"Relativity" by M.C. Escher, 1953.

Which way is up?

Frames of Reference, a film featuring two University of Toronto physicists, Pat Hume and Donald Ivey, 1960:

http://www.youtube.com/watch?v=pyBNImQkRuk

Which way is up?

Galilean Relativity

Back in Chapter 3 we studied relative motion, using a principle known as Galilean relativity. This idea originates from the 1600s with Galileo. At that time, one of the most important controversies in science was the question of whether the Earth moves around the Sun (a heliocentric model of the solar system) or whether the Sun moves around the Earth (a geocentric model). Galileo was one of the few who argued very forcefully that the Earth and the rest of the planets move around the Sun. However, an argument presented to counter Galileo can be phrased as a question: If the Earth moves around the Sun, then why don't we feel a "wind"

Chapter 28: Special RelativityTuesday, September 17, 2013 10:00 PM

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question: If the Earth moves around the Sun, then why don't we feel a "wind" because of our movement? When we ride a bicycle we feel a wind in our face, even if the air is still, just as we do if we are driving in a car and keep the windows open. Galileo counter-argued that the Earth carries its air along with it, so that the air is still relative to the Earth.

Aristotle had argued two millennia before Galileo that the Earth could not be rotating and moving, because then every time you dropped an object, it would land far away from you, not at your feet. (The Earth would rotate underneath it, according to Aristotle, whereas the object would fall "straight down.") Galileo, however, emphasized the need for careful observation and measurement to supplement reasoning. (I'm not suggesting that the ancient Greeks were not good observers, because they certainly were. However, in the following centuries, many thinkers merely took Aristotle's views as given without thinking carefully about whether there might be deeper insights.)

Galileo gave as an example a sailor who happens to be in the crow's nest at the top of a ship's mast. If the ship is moving in a straight line at a constant speed, and the sailor accidently drops something, where does the dropped object land? Experience shows that the object lands at the bottom of the mast.

Does this make sense? Think about your own experiences in a moving car or airplane. As long as the car or airplane moves in a straight line at a constant speed, a dropped or tossed object behaves in the same way as if the car or plane were not moving at all.

The conclusion is that all mechanical experiments (tossing a ball, moving about, etc.) that take place inside a closed vehicle that is moving in a straight line at a constant speed have the same results as the analogous mechanical experiments that take place "at rest." This is the principle of Galilean relativity.

This further means that if Newton's laws are valid "at rest" then they are also valid inside a closed vehicle that is moving in a straight line at a constant speed. But what do we mean by "at rest"? Well, a better way to say this is that if we happen to find a frame of reference in which Newton's laws are valid (such a reference frame is called an inertial reference frame), then any reference frame that moves in a straight line at a constant speed relative to the inertial reference frame is itself also an inertial reference frame. So once you find one inertial reference frame, there are an infinite number of other inertial reference frames, all in constant relative motion. And Newton's first law asserts that there exists an inertial reference frame, so Newton's laws and Galilean relativity together assert the existence of an infinite number of inertial reference frames.

Another way to summarize this complex of ideas is that Newtonian mechanics is valid in all inertial reference frames. Another way to say this is that the laws of Newtonian mechanics are frame-independent.

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The formula for the speed of a mechanical wave in a mechanical medium is analogous; the stiffer the medium, the faster the wave speed.

* * * * *

Despite the numerous successes of Newtonian mechanics, Einstein deduced in 1905 that Newtonian mechanics is only an approximation to a deeper physical theory, which we call relativistic mechanics (alternatively, the special theory of relativity). To understand a little about what led Einstein to develop special relativity, we need to discuss a bit about 19th-century electromagnetic theory. (We'll discuss this in more detail in second semester in PHYS 1P22/1P92.)

James Clerk Maxwell's theory of electromagnetism, reported by him in the 1860s, was an epochal advance, one of the most important physical theories we have. As a consequence of his theory, Maxwell concluded waves in the electromagnetic field were possible, and he calculated (based on electrical and magnetic parameters that were measured in laboratories) the speed at which these waves must travel. He was astonished at the result: the speed of electromagnetic waves is the same as the speed of light, which was known fairly accurately at that time.

Maxwell boldly concluded that light is an electromagnetic wave. Experimental confirmation came relatively soon, only about twenty years later, when Heinrich Hertz demonstrated that electromagnetic waves produced in his laboratory have all of the optical properties that light does.

The many successes of Maxwell's theory of electromagnetism, which helped usher in a new wave of industrialization, and especially electrification, led physicists of the late 19th century and early 20th century to have great confidence in the theory. But this era of physicists was deeply rooted in the mechanical perspective on the world. They thought that all waves must be mechanical waves; that is, a real wave such as light (an electromagnetic wave) must be a wave in a material medium. What is the material medium that supports the motion of electromagnetic waves?

They didn't know. They called this hypothetical medium the luminiferous ether (or aether, if you wish to spell it this way), and they set about trying to work out its properties theoretically and to perform experiments to detect it and to measure its properties.

This turned out to be a real puzzle, because for a mechanical wave, the faster the wave speed, the stiffer the material must be. Think of the formula for the angular frequency of a block on a spring that undergoes simple harmonic motion:

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analogous; the stiffer the medium, the faster the wave speed.

So the ether has to be a very unusual substance. It has to be present throughout all space, because light can presumably travel anywhere, so it has to be a fluid. It has to be extremely stiff (millions of times stiffer than steel!), because light travels extremely fast. But the planets seem to fly through the ether without any resistance whatsoever. How can this be? How can something be so stiff to light and yet so ephemeral to planets?

Despite its problematic contradictory properties, which were well-understood at the time, physicists of the day considered the ether essential. Maxwell's equations were very successful, and the mechanical viewpoint was very well-entrenched. One of the mathematical deductions made from Maxwell's equations is that they could only be valid in one reference frame, because according to Galilean relativity light would have different speeds in different reference frames, and Maxwell's theory was very clear that the speed of light in vacuum must have a particular value for the electromagnetic waves to satisfy Maxwell's equations. So it was convenient to think of this special reference frame in which Maxwell's equations were valid as being the frame in which the ether was at rest. They called it "the ether frame of reference."

And they had to do a sort of intellectual jujitsu to explain how Maxwell's equations could be valid in other reference frames. It was somewhat untidy, and esthetically unsatisfactory. But nevertheless, it was the best they could do at the time, and they didn't stop working on this problem; ideas were plentiful, but a resolution to this strange situation of the ether was not apparent.

