frame of reference

Dictionary: frame of reference

n., pl. frames of reference.

A set of coordinate axes in terms of which position or movement may be specified or with reference to which physical laws may be mathematically stated. Also called reference frame.
A set of ideas, as of philosophical or religious doctrine, in terms of which other ideas are interpreted or assigned meaning.

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Science of Everyday Things: Frame of Reference

Concept

Among the many specific concepts the student of physics must learn, perhaps none is so deceptively simple as frame of reference. On the surface, it seems obvious that in order to make observations, one must do so from a certain point in space and time. Yet, when the implications of this idea are explored, the fuller complexities begin to reveal themselves. Hence the topic occurs at least twice in most physics textbooks: early on, when the simplest principles are explained—and near the end, at the frontiers of the most intellectually challenging discoveries in science.

How It Works

There is an old story from India that aptly illustrates how frame of reference affects an understanding of physical properties, and indeed of the larger setting in which those properties are manifested. It is said that six blind men were presented with an elephant, a creature of which they had no previous knowledge, and each explained what he thought the elephant was.

The first felt of the elephant's side, and told the others that the elephant was like a wall. The second, however, grabbed the elephant's trunk, and concluded that an elephant was like a snake. The third blind man touched the smooth surface of its tusk, and was impressed to discover that the elephant was a hard, spear-like creature. Fourth came a man who touched the elephant's legs, and therefore decided that it was like a tree trunk. However, the fifth man, after feeling of its tail, disdainfully announced that the elephant was nothing but a frayed piece of rope. Last of all, the sixth blind man, standing beside the elephant's slowly flapping ear, felt of the ear itself and determined that the elephant was a sort of living fan.

These six blind men went back to their city, and each acquired followers after the manner of religious teachers. Their devotees would then argue with one another, the snake school of thought competing with adherents of the fan doctrine, the rope philosophy in conflict with the tree trunk faction, and so on. The only person who did not join in these debates was a seventh blind man, much older than the others, who had visited the elephant after the other six.

While the others rushed off with their separate conclusions, the seventh blind man had taken the time to pet the elephant, to walk all around it, to smell it, to feed it, and to listen to the sounds it made. When he returned to the city and found the populace in a state of uproar between the six factions, the old man laughed to himself: he was the only person in the city who was not convinced he knew exactly what an elephant was like.

Understanding Frame of Reference

The story of the blind men and the elephant, within the framework of Indian philosophy and spiritual beliefs, illustrates the principle of syadvada. This is a concept in the Jain religion related to the Sanskrit word syat, which means "may be." According to the doctrine of syadvada, no judgment is universal; it is merely a function of the circumstances in which the judgment is made.

On a complex level, syadvada is an illustration of relativity, a topic that will be discussed later; more immediately, however, both syadvada and the story of the blind men beautifully illustrate the ways that frame of reference affects perceptions. These are concerns of fundamental importance both in physics and philosophy, disciplines that once were closely allied until each became more fully defined and developed. Even in the modern era, long after the split between the two, each in its own way has been concerned with the relationship between subject and object.

These two terms, of course, have numerous definitions. Throughout this book, for instance, the word "object" is used in a very basic sense, meaning simply "a physical object" or "a thing." Here, however, an object may be defined as something that is perceived or observed. As soon as that definition is made, however, a flaw becomes apparent: nothing is just perceived or observed in and of itself—there has to be someone or something that actually perceives or observes. That something or someone is the subject, and the perspective from which the subject perceives or observes the object is the subject's frame of reference.

America and China: Frame of Reference in Practice

An old joke—though not as old as the story of the blind men—goes something like this: "I'm glad I wasn't born in China, because I don't speak Chinese." Obviously, the humor revolves around the fact that if the speaker were born in China, then he or she would have grown up speaking Chinese, and English would be the foreign language.

The difference between being born in America and speaking English on the one hand—even if one is of Chinese descent—or of being born in China and speaking Chinese on the other, is not just a contrast of countries or languages. Rather, it is a difference of worlds—a difference, that is, in frame of reference.

Indeed, most people would see a huge distinction between an English-speaking American and a Chinese-speaking Chinese. Yet to a visitor from another planet—someone whose frame of reference would be, quite literally, otherworldly—the American and Chinese would have much more in common with each other than either would with the visitor.

The View from Outside and Inside

Now imagine that the visitor from outer space (a handy example of someone with no preconceived ideas) were to land in the United States. If the visitor landed in New York City, Chicago, or Los Angeles, he or she would conclude that America is a very crowded, fast-paced country in which a number of ethnic groups live in close proximity. But if the visitor first arrived in Iowa or Nebraska, he or she might well decide that the United States is a sparsely populated land, economically dependent on agriculture and composed almost entirely of Caucasians.

A landing in San Francisco would create a falsely inflated impression regarding the number of Asian Americans or Americans of Pacific Island descent, who actually make up only a small portion of the national population. The same would be true if one first arrived in Arizona or New Mexico, where the Native American population is much higher than for the nation as a whole. There are numerous other examples to be made in the same vein, all relating to the visitors' impressions of the population, economy, climate, physical features, and other aspects of a specific place. Without consulting some outside reference point—say, an almanac or an atlas—it would be impossible to get an accurate picture of the entire country.

The principle is the same as that in the story of the blind men, but with an important distinction: an elephant is an example of an identifiable species, whereas the United States is a unique entity, not representative of some larger class of thing. (Perhaps the only nation remotely comparable is Brazil, also a vast land settled by outsiders and later populated by a number of groups.) Another important distinction between the blind men story and the United States example is the fact that the blind men were viewing the elephant from outside, whereas the visitor to America views it from inside. This in turn reflects a difference in frame of reference relevant to the work of a scientist: often it is possible to view a process, event, or phenomenon from outside; but sometimes one must view it from inside—which is more challenging.

Frame of Reference in Science

Philosophy (literally, "love of knowledge") is the most fundamental of all disciplines: hence, most persons who complete the work for a doctorate receive a "doctor of philosophy" (Ph.D.) degree. Among the sciences, physics—a direct offspring of philosophy, as noted earlier—is the most fundamental, and frame of reference is among its most basic concepts.

Hence, it is necessary to take a seemingly backward approach in explaining how frame of reference works, examining first the broad applications of the principle and then drawing upon its specific relation to physics. It makes little sense to discuss first the ways that physicists apply frame of reference, and only then to explain the concept in terms of everyday life. It is more meaningful to relate frame of reference first to familiar, or at least easily comprehensible, experiences—as has been done.

At this point, however, it is appropriate to discuss how the concept is applied to the sciences. People use frame of reference every day—indeed, virtually every moment—of their lives, without thinking about it. Rare indeed is the person who "walks a mile in another person's shoes"—that is, someone who tries to see events from the viewpoint of another. Physicists, on the other hand, have to be acutely aware of their frame of reference. Moreover, they must "rise above" their frame of reference in the sense that they have to take it into account in making calculations. For physicists in particular, and scientists in general, frame of reference has abundant "real-life applications."

