lecture is so interesting that it all seems to take place in a matter of moments, time "flies". But we all "know" that this is only a subjective experience, that the real time measured by an objective instrument such as a clock clicks along at the same rate whether or not we are interested in what is happening.

But do we really know this? Our knowledge of the objective world must be pieced together on the basis of our subjective experiences. For all we know at the moment we become engrossed in some interesting activity, at that precise moment everything in the universe "speeds up", including the clock! Or perhaps at the precise instant that we become bored, the universe decides to "slow down" everything, including the clock. So when we look at the clock and see that an hour has passed, we conclude that an objective hour has passed, but how do we really know if it was a fast hour or a slow hour? If everything in the universe could speed up or slow down, then the clock would either speed up or slow down as well, and there would be no way to tell.

Preposterous you say. All that is needed is for two people to be in the room, one bored and the other not, and then have both see that based upon the objective clock time, one hour has passed. Case closed. As Socrates and Plato noted long ago, it is impossible for a single individual to be objective. The experiences of others and communication with others about what they experience are necessary conditions for understanding the world. Part of what we mean by "objective" knowledge is public knowledge, a knowledge established by a community of observers. Also implied is a definite state of existence independent of our observations. In the case of time, it is what it is independent of our emotional states.

But wait. Suppose someday we finally discover another intelligent form of life separated from us by a great distance. Suppose that just by chance one day their entire planet is bored and everyone on our planet is not. Where will we now look to find an objective clock? How would we know that the universe does not slow down for them and speed up for us? We could synchronize two clocks on Earth and then transport one to this distant planet. How would we know that the two clocks stay synchronized? We would assume that if nothing is physically wrong with the clocks, they would stay synchronized. But unless we have a way of directly comparing the time these clocks are measuring, we would not know whether they are still synchronized.

This example is far-fetched, but its possibility demonstrates that unless a universal standard of time measurement exists, we cannot say we know that time flows on uniformly as our common sense dictates and does not speed up or slow down. Instead we must admit that we are assuming that time behaves throughout the universe the way we experience it here on Earth. We are assuming that on some other planet "now" is the same as "now" here on Earth, that at any given moment there is a "slice of simultaneity" throughout the universe.

This assumption is, of course, rather safe and reasonable for most practical purposes on Earth. But is it true? The history of science has shown us repeatedly that we should be very careful in projecting those views of reality that are practical as real. Most of what human beings do on this Earth can be accomplished by assuming the same set of beliefs accepted in the Middle Ages. We see the Sun moving everyday and we do not feel the Earth moving. We have learned, however, that our experience on this planet encompasses but a small portion of all that exists and that the universe is not required to conform to our view of things.

Strange indeed. The fact that we have no way of knowing if the universe is slowing down or speeding up at any given moment, that we must "assume" that it is not, is a paradox. It is the kind of thinking that the average person would not take seriously. With problems like this, small wonder that few people major in philosophy at universities. History shows us, however, that great thinkers have always taken such paradoxes seriously, seeing them as nature's way of waking us up to some possible secret and revealing the fallibility of our "normal" thinking. In this chapter we will see that Albert Einstein's realization that Newtonian scientists assumed, like the rest of us, that time stays normal stimulated a great discovery.

Einstein and a Philosophical Discovery

"It requires a very unusual mind to undertake the analysis of the obvious."

- Alfred North Whitehead

An important assumption we make about our universe is that a clock measures time the same regardless of our perspective of it. Suppose we synchronize two clocks and separate them. Suppose one clock stays on Earth and another is taken to Mercury. How do we know that they stay synchronized? Again following Newton, we assume that time flows on objectively, everywhere being the same, and that assuming that the clocks are both working correctly, they will objectively measure this flow the same way. If the clock on Earth says 2:00, I assume that this is also the time on Mercury according to the other clock. If two full hours have passed on Earth since I last looked at the clock, I assume that exactly two full hours have passed on Mercury as well.

