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The Blue Streak: A Hacker's Guide to Special Relativity
by Alexander Rein
268 pages; quality trade paperback (softcover); catalogue #03-0521; ISBN 1-4120-0153-6; US$24.00, C$31.00, EUR20.15, £13.96
Update your primitive notions of space, time and motion; brush up your rusty math and learn more about special relativity than you could in a college physics course.
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about the book about the author sample excerpts catalogue info
About the BookThe hacker's approach to Special Relativity grew out of an attempt to demystify the puzzling features of the theory to intelligent but intuition-blocked lay persons by a strategy aimed at this particular handicap:
Besides standard fare, two speculative topics are included: (1) a "Faster than Light" chapter dealing with its chief reputed consequence, the reversal of Time Arrow once the travel speed has "crashed" the "Light Barrier," and (2) a tentative description of a very-very fast moving object caught by our wide-open eyes or by a super-fast shutter speed camera. The book and its intended readership are described in the Preface. Basic concepts and a brief historical background of the theory are given in the Introduction. In Chapters I-XV, you'll find the main topics and in the Postscript, there are additional comments pertinent to, but reaching above and beyond, the contents of this book.
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About the AuthorAs a post-WWII refugee in the British Zone of Germany at the age 17, and still two years away from a complete high school education, I dreamed of a career in engineering. This seemed unrealistic at the time. Medical education seemed remotely but more attainable in the long haul. After settling in the USA, I finally graduated from one of the five medical schools in Philadelphia, PA., finished internship and three more years of residency training. I spent 34 undistinguished but rewarding years in medical practice and VA employment, retiring in 1996 to a "second career" in the pursuit of my abandoned but not neglected interests in the physical sciences. During the past years I kept up with physics, chemistry, astronomy, cosmology and expanded my mathematical horizons. The strange matters of relativity and the atom were special attractions and I gathered a lot of piecemeal facts and ideas about them. After I stumbled upon Herman Bondi's small paperback "Relativity and Common Sense," everything including the complex number mathematics suddenly fell into place and I progressed to the detailed technical literature, asked questions and worked out the answers. In my attempts to explain special relativity to a few friends I discovered that apart from the usual rustiness in math, the greatest obstacle to understanding the theory was a stubborn intuition block that could be overcome only by a focused attention to basic assumptions. It still took me a year to get a preliminary draft between covers with much more effort going into finalizing the exposition, detailed mathematical derivations written out in longhand and 276 illustrations inserted into a total of 247 pages. You'll find that the coverage of standard topics is completely orthodox. Personal opinions and philosophical ramifications are found only in the Postscript. Doppler acoustics is taken up in some detail because of its relevance to the Michelson-Morley experiment and its marked difference from the fundamentally relativistic Doppler optics. I doubt that you can find a better explanation of Lorentz contraction or any other concept anywhere else at the advanced high school or standard undergraduate level. Two speculative topics were included: The first one considers the reversal of the Time Arrow as the main consequences of the hypothetical faster-than-light motion. Nearly all math is omitted here in favor of an argument by spacetime mapping method. The second speculative topic is about the visual appearance of a near-light-velocity object as presented to our slow-response vision and to a picosecond shutter-speed camera.
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Sample Excerpts
From the Preface
The hacker's approach to Special Relativity grew out of an attempt to de-mystify the puzzling features of the theory to intelligent but intuition-blocked lay persons by a strategy tailored to their handicap. The results of this attempt are contained in this book and you, too, are invited to become a hacker. The invitation is extended to all those eager to dig into Special Relativity but lack confidence in their math skills and are intimidated by the complexities of the theory. Potential hackers come at all ages, in all walks of life and with varying backgrounds in the sciences. And there are those high school and college students who were exposed to Relativity in the usual, superficial manner but nevertheless became intrigued by its mind-bending features. Perhaps <>physics instructors can also be joined as honorary hackers, helped to recall their own initial puzzlement and made to quit talking over the heads of their students.