Real-Life Applications

Points and Graphs

There is no such thing as an absolute frame of reference—that is, a frame of reference that is fixed, and not dependent on anything else. If the entire universe consisted of just two points, it would be impossible (and indeed irrelevant) to say which was to the right of the other. There would be no right and left: in order to have such a distinction, it is necessary to have a third point from which to evaluate the other two points.

As long as there are just two points, there is only one dimension. The addition of a third point—as long as it does not lie along a straight line drawn through the first two points—creates two dimensions, length and width. From the frame of reference of any one point, then, it is possible to say which of the other two points is to the right.

Clearly, the judgment of right or left is relative, since it changes from point to point. A more absolute judgment (but still not a completely absolute one) would only be possible from the frame of reference of a fourth point. But to constitute a new dimension, that fourth point could not lie on the same plane as the other three points—more specifically, it should not be possible to create a single plane that encompasses all four points.

Assuming that condition is met, however, it then becomes easier to judge right and left. Yet right and left are never fully absolute, a fact easily illustrated by substituting people for points. One may look at two objects and judge which is to the right of the other, but if one stands on one's head, then of course right and left become reversed.

Of course, when someone is upside-down, the correct orientation of left and right is still fairly obvious. In certain situations observed by physicists and other scientists, however, orientation is not so simple. It then becomes necessary to assign values to various points, and for this, scientists use tools such as the Cartesian coordinate system.

Coordinates and Axes

Though it is named after the French mathematician and philosopher René Descartes (1596-1650), who first described its principles, the Cartesian system owes at least as much to Pierre de Fermat (1601-1665). Fermat, a brilliant French amateur mathematician—amateur in the sense that he was not trained in mathematics, nor did he earn a living from that discipline—greatly developed the Cartesian system.

A coordinate is a number or set of numbers used to specify the location of a point on a line, on a surface such as a plane, or in space. In the Cartesian system, the x-axis is the horizontal line of reference, and the y-axis the vertical line of reference. Hence, the coordinate (0, 0) designates the point where the x-and y-axes meet. All numbers to the right of 0 on the x-axis, and above 0 on the y-axis, have a positive value, while those to the left of 0 on the x-axis, or below 0 on the y-axis have a negative value.

This version of the Cartesian system only accounts for two dimensions, however; therefore, a z-axis, which constitutes a line of reference for the third dimension, is necessary in three-dimensional graphs. The z-axis, too, meets the x-and y-axes at (0, 0), only now that point is designated as (0, 0, 0).

In the two-dimensional Cartesian system, the x-axis equates to "width" and the y-axis to "height." The introduction of a z-axis adds the dimension of "depth"—though in fact, length, width, and height are all relative to the observer's frame of reference. (Most representations of the three-axis system set the x-and y-axes along a horizontal plane, with the z-axis perpendicular to them.) Basic studies in physics, however, typically involve only the x-and y-axes, essential to plotting graphs, which, in turn, are integral to illustrating the behavior of physical processes.

The Triple Point

For instance, there is a phenomenon known as the "triple point," which is difficult to comprehend unless one sees it on a graph. For a chemical compound such as water or carbon dioxide, there is a point at which it is simultaneously a liquid, a solid, and a vapor. This, of course, seems to go against common sense, yet a graph makes it clear how this is possible.

Using the x-axis to measure temperature and the y-axis pressure, a number of surprises become apparent. For instance, most people associate water as a vapor (that is, steam) with very high temperatures. Yet water can also be a vapor—for example, the mist on a winter morning—at relatively low temperatures and pressures, as the graph shows.

The graph also shows that the higher the temperature of water vapor, the higher the pressure will be. This is represented by a line that curves upward to the right. Note that it is not a straight line along a 45° angle: up to about 68°F (20°C), temperature increases at a somewhat greater rate than pressure does, but as temperature gets higher, pressure increases dramatically.

As everyone knows, at relatively low temperatures water is a solid—ice. Pressure, however, is relatively high: thus on a graph, the values of temperatures and pressure for ice lie above the vaporization curve, but do not extend to the right of 32°F (0°C) along the x-axis. To the right of 32°F, but above the vaporization curve, are the coordinates representing the temperature and pressure for water in its liquid state.

Water has a number of unusual properties, one of which is its response to high pressures and low temperatures. If enough pressure is applied, it is possible to melt ice—thus transforming it from a solid to a liquid—at temperatures below the normal freezing point of 32°F. Thus, the line that divides solid on the left from liquid on the right is not exactly parallel to the y-axis: it slopes gradually toward the y-axis, meaning that at ultra-high pressures, water remains liquid even though it is well below the freezing point.

Nonetheless, the line between solid and liquid has to intersect the vaporization curve somewhere, and it does—at a coordinate slightly above freezing, but well below normal atmospheric pressure. This is the triple point, and though "common sense" might dictate that a thing cannot possibly be solid, liquid, and vapor all at once, a graph illustrating the triple point makes it clear how this can happen.

Numbers

In the above discussion—and indeed throughout this book—the existence of the decimal, or base-10, numeration system is taken for granted. Yet that system is a wonder unto itself, involving a complicated interplay of arbitrary and real values. Though the value of the number 10 is absolute, the expression of it (and its use with other numbers) is relative to a frame of reference: one could just as easily use a base-12 system.

Each numeration system has its own frame of reference, which is typically related to aspects of the human body. Thus throughout the course of history, some societies have developed a base-2 system based on the two hands or arms of a person. Others have used the fingers on one hand (base-5) as their reference point, or all the fingers and toes (base-20). The system in use throughout most of the world today takes as its frame of reference the ten fingers used for basic counting.

Coefficients

Numbers, of course, provide a means of assigning relative values to a variety of physical characteristics: length, mass, force, density, volume, electrical charge, and so on. In an expression such as "10 meters," the numeral 10 is a coefficient, a number that serves as a measure for some characteristic or property. A coefficient may also be a factor against which other values are multiplied to provide a desired result.

For instance, the figure 3.141592, better known as pi (π), is a well-known coefficient used in formulae for measuring the circumference or area of a circle. Important examples of coefficients in physics include those for static and sliding friction for any two given materials. A coefficient is simply a number—not a value, as would be the case if the coefficient were a measure of something.

Standards of Measurement

Numbers and coefficients provide a convenient lead-in to the subject of measurement, a practical example of frame of reference in all sciences—and indeed, in daily life. Measurement always requires a standard of comparison: something that is fixed, against which the value of other things can be compared. A standard may be arbitrary in its origins, but once it becomes fixed, it provides a frame of reference.

Lines of longitude, for instance, are measured against an arbitrary standard: the "Prime Meridian" running through Greenwich, England. An imaginary line drawn through that spot marks the line of reference for all longitudinal measures on Earth, with a value of 0°. There is nothing special about Greenwich in any profound scientific sense; rather, its place of importance reflects that of England itself, which ruled the seas and indeed much of the world at the time the Prime Meridian was established.

The Equator, on the other hand, has a firm scientific basis as the standard against which all lines of latitude are measured. Yet today, the coordinates of a spot on Earth's surface are given in relation to both the Equator and the Prime Meridian.

Calibration

Calibration is the process of checking and correcting the performance of a measuring instrument or device against the accepted standard. America's preeminent standard for the exact time of day, for instance, is the United States Naval Observatory in Washington, D.C. Thanks to the Internet, people all over the country can easily check the exact time, and correct their clocks accordingly.