One of the great tasks of philosophy is to reveal the assumptions we make when we assert something to be true. Often this is very difficult precisely because it is so easy. Our assumptions are usually so obvious, so fundamentally embedded in our outlook, that we cannot recognize that we are making them. Philosophical analysis is a vital part of the scientific method. As we saw in "Philosophy and the Scientific Method", many ideas are involved in deducing possible results that can then be tested by experiment. Behind every experiment is a hypothesis set, consisting of the main hypothesis, many minor hypotheses and assumptions. This set, along with the conditions of the experiment, serves as a premise for inferring what should happen when nature is subjected to our probing. Logically, if the result of an experiment is negative, if what we expect to happen does not happen, then this proves only that at least one of the ideas of the hypothesis set must be false. Thus, it is important to identify as many as possible of the ideas that make up the premises for the predicted result.

Einstein recognized that to assume that two separated clocks stayed the same involved a philosophical bias. Why should they stay the same? If we are to be empirically honest and subject all our assumptions to tests, then we need some way to measure what time the clocks record. Because there is no big cosmic clock in the center of the universe for all to see simultaneously, the only way we can measure the time of our clocks and see if they stay synchronized is by directly comparing them, and at great distances this necessarily involves the speed of light. To be sure that a clock on Mercury is still keeping the same time as one on Earth, we must communicate with an observer by Mercury by sending an electromagnetic signal traveling at the speed of light. Also, because both planets are moving in relation to each other and because the speed of light is finite, we must take into account the speed of light and what effect, if any, the relative motion of the planets might have on this speed.

At the turn of the century when Einstein as a young man was thinking about such things, the speed of light had also become a paradox. It was known that the speed of light had a finite but very great velocity (186,000 miles per second), and it was assumed that the measurement of this velocity, like any other velocity, should change depending upon how fast the source of light is moving and in what direction. For instance, if we are on an object moving at a speed of 18 miles per second, the speed of our Earth around the Sun, and we send a signal to Mercury in the direction of the motion of the Earth around the Sun, then should not the speed of the electromagnetic signal be the regular speed of light plus the speed of our motion (total of 186,018 miles per second)?

If a train is moving at 60 miles per hour and a person on the train is walking at 3 miles per hour in the same direction as the train is moving, then that person's total speed, as measured by a stationary observer, is 63 miles per hour. The two velocities are added together. Should not light be the same?

By the turn of the century experiments showed that light did not behave the way our common sense says it should. No matter what speed or direction an object moved, a beam of light from that object was always the same--186,000 miles per second. Whether a beam of light was traveling in the direction of Earth's motion, at right angles to its motion, or in the opposite direction of its motion, the speed of the light beam was precisely the same.

The implications of this result were not easy to digest. If we could travel in a special vehicle, say at only 100 miles per second less than the speed of light (185,900 miles per second), and we turned on a flashlight and pointed it in the direction of our motion, the flashlight beam would move away from us at the regular speed of light. If we increased our speed to 185,999 miles per second, we would not gain on the beam of light. Finally, if our vehicle could reach the speed of light, the speed of our flashlight beam would still be the normal speed of light. It doesn't matter if we're holding the flashlight within the vehicle or we're watching someone holding the flashlight as they fly by, the beam travels at the speed of light in our reference frame! In the case of light the normal addition of velocities does not work; one plus one is one!

Einstein boldly accepted these paradoxes as axioms: Time must be "tested", and the speed of light is the same regardless of the speed of its source. Einstein then recognized that for science to establish universal laws of nature, laws that remain the same regardless of one's point of view, then a price had to be paid. We must accept the fact that when we test time, it will speed up or slow down relative to moving frames or reference. We must accept as commonplace that because Mercury and the Earth are moving in relation to each other, and any electromagnetic communication device will transmit a signal at the speed of light unaffected by the relative motion of the planets, the clocks on these planets will not be synchronized when they are compared.

Cosmic Trains

"The testimony of our common sense is supect at high velocities."

