Concepts Of Modern Physics

CONCEPTS OF MODERN PHYSICS

In 1905 a young physicist of twenty-six named Albert Einstein showed how measurements of time and space are affected by motion between an observer and what is being observed. To say that Einstein’s theory of relativity revolutionized science is no exaggeration. Relativity connects space and time, matter and energy, electricity and magnetism-links that are crucial to our understanding of the physical universe. From relativity have come a host of remarkable predictions, all of which have been confirmed by experiment. For all their profundity, many of the conclusions of relativity can be reached with only the simplest of mathematics.

SPECIAL RELATIVITY

All motion is relative; the speed of light in free space is the same for all observers.

When such quantities as length, time interval, and mass are considered in elementary physics, no special point is made about how they are measured. Since a standard unit exits for each quantity, who makes a certain determination would not seem to matter-everybody ought to get the same result. For instance, there is no question of principle involved in finding the length of an airplane when we are on board. All we have to do is put one end of a tape measure at the airplane’s nose and look at the number on the tape at the airplane’s tail.

But what if the airplane is in flight and we are on the ground? It is not hard to determine the length of a distant object with a tape measure to establish a baseline, a surveyor’s transit to measure angles, and knowledge of trigonometry. When we measure the moving airplane from the ground, though, we find it to be shorter than it is to somebody in the airplane itself. To understand how this unexpected difference arises, we must analyze the process of measurement when motion is involved.

Frames of Reference

The first step is to clarify what we mean by motion. When we say that something is moving, what we mean is that its position relative to something else is changing. A passenger moves relative to an airplane; the airplane moves relative to the earth; the earth moves relative to the sun; the sun moves relative to the galaxy of stars (the Milky Way) of which it is a member; and so on. In each case a frame of reference is part of the description of motion. To say that something is moving always implies a specific frame of reference.

An inertial frame of reference is one in which Newton’s first law of motion holds. In such a frame, an object at rest remains at rest and an object in motion continues to move at constant velocity (constant speed and direction) if no force acts on it. Any frame of reference that moves at constant velocity to an inertial frame is itself an inertial frame.

All inertial frames are equally valid. Suppose we see something changing its position with respect to us at constant velocity. It is moving or are we the laboratory moving? Suppose we are in a closed laboratory in which Newton’s first law holds. Is the laboratory moving or is it at rest? These questions are meaningless because all constant-velocity motion is relative. There is no universal frame of reference that can be used everywhere, no such thing as “absolute motion.”

The theory of relativity deals with the consequences of the lack of a universal frame of reference. Special relativity, which is what Einstein published in 1905, treats problems that involve inertial frames of reference. General relativity, published by Einstein a decade later, describes the relationship between geometry and the geometrical structure of space and time. The special theory has had an enormous impact on much of physics, and we shall concentrate on it here.

Postulates of Special Relativity

Two postulates underlie special relativity. The first, the principle of relativity, states:

The laws of physics are the same in all inertial frames of reference.

This postulate follows from the absence of a universal frame of reference. If the laws of physics were different for different observer in relative motion, the observers could find from these differences which of them were “stationary” in space and which were “moving.” But such a distinction does not exist, and the principle of relativity expresses this fact.

The second postulate is based on the results of many experiments:

The speed of light in free space has the same value in all inertial frames of reference.

This speed is 2.998 x 108 m/s to four significant figures.

To appreciate how remarkable these postulates are, let us look at a hypothetical experiment basically no different from actual ones that have been carried out in a number of ways. Suppose I turn on a searchlight just as you fly past in a spacecraft at a speed of 2 x 108 m/s (Fig 1.1). We both measure the speed of the light waves from the searchlight using identical instruments. From the ground I find their speed to be 3 x 108 m/s as usual. “Common sense” tells me that you ought to find a speed of (3-2) x 108 m/s, or only 1 x 108 m/s, for the same light waves. But you also find their speed to be 3 x 108 m/s, even though to me you seem to be moving parallel to the waves at 2 x 108 m/s.

Albert A. Michelson (1852-1931) was born in Germany but came to the United States at the age of two with his parents, who settled in Nevada. He attended the U.S. Naval Academy at Annapolis where, after two years of sea duty, he became a science instructor.

