by Isaac Asimov
M' = M
√1 − (2c)2/c2
= M = M
√1 − 4c2/c2 √–3
This works out to the fact that its mass while in motion would be some proper mass (M) divided by √–3. But √–3 is equal to √3 × √–1 and therefore to 1.74 √−1. The proper mass M is therefore equal to M' × 1.74 × √−1. Since any quantity that includes √−1 is called imaginary, we conclude that particles at superluminal velocities must have imaginary proper masses.
Ordinary particles in our ordinary universe always have masses that are zero or positive. An imaginary mass can have no imaginable significance in our universe. Does this mean that faster-than-light particles cannot exist?
Not necessarily. Allowing the existence of imaginary proper masses, we can make such faster-than-light particles fit all the equations of Einstein’s Special Theory of Relativity. Such particles, however, display an apparently paradoxical property: the more slowly they go, the more energy they contain. This is the precise reverse of the situation in our universe and is perhaps the significance of the imaginary mass. A particle with an imaginary mass speeds up when it meets resistance and slows down when it is pushed ahead by a force. As its energy declines, it moves faster and faster, until when it has zero energy it is moving at infinite speed. As its energy increases, it moves slower and slower until, as its energy approaches the infinite, it slows down to approach the speed of light.
Such faster-than-light particles have been given the name of tachyons from the Greek word for “speed,” by the American physicist Gerald Feinberg.
We may imagine, then, the existence of two kinds of universes. One, our own, is the tardyon-universe, in which all particles go at subluminal velocities and may accelerate to nearly the speed of light as their energy increases. The other is the tachyon-universe, in which all particles go at superluminal velocities and may decelerate to nearly the speed of light as their energy increases.
Between is the infinitely narrow luxon wall in which there are particles that go at exactly luminal velocities. The luxon wall can be considered as being held by both universes in common.
If a tachyon is energetic enough and therefore moving slowly enough, it might have sufficient energy and remain in one spot for a long enough period of time to give off a detectable burst of photons. (Tachyons would leave a wake of photons even in a vacuum as a kind of Cerenkov radiation.) Scientists are watching for those bursts, but the chance of happening to have an instrument in just the precise place where one of those (possibly very infrequent) bursts appears for a trillionth of a second or less, is not very great.
There are those physicists who maintain that “anything that is not forbidden is compulsory.” In other words, any phenomenon that does not actually break a conservation law must at some time or another take place; or, if tachyons do not actually violate special relativity, they must exist. Nevertheless, even physicists most convinced of this as a kind of necessary “neatness” about the universe, would be rather pleased (and perhaps relieved) to obtain some evidence for the non-forbidden tachyons. So far, they have not been able to.
EINSTEIN’S EQUATION
One consequence of the Lorentz equation was worked out by Einstein to produce what has become perhaps the most famous scientific equation of all time.
The Lorentz equation can be written in the form:
M' = M (1 − v2/c2)−½
since in algebraic notation l/√x can be written x−½. This puts the equation into a form that can be expanded (that is, converted into a series of terms) by a formula discovered by, of all people, Newton. The formula is the binomial theorem.
The number of terms into which the Lorentz equation can be expanded is infinite, but since each term is smaller than the one before, if you take only the first two terms you are approximately correct, the sum of all the remaining terms being small enough to be neglected. The expansion becomes:
> (1 − v2/c2)−½ = 1 + ½v2 …
c2
Substituting that in the Lorentz equation, we get:
M' = M (1 + ½v2 )
c2
= M + ½Mv2
c2
Now, in classical physics, the expression Y2My2 represents the energy of a moving body. If we let the symbol e stand for energy, the equation above becomes:
M' = M + e/c2
or:
M' − M = e/c2
The increase in mass due to motion (M' − M) can be represented as m, so:
m = e/c2
or:
e = mc2
It was this equation that for the first time indicated mass to be a form of energy. Einstein went on to show that the equation applies to all mass, not merely to the increase in mass due to motion.
