Speeding :: Light
Speeding :: Light
Key idea: Concepts useful in exploring the very fast help us to examine space time near very massive objects.
We use relativity to explore the boundaries of Nature. Special relativity describes the very fast. General relativity-the Theory of Gravitation describes matter and motion near massive objects: stars, galaxies, black-holes. General relativity also describes the Universe as a whole. This article discusses a few key concepts of special relativity useful in exploring general relativity. The treatment here is not designed to be an introduction to special relativity; for introductory treatments see Section 11, Readings in Special Relativity, and detailed references to our own introductory treatment at the end of each section.
Everyone agrees on the wristwatch time between two events.
What is the root of relativity? Is there a single, simple idea that launches us along the road to understanding? Alice's adventures in wonderland begin when a rabbit rushes past her carrying a pocket watch. Our adventure in relativity begins when a small stone flies past us wearing a wristwatch. Wristwatch ticks may be one second apart-or one microsecond. Measure the distance sand time t between these ticks in a particular free-float or inertial reference frame. (The free-float frame is described in Section 8. Briefly, it is one in which Newton's first law holds: a free particle at rest remains at rest and one in motion continues that motion at constant speed in a straight line.)Special relativity warns us that a different observer passing us in uniform relative motion typically records a different value of spatial separation 5 and a different value of time lapse t between these two ticks. That is the bad news. The good news is a central finding of special relativity. All inertial observers, whatever their state of relative motion and whatever values they measure for sand t, agree on the value of the time t between ticks as recorded on the wristwatch carried by the stone.
The fact that all free-float observers agree on the wristwatch time 't earns it the label invariant. Invariant means that all observers calculate the same value, independent of reference frame. In relativity every invariant quantity is a diamond, to be treasured. How fast does the stone travel between ticks? The stone's speed depends on the reference frame. For the frame of Figure 1, the speed (assumed to be constant) is v = sit. Measure distance s and time lapse t in the same unit.For example, a spaceship travels half a light-year of distance during one year of time; its speed is then 0.5 year and the units cancel. As another example, if an elementary particle moves 0.7 meter in one meter of light-travel time its speed is 0.7. Hence the speed v has no units. In this book the symbol v represents the speed of an object as a fraction of the speed of light.
Two firecrackers explode 1 meter apart and at the same time, as measured in a particular free-float frame. In this frame these explosions are simultaneous. No stone can travel fast enough to be present at both of these explosions without moving at an infinite velocity, which is impossible. Therefore equation is useless to define a wristwatch time 't between these two events. Simultaneous explosions are thus useless for measuring time. But they are perfect for measuring length. Question: How do you measure the length of a rod, whether it is moving or at rest in your frame? Answer: Set off two firecrackers at the two ends and at the same time (t = 0) in your frame. Then define the rod's length in your frame as the distance s between this pair of explosions.
Special relativity warns us that a different observer passing us in uniform relative motion typically will not agree that the two firecrackers exploded at the same time. That is the bad news (and the idea most difficult to understand in all of special relativity). But there is good news: All inertial observers, whatever their state of relative motion, can calculate the distance cr between explosions as recorded in the frame in which they do occur simultaneously. The new metric is a variation of the old metric.
The Principle of Extremal Aging
The Twin Paradox leads to a definition of natural motion.
To get ready for curved space time (whatever that may mean), look further at the motion of a free particle in flat space time, the arena of the free-float frame (Section 8) in which special relativity correctly describes motion. How does a free particle move in flat space time? We say: "What a ridiculous question! Everyone knows that a free particle moves with constant speed in a straight line-at least as observed in a free-float frame." Ah yes, but why does a free particle move straight with constant speed? What lies behind this motion? Our answer for flat space time will be a trial run for the description of motion in curved space time, the arena of general relativity. A deep description of motion arises from the famous Twin Paradox. Recall that one identical twin relaxes on Earth while her twin sister frantically travels to a distant star and returns. When the two meet again, the stay-at-home twin has aged more than her traveling sister. (This outcome can be predicted by extending Sample Problems 1 and 2 to include return of the traveler to the point of origin.) Upon being reunited, the "identical twins" are no longer identical. Very strange! But (almost) no one who has studied relativity doubts the difference in age, and experiments with fast moving particles verify it. Which twin has the motion we can call natural? Isaac Newton has a definition of natural motion. He would say, "A twin at rest tends to remain at rest." So it is the stay-at-home twin who moves in the natural way. In contrast, the out-and-back twin suffers the forces required to change her state of motion-from outgoing motion to incoming motion-so that the two sisters can meet again in person. The motion of the traveling twin is forced, not natural.