To improve his knowledge of optics, in which he wanted to specialize, Michelson went to Europe and studied in Berlin and Paris. Then he left the Navy to work first at the Case School of Applied Science in Ohio, then at Clark University in Massachusetts, and finally at the University of Chicago, where he headed the physics department from 1892 to 1929.Michelson’s specialty was high precision measurement, and for many decades his successive figures for the speed of light were the best available. He redefined the meter in terms of wavelengths of a particular spectral line and devised an interferometer that could determine the diameter of a star (stars appear as points of light in even the most powerful telescopes):

Michelson’s most significant achievement, carried out in 1887 in collaboration with Edward Morley, was an experiment to measure the motion of the earth trough “ether,” a hypothetical medium pervading the universe in which light waves were supposed to occur. The notion of the ether was hangover from the days before light waves were recognized as electromagnetic, but nobody at the time seemed willing to discard the idea that light propagates relative to some sort of universal frame of reference.

To look for the earth’s motion trough the ether, Michelson and Morley used a pair of light beams formed by a half-silvered mirror, as in Fig. 1.2. One light beam is directed to a mirror along a path perpendicular to the ether current, and the other goes to a mirror along a path parallel to the ether current. Both beams end up at the same viewing screen. The clear glass plate ensures that both beams pass through the same thicknesses of air and glass. If the transit times of the two beams are the same, they will arrive at the screen in phase and will interfere constructively. An ether current due to the earth’s motion parallel to one of the beams, however, would case the beams to have different transit times and the result would be destructive interference at the screen. This is the essence of the experiment.

Although the experiment was sensitive enough to detect the expected ether drift, to everyone’s surprise none was found. The negative result had two consequences. First, it showed that the ether does not exist and so there is no such thing as “absolute motion” relative to the ether: all motion is relative to a specified frame of reference, not to a universal one. Second, the result showed that the speed of light is the same for all observers, which is not true to waves that need a material medium in which to occur (such as sound and water waves).

The Michelson-Morley experiment set the stage for Einstein’s 1905 special theory of relativity, a theory that Michelson himself was reluctant to accept. Indeed, not long before the flowering of relativity and quantum theory revolutionized physics, Michelson announced that “physical discoveries in the future are a matter of the sixth decimal place.” This was common opinion of the time. Michelson received a Nobel Prize in 1907, the first American to do so.

There is only one way to account for these results without violating the principle of relativity. It must be true that measurements of space and time are not absolute but depend on the relative motion between an observer and what is being observed. If I were to measure from the ground the rate at which your clock ticks and the length of your meter stick, I would find that the clock ticks more slowly that it did at rest on the ground and that the meter stick is shorter in the direction of motion of the spacecraft. To you, your clock and the meter stick are the same as they were on the ground before you took off. To me they are different because of the relative motion, different in such a way that the speed of light you measure is the same 3 x 108 m/s I measure. Time intervals and lengths are relative quantities, but the speed of light in free space is the same to all observers.

Before Einstein’s work, a conflict had existed between the principles of mechanics, which were then based on Newton’s laws of motion, and those of electricity and magnetism, which had been developed into a unified theory by Maxwell. Newtonian mechanics had worked well for over two centuries. Maxwell’s theory not only covered all that was then known about electric and magnetic phenomena but had also predicted that electromagnetic waves exist and identified light as an example of them. However, the equations of Newtonian mechanics and those of electromagnetism differ in the way they relate measurements made in one inertial frame with those made in a different inertial frame.

Einstein showed that Maxwell’s theory is consistent with special relativity whereas Newtonian mechanics is not, and his modification of mechanics brought these branches of physics into accord. As we will find, relativistic and Newtonian mechanics agree for relative speeds much lower than the speed of light, which is why Newtonian mechanics seemed correct so long. At higher speeds Newtonian mechanics fails and must be replaced by the relativistic version.

TIME DILATION

A moving clock ticks slowly than a clock at rest

Measurements of time intervals are affected by relative motion between an observer and what is observed. As a result, a clock that moves with respect to an observer ticks more slowly than it does without such motion, and all process (including those of life) occur more slowly to an observer when they take place in different inertial frame.

If someone in a moving spacecraft finds that the time interval between two events in the spacecraft is t0, we on the ground would find that the same interval has the longer duration t. The quantity t0, which is determined by events that occur at the same place in an observer’s frame of reference, is called the proper time of the interval between the events. When witnessed from the ground, the events that mark the beginning and end of the time interval occur at different places, and in consequence the duration of the interval appears longer than the proper time. This effect is called time dilation (to dilate is to become larger).

To see how time dilation comes about, let us considerer two clocks, both of the particularly simple kind shown in Fig. 1.3. In each clock a pulse of light is reflected back and forth between two mirrors L0 apart. Whenever the light strikes the lower mirror, an electric signal is produced that marks the recording tape. Each mark corresponds to the tick of an ordinary clock.