Here again, most of the mathematics involved is only at the high-school level. Yet it presented the world with the beginnings of a view of the universe greater and broader even than that of Newton, and also pointed the way to concrete consequences. It pointed the way, for instance, to the nuclear reactor and the atom bomb.
Illustrations
* * *
I. The Solar System
Plate I.1. Our region of the universe—a drawing showing the other galaxies in our neighborhood. Courtesy Department of Library Services, American Museum of Natural History.
Plate I.2. Cornell University’s radio telescope. The reflector of this radio-radar telescope at Arecibo, Puerto Rico, is 1,000 feet in diameter and is suspended in a natural bowl. Courtesy of Cornell University and Air Force Office of Scientific Research. Courtesy of Arecibo Observatory, National Astronomy and Ionosphere Center (NAlC), Cornell University.
Plate I.3. Halley’s Comet, photographed 4 May 1910, with an exposure of 40 minutes. By permission of the Yerkes Observatory, Wisconsin.
Plate I.4. A spiral galaxy in broadside view—the “whirlpool nebula” in Canes Venatici, Courtesy of Palomar Observatory, California
Plate I.5. A globular cluster in Canes Venatici. Courtesy of Palomar Observatory, California.
Plate I.6. The Crab Nebula, the remains of a supernova, photographed in red light. Courtesy of Palomar Observatory, California.
Plate I.7. The Horsehead Nebula in Orion, south of Zeta Orionis, photographed in red light. Courtesy of Palomar Observatory, California.
Plate I.8 Saturn and its rings: a montage of photographs taken by Voyager 1 and Voyager 2. Here are pictured all of Saturn’s major satellites known before the Voyager launches in 1977. The satellites are (clockwise from upper right): Titan, Iapetus, Tethys, Mimas, and Rhea. Courtesy of the National Aeronautics and Space Administration.
Plate I.9 Jupiter and its moons in their relative positions: a montage of photographs made by Voyager 1 in 1977. The Galilean satellites are Io (upper left), Europa (center), and Ganymede and Callisto (lower right). Courtesy of the National Aeronautics and Space Administration.
Plate I.10. Mars, photographed 19 June from Viking 1. Clearly seen are the Tharsis Mountains, three huge volcanoes. Olympus Mons, Mars’s largest volcano, is toward the top of the photograph. Courtesy of the National Aeronautics and Space Administration.
Plate I.11. The sun’s corona. Courtesy of Mount Wilson Observatory, California.
Plate I.14. Solar prominences. Courtesy of Mount Wilson Observatory, California.
Plate I.15. Aurora borealis. Courtesy of the National Oceanic and Atmospheric Administration.
Plate I.16 Lunar map. Courtesy of the National Aeronautics and Space Administration.
Plate I.17 This view of the rising earth greeted the Apollo 8 astronauts as they came from behind the moon after orbiting it. On the earth 240,000 statute miles away, the sunset terminator bisects Africa. Courtesy of the National Aeronautics and Space Administration.
II. Earth and Space Travel
Plate II.1. Foucault’s famous experiment in Paris in 1851, which showed the rotation of the earth on its axis by means of the swing of a pendulum; the plane of its swings turned clockwise. By permission of the Bettmann Archive.
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br /> Plate II.2 The Montgolfier brothers’ hot-air balloon, launched at Versailles, 19 September 1783. By permission of the Bettmann Archive.
Plate II.3. Launching of the first U.S. satellite, Explorer 1 on 31 January 1958. Courtesy of the United States Army.
Plate II.4. The kneeling figure silhouetted by what seems to be a sparkling halo actually is a mechanic working inside the spacecraft fairing of a McDonnell Douglas Delta rocket. His portable lamp glints on thousands of facets of a triangular “isogrid” pattern milled into the shiny aluminum skin of a fairing to reduce weight while retaining maximum strength. The fairing, 8 feet in diameter and 26 feet long, protects the Delta’s payload as it is launched into orbit and from aerodynamic forces and heat during flight through the atmosphere. Courtesy of the McDonnell Douglas Astronautics Company, California.