After discussing the conceptual situation, let's now discuss the experimental attempts to measure properties of the ether. In summary, all experiments performed to detect the luminiferous ether failed; the most famous early ones are the series of experiments by A.A. Michelson and E. Morley (in Cleveland, Ohio) in the 1880s, but there were many, many such experiments, and they continued well past the middle of the 20th century, long after Einstein had decisively resolved the situation.

Here's the basic idea behind the Michelson-Morley experiment: In the same way that you feel a wind in your face when you travel on a bicycle through still air, the Earth should feel an "ether wind" as it passes through the still ether. This will influence the detection of light, because ether is the medium that supports light waves.

Michelson and Morley hoped to detect this effect by measuring the time difference for round-trip light journeys parallel to the ether wind and perpendicular to the ether wind. Here is the argument that there should be a time difference; the argument is based on Galilean relativity, which we studied in Chapter 3:

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The time needed for a round-trip light journey perpendicular to the ether wind is calculated in a similar way. First we need to determine the speed of light when it moves perpendicular to the ether wind. Just like the canoe travelling across the river current in an example we did back in Chapter 3, the light must be directed a little off the perpendicular direction for it to go directly across the current, and similarly on the way back. The speed of the light can be determined in the same way as in Chapter 3:

The total time needed for the round-trip journey is

Chapter 3:

The time needed for a round-trip light journey parallel to the ether wind is calculated as follows. A beam of light is emitted at the left and travels to the right a distance L to a mirror, is reflected there and travels to the left back to its origin. The ether wind is directed to the right and has speed v, and light has speed c.

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In summary:

Michelson and Morley intended to determine the ether wind speed, v. To do this, they set up their apparatus with a length L that they could measure, and then they used an ingenious method (based on wave interference) to measure the time difference

which leads to a single equation for v, which they could then solve to determine the ether wind speed:

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Once T and L are measured in the experiment, the value of can be calculated from the previous equation. Then the speed of the ether wind can be calculated as follows:

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The problem was that Michelson and Morley measured a time difference of zero,

to within experimental uncertainty. This leads to a calculated value of = 1, and a calculated value of the speed of the ether wind of v = 0.

Over time, the Michelson-Morley experiment was modified, improved, made more accurate, both by Michelson and Morley and by many other researchers. Every other experiment in the 125+ years since the 1887 Michelson-Morley experiment that was designed to measure the speed of the ether wind produced a null result. Every other experiment that was designed to measure some other property of the ether was also unsuccessful. Here are a few of the major ones:

This was puzzling. Light was thought to be a wave, and therefore was thought to require a medium to support its existence, and yet every attempt to detect this medium failed.

This was very puzzling.

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Other attempts were made to reconcile Maxwell's theory of electromagnetism with the ether and with the utter failure to observe the ether or measure any of its properties. Some of these attempts are now known as "emission theories," in which it was hypothesized that the speed of light emitted by an object depended on the state of motion of the emitting object. Such hypotheses were odd, as they contradicted the usual understanding of mechanical wave motion (where the wave speed depends on properties of the medium, not on the state of motion of the emitter), but times were desperate and desperate times sometimes require desperate measures.

Alas, the emission hypotheses were disproved by repeating experiments of

(Source: Introduction to Special Relativity, Robert Resnick, Wiley, 1968, page 29.)

How can it be that the Earth flies through the ether and yet we cannot detect the ether wind that ought to be "in our faces."

Physicists tried many ideas to resolve the puzzles of the ether. One idea proposed back then to explain why ether wind is not detected is "ether drag;" perhaps the Earth drags a "bubble" of ether along with it, so that at the surface of the Earth there is no ether wind. This is much like the difference between riding a bicycle (where you feel the wind in your face) and driving in a car. The car carries air along with it in its enclosed space, so passengers don't feel wind in their face. The wind is blocked by the windshield. Two variants of the ether drag theory were proposed by Fresnel in 1818 and Stokes in 1845. However, ether drag hypotheses fail because of observations of stellar aberration (first observed by Bradley in 1727):

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(Source: Classical Electricity and Magnetism, 2nd edition, by Wolfgang Panofsky and Melba Phillips, Addison-Wesley, 1962.)

Alas, the emission hypotheses were disproved by repeating experiments of the Michelson-Morley type, but using extraterrestrial light sources (for example, by Tomaschek using starlight and Miller using sunlight, and careful measurements of binary stars by de Sitter.

By the early years of the 20th century, this whole situation was quite a mess. Other attempts to salvage the ether involved inventing a fictitious "local time" (Lorentz) and the hypothesis that all material objects contracted in the direction of their motion (independently proposed by Lorentz and Fitzgerald). Although ingenious, all of these hypothesis give the impression of band-aid solutions to a situation that was severely broken.

The following table summarizes the major failed hypotheses of the time:

Enter Einstein

some biographical notes on Einstein

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some biographical notes on Einstein

Einstein's explanation for the Michelson-Morley experiment, and other similar null results was beyond bold: there is NO luminiferous ether. This is a sharp break from the mechanical view of the universe, that all waves are mechanical and therefore need a material medium to support their propagation. Einstein said, in effect, that if we can't measure the ether, can't detect it in any way, then it doesn't exist.

But what then is light? What bizarre sort of wave is it? Let's save that question for another day, and let's see what are the logical consequences of Einstein's bold proposition.

Einstein further proposed that the application of Galilean relativity to analyze experiments such as the Michelson-Morley experiment is incorrect, and must be modified. How he arrived at this conclusion is another important part of the relativity story, which we'll now discuss.

One of the primary motivations for Einstein's development of the theory of special relativity was his deep consideration of classical electromagnetic theory, which culminated in the work of Faraday in the early to mid 1800s, and the work of Maxwell in the 1860s. (It's notable that the portraits of three scientists decorated the walls of Einstein's study: Newton, Faraday, and Maxwell.)

Consider, for example, the relative motion of a magnet and an electrical conductor, such as a loop of wire. The effect is the same (i.e., electric current is induced in the wire) whether the magnet moves or the coil moves, but the standard explanations for the effect used at the time differed (as we'll see in PHYS 1P22/1P92); Einstein pondered this and was unsatisfied by the symmetry of the situations and the asymmetry of the explanations.