There are independent scientific laboratories responsible for the calibration of certain instruments ranging from clocks to torque wrenches, and from thermometers to laser beam power analyzers. In the United States, instruments or devices with high-precision applications—that is, those used in scientific studies, or by high-tech industries—are calibrated according to standards established by the National Institute of Standards and Technology (NIST).

The Value of Standardization to a Society

Standardization of weights and measures has always been an important function of government. When Ch'in Shih-huang-ti (259-210 B.C.) united China for the first time, becoming its first emperor, he set about standardizing units of measure as a means of providing greater unity to the country—thus making it easier to rule.

More than 2,000 years later, another empire—Russia—was negatively affected by its failure to adjust to the standards of technologically advanced nations. The time was the early twentieth century, when Western Europe was moving forward at a rapid pace of industrialization. Russia, by contrast, lagged behind—in part because its failure to adopt Western standards put it at a disadvantage.

Train travel between the West and Russia was highly problematic, because the width of railroad tracks in Russia was different than in Western Europe. Thus, adjustments had to be performed on trains making a border crossing, and this created difficulties for passenger travel. More importantly, it increased the cost of transporting freight from East to West.

Russia also used the old Julian calendar, as opposed to the Gregorian calendar adopted throughout much of Western Europe after 1582. Thus October 25, 1917, in the Julian calendar of old Russia translated to November 7, 1917 in the Gregorian calendar used in the West. That date was not chosen arbitrarily: it was then that Communists, led by V. I. Lenin, seized power in the weakened former Russian Empire.

Methods of Determining Standards

It is easy to understand, then, why governments want to standardize weights and measures—as the U.S. Congress did in 1901, when it established the Bureau of Standards (now NIST) as a nonregulatory agency within the Commerce Department. Today, NIST maintains a wide variety of standard definitions regarding mass, length, temperature, and so forth, against which other devices can be calibrated.

Note that NIST keeps on hand definitions rather than, say, a meter stick or other physical model. When the French government established the metric system in 1799, it calibrated the value of a kilogram according to what is now known as the International Prototype Kilogram, a platinum-iridium cylinder housed near Sévres in France. In the years since then, the trend has moved away from such physical expressions of standards, and toward standards based on a constant figure. Hence, the meter is defined as the distance light travels in a vacuum (an area of space devoid of air or other matter) during the interval of 1/299,792,458 of a second.

Metric Vs. British

Scientists almost always use the metric system, not because it is necessarily any less arbitrary than the British or English system (pounds, feet, and so on), but because it is easier to use. So universal is the metric system within the scientific community that it is typically referred to simply as SI, an abbreviation of the French Système International d'Unités—that is, "International System of Units."

The British system lacks any clear frame of reference for organizing units: there are 12 inches in a foot, but 3 feet in a yard, and 1,760 yards in a mile. Water freezes at 32°F instead of 0°, as it does in the Celsius scale associated with the metric system. In contrast to the English system, the metric system is neatly arranged according to the base-10 numerical framework: 10 millimeters to a centimeter, 100 centimeters to a meter, 1,000 meters to kilometer, and so on.

The difference between the pound and the kilogram aptly illustrates the reason scientists in general, and physicists in particular, prefer the metric system. A pound is a unit of weight, meaning that its value is entirely relative to the gravitational pull of the planet on which it is measured. A kilogram, on the other hand, is a unit of mass, and does not change throughout the universe. Though the basis for a kilogram may not ultimately be any more fundamental than that for a pound, it measures a quality that—unlike weight—does not vary according to frame of reference.

Frame of Reference in Classical Physics and Astronomy

Mass is a measure of inertia, the tendency of a body to maintain constant velocity. If an object is at rest, it tends to remain at rest, or if in motion, it tends to remain in motion unless acted upon by some outside force. This, as identified by the first law of motion, is inertia—and the greater the inertia, the greater the mass.

Physicists sometimes speak of an "inertial frame of reference," or one that has a constant velocity—that is, an unchanging speed and direction. Imagine if one were on a moving bus at constant velocity, regularly tossing a ball in the air and catching it. It would be no more difficult to catch the ball than if the bus were standing still, and indeed, there would be no way of determining, simply from the motion of the ball itself, that the bus was moving.

But what if the inertial frame of reference suddenly became a non-inertial frame of reference—in other words, what if the bus slammed on its brakes, thus changing its velocity? While the bus was moving forward, the ball was moving along with it, and hence, there was no relative motion between them. By stopping, the bus responded to an "outside" force—that is, its brakes. The ball, on the other hand, experienced that force indirectly. Hence, it would continue to move forward as before, in accordance with its own inertia—only now it would be in motion relative to the bus.

Astronomy and Relative Motion

The idea of relative motion plays a powerful role in astronomy. At every moment, Earth is turning on its axis at about 1,000 MPH (1,600 km/h) and hurtling along its orbital path around the Sun at the rate of 67,000 MPH (107,826 km/h.) The fastest any human being—that is, the astronauts taking part in the Apollo missions during the late 1960s—has traveled is about 30% of Earth's speed around the Sun.

Yet no one senses the speed of Earth's movement in the way that one senses the movement of a car—or indeed the way the astronauts perceived their speed, which was relative to the Moon and Earth. Of course, everyone experiences the results of Earth's movement—the change from night to day, the precession of the seasons—but no one experiences it directly. It is simply impossible, from the human frame of reference, to feel the movement of a body as large as Earth—not to mention larger progressions on the part of the Solar System and the universe.

From Astronomy to Physics

The human body is in an inertial frame of reference with regard to Earth, and hence experiences no relative motion when Earth rotates or moves through space. In the same way, if one were traveling in a train alongside another train at constant velocity, it would be impossible to perceive that either train was actually moving—unless one referred to some fixed point, such as the trees or mountains in the background. Likewise, if two trains were sitting side by side, and one of them started to move, the relative motion might cause a person in the stationary train to believe that his or her train was the one moving.

For any measurement of velocity, and hence, of acceleration (a change in velocity), it is essential to establish a frame of reference. Velocity and acceleration, as well as inertia and mass, figured heavily in the work of Galileo Galilei (1564-1642) and Sir Isaac Newton (1642-1727), both of whom may be regarded as "founding fathers" of modern physics. Before Galileo, however, had come Nicholas Copernicus (1473-1543), the first modern astronomer to show that the Sun, and not Earth, is at the center of "the universe"—by which people of that time meant the Solar System.

In effect, Copernicus was saying that the frame of reference used by astronomers for millennia was incorrect: as long as they believed Earth to be the center, their calculations were bound to be wrong. Galileo and later Newton, through their studies in gravitation, were able to prove Copernicus's claim in terms of physics.

At the same time, without the understanding of a heliocentric (Sun-centered) universe that he inherited from Copernicus, it is doubtful that Newton could have developed his universal law of gravitation. If he had used Earth as the center-point for his calculations, the results would have been highly erratic, and no universal law would have emerged.