Before you continue reading, it is only fair to point out that there is a hidden danger lurking in Chapter I and beyond. Once you master (or re-master) elementary algebra (and a little trig), fail to panic at the thought of imaginary numbers, become able to navigate around intuition blocks and manage marching in step to step-by-step instructions, you may find yourself mentally trapped for good in Spacetime. And there is no turning back !!!
A paragraph from Introduction to correct a common misconception
It is important to point out that unlike the later General Theory of Relativity, Special Relativity was NOT a brainstorm of a single scientist. The initial ideas were formulated clearly during renaissance by Galileo and were later extended by Newton with his laws of Motion. Considerable theoretical progress was driven by difficulties with cathode ray experiments and everything already discovered was generalized and spectacularly reduced to FIRST PRINCIPLES by Einstein. The 4-dimensional conceptualization by Minkowski further prepared the way for the General Theory. The latter, in turn, provided gravitation a better explanation than given by Newton and supplied the broad field of cosmology the first crucial theoretical tool that changed it from idle speculation into a respectable scientific discipline.
The beginning of Chapter I, The Invariant
An observable Event always happens at a certain LOCATION in Space and at a certain MOMENT in Time. It can be completely specified within four DIMENSIONS, three of SPACE and one of TIME. This broader definition of a SPACETIME LOCATION makes it possible to express all physical measurements in four dimensions instead of the single dimension featured in our conventional Distance and Time measurements. Attempting to force all our Distance and Time concepts into a four-dimensional mold may seem draconian but we'll find out soon enough that over one-dimensional measurements, Observers in high-speed relative Motion ALWAYS disagree. Four-dimensional measurements, on the other hand, are most reliable, completely dependable, exactly conserved and, as we'll see, absolutely the same to all Observers. These well-deserved superlatives refer to what is known as the INVARIANT, a true ABSOLUTE in this new world of "relative" quantities, the fundamental, 4-dimensional entity in Special Relativity.
In this chapter, we'll develop the Invariant first in a 2-D Space (surface) then in the 3-D Space and, finally, in the fully 4-D Spacetime. By this step-by-step process, the Invariant emerges as a geometric structure that will be our key concept in all subsequent chapters.
The beginning of Chapter II, Axioms and Operations
Let's start by defining the words in the chapter title. By AXIOMS, let us mean those basic statements that are simple, explicit and reasonable. These were taken up at length in the Introduction. It is most fortunate that Special Relativity has been reduced to axiomatic FIRST PRINCIPLES, also called BASIC POSTULATES, from which the entire theory can be derived. That's what Einstein did and that's what we'll adopt for our method. To keep everything as clear as possible, we'll do it slowly, paying special attention to concepts that have always been stumbling blocks to intuition. By OPERATIONS, we mean those rules of thumb with which we can use manipulate concepts, obtain results, get things done. But before getting down to Space, Time and Motion at close range, let us first summarize what we have learned in the first chapter about Minkowski's geometry in 4-D Spacetime.
The beginning of Chapter III, Mapping Spacetime
Without a street map, you can still find your way in a strange town. But once you enter Spacetime, home to Relativity, you are hopelessly lost without a handy SPACETIME MAP. In "Axioms And Operations," we got our feet firmly on the ground by blazing some Spacetime travel paths. What we need now is to first examine the surface features on the Map and then examine the outer reaches of the Realm.
The beginning of Chapter IV, Measuring Lengths
In the first chapter, we learned to simplify our equations by eliminating the Y and Z Coordinate values
leaving only one Space (X) and one Time (T) Dimension to work with. In our task in this chapter, we'll eliminate also the ever-present Time Segment and do battle with the meddlesome Relativity of Simultaneity when measuring Space Segments (Lengths).