These two considerations (i.e., his deep reflections on classical electrodynamics (and his deep confidence in Maxwell's theory), and the problem of the luminiferous ether) led Einstein to found a new theory of mechanics, what we now call Einstein's theory of special relativity.

He founded his theory on two postulates:

1. The speed of light in vacuum has the same value in all inertial reference frames, and does not depend on the state of motion of the light's source. (That is, the speed of light is absolute. It's not true that all is relative; some things are relative and some things are absolute.)

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

Einstein's first postulate was motivated by his deep considerations of Maxwell's equations, which summarized classical electromagnetic theory. A consequence of these equations is that light is an electromagnetic wave, and that it travels in

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these equations is that light is an electromagnetic wave, and that it travels in vacuum at a particular speed. At the time, it was generally accepted that Maxwell's theory was valid in the reference frame in which the luminiferous ether is at rest. However, it was also understood that Maxwell's theory was not consistent with Galileo's principle of relativity, because according to the latter the speed of light ought to be different in different inertial reference frames. This inconsistency was puzzling.

Einstein's second postulate resembles Galileo's relativity postulate, but in fact it generalizes Galileo's relativity principle. Galileo spoke only about mechanicalexperiments, but Einstein stated that ALL experiments will have the same results in all inertial reference frames.

Notice how bold Einstein's postulates were. He recognized that the generally accepted understanding of Maxwell's theory of electrodynamics at that time was not consistent with Newtonian mechanics. He considered deeply, and he figured that Maxwell's theory was OK and that Newtonian mechanics was not OK.

Einstein's first postulate contradicts Galileo's principle of relativity, and therefore contradicts Newtonian mechanics. Einstein's second postulate contradicts the idea of a luminiferous ether, by saying that Maxwell's theory of electromagnetism is valid in all inertial reference frames. In effect, Einstein has declared that the luminiferous ether DOES NOT EXIST!

Here is the introduction to his revolutionary relativity paper "On the Electrodynamics of Moving Bodies," first published in the journal Annalen der Physik, volume 17, 1905; I have quoted this passage from a version of the paper translated into English by W. Perrett and W.B Jeffery and published in The Principle of Relativity, Dover, 1952:

"It is known that Maxwell's electrodynamics as usually understood at the

present time when applied to moving bodies, leads to asymmetries that do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. Bur if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in

itself there is no corresponding energy, but which gives rise assuming equality

of relative motion in the two cases discussed to electric currents of the same path and intensity as those produced by the electric forces in the former case.

"Examples of this sort, together with the unsuccessful attempts to discover any Ch28 Page 12

"Examples of this sort, together with the unsuccessful attempts to discover any motion of the Earth relatively to the "light medium," suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the "Principle of Relativity") to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies. The introduction of a "luminiferous ether" will prove to be superfluous inasmuch as the view here to be developed will not require an "absolutely stationary space" provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place."

Consequences of Einstein's postulates

Time dilation1.

Einstein's two relativity postulates have shocking consequences, as the following argument shows (and as has been verified countless times by experiments of many different types).

The first, and perhaps most shocking deduction that Einstein makes in the paper referred to above, is that time is not absolute, but rather relative to the state of motion of the observer relative to the observed phenomenon. Here is the essence of Einstein's argument: Consider a spacecraft that moves from left to right at speed v. Consider a source of light in the floor of the spacecraft that emits a beam of light that travels upwards, reflects from a mirror in the ceiling, and then is reabsorbed in the floor.

Question: How long does it take for the light beam to make its round-trip journey?the time from inside the spacecraft, and then again from the perspective of an observer "at rest;" that is, the second observer sees the spacecraft moving to the right at speed v.

Answer: We'll consider the journey from the perspective of an observer measuring From the perspective of an observer inside the spacecraft, the time interval between the emission of the light pulse and its return is:

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Now consider the perspective of an observer that sees the spacecraft move to the right at speed v:

How shall we calculate the time interval between emission and absorption of the light ray? Suppose we use Galilean relativity; then the calculation is just like the canoe problem back in Chapter 3. The current takes the canoe to the right, downriver, but the time needed for the round trip depends only on the "northward" component of the canoe's speed, which is c, and so we get the same result as the previous calculation. However, note that in this perspective, the resultant speed of light (which is analogous to the speed of the canoe with respect to the ground) can be calculated as follows:

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The upward component of the velocity of light is:

Thus, using Einstein's relativity, the time needed for the round-trip light journey is

Hold on a second. All of this calculation from the perspective of the "ground observer" is inconsistent with Einstein's principles of relativity. Einstein postulated that the speed of light in vacuum is always measured to be c, no matter the state of motion of the object emitting the light, and no matter the state of motion of the observer. Using Galilean relativity, we conclude that the speed of light depends on the state of motion of the observer.

So which perspective is correct? Well, as usual, experiment decides. Both the classical Galilean perspective and the Einsteinian perspective have their own consistent inner logic, and we can decide on whether either of them is useful by predicting consequences based on their theory and then do experiments to test the predictions. So let's recalculate the time interval from the "ground observer" perspective using Einstein's postulates and compare to the classical, Galilean prediction. From Einstein's perspective, the speed of light is c, so:

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Here are some typical values of the gamma-factor; note that the value of gamma is always greater than or equal to 1:

v/c

0.01 1.00005

0.1 1.005

0.5 1.155

0.866 2

0.9 2.3

Recall that the time interval from the perspective of an observer on the spacecraft (which is called the "proper time" interval) is

And this is the first of Einstein's revolutionary deductions: Time is NOT absolute, but a relative quantity. The time interval between two events measured by two different observers may be different, and depends on the relative motion between the observers.

A slogan that is often used is: "Moving clocks run slow."

How large is this effect? Consider the ratio of the two time intervals just calculated:

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0.5 1.155

0.866 2

0.9 2.3

0.95 3.2

0.99 7.1

This phenomenon is called "time dilation." The time interval measured by an observer moving relative to you is always longer than the corresponding time interval measured in your own reference frame (i.e., the proper time). The term "proper" doesn't mean "correct," it's a translation of a term (from German) that means something along the lines of "your own reference frame."

Also, notice the resemblance between the two time intervals just calculated and the time intervals that were calculated for the Michelson-Morley experiment. And indeed, Einstein's postulates and time dilation do provide a neat explanation for the null results of the Michelson-Morley experiment (see the textbook for details).

_________________________________________________________________

_________________________________________________________________

Example: A proton in a particle accelerator travels a distance of 285 m in a time of 4.1 µs as measured by an observer in the accelerator's frame of reference. Determine the time needed for this journey in the proton's frame of reference.