Relativity

For centuries, the model of the universe developed by Newton stood unchallenged, and even today it identifies the basic forces at work when speeds are well below that of the speed of light. However, with regard to the behavior of light itself—which travels at 186,000 mi (299,339 km) a second—Albert Einstein (1879-1955) began to observe phenomena that did not fit with Newtonian mechanics. The result of his studies was the Special Theory of Relativity, published in 1905, and the General Theory of Relativity, published a decade later. Together these altered humanity's view of the universe, and ultimately, of reality itself.

Einstein himself once offered this charming explanation of his epochal theory: "Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. That's relativity." Of course, relativity is not quite as simple as that—though the mathematics involved is no more challenging than that of a high-school algebra class. The difficulty lies in comprehending how things that seem impossible in the Newtonian universe become realities near the speed of light.

Playing Tricks With Time

An exhaustive explanation of relativity is far beyond the scope of the present discussion. What is important is the central precept: that no measurement of space or time is absolute, but depends on the relative motion of the observer (that is, the subject) and the observed (the object). Einstein further established that the movement of time itself is relative rather than absolute, a fact that would become apparent at speeds close to that of light. (His theory also showed that it is impossible to surpass that speed.)

Imagine traveling on a spaceship at nearly the speed of light while a friend remains stationary on Earth. Both on the spaceship and at the friend's house on Earth, there is a TV camera trained on a clock, and a signal relays the image from space to a TV monitor on Earth, and vice versa. What the TV monitor reveals is surprising: from your frame of reference on the spaceship, it seems that time is moving more slowly for your friend on Earth than for you. Your friend thinks exactly the same thing—only, from the friend's perspective, time on the spaceship is moving more slowly than time on Earth. How can this happen?

Again, a full explanation—requiring reference to formulae regarding time dilation, and so on—would be a rather involved undertaking. The short answer, however, is that which was stated above: no measurement of space or time is absolute, but each depends on the relative motion of the observer and the observed. Put another way, there is no such thing as absolute motion, either in the three dimensions of space, or in the fourth dimension identified by Einstein, time. All motion is relative to a frame of reference.

Relativity and Its Implications

The ideas involved in relativity have been verified numerous times, and indeed the only reason why they seem so utterly foreign to most people is that humans are accustomed to living within the Newtonian framework. Einstein simply showed that there is no universal frame of reference, and like a true scientist, he drew his conclusions entirely from what the data suggested. He did not form an opinion, and only then seek the evidence to confirm it, nor did he seek to extend the laws of relativity into any realm beyond that which they described.

Yet British historian Paul Johnson, in his unorthodox history of the twentieth century, Modern Times (1983; revised 1992), maintained that a world disillusioned by World War I saw a moral dimension to relativity. Describing a set of tests regarding the behavior of the Sun's rays around the planet Mercury during an eclipse, the book begins with the sentence: "The modern world began on 29 May 1919, when photographs of a solar eclipse, taken on the Island of Principe off West Africa and at Sobral in Brazil, confirmed the truth of a new theory of the universe."

As Johnson went on to note,"…for most people, to whom Newtonian physics… were perfectly comprehensible, relativity never became more than a vague source of unease. It was grasped that absolute time and absolute length had been dethroned…. All at once, nothing seemed certain in the spheres…. At the beginning of the 1920s the belief began to circulate, for the first time at a popular level, that there were no longer any absolutes: of time and space, of good and evil, of knowledge, above all of value. Mistakenly but perhaps inevitably, relativity became confused with relativism."

Certainly many people agree that the twentieth century—an age that saw unprecedented mass murder under the dictatorships of Adolf Hitler and Josef Stalin, among others—was characterized by moral relativism, or the belief that there is no right or wrong. And just as Newton's discoveries helped usher in the Age of Reason, when thinkers believed it was possible to solve any problem through intellectual effort, it is quite plausible that Einstein's theory may have had this negative moral effect.

If so, this was certainly not Einstein's intention. Aside from the fact that, as stated, he did not set out to describe anything other than the physical behavior of objects, he continued to believe that there was no conflict between his ideas and a belief in an ordered universe: "Relativity," he once said, "teaches us the connection between the different descriptions of one and the same reality."

Where to Learn More

Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.

Fleisher, Paul. Relativity and Quantum Mechanics: Principles of Modern Physics. Minneapolis, MN: Lerner Publications, 2002.

"Frame of Reference" (Web site). <http://www.physics.reading.ac.uk/units/flap/glossary/ff/frameref.html> (March 21, 2001).

"Inertial Frame of Reference" (Web site). <http://id.mind.net/~zona/mstm/physics/mechanics/framesOfReference/inertialFrame.html> (March 21, 2001).

Johnson, Paul. Modern Times: The World from the Twenties to the Nineties. Revised edition. New York: HarperPerennial, 1992.

King, Andrew. Plotting Points and Position. Illustrated by Tony Kenyon. Brookfield, CT: Copper Beech Books, 1998.

Parker, Steve. Albert Einstein and Relativity. New York: Chelsea House, 1995.

Robson, Pam. Clocks, Scales, and Measurements. New York: Gloucester Press, 1993.

Rutherford, F. James; Gerald Holton; and Fletcher G. Watson. Project Physics. New York: Holt, Rinehart, and Winston, 1981.

Swisher, Clarice. Relativity: Opposing Viewpoints. San Diego, CA: Greenhaven Press, 1990.

Sci-Tech Encyclopedia: Frame of reference

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A base to which to refer physical events. A physical event occurs at a point in space and at an instant of time. Each reference frame must have an observer to record events, as well as a coordinate system for the purpose of assigning locations to each event. The latter is usually a three-dimensional space coordinate system and a set of standardized clocks to give the local time of each event. For a discussion of the geometrical properties of space-time coordinate systems See also Space-time; Relativity.

In the ordinary range of experience, where light signals, for all practical purposes, propagate instantaneously, the time of an event is quite distinct from its space coordinates, since a single clock suffices for all observers, regardless of their state of relative motion. The set of reference frames which have a common clock or time is called newtonian, since Isaac Newton regarded time as having invariable significance for all observers.

For discussion of other types of reference frames

Thesaurus: frame of reference

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noun

See

Britannica Concise Encyclopedia: reference frame

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Coordinate system that allows description of time and position of points relative to a body. The axes, or lines, emanate from a position called the origin. As a point moves, its velocity can be described in terms of changes in displacement and direction. Reference frames are chosen arbitrarily. For example, if a person is sitting in a moving train, the description of the person's motion depends on the chosen frame of reference. If the frame of reference is the train, the person is considered to be not moving relative to the train; if the frame of reference is the Earth, the person is moving relative to the Earth.

For more information on reference frame, visit Britannica.com.

Sports Science and Medicine: frame of reference

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1. In sociology, a set of standards that determines and sanctions behaviour.

2. Any set of planes or curves, such as the coordinates and axes used to define a set of points.

Wikipedia: Frame of reference

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Different aspects of "frame of reference"

The need to distinguish between the various meanings of "frame of reference" has led to a variety of terms. For example, sometimes the type of coordinate system is attached as a modifier, as in Cartesian frame of reference. Sometimes the state of motion is emphasized, as in rotating frame of reference. Sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in macroscopic and microscopic frames of reference.^[2]

In this article the term observational frame of reference is used when emphasis is upon the state of motion rather than upon the coordinate choice or the character of the observations or observational apparatus. In this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a coordinate system may be employed for many purposes where the state of motion is not the primary concern. For example, a coordinate system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, of course, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors, which are only indirectly related to space and time. It seems useful to divorce the various aspects of a reference frame for the discussion below. We therefore take observational frames of reference, coordinate systems, and observational equipment as independent concepts, separated as below:

An observational frame (such as an inertial frame or non-inertial frame of reference) is a physical concept related to state of motion.