One clock is at rest in a laboratory on the ground and the other is in spacecraft that moves at the speed v relative to the ground. An observer in the laboratory watches both: does she find that they tick at the same rate?

Figure 1.4 shows the laboratory clock in operation. The time interval between ticks is the proper time t0 and the time needed for the light pulse to travel between the mirrors at the speed of light c is t0/2. Hence t0/2 = L0/c and

t_0=〖2L〗_0/c (1.1)

Figure 1.5 shows the moving clock with its mirrors perpendicular to the direction of motion relative to the ground. The time Interval between ticks is t. Because the clock is moving, the light pulse, as seen from the ground, follows a zigzag path. On its way from the lower mirror to the upper one in the time t/2, the pulse travels a horizontal distance of v(t/2) and a total distance of c(t/2). Since L0 is the vertical distance between -the mirrors,

(ct/2)^2= 〖L_0〗^2+(vt/2)^2

t^2/4 (c^2-v^2 )=〖L_0〗^2

t^2=〖〖4L〗_0〗^2/((c^2-v^2 ) )=〖〖2L〗_0〗^2/(c^2 (1-v^2⁄c^2 ) )

t=(2L_0/c)/√(1-v^2⁄c^2 ) (1.2)

But 2L0/c is the time interval t0 between ticks on the clock on the ground, as in Eq. (1.1), and so time dilation

t=t_0/√(1-v^2⁄c^2 ) (1.3)

Here is a reminder of what the symbols in Eq. (1.3) represent:

t0 = time interval on clock at rest relative to an observer = proper time

t = time interval on clock in motion relative to an observer

v = speed of relative motion

c = speed of light

Because the quantity √(1-v^2/c^2 ) is always smaller than 1 for a moving object, t is always greater than t0. The moving clock in the spacecraft appears to tick at a slower rate than the stationary one on the ground, as seen by an observer on the ground.

Exactly the same analysis holds for measurements of the clock on the ground by the pilot of the spacecraft. To him, the light pulse of the ground clock follows a zigzag path that requires a total time t per round trip. His own clock, at rest in the spacecraft, ticks at intervals of t0. He too finds that

t=t_0/√(1-v^2⁄c^2 )

so the effect is reciprocal: every observer finds that clocks in motion relative to him tick more slowly than clocks at rest relative to him.

Our discussion has been based on a somewhat unusual clock. Do the same conclusions apply to ordinary clocks that use machinery—spring-controlled escapements, tuning forks, vibrating quartz crystals, or whatever—to produce ticks at constant time intervals? The answer must be yes, since if a mirror clock and a conventional clock in the spacecraft agree with each other on the ground but not when in flight, the disagreement between then could be used to find the speed of the spacecraft independently of any outside frame of reference—which contradicts the principle that all motion is relative.

THE ULTIMATE SPEED LIMIT

The earth and the other planets of the solar system seem to be natural products of the evolution of the sun. Since the sun is a rather ordinary star in other ways, it is not surprising that other stars have been found to have planetary systems around them as well. Life developed here on earth, and there is no known reason why it should not also have done so on some of these planets. Can we expect ever to be able to visit them and meet our fellow citizens of the universe? The trouble is that nearly all stars are very far away—thousands or millions of light-years away (A light-year, the distance light travels in a year, is 9.46 X 1015 m.) But if we can build a spacecraft whose speed is thousands or millions of times greater than the speed of light c, such distances would not be an obstacle.

Alas, a simple argument based on Einstein’s postulates shows that nothing can move faster than c. Suppose you are in a spacecraft traveling at a constant speed v relative to the earth that is greater than c. As I watch from the earth, the lamps in the spacecraft suddenly go out. You switch on a flashlight to find the fuse box at the front of the spacecraft and change the blown fuse (Fig. 1.6a). The lamps go on again.

From the ground, though, I would see something quite different. To me, since your speed v is greater than c, the light from your flashlight illuminates the back of the spacecraft (Fig. 1.6b).I can only conclude that the laws of physics are different in your inertial frame from what they are in my inertial frame—which contradicts the principle of relativity. The only way to avoid this contradiction is to assume that nothing cann move faster than the speed of light. This assumption has been tested experimentally many times and has always been found to be correct.