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Example: An unstable particle moves at a speed of 0.94c as measured with respect to a laboratory frame of reference. It decays 27 ns after it was created as measured in its own reference frame. How long does the particle last in the laboratory reference frame?

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Notice that for speeds that one is likely to encounter in every-day life, the gamma-factor is so close to 1 that one would never notice any difference between the two time intervals. This explains why the theory of special relativity was not discovered back in the time of Galileo and Newton.

Also notice that if v = c, then the gamma factor does not exist, because division by zero does not make any sense. Furthermore, if v is greater than c, then the gamma factor is a complex number, which is problematic. This means that special relativity is not a useful description when the speed of a reference frame is equal to or greater than the speed of light. A conclusion that can be drawn from this observation (and which Einstein did state) is that material particles cannot be accelerated to speeds equal to or greater than the speed of light. Nevertheless, some physicists have toyed with the idea of material particles that always move faster than c, which they have called tachyons. This is pure speculation at this point, as nobody has ever observed a tachyon. Nevertheless, it's good to speculate sometimes, as occasionally wild speculations lead to interesting ideas, even though most of the time they don't lead anywhere. (Of course, this is only healthy if you recognize a wild speculation for what it is, and don't take it too seriously.)

So which theory is correct, Newtonian mechanics or Einstein's theory of special relativity? After all, Newton's theory of mechanics has had numerous amazing successes in accurately predicting and describing every-day phenomena. If you dig deeper into the subject, you'll understand that Newtonian mechanics is a special case of Einstein's theory of special relativity, accurate for speeds that are very small

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case of Einstein's theory of special relativity, accurate for speeds that are very small relative to the speed of light.

So we don't throw Newtonian mechanics into the garbage heap, but rather we recognize that it is an excellent theory for mechanical phenomena that occur at speeds that are quite slow relative to the speed of light. For phenomena that occur at speeds that are significant fractions of the speed of light, Einstein's theory of relativity is much more accurate and remains the gold standard.

Experimental evidence for time dilation

Hafele-Keating experiment, 1971 (4 atomic clocks flown around the world on commercial aircraft, one set flying east and the other flying west, and then compared with atomic clocks that remained on the ground)

•

Ives-Stillwell experiments, 1938, 1941 (first direct confirmation of time dilation, done by measuring the relativistic Doppler effect)

•

decay of unstable particles; numerous experiments could be quoted here, but the first one was done by Rossi and Hall in 1941; they made measurements on the half-lifes of muons (unstable particles discovered by Anderson in 1935) and found that faster ones lasted longer before transforming into other particles

•

operation of modern particle accelerators•many other high-precision experiments•GPS•

Length contraction 1.

Another consequence of Einstein's postulates is length contraction. One can construct an argument for length contraction based directly on Einstein's postulates, or one can argue for length contraction based on the conclusions that we just made concerning time dilation. The conclusion is that spatial distances (i.e., lengths) are measured differently depending on the relative motion of the observer and the spatial distance being measured.

A formula for length contraction can be derived by noting that the magnitude of the relative velocity is the same from the perspective of either reference frame. Consider the following problem._________________________________________________________________

Example: A particle moves at a speed of 0.85c directly towards Earth. A hypothetical observer moving along with the particle measures a time of 0.8 s until the particle hits the Earth. (I.e., the time needed for the particle to hit the Earth, as measured in the particle's frame of reference is 0.8 s.)(a) How long does the motion take as measured by an observer on Earth?(b) How far does the particle move as measured by an observer on Earth?(c.) How far does the particle move as measured by the hypothetical observer moving along with the particle?

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_________________________________________________________________

Notice that in the previous example, the "moving" observer measures a shorter time interval, and therefore also a shorter corresponding distance interval.

From the perspective of the hypothetical observer moving along with the particle, it is at rest, and the Earth moves towards it. Thus the "moving" particle considers itself to be at rest and the Earth moving; in this sense, the distance of the moving object is judged to be less than the distance "at rest;" hence the term "length contraction."

Reviewing the previous example, one can't help but notice that

and therefore

Solution: First calculate the value of the gamma-factor:

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______________________________________________________________

Example: How fast must a metre-stick be moving so that its length will be observed to be half a metre?

Solution:

and therefore

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Note that the statement of the problem in the previous example is vague. It would be more precise to ask the question in this form: "How fast must a metre-stick be moving relative to reference frame X so that its length will be measured to be half a metre by an observer in reference frame X?

Note also that length contraction takes place only in the direction of motion.

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Exactly the same reasoning can be applied by an observer in each rocket, so an observer in each rocket measures that the other rocket is shorter than her own rocket!_________________________________________________________________

A friendly paraphrase from the bottom of page 877 of your textbook:

Remember the meanings of the different time and length variables. The "proper" (i.e., with the subscript 0) time interval between two events is measured by an observer who is at rest relative to the events and sees them occurring at the same place. The "proper" (i.e., with the subscript 0) length of an object is the length measured by an observer who is at rest with respect to the object.

Thus, time and space are relative, not absolute as they were thought to be in Newtonian mechanics.

Because the concepts of space and time are revised in Einstein's theory, one might expect that Newtonian mechanics must be revised in light of Einstein's revolutionary ideas; this is in fact the case, and so a new theory of mechanics has been developed that is more accurate than Newtonian mechanics for high-speed phenomena. For more details on this new "relativistic mechanics," you might like to attend Physics 2P50 next year.

In this sense special relativity is a metatheory in addition to being a physical theory; that is, special relativity places restrictions on physical theories. It specifies certain conditions that physical theories must satisfy in order that they even have a chance to be valid physical theories. Newtonian mechanics did not meet these conditions, and so was modified. Maxwell's theory of electromagnetism did indeed meet the conditions of special relativity, and so did not require any modifications.__________________________________________________________________

Example: Two rockets moving parallel to each other and in opposite directions pass in the night. Each rocket has a length of 100 m as measured in its own rest frame. Each rocket has a speed of 0.9c as measured relative to the other rocket. Determine the length of each rocket as measured by an observer in the other rocket.

Solution: It's usually a good idea to calculate gamma from the start:

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length measured by an observer who is at rest with respect to the object.