- Carl Sagan


In the figure shown imagine a special train moving toward point A and away from point B. On the train is a person we will designate as X. As the train passes by, imagine a person, Y, midway between points A and B. Suppose as Y watches the train pass, at exactly the moment X is opposite Y, two bolts of lightning strike the ground from Y's point of view simultaneously at points A and B. How would X view the two bolts of lightning?

Let's imagine that the train is moving very fast, at about three-fifths the speed of light. Since X is moving toward point A at such a great speed, the light from A will be received significantly before the light from point B, which will have to catch up to the swiftly moving train. Thus, whereas from Y's point of view the two events were simultaneous (happening at the same time), from X's point of view they were not simultaneous at all. The bolt of lightning struck A before B. Who is right?

It is tempting to respond immediately that Y is right because X is moving. It is comfortable to think that Y's reference frame is the right place from which to view the "actual truth of the matter". Because X is moving so fast, it is easy to believe that his experience is an illusion due to his motion. If X got in the "right place", if he slowed down, then both observers would see the same thing, the bolts of lightning striking the ground at the same time. But wait. Y is also moving. Y is, in fact, moving many different ways, depending upon which reference frame is adopted. If Y is close to the equator of the Earth, he is moving at about 1,000 miles per hour. From the point of view of the Sun, Y is moving at approximately 66,600 miles per hour as the Earth orbits the Sun. And from the point of view of the center of our galaxy, he is moving at a speed of over 500,000 miles per hour as the Sun orbits the galactic center. Where is the right place? Why can't X assume that Y is the one who is moving?

A Newtonian might object that X could use simple mathematics to detect his motion relative to the lightning flashes, and on this basis he could calculate the simultaneity of the flashes. In other words, perhaps X could measure the incoming speed of light from A and find that it is approaching at a speed equal to the speed of light plus X's own speed, three-fifths the speed of light. The opposite, of course, is true when X looks at light coming from point B.

Unfortunately, in the case of light, nature does not cooperate. If X had the proper equipment to measure the speed of the incoming light signals from A and B, he would find that the speeds of each beam are the same, the normal speed of light. Similarly, if Y had the proper equipment, he would also find the beams coming from A and B to have the same speed. Thus, both observers are entitled to adopt the perspective that they are at rest and the other is moving.

Einstein's solution states that we must obey the facts. The speed of light, as a law of nature, is the same everywhere for every observer, and this is true no matter how each observer is moving relative to another. Furthermore, unless we are willing to make an unwarranted metaphysical assumption, time must be tested; that there is a "right" place where time is absolute is just an assumption for which there is no evidence.

Time is very much related to our relative place in space. Our time measurements (clocks, calendars) are actually spatially "local" things. On Earth when we measure one hour, we are really measuring a portion of our space, a portion of the rotation of the Earth (approximately 15 degrees). On Mercury this convention would be inconvenient, because that planet rotates once every 59 Earth days and revolves around the Sun in 88 Earth days. The combination of these rotation and revolution periods makes one Mercury day equal to two of its years!

Einstein's theory of relativity teaches us that when we are fairly close together and moving together - when, as on Earth, the speed of our relative motions is very small in comparison to the speed of light - then time will behave itself. In our train example, if the train is moving at a normal train speed, then X also measures the bolts of lightning as simultaneous. But when astronomical objects are widely separated and move at great speeds relative to each other, time does not obey Earthly standards.

Our assumption of an absolute time, the intuitive feeling that time clicks along at a steady rate throughout the universe, that "now" on Earth is the same as "now" at all locations, is due to the fact that we normally do not move at such great speeds relative to the speed of light. These relative measurements of time show up only when relative speeds are attained appreciably close to the speed of light. This is the essence of Einstein's great discovery: partly a philosophical discovery (time must be tested) and partly an empirical discovery (the speed of light is the same in all reference frames). For the most part, the rest was logic and mathematical deduction.

