Particle momentum in special relativity

Equation [28] gives the momentum per unit mass for a particle moving with constant speed. If the particle changes speed, then once again we use calculus notation:

Equation [29] has the same form as in Newton's nonrelativistic mechanics, except here the incremental wristwatch time dx replaces the Newtonian lapse dt of "universal time."

Find mass from energy and momentum

Fuller Explanations: Momentum in flat spacetime: Spacetime Physics, Chapter 7, Momenergy.

7 Mass in Relativity

Everyone agrees on the value of the mass m of the stone.

An important relation among mass, energy, and momentum follows from the metric and our new expressions for energy and momentum. Suppose a moving stone emits two flashes very close together in space ds and in time dt. Then equation [1] gives the increase of wristwatch time dx:

Divide through by dx1 and multiply through by m2 to obtain m

2 2(dty 2(ds\2 ( dtY ( ds = m{dx)-m [dx) = lmdx) Hmdx.


or, substituting expressions [25] and [29] for energy and momentum, m2 = E2-p2 [32]

In equation [32], mass, energy, and momentum are all expressed in the same units, such as kilograms or electron-volts. In conventional units, the equation has a more complicated form:


Energy (also momentum) may be different for different observers where the subscript "conv" means "conventional units."

Equations [32] and [33] are central expressions in special relativity. The particle energy E will typically have a different value when measured in different frames that are in uniform relative motion. Also the particle momentum p will typically have a different value when measured in different frames that are in uniform relative motion. However, the values of these two quantities in any given free-float frame can be used to determine the value of the particle mass m, which is independent of the reference frame. Particle mass m is an invariant, independent of reference frame, just as the time dx recorded on the wristwatch between ticks in equation [1] is an invariant, independent of the reference frame.

The mass m of key, car, or coffee cup defined in equation [32] is the one we use throughout our study of both special and general relativity. Such a test particle responds to the structure of spacetime in its vicinity but has small enough mass not to affect this spacetime structure. (In contrast, the large mass M of a planet, star, or black hole does affect spacetime in its vicinity.) Wherever we are, we can always climb onto a local free-float frame (Section 8) and apply special-relativity expression [32] or some other standard method to measure the mass m of our test particle.

Fuller Explanations: Mass and momentum-energy in flat spacetime: Spacetime Physics, Chapter 7, Momenergy.

but mass is an invariant, the same for every observer

No Mass Change with Velocity!

The fact that no object moves faster than the speed of light is sometimes "explained" by saying that "the mass of a particle increases with speed " This interpretation can be applied consistently, but what could it mean in practice? Someone riding along with a faster-moving stone detects no change in the number of atoms in the stone, nor any change whatever in the individual atoms, nor in the binding energy between atoms Our viewpoint in this book is that mass is an invariant, the same for all free-float observers when they use equations [32] or [33] to reckon the mass In relativity, invariants are diamonds Do not throw away diamonds' For more on this subject, see Spacetime Physics, Dialog: Use and Abuse of the Concept of Mass, pages 246-251

8 The Free-Float Frame Is Local

In practice there are limits on the space and time 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 unpowered spaceship in orbit around Earth or Sun or (2) an elevator whose cables have been cut or (3) an unpowered spaceship in interstellar space. Riding in these free-float frames for a short time, we find no evidence of gravity.

Free-float frame cannot be Well, almost no evidence. The enclosure in which we ride cannot be too too large large or fall for too long a time without some unavoidable changes in rela tive 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 (Figure 4, left). 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 (Figure 4, right). 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-

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Limits of local free-float frames imply the need for general relativity

Figure 4 Einstein's old-fashioned railway coach in free fatL Left horizontal orientation Right vertical orientation

ther apart as observed inside the falling coach. Conclusion: The large enclosure is not a free-float frame,

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 roomi

Let someone have instruments for detection of relative motion with any Reduce space or time erven degree of sensitivity. No matter how fine that sensitivity, the room extension to preserve free-

1 U J -,-t -v . ji , I- ' . float frame 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

Test for free-float property within the frame itself

General relativity requires more than one free-float frame

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 spacetime.

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 spacetime. 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. Special relativity has reached its limit! To describe accurately motion that oversteps a single free-float frame, we must turn to general relativity—the Theory of Gravitation—as we do in Chapter 2.

Detect each event locally, using a latticework of clocks

Fuller explanations: Spacetime Physics, Chapter 2, Floating Free, and Chapter 9, Gravity: Curved Spacetime in Action.

9 The Observer

Ten thousand local witnesses

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 metersticks and clocks into a cubical latticework similar to a playground jungle gym (Figure 5). At every intersection of the latticework fix a clock. These clocks are identical and measure time in meters of light-travel time.

Figure 5 Latticework of metersticks and docks

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 clock. 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 midnight 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 dock. The location of this nearest clock is measured along three directions, northward, eastward, and upward from the reference clock. The spacetime 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.

Synchronize docks in the lattice

Measuring the space and time location of an event

The "observer" is all the recording clocks in one frame

The far-away lattice is not free float when extended to near Earth or black hole.

Many local frames are required near Earth or black hole

Specify the location of an event as the location of the clock nearest to it. With a latticework made of metersticks, 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 clock 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 clock. The memory of all clocks 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 clocks associated with one free-float frame. This is the sophisticated sense in which we hereafter use the phrase "the observer measures such-and such."

What happens to our latticework of clocks 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 general relativity we often speak of a far-away observer.