The beginning of Chapter VI, Doppler Optics
Doppler behavior of Light is in many ways similar to that of Sound. By habit, we consider them on equal terms except for their markedly unequal Velocities of propagation. But the relativistic character of Light, not seen with Sound, is best explained by a single feature unique to Light which happens to be the absence of a medium (or the non-dependence on it) for transmission thereby explaining also the ABSENCE of the Wind Effect. The question about the existence of such a medium was settled with finality by the historic Michaelson-Morley experiment (Chapter X).
In Doppler Optics, the absence of the Wind Effect makes the Motion of the Signal Source and of the Receiver through the intervening Space individually irrelevant. Only the relative Motion, the simple intervening Distance change between them, is all that matters. So the Light (or Radar) Signal is unique not only by the (1) absence of a known medium for transmission but also by (2) possessing the same Velocity in all Inertial Systems. In addition, the Signal is (3) the only modality that can reach across the wide open spaces and interrogate its dimensions that are calibrated only by the Velocity of Light as to Distance and are totally inaccessible to direct measurement.
The uniqueness of Light as an exclusive yardstick further underscores the impossibility of performing classical measurements in Space. As we already (should) know, one-dimensional Distance, Length and Time measurements do not yield the same values to all Inertial Observers in Motion relative to each other. Remember, the new Absolute is the INVARIANT which has the same value to all Observers. And it's all due to RELATIVITY OF SIMULTANEITY, itself a consequence of the unique properties of Light.
The beginning of Chapter VII, Lorentz Transformation
In Chapter II we used Lorentz Multiplier to calculate one Observer's measurements from those of another in situations where the Spacetime invariant "Distance" was marked by two Events: (1) the O, located at the crossing of their Worldlines and (2) the Q, located on one of the Observer's Worldline. In this chapter, the Q-Event is no longer on anyone's Worldline but is placed at various locations away from both Observers' WLs. With our earlier methods no longer adequate, the Lorentz Transformation we'll develop for new Q-Locations enables us to derive X and T Segment measurements of one Observer (such as <A>) from those of another (such as <B> and vice versa) in a much wider variety of Spacetime situations including those we have already mastered. In its capacity, Lorentz Transformation can additionally reconcile unequal coordinate measurements of any Space or Time Segment in a number of situations. The key component linking the two Systems such as <A>'s and <B>'s in our exercises is their shared relative Velocity, v along with the versatile Lorentz Multiplier, ß. As usual, a ST Map gives us the best hint about how to proceed.
The beginning of Chapter VIII, Adding Velocities
In classical mechanics the addition of velocities is a simple arithmetical addition of the given velocities. For example, if a railroad train is passing a bystander who is very close to the tracks and at the same time a stowaway on top of a railroad car is running in the forward direction, the velocity of the stowaway relative to the bystander is simply the sum of the two velocities: the ground velocity of the train and the velocity of the stowaway relative to the train.
Let v be the ground velocity of the train and u the velocity of a stowaway on top of the train. The sum of these velocities, w would then be w = v + u. If v = 50 mi/hr and u = 5 mi/hr, then w = v + u = 55 mi/hr.
The beginning of Chapter IX, Faster Than Light
Like a once popular televison series, this chapter can also be called "Mission Impossible." On TV, the challenges were always met with the most ingenious solutions. But what we are attempting here is absolutely the most impossible task: describing Motion faster than Light.
According to Special Relativity, nothing goes faster than Light and that's that. Agreed! Settled! Final!...Or is it? Well, one exception does suddenly come to mind. Do you know what CHERENKOV RADIATION is? And while we are on the subject, would anyone, please, tell us what professor Feynman meant when he said that POSITRONS are backward electrons, that is, "renegade" electrons going backward in Time. Also, why did professor Guth propose a SUPERLUMINAL INFLATIONARY STAGE during which the original Big Bang expanded thousands or, perhaps, more than millions or billions of Lightyears within an extremely small fraction of a second right after, or as part of, the Big Bang? Was he there to see it? And, finally, what about this NONLOCALITY, that "Spooky Action at a Distance," the thorn in Einstein's mind, the 30 year argument with Niels Bohr he lost? The topics listed do not belong to Special Relativity and will not be taken up here but the mere fact that these puzzling issues have been raised does suggest that perhaps some things just might be going faster than Light after all. And this possibility makes it worth our while to explore the KINEMATICS of SUPERLUMINAL VELOCITIES while leaving their actual existence undecided.