It is important to understand that the relativity of time measurements is necessary to preserve the laws of nature. Although two observers will measure the temporal occurrence of events differently, from their respective reference frames they will not notice anything unusual. The theory of relativity does not prove that everything is relative. Both observers in our train example will find that the laws of nature apply normally, regardless of what they might think or wish to be true. If observer X conducted experiments in a laboratory on the train, he would obtain the same results as Y would in a laboratory located in his reference frame.

Einstein took this thought logically to its limits in his Autobiographical Notes:

After ten years of reflection such a principle resulted from a paradox. If I pursue a beam of light with a velocity equal to the speed of light, I should observe such a beam at rest. However, there seems to be no such thing [light has always been measured with a speed equal to the speed of light, regardless of the reference frame]. From the very beginning it appeared to me intuitively clear that, judged from the standpoint of such an observer, everything would have to happen according to the same laws as for an observer who, relative to the Earth, was at rest.

Time Dilation

"The idea that time can vary from place to place is a difficult one, but it is the idea Einstein used, and it is correct - believe it or not."

- Richard Feynman

Einstein then realized that if the laws of nature are the same from every standpoint, then for physicists to be able to continue to do physics in a universe of relative moving objects, they will need to mathematically transform how time will be viewed from different reference frames. Einstein used a mathematical equation called the Lorentz-Fitzgerald transformation. It worked perfectly. The equation is a relatively simple algebraic relationship:


In this equation, T is the time of an event in a reference frame moving at a velocity v in relation to an observer is measured by the variable T'. The c is a constant, the speed of light. Let's see how it works.

Suppose in our train example that X and Y both possessed clocks that some time ago were synchronized when the train was stationary. Y then moved at a normal speed, very slowly compared to the speed of light, to a point between A and B. Suppose at a prearranged time X departs and that for the past 15 minutes by Y's reckoning the train has left the initial starting point and been moving to the point where Y is at an average speed of three-fifths the speed of light. Y would need to be 100,440,000 miles away, and it would have taken him a little over 191 years to travel to this point at 60 miles per hour. What time will the clock of X read according to the equation? If we plug in the data,


When X pases Y, X's clock will show that the train has been moving for only 12 minutes! From X's point of view, time has slowed down relative to Y. It would not just seem to slow down; real physical measurable effects would be seen when X and Y compare their clocks. If X and Y both lit cigars at the prearranged time, X would find that his cigar has burned less than Y's as they pass each other.

X will experience nothing unusual. The laws of nature are the same, including those for burning cigars. For X everything will appear normal including the movement of his clock. At no point will he see the clock suddenly slow down dramatically. It will appear normal. Likewise, Y will not suddenly see the last few minutes of his expected 15 minutes fly by like the clocks in bad aspirin commercials. It will also appear normal. Nor will either notice anything strange about the rate their cigars burn. Time slows down in reference frame X only in relation to reference frame Y. Within their respective reference frames, everything is normal.

This slowing down of time of a reference frame relative to another reference frame is called time dilation. As unbelievable as it may seem, it is one of the most accepted scientific facts of our time. It has been tested in numerous ways. Very precise atomic clocks have been synchronized and then compared again after one was flown around the world in a speeding jet. The clock on the speeding jet slowed down in relation to the clock that stayed on the ground. A similar test was conducted using one of the U. S. space shuttles with the same result.

Scientists now apply time dilation routinely in sophisticated laboratory situations. In the billion-dollar particle accelerator laboratories all over the world, physicists keep special particles of matter "alive" far longer than would normally be expected because of the time dilation effects that result by accelerating particles to speeds close to the speed of light. In this way special forms of energy can actually be stored for use in crucial experiments.

Trains, of course, do not move at three-fifths the speed of light (or 111,600 miles per second); hence, it is easy to see why time dilation effects were not noticed by common sense observations. Theoretically, spaceships could travel this fast. What kind of astronautical scenarios are possible? Suppose we know twins 20 years of age, one an astronaut who will take a space voyage that will take 20 years by Earth time. Suppose that the astronaut twin averages in his rocketship a speed of three-fifths the speed of light. How old will each twin be when they meet again 20 Earth years from now? Using the time dilation equation we have:


which would equal 36. The twin who stays on Earth will be, of course, 40 years of age. And yet his brother will now be 36!