Read the full paragraph, it's good.__________________________________________________________________

In the previous example, each observer measures that the other rocket is shorterthan her own rocket! It's a symmetrical situation. Here's one way to look at it, from the reference frame of an observer on Rocket A:

And here it is from the reference frame of an observer on Rocket B:

The situation is symmetrical; each measures the length of the other rocket as smaller. I suppose it would have been natural to expect that if A measures the length of B to be smaller, then B would measure the length of A to be larger, but the world is a stranger place than this. Each measures the other's rocket to be smaller!

Evidently time is at the root of this strange situation. The way we measure time has to do with the relative state of motion of the object that we are measuring and us. Contrary to Newton, who spoke of time as follows:

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Another consequence of Einstein's theory of relativity is that this relative velocity formula is incorrect. Despite being incorrect, it's an extremely good approximation to the relativistic formula at low speeds, which explains why the correct formula was not discovered by Galileo, Newton, or any of their illustrious contemporaries and followers. The relativistic formula is

Relative velocity formula ("velocity addition formula")3.

Think back to Chapter 3, where we developed a formula for relative velocity:

and us. Contrary to Newton, who spoke of time as follows:

"Absolute, true, and mathematical time, in and of itself and of its own nature, without reference to anything external, flows uniformly and by another name is called duration." (http://www.relativity.li/en/epstein2/read/a0_en/a1_en/ )

Newton was very clear that time is absolute; Einstein realized that time is relative, and used logic to deduce remarkable consequences, some of which are described in these notes. The immediate responses to Einstein's work were typical: The very greatest physicists in the world instantly recognized the work of a genius, many more ignored it, and many just did not understand it. The responses of some great mathematicians and physicists, such as Poincare, was also understandable: Despite their greatness, they were so rooted in classical ideas that they could not quite wrap their minds around Einstein's new perspectives, and died without ever (apparently) truly understanding Einstein's theory of relativity.

Experimental confirmation of length contraction:

A key experiment was the one of Kennedy and Thorndike, first performed in 1932. It was similar to the Michelson-Morley experiment, only with perpendicular "arms" of different lengths. One could interpret the Michelson-Morley effect as confirming only length contraction, but to explain the results of the Kennedy-Thorndike experiment one needs to invoke both time dilation and length contraction. Thus, the Kennedy-Thorndike experiment was powerful confirmation of the reality of both effects, time dilation and length contraction.

Experimental work to confirm the predictions of special relativity are ongoing, and become more and more precise over time. If you would like to learn more about this, do a search for "experimental tests of Lorentz invariance."

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Example: An airplane moves at a speed of 200 m/s relative to the ground. A ball is tossed inside the plane towards the front of the plane at a speed of 2 m/s relative to the plane. Use the relativistic velocity addition formula to determine the speed of the ball with respect to the ground.

Solution:

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Although we won't derive this formula from Einstein's basic principles, we'll check it with a number of examples, starting with the following one:

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The result in the previous example is so close to 202 m/s (which is the expected result based on Newtonian mechanics) that it is unmeasurable except in the most careful experiments. The conclusion is that at speeds very low relative to the speed of light, the difference between the relativistic formula and the Newtonian formula is minuscule.

However, for speeds that are an appreciable fraction of the speed of light, the difference between the two formulas is quite significant, as the next example shows.____________________________________________________________

Example: A rocket moves at a speed of 0.5c relative to the Earth. The rocket emits a particle moving at a speed of 0.6c relative to the rocket in the same direction as the rocket. Determine the speed of the particle with respect to the Earth.

Solution:

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Example: A rocket moves at a speed of 0.5c relative to the Earth. The rocket emits a beam of light moving at speed c relative to the rocket in the same direction as the rocket. Determine the speed of the light beam with respect to the Earth.

Solution:

____________________________________________________________

The previous example embodies the fact that according to Einstein's theory of relativity, light has the same speed no matter which reference frame it is measured in. This is one of the basic postulates of special relativity, and it has been verified in numerous experiments.

Matter can be converted into energy and vice-versa4.

Another consequence of Einstein's postulates is that matter can be converted into energy, and energy can be converted into matter. The conversion factor is the square of the speed of light:

E = mc2

In Einstein's words, the mass of an object is a measure of its energy content. Because the square of the speed of light is so enormous in SI units, a tiny amount

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Because the square of the speed of light is so enormous in SI units, a tiny amount of matter can be converted into an enormous amount of energy.

The derivation of the energy-mass equivalence formula is a little beyond where we want to go in this course. However, it may be useful to note that some concepts in Newtonian mechanics are still useful in relativistic mechanics, although their definitions are different. (It turns out the Newton's law is not valid in relativistic mechanics, and in fact the concept of force is not very useful in Newtonian mechanics either.) For example, the correct formulas for kinetic energy and momentum in relativistic mechanics are

In other words, for an object of mass m moving with speed v, the total energy is

Using a standard approximation trick (search for "binomial theorem" or "binomial series"), Einstein argued for the mass-energy equivalence as follows:

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Einstein used this low speed approximation to argue that an object with mass has a certain energy content just by virtue of its mass; this is called the object's rest energy to distinguish it from its kinetic energy. More sophisticated arguments show that this relation is exact, not just an approximation. As usual, the final test is experiment, and an enormous number of high-precision experiments have verified the energy-mass equivalence relation to a very high degree of accuracy.

Examples of conversion of matter into energy:

radioactivity•nuclear reactions; nuclear power plants make use of the process of nuclear fission to convert minute amounts of matter to significant amounts of energy; the Sun uses nuclear fusion to combine Hydrogen atoms into Helium atoms (and via other processes) with the result that a minute amount of mass is converted into a significant amount of energy

•

Example of conversion of energy into matter: "pair production"

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Example: If it were possible to convert a kilogram of matter entirely into energy, how much energy would be obtained? Compare this to the amount of electrical energy needed to power the average Canadian home. (The average Canadian

For speeds that are low compared to the speed of light, just the first two terms of the infinite series form a good approximation. Thus,

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_________________________________________________________________

energy needed to power the average Canadian home. (The average Canadian

household consumes 12 MWh per year.)

Solution: Converting 1 kg of matter into energy results in

Example: Determine the amount of energy radiated away from the Sun per year. Use this value to determine how much mass the Sun loses per year, assuming that all of the power output results from conversion of mass to energy. What percentage of its mass will the Sun lose in this way in 1 billion years? (The Sun's luminosity is about 3.85 × 1026 W, and its mass is about 1.99 × 1030 kg.)