But there are problems in extending the far-away latticework of clocks 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 spacetime near nonrotating Earth or black hole, the task of binding together individual localized free-float frames is carried out by the Schwarzschild metric, introduced in Chapter 2. The Schwarzschild metric frees us from limitation to a single free-float frame and introduces us to curved spacetime.

Fuller Explanations: Spacetime Physics, Chapter 2, Section 2.7, Observer.

10 Summary

The wristwatch time x between two events, the time recorded on a watch that moves uniformly from one event to the other, is related to the separation s between the events and the time difference t between them as measured in a given free-float (inertial) frame. For space and time measured in the same units, this relation is given by the equation

The wristwatch time x is an invariant, the same calculated by all observers, even though t and s may have different values, respectively, as measured in different reference frames. Equation [1] is an example of the metric.

Of all possible paths between an initial event and a final event, a free particle takes the path that makes the wristwatch time along the path an extre-mum. This is called the Principle of Extremal Aging.

From the metric and the Principle of Extremal Aging one can derive two quantities that are constants of the motion for a free particle. One constant of the motion is the energy per unit mass E/m:

The second constant of the motion is the momentum per unit mass p/m:

m ax

The spacetime arena for special relativity is the free-float (inertial) frame, one in which a free test particle at rest remains at rest and a free test particle in motion continues that motion unchanged. We call a region of spacetime flat if a free-float frame can be set up in it.

In principle one can set up a latticework of synchronized recording clocks in a free-float frame. The position and time of any event is then taken to be the location of the nearest lattice clock and the time of the event recorded on that clock. The observer is the collection of all such recording clocks in a given reference frame.

Most regions of spacetime are flat over only a limited range of space and time. Evidence that a frame is not inertial (so that its region of spacetime is not flat) is the relative acceleration ("tidal acceleration") of a pair of free test particles with respect to one another. If tidal accelerations affect an experiment in a region of space and time, then we say that spacetime region is curved, and special relativity cannot validly be used to describe this experiment. In that case we must use general relativity, the theory of gravitation, which correctly describes the relations among events spread over regions of space and time too large for special relativity.

Note on terminology: In this book we use the convention recommended by the International Astrophysical Union that names for objects in the solar system be capitalized and used without the article. For example, we say "orbits around Sun" or "the mass of Moon." This provides a consistent convention; one would not say "orbits around the Mars." We also capitalize the words Nature and Universe out of respect for our cosmic home.

11 Readings in Special Relativity

Spacetime Physics, Introduction to Special Relativity, Second Edition, Edwin F. Taylor and John Archibald Wheeler, W. H. Freeman and Co., New York, 1992, ISBN 0-7167-2327-1. Our own book, to which reference is made at the end of several sections in Chapter 1 and elsewhere in the present book.

Special Relativity, A. P. French, W. W. Norton & Co., New York, 1968, Library of Congress 68-12180. An introduction carefully based on experiment and observation.

A Traveler's Guide to Spacetime, An Introduction to the Special Theory of Relativity, Thomas A. Moore, McGraw-Hill, Inc., News York, 1995, ISBN 0-07-043027-6. A concise treatment by a master teacher.

Flat and Curved Space-Times by George F. R. Ellis and Ruth M. Williams, Clarendon Press, Oxford, 1988, ISBN 0-19-851169-8. A leisurely, informative, and highly visual trip through special relativity is followed by treatment of curved spacetime. See more on this book in the section Readings in General Relativity at the end of the present book.

Space and Time in Special Relativity, N. David Mermin, Waveland Press, Inc., Prospect Heights, IL, 1989, ISBN 0-8813-420-0. Rigorous and mildly eccentric.

Understanding Relativity: A Simplified Approach to Einstein's Theories, Leo Sartori, University of California Press, Berkeley, 1996, ISBN 0-520-20029-2. Thoughtful and complete.

Relativity, The Special and General Theory, Albert Einstein, Crown Publishers, New York, 1961, ISBN 0-517-025302. A popular treatment by the Old Master himself. Published originally in 1916. Enjoyable for the depth of physics, the humane viewpoint, and the charm of old-fashioned trains racing past embankments.

Relativity Visualized, Lewis Carroll Epstein, Insight Press, San Francisco, 1997, ISBN 0-953218-05-X. An enjoyable and eccentric presentation of special and general relativity, done primarily with figures and graphics. Available in some bookstores, or send $19.95 plus $2 handling to Insight Press, 614 Vermont Street, San Francisco, CA 94107-2636, USA.

Of historical interest

Relativity and Its Roots, Banesh Hoffmann, Scientific American Books, New York, 1983, ISBN 0-7167-1510-4. History of the subject by one of Einstein's collaborators.

The Principle of Relativity, A. Einstein, H. A. Lorentz, H. Weyl, H.

Minkowski, Dover Publications, Inc., New York, 1952, Standard Book Number 486-60081-5. Translations of many of the original papers. See the following reference for a more recent translation of Einstein's special relativity paper.

Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905-1911), Arthur I. Miller, Addison-Wesley Publishing Co., Inc., 1981, ISBN 0-201-04680-6. Careful historical analysis of Einstein's original special relativity paper "On the Electrodynamics of Moving Bodies," the setting in which it was produced, and early consequences for the scientific community. Includes a modern, corrected translation of the paper itself.

12 Reference

Initial quote: Personal memoir of William Miller, an editor of Life magazine, quoted in the issue of May 2,1955. See The Quotable Einstein, edited by Alice Calaprice, Princeton University Press, 1996, page 199.

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