This example is not science fiction. We believe, based on the best laboratory data available and other corroborating evidence, that this effect on an astronaut would indeed happen. If it is so hard to believe, it is because we have difficulty realizing that the things we take for granted on Earth do not necessarily apply throughout the cosmos. With Einstein, the cosmos is now our laboratory, and we must adjust to the conditions of this new laboratory.

Stranger still, consider this scenario. The star Vega, a star very much like our Sun and a possible candidate for a planetary system, is approximately 32 light years away. Suppose in the near future a 30-year-old mother, who has a 5-year-old son, went on a space voyage to explore this star, averaging the colossal speed of 99.5 percent the speed of light. At such a speed, it would take her about 64 years Earth time to make the trip. When she returns, her son, who remained on Earth, would be into his 69th year. Both would be in for a great shock. The 69-year-old son would embrace his long-awaited 36-year-old mother! Their personal histories would have seemed to be normal in all other respects, but it would now be clear that travelling at great speed slows our histories down from one point of view and allows us to speed to the future from another. If such a voyage were taken, the mother astronaut, if she left in the year 1990, could get to the year 2054 in a little over 6 years.

An event requiring only 6 years for the mother would require 64 years for the son. In our train example, an event that happened before another event for one observer (lightning striking point A before point B for the observer X on the train) happened at the same time for another observer (the stationary observer, Y). If would also be possible then for another observer moving in the opposite direction of reference frame X at a great speed to record the lightning striking at point B first. Thus, one person's past could be another's future. Would it then be possible for the mother to return at an age before her son was born?

Not according to Einstein's theory, not if the speed of light is a law of nature. Because the speed of light is an absolute that cannot be exceeded, causal connections, such as mothers' causing the birth of babies, are preserved in their normal sequences. According to Einstein's theory, the measurement of "before" and "after" may involve a wide latitude, but the order of events will not be changed. The time between the mother's "before" and "after" of her space voayge is much shorter than that experienced by her son, but both would experience her leaving before she came back.

If and only if the speed of light can be exceeded will the sequencing of causal events be changed, and it is a basic consequence of Einstein's theory that the speed of light cannot be exceeded. According to his theory, it would take an infinite amount of energy to accelerate any object (even an electron) up to the speed of light and thus require more than an infinite amount of energy to exceed the speed of light. Note, however, that the speed of light would only need to be exceeded by 0.004358 percent for the mother, if she left on the day of her 30th birthday, to return on the beginning of the second day after her 24th birthday, at least one year before her son was born and a few months before the conception! What would happen if she were then involved in a fatal car accident?

Epistemological Implications

"The universe plays fair. Its tricks may operate by principles of incredible subtlety, and we may never discover all of them, but it keeps performing its illusions over and over again, always by the same method."

- Martin Gardner

The epistemological implications of relativity theory are very significant. The role of the observer is much different from that of Newtonian science. In Newtonian science the variety of perspectives of human observation, and the observer himself, could be ignored, excused as irrelevant to our descriptions of the real world. The different results of observation due to different reference frames were considered to be simply practical inconveniences that could be reconciled by Galilean transformations. But in relativity theory, the observer is intimately involved in scientific measurement, and what is measured can be different depending on one's reference frame. Our knowledge of the world must unfold from empirical measurement of it. In the destruction of absolute space and time, Einstein showed that an honest empiricism must involve the observer, and that to some extent what is real does depend on us.

However, Einstein did not think he had proved that each observer is involved in creating reality. Einstein did not doubt the existence of an independent physical reality, or whether there must be some absolutes. In fact, the intent of his theory was to preserve absolutes, the laws of nature. Einstein thought the secret structure of nature resembled the internal mechanism of a special, mysterious, cosmic clock. We are forever limited to seeing the outside motions of the hands and can only submit hypotheses about how the internalmechanism produces the movements of the hands. But limited as we are, we can judge which hypotheses are better on the basis of which ones predict best the motions that we observe.