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The fraction of the Sun's mass that is converted in a billion years is

The Sun's mass is so enormous that even losing mass at such a huge rate means that it loses only a tiny fraction of 1% of its total mass in a billion years. (Of course, this calculation is based on the assumption that the Sun will continue to convert matter to energy at a constant rate, which is not correct. However, the calculation can be considered an approximation that gives us a sense for the time scales involved.)

Historically, the discovery of Einstein's energy-mass relation helped resolve a puzzle; the means by which the Sun produced radiant energy was unclear a hundred years ago, and it was thought that the Sun was younger than the Earth because of new discoveries in geology. This was confusing. The Einstein relation raised the possibility that the Sun could exist for a lot longer, and so removed the contradiction, although it wasn't until the mid- to late- twentieth century that a reasonably detailed understanding of energy conversion in the Sun was developed by many people, culminating with the work of Hans Bethe.

* * * * *

_________________________________________________________________

In summary, the classical principle of conservation of energy is seen to be incomplete, and mass must be included as a kind of energy; if we do this, then it appears that the principle of conservation of energy is still valid, in this expanded sense, as a principle of conservation of mass-energy. (In chemistry class we learn

This is an enormous amount of mass, but a minuscule amount compared to the truly enormous mass of the Sun. In a billion years, the amount of mass converted is

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sense, as a principle of conservation of mass-energy. (In chemistry class we learn about a principle of conservation of mass, which is used to help us determine the results of chemical reactions.)

So, the modern understanding of the principles of conservation of mass and energy is that they are not separately conserved, but their combination is conserved. What was thought to be separate is actually part of a unified whole, which is a theme in modern physics.

Minkowski's idea of spacetime

Similarly, space and time are not separately absolute; rather they are relative quantities, and the measure of a displacement or a time interval depends on the state of relative motion of the measured quantity and the observer doing the measuring.

However, a certain combination of spatial and time intervals IS invariant, and is the same for a given pair of events, no matter which observer measures it.

There is a truism that "all is relative," but this is not what Einstein's theory of relativity proposes. Rather, special relativity revises our understanding of what is relative and what is absolute. Previously it was thought that space and time are absolute, whereas we now understand that they are relative. On the other hand, it was previously that that the speed of light in vacuum is relative, but we now understand that it is absolute.

Here is the introduction to Hermann Minkowski's address to the 80th Assembly of German Natural Scientists and Physicians, "Space and Time," delivered at Cologne, German, 21 September 1908; I have quoted this from a version of the address translated into English by W. Perrett and W.B Jeffery and published in The Principle of Relativity, Dover, 1952:

"The views of space and time that I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

spacetime diagrams; we didn't have time to study spacetime diagrams in this course, but interested students can look up this beautiful subject in a good relativity book (such as Special Relativity, by A.P. French, or Spacetime Physics, by E.F. Taylor and J.A. Wheeler, see http://www.eftaylor.com/special.html ) or take PHYS 2P50 next Fall!

* * * * *

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Application: Global Positioning System (GPS)

GPS satellites move in orbits that have radii of about 23,000 km. Therefore their speeds can be calculated as follows (recall Chapter 6):

Because the speed of light is about 1 foot per nanosecond, if you would like to measure distances accurately to within, say, a few metres, then you will need to measure time intervals accurately to within about 10 ns or so. Because the gamma-factor is not equal to 1, there is a difference between time intervals as measured by the satellite clocks and those measured by the ground clocks. The discrepancy is calculated as follows:

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This is about 1000 times the allowable error if we wish to use GPS to measure locations accurately to within a few metres. As you can see, if we didn't take special relativity into account, GPS would become totally unusable within a day of being set up.

(The real story is even worse, because there is an additional (and even greater) effect that fouls up the time, but that depends on general relativity, so we'll leave that part of the story to another time.)

The conclusion we can draw from this calculation is that special relativity (and the more advanced general relativity) are not esoteric subjects that have no practical importance. On the contrary, GPS would not work without careful consideration of both special relativity and general relativity. It's all about time!

* * * * *

Some comments on general relativity

As we said earlier, special relativity is both a theory of physics and also a metatheory. The latter means that special relativity places restrictions that must be satisfied by all physical theories in order that they be valid. That is, the formulas of a physical theory must transform appropriately so that the formulas have the same structure in all inertial reference frames. This restriction can be stated in short-form as "frame-invariance," or equivalently "Lorentz-invariance;" so the slogan is that all laws of physics must be frame invariant, or in longer form, all laws of physics must be invariant with respect to transformations to different inertial reference frames.

Einstein immediately began to assess all the fundamental theories of physics that were known at his time, and he found that Newtonian mechanics did not satisfy Lorentz invariance, and so needed modification. He accomplished this in his initial paper on relativity, cited earlier, and so we now have relativistic mechanics, which is the successor

This means that even if the ground clocks and satellite clocks are synchronized to begin with, they will gradually drift apart as time goes on. After a day, the time difference is

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relativity, cited earlier, and so we now have relativistic mechanics, which is the successor and generalization of Newtonian mechanics. Maxwell's theory of electromagnetism needed no modification whatsoever; it was already Lorentz invariant in its original form. However, Einstein added his own deeper insights into Maxwell's theory, so that we now realize that magnetic fields and electric fields are aspects of a single underlying reality, which we now term the electromagnetic field. In the same way that space and time are now unified (Einstein's insight, as supplemented by Minkowski's beautiful geometric perspective), and it is seen that there is a single underlying reality (spacetime), and that space and time are simply aspects of this single underlying reality, Einstein also unified electric and magnetic fields, so that we see that electric fields and magnetic fields are simply aspects of the underlying reality of an electromagnetic field. This theme of unification has been a fruitful one, and has been a central guiding idea in physics throughout the 20th century and into the 21st century.

Next to be assessed was gravity, and by about 1909 Einstein realized that it would be impossible to modify Newton's law of gravity in a way similar to his modification of Newton's laws of motion to make it Lorentz invariant. Something far more drastic would be needed. Einstein was originally unenthusiastic about Minkowski's work (he made a snide comment about mathematicians getting hold of his theory and twisting it around so that not even he could understand it anymore), but he quickly realized that Minkowski's perspective was valuable and that he could build on it. He hoped to show that gravity could be considered to be a manifestation of the curvature of spacetime, but the mathematics of curved spaces and curved spacetimes was fairly new at that time, and was formidable.