Plate II.5. Astronaut Edwin E. Aldrin, Jr., lunar module pilot, is photographed walking near the lunar module during Apollo 11 extravehicular activity. Courtesy of the National Aeronautics and Space Administration.
Plate II.6. Apollo 11 astronaut Edwin E. Aldrin, Jr., deploys Solar Wind Composition experiment on the moon’s surface. Courtesy of the National Aeronautics and Space Administration.
Plate II.7. The Orientale Basin photographed from 1,690 miles above the moon’s surface by Lunar Orbiter IV. Courtesy of the National Aeronautics and Space Administration.
Plate II.8. This photograph of the crater Copernicus was taken from 28.4 miles above the surface of the moon by Lunar Orbiter II. Courtesy of the National Aeronautics and Space Administration.
Plate II.9. The Apollo 17 space vehicle, 28 August 1972. Courtesy of the Natiou.il Aeronautics and Space Administration.
Plate II.10. The moon’s surface: scientist-astronaut Harrison F. Schmitt stands next to a huge, split lunar boulder, during the Apollo 17 expedition. This scene is a composite of three views. Courtesy of the National Aeronautics and Space Administration.
Plate II.11. Model of a future space station. Courtesy of the National Aeronautics and Space Administration.
Plate II.12. Earth, from sunrise to sunset. This sequence was taken by the ATS-III satellite from a point about 22,300 miles above South America, November 1967. The photos show all of that continent and portions of North America, Africa, Europe, and Greenland; clouds cover Antarctica. Courtesy of the National Aeronautics and Space Administration.
Plate II.13. Weather photograph of Earth, showing storms over the Pacific and Caribbean oceans. Courtesy of the United States Department of Commerce.
Plate II.14. Sally Ride, the first woman astronaut, preparing for the STS 7 Space Shuttle launch of 18 June 1983. By permission of United Press International.
Plate II.15. The first free-floating space walk, February 1984: Astronaut Bruce McCandless II, without the use of a tether, at maximum distance from the Challenger. Courtesy of the National Aeronautics and Space Administration.
Plate II.16. Earth, as photographed from Apollo 17 during the final lunar landing mission. Visible is almost the entire coastline of Africa and the Arabian peninsula. A heavy cloud covers the Antarctic icecap. Courtesy of the National Aeronautics and Space Administration.
III. Aspects of Technology
Plate III.1. Stone tools of early man. The oldest, from the Miocene period, are at the lower left; the most recent, at the lower right. Neg. No. 411257. (Photo: J. Kirschner) Courtesy Department of Library Services, American Museum of Natural History.
Plate III.2. Galvani’s experiment, which led to the discovery of electric currents. Electricity from his static-electricity machine made the frog’s leg twitch; he found that touching the nerve with two different metals also caused the leg to twitch. By permission of the Bettmann Archive.
Plate III.3. A single ice crystal photographed by X-ray diffraction, showing the symmetry and balance of the physical forces holding the structure together. From Franklyn Branley, ed., Scientist’s Choice (New York: Basic Books, n.d.).
Plate III.4 Electric field around a charged crystal is photographed with the electron microscope by means of a shadow technique. The method uses a fine wire mesh; the distortion of the net, caused by deflection of electrons, shows the shape and strength of the electric field. Courtesy of the National Bureau of Standards.
Plate III.5. Molecular model of titanium oxide in crystalline form, which can serve as a transistor. Removal of one of the oxygen atoms (light balls) will make the material semiconducting. Courtesy of the National Bureau of Standards.
Plate III.6. In a cyclotron, magnets are used to bend a beam of electrically charged particles into a circular path. With the ever-increasing scientific need for beams of higher energies, these accelerators and their magnets have grown in size. Pictured is a super-conducting magnet model developed by Clyde Taylor and co-workers at Lawrence Berkeley Laboratory. Courtesy of Lawrence Berkeley Laboratory, University of California, Berkeley.