A coordinate system is a mathematical concept, amounting to a choice of language used to describe observations.^[3] Consequently, an observer in an observational frame of reference can choose to employ any coordinate system (Cartesian, polar, curvilinear, generalized, …) to describe observations made from that frame of reference. A change in the choice of this coordinate system does not change an observer's state of motion, and so does not entail a change in the observer's observational frame of reference. This viewpoint can be found elsewhere as well.^[4] Which is not to dispute that some coordinate systems may be a better choice for some observations than are others.

Choice of what to measure and with what observational apparatus is a matter separate from the observer's state of motion and choice of coordinate system.

Here is a quotation applicable to moving observational frames $\mathfrak{R}$ and various associated Euclidean three-space coordinate systems [R, R' , etc.]: ^[5]

“

We first introduce the notion of reference frame, itself related to the idea of observer: the reference frame is, in some sense, the "Euclidean space carried by the observer". Let us give a more mathematical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted $\mathfrak{R}$ , is said to move with the observer.… The spatial positions of particles are labelled relative to a frame $\mathfrak{R}$ by establishing a coordinate system R with origin O. The corresponding set of axes, sharing the rigid body motion of the frame $\mathfrak{R}$ , can be considered to give a physical realization of $\mathfrak{R}$ . In a frame $\mathfrak{R}$ , coordinates are changed from R to R' by carrying out, at each instant of time, the same coordinate transformation on the components of intrinsic objects (vectors and tensors) introduced to represent physical quantities in this frame.

”

and this on the utility of separating the notions of $\mathfrak{R}$ and [R, R' , etc.]:^[6]

“

As noted by Brillouin, a distinction between mathematical sets of coordinates and physical frames of reference must be made. The ignorance of such distinction is the source of much confusion… the dependent functions such as velocity for example, are measured with respect to a physical reference frame, but one is free to choose any mathematical coordinate system in which the equations are specified.

”

and this, also on the distinction between $\mathfrak{R}$ and [R, R' , etc.]:^[7]

“

The idea of a reference frame is really quite different from that of a coordinate system. Frames differ just when they define different spaces (sets of rest points) or times (sets of simultaneous events). So the ideas of a space, a time, of rest and simultaneity, go inextricably together with that of frame. However, a mere shift of origin, or a purely spatial rotation of space coordinates results in a new coordinate system. So frames correspond at best to classes of coordinate systems.

”

and from J. D. Norton:^[8]

“

In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas. The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The second, the frame of reference, refers to an idealized system used to assign such numbers … To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. … Of special importance for our purposes is that each frame of reference has a definite state of motion at each event of spacetime.…Within the context of special relativity and as long as we restrict ourselves to frames of reference in inertial motion, then little of importance depends on the difference between an inertial frame of reference and the inertial coordinate system it induces. This comfortable circumstance ceases immediately once we begin to consider frames of reference in nonuniform motion even within special relativity.…More recently, to negotiate the obvious ambiguities of Einstein’s treatment, the notion of frame of reference has reappeared as a structure distinct from a coordinate system.

”

The discussion is taken beyond simple space-time coordinate systems by Brading and Castellani.^[9] Extension to coordinate systems using generalized coordinates underlies the Hamiltonian and Lagrangian formulations^[10] of quantum field theory, classical relativistic mechanics, and quantum gravity.^[11]^[12]^[13]^[14] ^[15]

Coordinate systems

Main article: Coordinate systems

Observational frames of reference

Main article: Inertial frame of reference

An observational frame of reference, often referred to as a physical frame of reference, a frame of reference, or simply a frame, is a physical concept related to an observer and the observer's state of motion. Here we adopt the view expressed by Kumar and Barve: an observational frame of reference is characterized only by its state of motion.^[26] However, there is lack of unanimity on this point. In special relativity, the distinction is sometimes made between an observer and a frame. According to this view, a frame is an observer plus a coordinate lattice constructed to be an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector. See Doran.^[27] This restricted view is not used here, and is not universally adopted even in discussions of relativity.^[28]^[29] In general relativity the use of general coordinate systems is common (see, for example, the Schwarzchild solution for the gravitational field outside an isolated sphere^[30]).

There are two types of observational reference frame: inertial and non-inertial.

An inertial frame of reference is defined as one in which all laws of physics take on their simplest form. In special relativity these frames are related by Lorentz transformations. In Newtonian mechanics, a more restricted definition requires only that Newton's first law holds true; that is, a Newtonian inertial frame is one in which a free particle travels in a straight line at constant speed, or is at rest. These frames are related by Galilean transformations. These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms of representations of the Poincaré group and of the Galilean group.

In contrast to the inertial frame, a non-inertial frame of reference is one in which fictitious forces must be invoked to explain observations. An example is an observational frame of reference centered at a point on the Earth's surface. This frame of reference orbits around the center of the Earth, which introduces a fictitious force known as the Coriolis force (among others).

Measurement apparatus

A further aspect of a frame of reference is the role of the measurement apparatus (for example, clocks and rods) attached to the frame (see Norton quote above). This question is not addressed in this article, and is of particular interest in quantum mechanics, where the relation between observer and measurement is still under discussion (see measurement problem).

In this connection it may be noted that the clocks and rods often used to describe observers' measurement equipment in thought, in practice are replaced by a much more complicated and indirect metrology that is connected to the nature of the vacuum, and uses atomic clocks that operate according to the standard model and that must be corrected for gravitational time dilation.^[31] (See second, meter and kilogram).

In fact, Einstein felt that clocks and rods were merely expedient measuring devices and they should be replaced by more fundamental entities based upon, for example, atoms and molecules. ^[32]

Examples of inertial frames of reference

Simple example

Figure 1: Two cars moving at different but constant velocities observed from stationary inertial frame S attached to the road and moving inertial frame S' attached to the first car.

Consider a situation common in everyday life. Two cars travel along a road, both moving at a constant velocity. See Figure 1. At some particular moment, they are separated by 200 meters. The car in front is traveling at 22 meters per second and the car behind is traveling at 30 meters per second. If we want to find out how long it will take the second car to catch up with the first, there are three obvious "frames of reference" that we could choose.

First, we could observe the two cars from the side of the road. We define our "frame of reference" S as follows. We stand on the side of the road and start a stop-clock at the exact moment that the second car passes us, which happens to be when they are a distance d = 200 m apart. Since neither of the cars are accelerating, we can determine their positions by the following formulas, where $x 1 (t)$ is the position in meters of car one after time t seconds and $x 2 (t)$ is the position of car two after time t.