Viewed from a second relatively moving free-float frame, the stay-at-home twin moves with constant speed in a straight line. Hers is also natural motion. Newton would say, "A twin in motion tends to continue this motion at constant speed in a straight line." So the motion of the stay-on Earth twin is also natural from the viewpoint of a second frame in uniform relative motion-or from any frame moving uniformly with respect to the original frame. In any such frame, the time lapse on the wristwatch of the stay-at-home twin can be calculated from the metric.
The lesson of the Twin Paradox is that the natural motion of a free object between two events in flat space time is the one for which the wristwatch worn by the object has a maximum time reading between those two events. Purists insist that we say not maximum reading but rather extremal reading: either maximum or minimum. This book contains only examples of maximum wristwatch time for natural motion. Still, let's try to keep the purists happy! Replace the two words maximum and minimum with the single word extremal. The result is the Principle of Extremal Aging.
Principle of Extremal Aging: The path a free object takes between two events in space time is the path for which the time lapse between these events, recorded on the object's wristwatch, is an extreme.
It turns out that the Principle of Extremal Aging describes motion even when space time is not flat. The Principle of Extremal Aging accompanies us into curved space time, into the realm of general relativity. But for now we stay in flat space time and use the Principle of Extremal Aging to derive relativistic expressions for energy and momentum.
The Free-Float Frame Is Local
In practice there are limits on the space and tirr.e extent of the free-float (inertial) frame
The free-float (inertial) frame is the arena in which special relativity describes Nature. The power of special relativity applies strictly only in a frame-or in each one of a collection of overlapping frames in uniform relative motion-in which a free particle released from rest stays at rest and a particle launched with a given velocity maintains the magnitude and direction of that velocity. If it were possible to embrace the Universe with a single free-float (inertial) frame, then special relativity would describe that Universe, and general relativity would not be needed. But general relativity is needed precisely because typically inertial frames are inertial in only a limited region of space and time. Inertial frames are local. The free-float frame can be realized, for example, inside various "containers," such as (1) an underpowered spaceship in orbit around Earth or Sun or (2) an elevator whose cables have been cut or (3) an underpowered spaceship in interstellar space. Riding in these free-float frames for a short time, we find no evidence of gravity.
Well, almost no evidence. The enclosure in which we ride cannot be too large or fall for too long a time without some unavoidable changes in relative motion being detected between particles in the enclosure. Why? Because widely separated test particles within a large enclosed space are differently affected by the nonuniform gravitational field of Earth-to use the Newtonian way of speaking. For example, two particles released side by side are both attracted toward the center of Earth, so they move closer together as measured inside a falling long narrow horizontal railway coach, Moving toward one another has nothing to do with gravitational attraction between these test particles, which is entirely negligible. As another example, think of two test particles released far apart vertically but one directly above the another in a long narrow vertical falling railway coach. For vertical separation, their gravitational accelerations toward Earth are in the same direction, according to the Newtonian analysis. However, the particle nearer Earth is more strongly attracted to Earth and gradually leaves the other behind: the two particles move far the apart as observed inside the falling coach.
A rider in either railway car shown in Figure 4 sees the pair of test particles accelerate toward one another or away from one another. These relative motions earn the name tidal accelerations, because they arise from the same kind of nonuniform gravitational field-this time the field of Moon-that account for ocean tides on Earth. Now, we want the laws of motion to look simple in our free-float frame. Therefore we want to eliminate all relative accelerations produced by external causes. "Eliminate" means to reduce them below the limit of detection so that they do not affect measurements of, say, the velocity of a particle in an experiment. We eliminate the problem by choosing a room that is sufficiently small. Smaller room? Smaller relative motions of objects at different points in the room!
Let someone have instruments for detection of relative motion with any given degree of sensitivity. No matter how fine that sensitivity, the room can always be made so small that these perturbing relative motions are too small to be detectable in the time required for the experiment. Or, instead of making the room smaller, shorten the time duration of the experiment to make the perturbing motions undetectable. For example, very fast particles emitted by a high-energy accelerator on Earth traverse the few-meter span of a typical experiment in so short a time that their deflection in Earth's gravitational field is negligible. The result: The frame of the laboratory at rest on Earth's surface is effectively free-float for purposes of analyzing these experiments. Both space and time enter into the specification of the limiting dimensions of a free-float frame. Therefore--for a given sensitivity of the measuring devices-a reference frame is free-float only within a limited region of space time.