This represents a revolution in our conception of force; Newton had no idea how gravitational force was transmitted from the Sun to the Earth, for example, but Einstein now provided a totally new conception. In Einstein's view, there are no gravitational forces per se. Rather, the presence of mass and energy curves spacetime; then other things in the vicinity just follow the curvature of spacetime in their motions. For example, the Sun curves spacetime in its vicinity, and then the Earth just responds to the curvature of spacetime as it moves. There is no force in this perspective; or rather, force is not a primary concept, but rather a derived concept that is not very useful in the Einsteinian perspective.

To work out the formidable mathematics needed for his general theory of relativity (in final form late in 1915), Einstein relied on his good friend from student days Marcel Grossman to help him learn tensor analysis and Riemannian geometry. Einstein then applied this high-powered mathematics to successfully create a theory of gravity that we call Einstein's theory of gravity, or equivalently, the general theory of relativity. For an introduction to general relativity, see the book of Taylor and Wheeler mentioned earlier, and consider taking MATH 4P94/PHYS 4P94, Relativity Theory and Black Holes. You'll need to take advanced calculus (MATH 2P03), a little differential equations (MATH 2P08), second-year mechanics (PHYS 2P20), and modern physics (PHYS 2P50) as preparation.

Some concluding remarks

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Laplace's dream slain by special relativity; we didn't have time to discuss this in class•

causality; we didn't have time to discuss this in class•

comments on the relations of Newtonian mechanics and relativity; the idea of physical theories being approximations, and being special cases of more advanced theories; thus, we can't say that a physical theory is "true" only that it's useful or not; to be more specific, we should specify the realm of validity of useful physical theories; we didn't have time to discuss this in class

•

reprise of the core theories of mechanics:•

Core theories of mechanics:

QuantumField Theory(1920s ---1970s)

RelativisticMechanics(1905)

QuantumMechanics(1926)

Classical Mechanics(1687)

increasingdistance

(The table is adapted from a table found on Page 1 of Introduction to Electrodynamics, by David J. Griffiths, Prentice-Hall, 1989. Specific dates for each theory are a little misleading, because they were shaped over many years, with many workers involved in their creation, criticism, experimental testing, refinement, and development.)

So in conclusion, the special theory of relativity is a theory of mechanics (we've only scratched the surface of the full theory) in that it describes the motion of objects and makes predictions about their motions that is accurately verified by numerous experiments. But it is also a metatheory, in a sense that we have only described briefly in these notes: It places restrictions that must be satisfied by any proposed physical theory. To be an acceptable physical theory, the basic laws of the theory must transform properly from one inertial reference frame to another inertial reference frame, in order that the laws will predict values for quantities such as displacements and time intervals (and energies, and momenta, and …) that are consistent with what we know must be true from our analysis of special relativity. Search on the term "Lorentz invariance" if you wish to explore this more deeply.

This deeper understanding about basic physical theory that we have obtained through special relativity originated from the consideration of a very specific physical problem: What is the luminiferous ether? This is pretty typical in physics; theoretical physicists

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What is the luminiferous ether? This is pretty typical in physics; theoretical physicists typically don't sit around thinking about developing the next great theory. They analyze existing theories, find points that are weak, puzzling, strange, and so on, and try to fix the problematic points. They focus on specific problems, and resolving them often requires some creativity, which often involves genuinely new ideas. Once they generate new ideas, which is typically not a logical process (as it involves intuition, guesswork, artistry, imagination, etc.), they then use logical deduction to work out the consequences of the theory. They check extremely carefully to make sure that the new bits of the theory are logically consistent with the rest of physics, and if not, then modifications must be made or something has to be tossed onto the (very large) garbage heap. After all that, the final arbiter is experiment: Experiments are done, or observations of natural phenomena are made, and the predictions of the theory are tested.

Foundational theories are not necessarily discarded when new and improved theories are invented, although sometimes this occurs. The old impetus theories of motion bit the dust once Newtonian mechanics was invented, because the old theories were recognized as being misleading and having no predictive power or offering any deeper insight into motion whatsoever. However, with the establishment of special relativity, Newtonian mechanics was not discarded, because it still provides tremendous insight into low-speed phenomena, and very accurate predictions for the same. In a very well-defined mathematical sense, Newtonian mechanics is a good approximation to special relativity for low speeds. Physicists consider Newtonian mechanics to be a foundational theory, and special relativity is "built" on these foundations.

Some of the questions that beginners often ask about special relativity concern how real are the effects predicted by the theory. Is time dilation a real effect? Is length contraction a real effect? Does a rocket really shrink in the direction of its motion?

Yes, the effects are real, but they are subtle as well. Lorentz, Poincare, Fitzgerald, and others were toying with such ideas before Einstein, but what distinguishes Einstein from the earlier workers was their perspectives. Lorentz and Fitzgerald independently proposed that every object shrinks in the direction parallel to its motion relative to the ether, and they made this proposal as a way of salvaging the ether hypothesis. Lorentz thought of time dilation as fictitious; he assigned a fictitious "local" time to each moving reference frame to make the mathematics work out correctly, while still accepting that there was only one "real" time, which was absolute.

Einstein's perspective was much more subtle. First, he dispensed with the ether as an unnecessary concept, and one for which there was absolutely no experimental evidence for. Next, he treated time as relative, not absolute. Finally, he treated length contraction not as an absolute phenomenon, but rather as a relative one. That is, it's not quite right to ask the question, "What is the length of the rocket ship?" This is what, in effect, we are asking when we ask whether length contraction is a real effect. There is no such definite reality; the length of the rocket ship depends on the state of relative motion between the rocket ship and the reference frame of the observer who is doing the measuring.

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This quantity is called the invariant interval for the two events, and it resembles Pythagoras's theorem a bit, except for the negative sign. It's the mixture of positive and negative signs in the expression that is responsible for the departure from Euclidean geometry; the resulting geometry is called hyperbolic geometry. If you wish to learn more about the geometry behind special relativity, a good place to start is The Geometry of Special Relativity, by Tevian Dray. You'll find a prepublication version of his book here:

http://physics.oregonstate.edu/coursewikis/GSR/

Thus, the proposals of Poincare, Lorentz, Fitzgerald, and others, were ad hoc band-aids intended to patch up existing theory. Einstein did something much more radical, in that he created an entirely new theory, with a completely revolutionary perspective.