The beginning of Chapter X, The Michelson-Morley Experiment
In the first half of Doppler Acoustics, the moving train experiment carried both the Source (of the Signal) and also the Receiver (Observer) of Sound Signals, the Receiver standing proxy for us. In the virtual experiments, both participants were made to move through the air at a constant Velocity. It was pointed out that the Air Wind was responsible for changing the True Velocity of the signal through that medium, the air, into an observable "Effective" Velocity measured across a fixed Space Segment (Distance) that also was moving through the same medium, the air. The measured Sound Velocity was found to be either the sum or difference of the two Velocities: the Sound Velocity and the train Velocity. The virtual setup of the experiment was then used to challenge us to imagine that the air around the train was rendered undetectable. The analogy with the Michelson-Morley Experiment (MME) was duly pointed out. Now we have finally arrived at the historic experiment and our task here is to find out what happened to the ETHER WIND (and the "Effective" Velocity of Light).
The beginning of Chapter XI, Acceleration from 2-D Motion
Except for a little cheating at the O-Event to avoid collisions and to synchronize clocks, our virtual experiments were always staged in a Space of just one dimension. And true to promises, elementary algebra was all we needed. Enjoying just a little more cosmic elbow room was nice but we totally overlooked how dangerous a place it was? As pointed out in the Preface, a 1-D Lineland is like a one-lane highway featuring two-way traffic. This 1-D Space is a strange place. It is where the freedom of choice as to location is confined along a straight, thin line and anything that moves is also limited in the choice of direction and inevitably finds itself on a collision course with everything in its forward path. An Observer making a study of the natural events in this restricted setting is always in grave danger of being hit, smashed into, run over or through by the objects of his attention. Only by limiting our dangerous experiments to the imaginary world of Virtual Reality have we not only saved our own lives but also those of all the other participants.
The beginning of Chapter XII, Time In Acceleration
In the task of extracting Constant Acceleration, a from 2-D Motion, we measured Time with a stationary clock in <A>'s Home Frame. With <A>'s Time (T) and Passing Distance (R), we had all we needed to calculate both Instantaneous Velocities and Acceleration. The rate of Time progression in the Accelerating System was not needed for anything. But Time clocked in an accelerating system with constantly increasing Velocity is OBSERVED going slower and slower (and slower still). So what? Can we use it for anything? Not really. We didn't need it for calculating Accelerating Velocities or Acceleration. Nevertheless, it is a subject of curiosity begging to be scrutinized. The path to wisdom here is much more complicated so be prepared for a lot more difficult lines of argument. Fortunately, our 2-D scenario has served us well so far and will be useful also here as a convenient starting point.
Let us review the data available to <A>, the stationary Observer at Moment, T1 on our usual ST Map. The Accelerating Observer, <Ba> has moved from P to Q after decelerating to a momentary stop at the Passing Distance, R from <A>:
The beginning of Chapter XIII, Relativistic Mechanics
To understand the driving force in nature, we need more than Space, Time and (Uniform) Motion so far at the focus of our attention. It is fortunate that Classical Mechanics still provides many of the answers we seek. Reviewing it briefly would, therefore, be the best introduction to our limited exploration of Relativistic Mechanics presented here as an extension of the topic, Acceleration From 2-D Motion (Chapter XI).