An observer tests for a free-float frame by releasing particles from rest throughout the space and noting whether they remain effectively at rest during the time set aside for our particular experiment. Wonder of wonders! Testing for free float can be carried out entirely within the frame itself. The observer need not look out of the room or refer to any measurements made external to the room. A free-float frame is "local" in the sense that it is limited in space and time--and also "local" in the sense that its free-float character can be determined from within, locally. One way to get rid of "gravitational force" is to jump from a high place toward a trampoline below. That is to say, a locally free-float frame is always available to us. But no contortion or gyration whatsoever will eliminate the relative accelerations of test particles that indicate the limits of the free-float frame. These relative accelerations are the central indicators of the curvature of space time. They stand as warning signs that we are reaching the limits of special relativity. How can we analyze a pair of events widely separated near Earth, near Sun, or near a neutron star, events too far apart to be enclosed in a single free-float frame? For example, how do we describe the motion of an asteroid whose orbit completely encircles Sun, with an orbital period of many years? The asteroid passes through many free-float frames but cannot be tracked using a single free-float frame.
How, in principle, do we record events in space and time? Nature puts an unbreakable speed limit on signals-the speed of light. This speed limit causes problems with the recording of widely separated events, because we do not see a remote event until long after it has occurred. To avoid the light-velocity delay, adopt the strategy of detecting each event using equipment located right next to that event. Spread event-detecting equipment over space as follows. Think of assembling meter sticks and clocks into a cubical latticework similar to a playground jungle gym. At every intersection of the latticework fix a clock. These clocks are identical and measure time in meters of light-travel time. These clocks should read the same time. That is, the clocks need to be synchronized in this frame. There are many valid ways to synchronize clocks. Here is one: Pick one clock as the standard, the reference dock. At midnight the reference clock sends out a synchronizing flash of light in all directions. Prior to emission of the synchronizing flash, every other clock in the lattice has been stopped and set to a time (in meters) later than mid- night equal to the straight-line distance (in meters) of that clock from the reference clock. Each clock is then started when it receives the reference flash. The clocks in the latticework are then said to be synchronized. Use the latticework of synchronized clocks to determine the location and time at which any given event occurs. The spatial position of the event is taken to be the location of the clock nearest the event and the time of the event is the time recorded on that clock. The location of this nearest clock is measured along three directions, northward, eastward, and upward from the reference clock. The space time location of an event then consists of four numbers, three numbers that specify the space position of the clock nearest the event and one number that specifies the time the event occurs as recorded by that clock.
Specify the location of an event as the location of the dock nearest to it. With a latticework made of meter sticks, the location of the event will be uncertain to some substantial fraction of a meter. For events that must be located with greater accuracy, a lattice spacing of 1 centimeter or 1 millimeter would be more appropriate. To track an Earth satellite, lattice spacing of 100 meters might be adequate. The lattice clocks, when installed by a foresighted experimenter, will be recording clocks. Each dock is able to detect the occurrence of an event (collision, passage of light flash or particle). Each reads into its memory the nature of the event, the time of the event, and the location of the dock.The memory of all docks can then be read out and analyzed later at some command center. In relativity we often speak about the observer. Where is this observer? At one place or all over the place? Answer: The word observer is a shorthand way of speaking about the whole collection of recording docks associated with one free-float frame. This is the sophisticated sense in which we here after use the phrase "the observer measures such-and such." What happens to our latticework of docks in the vicinity of Earth or Sun or neutron star or black hole? Suppose one of these centers of attraction is isolated in space and we stay far away from it. Then there is no problem in setting up an extensive latticework that starts far from the center and stretches even farther away in all directions. Such an extensive far-away lattice can represent a single valid free-float frame. And in studying gen- eral relativity we often speak of a far-away observer. But there are problems in extending the far-away latticework of docks down toward the surface of any of these structures. A free particle released from rest near that center does not remain at rest with respect to the far-away lattice. A single free-float frame no longer provides a simple description of motion. To describe motion near a center of gravitational attraction we must give up the idea of a single global free-float frame, one that covers all space and time around Earth or black hole. Replace it with many local frames, each of which provides only a small part of the global description. A world atlas binds together many overlapping maps of Earth. Individual maps in the atlas can depict portions of Earth's surface small enough to be essentially flat. Taken together, the collection of maps bound together in the world atlas correctly describes the entire spherical surface of Earth, a task impossible using a single large flat map for the entire Earth. For space time near non rotating Earth or black hole, the task of binding together individual localized free-float frames is carried out by the Schwarzschild metric.
The important thing is not to stop questioning. Curiosity has its own reason for existing. One cannot help but be in awe when he [or she] contemplates the mysteries of eternity, of life, of the marvelous structure of reality. It is enough if one tries merely to comprehend a little of this
mystery every day. Never lose a holy curiosity.
I have no special talent. I am only passionately curious.