In PHYS 1P22/1P92 we will learn a little about quantum mechanics, the other revolutionary theory of the early 20th century, and we will see that there are quite a few other questions that are quite natural in classical physics but have no meaning in quantum physics. The worlds of the very fast (described very well by special relativity) and the subatomically small (described very well by quantum mechanics) are quite outside our normal every-day experience, and therefore appear very strange indeed. But it's mind-expanding and exciting to study these subjects, and it's another motivation for learning a little bit of physics.

Don't make the mistake of thinking that all is relative according to special relativity. (Many people who don't know much about science still ignorantly reject special relativity because they think it encompasses moral relativism, which is quite another thing.) Special relativity doesn't say that all is relative, it just says that what we thought was absolute isn't, and what we thought was relative isn't. In Newtonian mechanics, time and space are absolute, but the speed of light ought to be relative. In special relativity, it's the speed of light in vacuum that is absolute, whereas measurements of time and space are relative.

The other very significant conclusion from special relativity is that the theory can be described in geometrical terms, albeit with a strange geometry, one that is quite different from the usual Euclidean geometry. In Minkowski's perspective, space and time are not distinct, but rather form an integrated continuum, which we call spacetime. Observers moving along different inertial reference frames see space and time differently; they see different aspects of spacetime. This can be expressed mathematically in a number of ways; one conclusion that can be stated briefly is as follows: If two events are measured by observers in reference frames in relative motion, then the time interval and distance between the events will be judged to be different, but a certain combination of the measurements is invariant, absolute (that is, is the same for all observers):

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As mentioned earlier in these notes, Einstein initially scoffed at Minkowski's geometrical perspective on special relativity, but he soon warmed up to it, and he made decisive use of it in creating general relativity, Einstein's theory of gravity. General relativity has the same geometrical perspective, but works with curved spacetimes rather than the flat spacetime of special relativity.

By now, all foundational modern physical theories have been expressed using the geometric perspective pioneered relativity theory, and have been further developed over the last century. To learn more about this aspect of modern physical theory, learn about differential geometry, which you will be ready for once you've got advanced calculus and some linear algebra under your belt.______________________________________________________________________

Additional solved problems

Example: A rocket of rest length 100 m passes Earth in a time of 0.2 microseconds as measured by an observer on Earth. (a) Determine the speed of the rocket relative to Earth.(b) Determine the length of the rocket as measured by an observer on Earth.

Solution: (a) Let L represent the length of the rocket relative to the Earth, and let L0

represent the length of the rocket in its own rest frame. Thus,

Let v represent the speed of the rocket with respect to Earth. Because the rocket passes Earth in 0.2 microseconds,

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Example: Muons are emitted at a speed of 0.95c relative to a laboratory. The half-life of muons in their rest frame is 1.5 microseconds. (a) Determine the half-life of muons in the laboratory reference frame.(b) Determine the distance (relative to the laboratory) that muons travel after one half-life.

Solution: (a) First determine the value of gamma:

(b) First determine the value of gamma; then use it to determine L.

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Example: Two galaxies recede from Earth in opposite directions at the same relative speed of 0.4c as measured relative to Earth. Determine the recession speed of one galaxy relative to the other.

Solution: Label the galaxies by A and B. Then

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Example: An electron has a kinetic energy of 4.05 × 10-13 J. Determine the total energy and momentum of the electron.

Solution: The mass of an electron is

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_________________________________________________________________

Example: Determine the increase in total energy of an electron as its speed increases from (a) 0.01c to 0.02c, and (b) 0.98c to 0.99c.

From the given information we can determine the value of gamma, from which we can then determine the total energy and momentum of the electron.

The total energy of the electron is

To determine the momentum of the electron, we first need to determine its speed.

Thus, the electron's momentum is

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from (a) 0.01c to 0.02c, and (b) 0.98c to 0.99c.

Solution: First calculate the four values of gamma:

____________________________________________________________________

Comparing the results of Parts (a) and (b), note that the change in speed is the same in each case (0.01c) and yet the cost in energy is about 10,000 times greater in Part (b). This is true in general: As the speed increases, the amount of work needed to increase it further becomes greater and greater.

If you calculate the amount of work needed to increase the speed of a particle that has mass up to the speed of light, you'll determine that the value is infinite! Apparently it's not possible to accelerate a particle with non-zero mass up to the speed of light. This is a stark difference between Newtonian mechanics and special relativity; according to Newtonian mechanics the amount of work needed to increase the speed of a particle by a given amount does not depend on its speed, so it's possible to increase the speed of a massive particle up to an arbitrary value.

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possible to increase the speed of a massive particle up to an arbitrary value.

Which theory makes the correct prediction? This is a matter of experiment, and experiments with modern particle accelerators most definitely and repeatedly confirm the predictions of special relativity, not the predictions of Newtonian mechanics.___________________________________________________________________

Some good books for those who would like to learn more about relativity:

It's About Time, by N. David Mermin, Princeton University Press, 2005.

How to Teach Relativity to Your Dog, by Chad Orzel, Basic Books, 2012.

Special Relativity, by A.P. French, W.W. Norton, 1968.

Spacetime Physics, by E.F. Taylor and J.A. Wheeler, W.H. Freeman, 1992. (Taylor's website is here: www.eftaylor.com/

Some links you might like to check out:

Einstein Online: www.einstein-online.info

Special Relativity, by Tatsu Takeuchi, Virginia Tech: http://www.phys.vt.edu/~takeuchi/relativity/

The Geometry of Special Relativity, Tevian Dray, Oregon State University:http://physics.oregonstate.edu/coursewikis/GSR/

There is a very long list of books on relativity, with comments by John Baez (University of California, Riverside), at his site: http://math.ucr.edu/home/baez/physics/Administrivia/rel_booklist.html

Good biographies of Einstein:

Albert Einstein, Creator and Rebel, by Banesh Hoffmann (with Helen Dukas), Viking Press, 1972.

Albert Einstein, The Human Side, by Albert Einstein, Banesh Hoffman (Editor), Helen Dukas (Editor), Princeton University Press, 1981.

Subtle is the Lord …, by Abraham Pais, Oxford University Press, 2005.

The Other Einstein, by Lee Smolin, The New York Review of Books, http://www.nybooks.com/articles/archives/2007/jun/14/the-other-einstein/____________________________________________________________________

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"One of the things I have learned in a long life: that all our science, measured against reality, is primitive and childlike—and yet is it the most precious thing we have." — Albert Einstein

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