Aristotle taught that to KEEP things in Motion, you had to KEEP PUSHING or PULLING them. It is quite descriptive of the ancient and not-so-ancient worlds of chariots, ox carts and sailing ships. Newton's laws of Motion revised all that, brought it up-to-date, set it straight. So, let us restate his most important Law of Motion: An object will stay at rest or in Uniform Motion until (and only until) it is acted upon by a Force. Now, that FORCE, F will cause a body with MASS, m to undergo ACCELERATION, a, while (and only while) that Force is being applied. What a neat package of insight! Let's put it into symbolic, mathematical language: F = ma
Here we have an equation with three components. There is a description for F but the other two elements have separate identities of their own, expressed as follows:
m = F/a
a = F/m
The last equation tells us that Acceleration is proportional to Force and inversely proportional to Mass. This agrees well with our everyday experience as things respond by Acceleration more willingly to greater Force and more stubbornly if there is more Mass.
The beginning of Chapter XIV, Aberration Of Light
When viewing objects through complex optical devices, we no longer notice colored halos around them. It used to be a rather annoying problem caused by the off-center prism-like action of lenses and was appropriately called CHROMATIC ABERRATION. Ingenuity in the design of compound lenses and other optical components has practically eliminated it in cameras, microscopes and refracting astronomical equipment. The introduction of concave (Newtonian) mirrors avoided it in telescopes early in the history of modern astronomy. But a subtle shift in the observed location of stars, now known as GEOMETRIC ABERRATION, came to light gradually as increasingly accurate instruments were put into use by the early 18th century. A British astronomer, Bradley (1727) was the first to point out that a part of the seasonably variable star locations could be distinguished from parallax and argued that the difference was due to Motion of the telescope along with its platform, the Earth, through Space. That this was less than an arc second is why it was not noticed earlier.
The beginning of Chapter XV, Sighting Blue Streaks
Sighting Blue Streaks is all about VISUAL KINEMATICS, or how objects moving real fast would appear to us. The problem is not only about their looks at high Velocities but also how to catch a glimpse of those elusive birds in flight.
The time limitations are severe. For example, only 0.1 second is needed for a near-Light-velocity object to approach us from a Distance of 15,000 km, pass us at close range and disappear behind another 15,000 km giving us hardly enough time for a picture opportunity. Using a wide-angle camera, all we would get is a big blur with outlines of the moving object smeared beyond recognition. With a fast-action camera, the image caught on film or magnetic memory would be shaped by Lorentz-contraction, Doppler Effects (on image and color), Bradley and Headlight Effects and still another Transformation we need to add to our list. What all this adds up to is a new territory in Spacetime we need to explore by the step-by-step method that has served us well with other topics. We'll revisit a few adventure trails in Lineland (4-D Spacetime "compressed" into 2-D) already familiar to us as we finally charge head-on into the full 4-D Spacetime for a closer look at a few less known applications of Special Relativity.
The beginning of Postscript
After considerable detail about Relativity, the time has finally come to reflect upon our learning experience. A realization that understanding is profoundly intuitive is, hopefully, of lasting benefit. Actually, understanding is possible only by INTUITION, so intuition is the ultimate faculty of comprehension, the QUEEN of UNDERSTANDING. And it should now be clear that RELATIVITY is NOT ANTI-INTUITIVE. It is more ANTI-COMMON-SENSE than anti-intuitive. But does it mean that we must distrust common sense? Far from it. It took a good deal of intuition to develop it in the first place. We can improve upon it by additional doses of intuition until it becomes "UNCOMMON SENSE." Once acquired, we are not forced to abandon common sense. This is not an either-or choice. Common sense and uncommon sense can live in peaceful coexistence in our minds. An open mind can accommodate both.
I have hardly shown you any mathematics and none of the illustrations. Don't worry. The introduction to math will be painless although you might have to, perhaps, struggle a little with it. The key equations will not be planked down the way you always see it done in textbooks. These will be all DERIVED, given to you the slow, painless way, all written out in longhand. See if you can duplicate the derivations without peeking. It'll be a good review and a help to "brush up" your rusty math. On the other hand, the illustrations will be a piece of cake (chocolate or vanilla?).