$x_1(t)= d + v_1 t = 200\ + \ 22t\ ; \quad x_2(t)= v_2 t = 30t$

Notice that these formulas predict at t = 0 s the first car is 200 m down the road and the second car is right beside us, as expected. We want to find the time at which $x 1 = x 2$ . Therefore we set $x 1 = x 2$ and solve for $t$ , that is:

$200 + 22 t = 30t \quad$

$8t = 200 \quad$

$t = 25 \quad \mathrm{seconds}$

Alternatively, we could choose a frame of reference S' situated in the first car. In this case, the first car is stationary and the second car is approaching from behind at a speed of v₂ − v₁ = 8 m / s. In order to catch up to the first car, it will take a time of d /( v₂ − v₁) = 200 / 8 s, that is, 25 seconds, as before. Note how much easier the problem becomes by choosing a suitable frame of reference. The third possible frame of reference would be attached to the second car. That example resembles the case just discussed, except the second car is stationary and the first car moves backward towards it at 8 m / s.

It would have been possible to choose a rotating, accelerating frame of reference, moving in a complicated manner, but this would have served to complicate the problem unnecessarily. It is also necessary to note that one is able to convert measurements made in one coordinate system to another. For example, suppose that your watch is running five minutes fast compared to the local standard time. If you know that this is the case, when somebody asks you what time it is, you are able to deduct five minutes from the time displayed on your watch in order to obtain the correct time. The measurements that an observer makes about a system depend therefore on the observer's frame of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three).

Additional example

Figure 2: Simple-minded frame-of-reference example

For a simple example involving only the orientation of two observers, consider two people standing, facing each other on either side of a north-south street. See Figure 2. A car drives past them heading south. For the person facing east, the car was moving toward the right. However, for the person facing west, the car was moving toward the left. This discrepancy is because the two people used two different frames of reference from which to investigate this system.

For a more complex example involving observers in relative motion, consider Alfred, who is standing on the side of a road watching a car drive past him from left to right. In his frame of reference, Alfred defines the spot where he is standing as the origin, the road as the x-axis and the direction in front of him as the positive y-axis. To him, the car moves along the x axis with some velocity v in the positive x-direction. Alfred's frame of reference is considered an inertial frame of reference because he is not accelerating (ignoring effects such as Earth's rotation and gravity).

Now consider Betsy, the person driving the car. Betsy, in choosing her frame of reference, defines her location as the origin, the direction to her right as the positive x-axis, and the direction in front of her as the positive y-axis. In this frame of reference, it is Betsy who is stationary and the world around her that is moving - for instance, as she drives past Alfred, she observes him moving with velocity v in the negative y-direction. If she is driving north, then north is the positive y-direction; if she turns east, east becomes the positive y-direction.

Finally, as an example of non-inertial observers, assume Candace is accelerating her car. As she passes by him, Alfred measures her acceleration and finds it to be a in the negative x-direction. Assuming Candace's acceleration is constant, what acceleration does Betsy measure? If Betsy's velocity v is constant, she is in an inertial frame of reference, and she will find the acceleration to be the same as Alfred - in her frame of reference, a in the negative y-direction. However, if she is accelerating at rate A in the negative y-direction (in other words, slowing down), she will find Candace's acceleration to be a' = a - A in the negative y-direction - a smaller value than Alfred has measured. Similarly, if she is accelerating at rate A in the positive y-direction (speeding up), she will observe Candace's acceleration as a' = a + A in the negative y-direction - a larger value than Alfred's measurement.

Frames of reference are especially important in special relativity, because when a frame of reference is moving at some significant fraction of the speed of light, then the flow of time in that frame does not necessarily apply in another reference frame. The speed of light is considered to be the only true constant between moving frames of reference.

Remarks

It is important to note some assumptions made above about the various inertial frames of reference. Newton, for instance, employed universal time, as explained by the following example. Suppose that you own two clocks, which both tick at exactly the same rate. You synchronize them so that they both display the exact same time. The two clocks are now separated and one clock is on a fast moving train, traveling at constant velocity towards the other. According to Newton, these two clocks will still tick at the same rate and will both show the same time. Newton says that the rate of time as measured in one frame of reference should be the same as the rate of time in another. That is, there exists a "universal" time and all other times in all other frames of reference will run at the same rate as this universal time irrespective of their position and velocity. This concept of time and simultaneity was later generalized by Einstein in his special theory of relativity (1905) where he developed transformations between inertial frames of reference based upon the universal nature of physical laws and their economy of expression (Lorentz transformations).

It is also important to note that the definition of inertial reference frame can be extended beyond three dimensional Euclidean space. Newton's assumed a Euclidean space, but general relativity uses a more general geometry. As an example of why this is important, let us consider the non-Euclidean geometry of an ellipsoid. In this geometry, a "free" particle is defined as one at rest or traveling at constant speed on a geodesic path. Two free particles may begin at the same point on the surface, traveling with the same constant speed in different directions. After a length of time, the two particles collide at the opposite side of the ellipsoid. Both "free" particles traveled with a constant speed, satisfying the definition that no forces were acting. No acceleration occurred and so Newton's first law held true. This means that the particles were in inertial frames of reference. Since no forces were acting, it was the geometry of the situation which caused the two particles to meet each other again. In a similar way, it is now believed that we exist in a four dimensional geometry known as spacetime. It is believed that the curvature of this 4D space is responsible for the way in which two bodies with mass are drawn together even if no forces are acting. This curvature of spacetime replaces the force known as gravity in Newtonian mechanics and special relativity.

Non-inertial frames

Main articles: Fictitious force, Non-inertial frame, and Rotating frame of reference

Here we consider the relation between inertial and non-inertial observational frames of reference. The basic difference between these frames is the need in non-inertial frames for fictitious forces, as described below.

An accelerated frame of reference is often delineated as being the "primed" frame, and all variables that are dependent on that frame are notated with primes, e.g. x' , y' , a' .

The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as R. Given a point of interest that exists in both frames, the vector from the inertial origin to the point is called r, and the vector from the accelerated origin to the point is called r'. From the geometry of the situation, we get

$\mathbf r = \mathbf R + \mathbf r'$

Taking the first and second derivatives of this, we obtain

$\mathbf v = \mathbf V + \mathbf v'$

$\mathbf a = \mathbf A + \mathbf a'$

where V and A are the velocity and acceleration of the accelerated system with respect to the inertial system and v and a are the velocity and acceleration of the point of interest with respect to the inertial frame.

These equations allow transformations between the two coordinate systems; for example, we can now write Newton's second law as

$\mathbf F = m\mathbf a = m\mathbf A + m\mathbf a'$

When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decelerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or deceleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating frame of reference the manifestation of inertia appears to exert a force (either in centrifugal direction, or in a direction orthogonal to an object's motion, the Coriolis effect).