Einstein's basic insight concerning space and time served as only the first premises in the development of the many marvels of 20th-century physics. From these insights, known as the Special Theory, Einstein later showed in his General Theory that if his ideas on space and time were true, many hard-to-believe things must be true of our cosmic laboratory, as one sees in the study of quantum mechanics, for example.

Although Einstein did not believe that his theory proved that human observers create reality, he did show that the observer beings to play a crucial role in what is real and that Copernicanism had gone too far, or in a sense not far enough. The completion of the Copernican revolution in Newtonianism fulfilled the dream of a unified science; the laws that governed terrestrial motion were the same that governed celestial motion. This enabled other ideas to get in through the back door: Our common-sense notions of space and time, which worked so well on Earth, were assumed to be true for the entire universe. This assumption was so pervasive that it was not recognized as an assumption, especially given the success of Newtonian physics. In revealing this assumption as a metaphysical postulate and not an empirical fact Einstein showed that what was masquerading as a removal of humankind and subjectivism from science was actually a projection of a subjective human point of view, which is close enough to the truth at a certain level to enable us to fail to recognize it as a human point of view.

Think about this last statement carefully. Think about it for a long time. It is crucial for understanding the paradox of 20th-century science. Our intuitive feeling that there must be an objective time and space is actually just a projection of a human point of view. We can accurately calculate the motions of the planets within our solar system using a perspective of uniform space and uniform time. We can send our robot spacecraft to the outer planets and beyond. Our equations work. But that they work in this domain does not prove that the concepts we assume in applying the equations are valid for other domains. As we will see in our discussion next of quantum physics, this realization is only the beginning. As in a long romantic relationship, one's partner may eventually reveal a totally unexpected set of personality traits. After Newton we thought we knew what the universe was like. Little did we suspect the unnerving surprises it had in store for us in the 20th century...

Concept Summary

Newtonianism forces us to assume that space and time are absolute--that each event has one spatial and temporal location. In this chapter we have seen that part of Einstein's discovery involved a philosophical insight, that Newtonianism involved this assumption, and that scientific knowledge of space and time would require empirical tests. It seems paradoxical that we cannot safely assume and know upon reflection alone that time flows on uniformly throughout the universe as our common sense dictates. It never occurs to me that if I leave my home at 8:00 and arrive at my office at 8:30 that I am assuming that it is not then 9:00 at home.

Einstein recognized that one could not assume that two widely separated, initially synchronized clocks kept the same time. He recognized that time must be tested by measuring and comparing what time each clock records. To compare such clocks, one must take into account the speed of light and the relative motions of the reference frames of each clock. Einstein showed that when this is actually done, because the speed of light had been discovered to be the same regardless of its direction and the speed of its source, our intuitive sense that time is something that just clicks along independently of moving objects will be violated when we compare initially synchronized clocks. Simultaneity is relative to a reference frame, and time dilation, the slowing of time relative to another reference frame, is a fact of life. If my home and office were separated by many light years, not only could I not assume the times to be the same, but my home (depending on the distance and the speed of my travel) could be many thousands of years in the future when I arrive at work.

Although the success of Einstein's theories does not imply that everything is relative or that scientists create reality, it does show that an honest empiricism, one that tests fundamental assumptions, has brought the observer into 20th century physics. To some extent what is real does depend on us. Einstein's theories also set the stage for the great paradox of 20th century science. We must be careful about what we assume is objective; what seems to be obviously an objective property of reality (say, what time it is) may be a subjective projection of a merely human point of view, one applicable to only a limited range of experience (to velocities far less than the speed of light). Nothing seems to me more certainly objective and independent of my wishes than the thought that it is the same time at my home now as at my office regardless of the distance between the two, but Einstein has shown that reality need not obey my sense of certainty or the workings of the human mind.

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