Plate III.7. Spinning protons in this schematic drawing are oriented in random directions. The white arrow shows the direction of the spin. Courtesy of the National Bureau of Standards.
Plate III.8. Protons lined up by a steady magnetic field. Those oriented in the opposite-to-normal direction (arrows pointed downward) are in the excited state. Courtesy of National Bureau of Standards.
Plate III.9. Tracks of electrons and positrons formed in a bubble chamber by high-energy gamma rays. The circular pattern was made by an electron revolving in a magnetic field. Courtesy of the University of California, Berkeley.
Plate III.10. Fission of a uranium atom. The white streak in the middle of this photographic plate represents the tracks of two atoms flying apart from the central point where the uranium atom split in two. The plate was soaked in a uranium compound and bombarded with neutrons, which produced the fission caught in this picture. The other white dots are randomly developed silver grains. The picture was made in the Eastman Kodak Research Laboratories. By permission of United Press International.
Plate III.11. Radioactivity made visible. On the tray is some tantalum made radioactive in the Brookhaven reactor; the glowing material is shielded here under several feet of water. The radioactive tantalum will be placed in the pipe shown and then transferred to a large lead container for use as a 1,OOO-curie source of radioactivity for industrial purposes. Courtesy of Brookhaven National Laboratory, New York.
Plate III.12. Drawing of the first chain reactor, built under the Chicago football stadium. Courtesy of Argonne National Laboratory, Illinois.
Plate III.13. The Chicago reactor under construction. This was one of only a few photographs made during the building of the reactor. The rods in the holes are uranium, and the reactor’s nineteenth layer, consisting of solid graphite blocks, is being laid on. Courtesy of Argonne National Laboratory, Illinois.
Plate III.14. The mushroom cloud from the atomic bomb dropped by the United States on Hiroshima, Japan, 6 August 1945. This picture was taken by Seizo Yamada, a middle-school student at the time. By permission of the photographer.
Plate III.15. A silo at the nuclear reactor at Three Mile Island, Pennsylvania. Photograph by Sylvia Plachy. Used by permission.
Plate III.16. The life and death of a pinch. This series of pictures shows the brief history of a wisp of plasma in the magnetic field of the Perhapsatron. Each photograph gives two views of the plasma, one from the side and one from below through a mirror. The pinch broke down in a millionths of a second; the number on each picture is in microseconds. Courtesy of Los Alamos Scientific Laboratory, New Mexico.
Plate III.17. The dark streaks are the tracks left by some of the first uranium nuclei ever to be accelerated to near the speed of light. Here you see the last ½ millimeter of three tracks as theycame to rest in a special photographic emulsion. The bottom track shows a nucleus splitting into two lighter nuclei. The work was done at the Bevalac, the only accelerator facility in the world that provides ions as heavy as uranium at relativistic energies. The accelerator is located at the University of California’s Lawre
nce Berkeley Laboratory. Courtesy of the Lawrence Berkeley Laboratory, University of California, Berkeley.
Plate III.18. An engineer with the rectifier decks that are part of the high-voltage power supply for a new ion injector system at the Lawrence Berkeley Laboratory’s Super-HILAC. The new injector, called Abel, extends the accelerator’s capabilities to include high-intensity beams of heavy ions such as uranium. Courtesy of the Lawrence Berkeley Laboratory, University of California, Berkeley.
Plate III.19. The aluminum shell that sits on top of the rectifier decks contains a magnet, and an ion source, where an electrical arc strips away electrons from the atoms. The stripped atoms (called ions) now have a positive charge and can be accelerated by the electrical field in the accelerating columns (visible in its lucite enclosure). The ion beam then receives further acceleration, in a new Wideroe accelerator, by the SuperHILAC before being sent down to the Bevatron. When the Super HILAC and the Bevatron act in tandem, as they often do, we refer to the combination as the Bevalac. Courtesy of the Lawrence Berkeley Laboratory, University of California, Berkeley.