A common sort of accelerated reference frame is a frame that is both rotating and translating (an example is a frame of reference attached to a CD which is playing while the player is carried). This arrangement leads to the equation (see Fictitious force for a derivation):

$\mathbf a = \mathbf a' + \dot{\boldsymbol\omega} \times \mathbf r' + 2\boldsymbol\omega \times \mathbf v' + \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') + \mathbf A_0$

or, to solve for the acceleration in the accelerated frame,

$\mathbf a' = \mathbf a - \dot{\boldsymbol\omega} \times \mathbf r' - 2\boldsymbol\omega \times \mathbf v' - \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') - \mathbf A_0$

Multiplying through by the mass m gives

$\mathbf F' = \mathbf F_\mathrm{physical} + \mathbf F'_\mathrm{Euler} + \mathbf F'_\mathrm{Coriolis} + \mathbf F'_\mathrm{centripetal} - m\mathbf A_0$

where

$\mathbf F'_\mathrm{Euler} = -m\dot{\boldsymbol\omega} \times \mathbf r'$ (Euler force)

$\mathbf F'_\mathrm{Coriolis} = -2m\boldsymbol\omega \times \mathbf v'$ (Coriolis force)

$\mathbf F'_\mathrm{centrifugal} = -m\boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r')=m(\omega^2 \mathbf r'- (\boldsymbol\omega \cdot \mathbf r')\boldsymbol\omega)$ (centrifugal force)

Particular frames of reference in common use

International Terrestrial Reference Frame
International Celestial Reference Frame
In fluid mechanics, Lagrangian and Eulerian specification of the flow field