The speed of light c in relativity is always its value in free space of 3.00 X 108 m/s. ln all material media, such as air, water, or glass, light travels more slowly than this, and atomic particles are able to move faster in such media than does light. When an electrically charged particle moves through a transparent substance at a speed exceeding that of light in the substance, a cone of light waves is emitted that corresponds to the bow wave produced by a ship moving through the water faster than water waves do. These light waves are known as Cerenkov radiation and form the basis of a method of determining the speeds of such particles. The minimum speed a particle must have to emit Cerenkov radiation is c/n in a medium whose index of refraction is n. Cerenkov radiation is visible as a bluish glow when an intense beam of particles is involved.

Albert Einstein (1879-1955), bitterly unhappy with the rigid discipline of the schools of his native Germany, went at sixteen to Switzerland to complete his education, and later got a job examining patent applications at the Swiss Patent Office. Then, in 1905, ideas that had been germinating in his mind for years when he should have been paying attention to other matters (one of his math teachers called Einstein a “lazy dog”) blossomed into three short papers that were to change decisively the course not only of physics but of modem civilization as well.

The first paper, on the photoelectric effect, proposed that light has a dual character with both particle and wave properties. The subject of the second paper was Brownian motion, the irregular zigzag movement of tiny bits of suspended matter, such as pollen grains in water. Einstein showed that Brownian motion results from the bombardment of the particles by randomly moving molecules in the fluid in which they are suspended. This provided the long-awaited definite link with experiment that convinced the remaining doubters of the molecular theory of matter. The third paper introduced the special theory of relativity-

Although much of the world of physics was originally either indifferent or skeptical, even the most unexpected of Einstein’s conclusions were soon confirmed and the development of what is now called modem physics began in earnest. After university posts in Switzerland and Czechoslovakia, in 1913 he took up an appointment at the Kaiser Wilhelm Institute in Berlin that left him able to do research free of financial worries and routine duties. Einstein’s interest was now mainly in gravitation, and he started here Newton had left off more than two centuries earlier.

Einstein’s general theory of relativity, published in 1916, related gravity to the structure of space and time. In this theory the force of gravity can be thought of as arising from a warping of spacetime around a body of matter so that a nearby mass tends to move toward it, much as a marble rolls toward the bottom of a saucer-shaped hole. From general relativity came a number of remarkable predictions, such as that light should be subject to gravity, all of which were verified experimentally. The later discovery that the universe is expanding fit neatly into the theory. In 1917 Einstein introduced the idea of stimulated emission of radiation, an idea that bore fruit forty years later in the invention of the laser.

The development of quantum mechanics in the 1920s disturbed Einstein, who never accepted its probabilistic rather than deterministic view of events on an atomic scale. “God does not play dice with the world,” he said, but for once his physical intuition seemed to be leading him in the wrong direction.

Einstein, by now a world celebrity, left Germany in 1933 after Hitler came to power and spent the rest of his life at the Institute for Advanced Study in Princeton, New Jersey, thereby escaping the fate of millions of other European Jews at the hands of the Germans. His last years were spent in an unsuccessful search for a theory that would bring gravitation and electromagnetism together into a single picture, a problem worthy of his gifts but one that remains unsolved to this day.

Example 1.1

A spacecraft is moving relative to the earth. An observer on the earth finds that, between 1 P.M. and 2 P.M. according to her clock, 3601 s elapse on the spacecraft’s clock. What is the spacecraft’s speed relative to the earth?

Solution

Here t0 =3600 s is the proper time interval on the earth and t= 3601 s is the time interval in the moving frame as measured from the earth. We proceed as follows:

t=t_0/√(1-v^2⁄c^2 )

1-v^2/c^2 =(t_0/t)^2

v=c√(1-(t_0/t)^2 )= (2.998 x 108 m/s) √(1-((3600 s)/(3601 s))^2 )

= 7.1 x 106 m/s

Today’s spacecraft are much slower than this. For instance, the highest speed of the Apollo 11 spacecraft that went to the moon was only 10,840 m/s, and its clocks differed from those on the earth by less than one part in 109. Most of the experiments that have confirmed time dilation made use of unstable nuclei and elementary particles which readily attain speeds not far from that of light.

Although time is a relative quantity, not all the notions of time formed by everyday experience are incorrect. Time does not run backward to any observer, for instance. A sequence of events that occur at some particular point at t1, t2, t3, . . . will appear in the same order to all observers everywhere, though not necessarily with the same time intervals t2 – t1, t3 –t2, . . . between each pair of events. Similarly, no distant observer, regardless of his or her state of motion, can see an event before it happens—more precisely, before a nearby observer sees it—since the speed of light is finite and signals require the minimum period of time L/c to travel a distance L.