Other frames

Footnotes

^ Sybil P. Parker (editor) (1997). McGraw-Hill Encyclopedia of Science and Technology. Vol. 7 of 20 (8th ed.). p. 470. ISBN 0079115047. http://books.google.com/books?lr=&as_brr=0&q=frame+must++have+an+observer++to+record++McGraw&btnG=Search+Books.
^ The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where constitutive relations of various time and length scales are used to determine the current and charge densities entering Maxwell's equations. See, for example, Kurt Edmund Oughstun (2006). Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media. Springer. p. 165. ISBN 038734599X. http://books.google.com/books?id=behRnNRiueAC&pg=PA165&dq=macroscopic+frame++electromagnetism&lr=&as_brr=0&sig=ACfU3U3J6a2ZwvTOx7T3S6Zunptf9E9nxQ. . These distinctions also appear in thermodynamics. See Paul McEvoy (2002). Classical Theory. MicroAnalytix. p. 205. ISBN 1930832028. http://books.google.com/books?id=dj0wFIxn-PoC&pg=PA206&dq=macroscopic+frame&lr=&as_brr=0&sig=ACfU3U2JFNgXVpz6Ew6hmp2rmdL6p9O7Ng#PPA205,M1. .
^ In very general terms, a coordinate system is a set of arcs xⁱ = xⁱ (t) in a complex Lie group; see Lev Semenovich Pontri͡agin. L.S. Pontryagin: Selected Works Vol. 2: Topological Groups (3rd Edition ed.). Gordon and Breach. p. 429. ISBN 2881241336. http://books.google.com/books?id=JU0DT_wXu2oC&pg=PA429&dq=algebra+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U07j7JfzfwMigYTa2iDVygAb0WKCA. . Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {e₁, e₂,… e_n}; see Edoardo Sernesi, J. Montaldi (1993). Linear Algebra: A Geometric Approach. CRC Press. p. 95. ISBN 0412406802. http://books.google.com/books?id=1dZOuFo1QYMC&pg=PA95&dq=algebra+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U1TD6WmY73w4hEYlVFcXK5NxtKSDQ. As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion.
^ J X Zheng-Johansson and Per-Ivar Johansson (2006). Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces. Nova Publishers. p. 13. ISBN 1594542600. http://books.google.com/books?id=I1FU37uru6QC&pg=PA13&dq=frame+coordinate+johansson&lr=&as_brr=0&sig=ACfU3U1VAkGbfRjt_GTknoX6WRLWP-AVZw.
^ Jean Salençon, Stephen Lyle (2001). Handbook of Continuum Mechanics: General Concepts, Thermoelasticity. Springer. p. 9. ISBN 3540414436. http://books.google.com/books?id=H3xIED8ctfUC&pg=PA9&dq=physical+%22frame+of+reference%22&lr=&as_brr=0&sig=ACfU3U1tEWQICZdsXeuLyfmH2PoLgZnMGA.
^ Patrick Cornille (Akhlesh Lakhtakia, editor) (1993). Essays on the Formal Aspects of Electromagnetic Theory. World Scientific. p. 149. ISBN 9810208545. http://books.google.com/books?id=qsOBhKVM1qYC&pg=PA149&dq=coordinate+system+%22reference+frame%22&lr=&as_brr=0&sig=ACfU3U0xhpZ2lI99UPiYQCOL6oJ0ALO5uA.
^ Graham Nerlich (1994). What Spacetime Explains: Metaphysical essays on space and time. Cambridge University Press. p. 64. ISBN 0521452619. http://books.google.com/books?id=fKK7rKOpc7AC&pg=PA64&dq=%22idea+of+a+reference+frame%22&lr=&as_brr=0&sig=ACfU3U2wsO42pqLOJ453eeIzk7ztXTa6uQ.
^ John D. Norton (1993). General covariance and the foundations of general relativity: eight decades of dispute, Rep. Prog. Phys., 56, pp. 835-7.
^ Katherine Brading & Elena Castellani (2003). Symmetries in Physics: Philosophical Reflections. Cambridge University Press. p. 417. ISBN 0521821371. http://books.google.com/books?id=SnmBN64cAdYC&pg=PA417&dq=%22idea+of+a+reference+frame%22&lr=&as_brr=0&sig=ACfU3U1PdXJdmFyMRiDb7xPDAI_dy9MgJg.
^ Oliver Davis Johns (2005). Analytical Mechanics for Relativity and Quantum Mechanics. Oxford University Press. Chapter 16. ISBN 019856726X. http://books.google.com/books?id=PNuM9YDN8CIC&pg=PA318&dq=coordinate+observer&lr=&as_brr=0&sig=ACfU3U3TRrg4EVCiIW8btVgFdR49PD9RUg#PPA276,M1.
^ Donald T Greenwood (1997). Classical dynamics (Reprint of 1977 edition by Prentice-Hall ed.). Courier Dover Publications. p. 313. ISBN 0486696901. http://books.google.com/books?id=x7rj83I98yMC&pg=RA2-PA314&dq=%22relativistic+%22+Lagrangian+OR+Hamiltonian&lr=&as_brr=0&sig=ACfU3U3l7hGibLCCG40qWuiO3A5sN7E7lg#PRA2-PA313,M1.
^ Matthew A. Trump & W. C. Schieve (1999). Classical Relativistic Many-Body Dynamics. Springer. p. 99. ISBN 079235737X. http://books.google.com/books?id=g2yfLOp0IzwC&pg=PA99&dq=relativity+%22generalized+coordinates%22&lr=&as_brr=0&sig=ACfU3U230ux_i1Ov3QHf_xy1dAA_oLSKpw#PPA99,M1.
^ A S Kompaneyets (2003). Theoretical Physics (Reprint of the 1962 2nd Edition ed.). Courier Dover Publications. p. 118. ISBN 0486495329. http://books.google.com/books?id=CQ2gBrL5T4YC&pg=PA118&dq=relativity+%22generalized+coordinates%22&lr=&as_brr=0&sig=ACfU3U3OOAmAAh2wV46vG0gUN0wSIPiIww.
^ M Srednicki (2007). Quantum Field Theory. Cambridge University Press. Chapter 4. ISBN 978-0-521-86449-7. http://books.google.com/books?id=5OepxIG42B4C&pg=PA266&dq=isbn=9780521864497&sig=ACfU3U2J9PylxA2eptc48_TBT2u2GPfnVA#PPA31,M1.
^ Carlo Rovelli (2004). Quantum Gravity. Cambridge University Press. p. 98 ff. ISBN 0521837332. http://books.google.com/books?id=HrAzTmXdssQC&pg=PA179&dq=%22relativistic+%22+Lagrangian+OR+Hamiltonian&lr=&as_brr=0&sig=ACfU3U3TLyr3CXsHYKFUGDe1dpq5ZWm_kg#PPA98,M1.
^ William Barker & Roger Howe (2008). Continuous symmetry: from Euclid to Klein. p. 18 ff. ISBN 0821839004. http://books.google.com/books?id=NIxExnr2EjYC&pg=PA17&dq=geometry++axiom+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U1-Tq9AjjikB_HSYJ1Xn0fFdHWY0g#PPA18,M1.
^ Arlan Ramsay & Robert D. Richtmyer (1995). Introduction to Hyperbolic Geometry. Springer. p. 11. ISBN 0387943390. http://books.google.com/books?id=UVozmKVh7GsC&pg=PA202&dq=geometry++axiom+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U0Nr0kKmV8XtorU41jRiiEo2wil3Q#PPA11,M1.
^ According to Hawking and Ellis: "A manifold is a space locally similar to Euclidean space in that it can be covered by coordinate patches. This structure allows differentiation to be defined, but does not distinguish between different coordinate systems. Thus, the only concepts defined by the manifold structure are those that are independent of the choice of a coordinate system." Stephen W. Hawking & George Francis Rayner Ellis (1973). The Large Scale Structure of Space-Time. Cambridge University Press. p. 11. ISBN 0521099064. http://books.google.com/books?id=QagG_KI7Ll8C&pg=PA59&dq=manifold+%22The+Large+Scale+Structure+of+Space-Time%22&lr=&as_brr=0&sig=ACfU3U1q-iaRTBDo6J8HMEsyPeFi8cJNWg#PPA11,M1. A mathematical definition is: A connected Hausdorff space M is called an n-dimensional manifold if each point of M is contained in an open set that is homeomorphic to an open set in Euclidean n-dimensional space.
^ Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore. p. 12. ISBN 0821810456. http://books.google.com/books?id=5N33Of2RzjsC&pg=PA12&dq=geometry++axiom+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3Vi7xsLiYiWCK0erF6X2gczHOkJA#PPA12,M1.
^ Granino Arthur Korn, Theresa M. Korn (2000). Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review. Courier Dover Publications. p. 169. ISBN 0486411478. http://books.google.com/books?id=xHNd5zCXt-EC&pg=PA169&dq=curvilinear+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3psSqwpBtA3U40e46VPPaMNMEw4g#PPA169,M1.
^ See Encarta definition
^ Katsu Yamane (2004). Simulating and Generating Motions of Human Figures. Springer. p. 12-13. ISBN 3540203176. http://books.google.com/books?id=tNrMiIx3fToC&pg=PA12&dq=generalized+coordinates+%22kinematic+chain%22&lr=&as_brr=0&sig=ACfU3U3LRGJJTAHs21CHdOvuu08vw0cAuw#PPA13,M1.
^ Achilleus Papapetrou (1974). Lectures on General Relativity. Springer. p. 5. ISBN 9027705402. http://books.google.com/books?id=SWeOggyp1ZsC&pg=PA3&dq=relativistic++%22general+coordinates%22&lr=&as_brr=0&sig=ACfU3U3sPmqPV3oEzbV5zHqErtZrqcx4bg#PPA5,M1.
^ Wilford Zdunkowski & Andreas Bott (2003). Dynamics of the Atmosphere. Cambridge University Press. p. 84. ISBN 052100666X. http://books.google.com/books?id=GuYvC21v3g8C&pg=RA1-PA84&dq=%22curvilinear+coordinate+system%22&lr=&as_brr=0&sig=ACfU3U2g2k7kY5u-CVcJ1pH5ZxsbEb9Rig.
^ A. I. Borisenko, I. E. Tarapov, Richard A. Silverman (1979). Vector and Tensor Analysis with Applications. Courier Dover Publications. p. 86. ISBN 0486638332. http://books.google.com/books?id=CRIjIx2ac6AC&pg=PA86&dq=coordinate+metric&lr=&as_brr=0&sig=ACfU3U1osXaT2hg7Md57cJ9katl3ttL43Q.
^ See Arvind Kumar & Shrish Barve (2003). How and Why in Basic Mechanics. Orient Longman. p. 115. ISBN 8173714207. http://books.google.com/books?id=czlUPz38MOQC&pg=PA115&dq=%22characterized+only+by+its+state+of+motion%22+inauthor:Kumar&lr=&as_brr=0&sig=ACfU3U36HY3RerJYLWRlfJaGxRw7EqzIeA.
^ Chris Doran & Anthony Lasenby (2003). Geometric Algebra for Physicists. Cambridge University Press. p. §5.2.2, p. 133. ISBN 978-0-521-71595-9. http://www.worldcat.org/search?q=9780521715959&qt=owc_search. .
^ For example, Møller states: "Instead of Cartesian coordinates we can obviously just as well employ general curvilinear coordinates for the fixation of points in physical space.…we shall now introduce general "curvilinear" coordinates xⁱ in four-space…." C. Møller (1952). The Theory of Relativity. Oxford University Press. p. 222 and p. 233.
^ Alan P. Lightman, R. H. Price & William H. Press (1975). Problem Book in Relativity and Gravitation. Princeton University Press. p. 15. ISBN 069108162X. http://books.google.com/books?id=YtxGYnnP1PEC&pg=PA15&dq=relativistic++%22general+coordinates%22&lr=&as_brr=0&sig=ACfU3U28SuvzlWcCJ7PXCY71TVn33GeBvA.
^ Richard L Faber (1983). Differential Geometry and Relativity Theory: an introduction. CRC Press. p. 211. ISBN 082471749X. http://books.google.com/books?id=ctM3_afLuVEC&pg=PA149&dq=relativistic++%22general+coordinates%22&lr=&as_brr=0&sig=ACfU3U1_zMogoRkH1OhzpC77ULTkMN0ihg#PPA211,M1.
^ Richard Wolfson (2003). Simply Einstein. W W Norton & Co.. p. 216. ISBN 0393051544. http://books.google.com/books?id=OUJWKdlFKeQC&pg=PA216&dq=%22gravitational+time+dilation+%22&lr=&as_brr=0&sig=ACfU3U0_wc8IuNJdGCLnsaO-SyqXYaRapw.
^ See Guido Rizzi, Matteo Luca Ruggiero (2003). Relativity in rotating frames. Springer. p. 33. ISBN 1402018053. http://books.google.com/books?id=_PGrlCLkkIgC&pg=PA226&dq=centrifugal+%22+%22+relativity+OR+relativistic&lr=&as_brr=0&sig=ACfU3U038RpTaZOnfjYn6zH9umefW7y_-Q#PPA33,M1. .

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