There is no way to peer into the future, although past events may appear different to different observers.

DOPPLER EFFECT

Why tire universe is believed to be expanding

We are all familiar with the increase in pitch of a sound when its source approaches us (or we approach the source) and the decrease in pitch when the source recedes from us (or we recede from the source). These changes in frequency constitute the doppler effect, whose origin is straightforward. For instance, successive waves emitted by a source moving toward an observer are closer together than normal because of the advance of the source; because the separation of the waves is the wavelength of the sound, the corresponding frequency is higher. The relationship between the source frequency ν0 and the observed frequency ν is

Doppler effect in sound

ν= ν_0 ((1+ v⁄c)/(1-V⁄c))

Where c = speed of sound

v = speed of observer (+ for motion toward source, - for motion away from it)

V = speed of the source (+ for motion toward the observer, - for motion away from him)

If the observer is stationary, v = 0, and if the source is stationary, V = 0.

The doppler effect in sound varies depending on whether the source, or the observer, or both are moving. This appears to violate the principle of relativity: all that should count is the relative motion of source and observer. But sound waves occur only in a material medium such as air or water, and this medium is itself a frame of reference with respect to which motions of source and observer are measurable. Hence there is no contradiction. In the case of light, however, no medium is involved and only relative motion of source and observer is meaningful. The doppler effect in light must therefore differ from that in sound.

We can analyze the doppler effect in light by considering a light source as a clock that ticks v0 times per second and emits a wave of light with each tick. We will examine the three situations shown in Fig. 1.7.

Observer moving perpendicular to a line between him and the light source. The proper time between ticks is t0 = 1/v0, so between one tick and the next the time t=t_0/√(1-v^2⁄c^2 ) elapses in the reference frame of the observer. The frequency he finds is accordingly

ν (transverse)=1/t=√(1-v^2⁄c^2 )/t_0

Transverse Doppler effect in light

ν=ν_0 √(1-v^2⁄c^2 ) (1.5)

The observed frequency ν is always lower than the source frequency ν0.

Observer receding from the light source. Now the observer travels the distance vt away from the source between ticks, which means that the light wave from a given tick takes vt/c longer to reach him than the previous one. Hence the total time between the arrival of successive waves is

T=t+vt/c=t_0 (1+v⁄c)/√(1-v^2⁄c^2 )=t_0 (√(1+v⁄c) √(1+v⁄c))/(√(1+v⁄c) √(1-v⁄c))=t_0 √((1+v⁄c)/(1-v⁄c))

and the observed frequency is

ν(receding)=1/T=1/t_0 √((1-v⁄c)/(1+v⁄c))=ν_0 √((1-v⁄c)/(1+v⁄c))

The observed frequency ν is lower than the source frequency ν0. Unlike the case of sound waves, which propagate relative to a material medium it makes no difference whether the observer is moving away from the source or the source is moving away from the observer.

3 Observer approaching the light source. The observer here travels the distance vt toward the source between ticks, so each light wave takes vt/c less time to arrive than the previous one. In this case T = t- vt/c and the result is

ν(approaching)=ν_0 √((1+v⁄c)/(1-v⁄c)) (1.7)

The observed frequency is higher than the source frequency. Again, the same formula holds for motion of the source toward the observer.

Equations (1.6) and (1.7) can be combined in the single formula

Longitudinal Doppler effect in light ν= v_0 √((1+v⁄c)/(1-v⁄c)) (1.8)

By adopting the convention that ν is + for source and observer approaching each other and – for source and observer receding from each other.

Example 1.2

A driver is caught going through a red light. The driver claims to the judge that the color she actually saw was green (ν = 5.60 X 1014 Hz) and not red (vo = 4.80 X 10“ Hz) because of the doppler effect. The judge accepts this explanation and instead fines her for speeding at the rate of $1 for each km/h she exceeded the speed limit of 80 km/h. What was the fine?

Solution

Solving Eq. (1.8) for ν gives

ν=c((ν^2-〖ν_0〗^2)/(ν^2+〖ν_0〗^2 ))=(3.00 x 〖10〗^8 m/s)[((5.60)^2-(4.80)^2)/((5.60)^2+(4.80)^2 )]

= 4.59 x 107 m/s = 1.65 x 108 km/h

Since 1 m/s = 3.6 km/h. The fine is therefore $(1.65 x 108 – 80) = $164,999,920.

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