Time is Relative

十月 7, 2012

http://www.ram.org/ramblings/ramblings.html


I recently had an argument about why time is relative. My friend found it hard to conceive how time could not be absolute. At that time, when I tried to explain it, I found that it was hard for me see WHY time really is relative. All I could do was state experimental results that indicate it is the case. This shows that thinking you can understand a deep concept at a very shallow, philosophical level doesn’t really mean much, but at least it’s better than no understanding at all.

Newton’s laws put an end to the idea of absolute motion. Two people playing ping pong on a moving train would measure the distance the ball bounces on the table (between consequecutive bounce) as lesser than a person standing along the track watching the ball bounce. Both measurements are equally valid since there is no absolute standard of rest.

Maxwell’s equations predicted that light waves travel at a fixed speed. This is something we have to believe, and it arises out of the fact that his equations predicted “there would be wavelike disturbances in the combined electromagnetic field, and that these would travel at a fixed speed, like ripples on a pond.”

But Newton’s theory got rid of the idea of absolute rest and if we believe that there’s no ambiguity about time, then if light is supposed to travel at a fixed speed, we need say what the fixed speed is measured relative to. It was suggested that there was a substance called “ether” that was present everywhere (even in “empty space”) and that light waves travelled through this ether at a fixed speed. However, different observers, moving relative to the ether, would see light coming at them at different speeds.

The way to verify this is to measure the speed of light as the earth moves through the ether on its orbit around the sun. So, light speed measured in the direction of earth’s orbit (when we are moving towards the source of light) should be higher than the speed of light measured at right angles from the direction of earth’s motion (when we are not moving towards the source of light). This experiment, called the Michaelson-Morley experiment, was performed in 1887, and to their great surprise, it was shown that the speed of light was exactly the same in both cases!

Einstein then pointed out that the whole idea of ether was unnecessary as long as one was willing to abandon the idea of absolute time. He postulated that the laws of science should be the same for all freely moving observers regardless of the speed. This was true for Newton’s laws of motion, but it also now applied to Maxwell’s theory and the speed of light: all observers should measure the same speed of light, no matter how fast they are moving.

One of the remarkable consequences of this theory is that how it has changed our thinking regarding space and time. In Newton’s theory, if a pulse of light is sent from one place to another, different observers would agree on the time the journey took (since time is absolute) but will not agree on how far the light travelled (since space is not absolute). Since the speed of light is measured as distance travelled/time taken, different observers would measure different speeds for the speed of light. But this clearly violates what Maxwell’s equations say and what has been shown by experiment. In relativity, all observers MUST agree on how fast light travels, but they don’t have to agree regarding the distance and consequently, they must disagree over the time it has taken! (The time taken is the distance (which they don’t agree on) divided by the speed of light (which they do agree on).) That is, observers must have their own measure of time, as recorded by a clock carried by them and even identical clocks carried by different observers would not necessarily agree.

From all this we can make the statement that time MUST be relative, because if it weren’t, it would contradict experimental results. But this doesn’t, at least to me, shed light on WHY this is the case. This might be a question more metaphysical than physical (i.e., time is relative just as Newton’s laws are the way they are). I shall attempt to address my question of “WHY” in greater detail now. (Though ultimately I don’t have a good answer I think—anyone?)

This theory (special relativity) and the more general form (which takes gravitional effects into account) of it makes several other important predictions which have withstood empirical observations. But one of the prediction of general relativity is time should appear to run slower near a massive body like the earth. This is because the energy of light is proportional its frequency (waves/second). As light travels upward in earth’s gravitional field, it loses energy, and therefore its frequency goes down. (The length of time between one wave crest and the next goes up.) To someone high up, it would appear that that everything down below was taking longer to happen. Apparently, this prediction can be tested by using a pair of very accurate clocks, one at the bottom nearer to the earth, and the other at the top. This was done in 1962, and the clock at the bottom was found to run slower, in exact agreement with general relativity! This is of practical importance, if you wish to trust navigational signals from satellites. If you ignore relativistic effects, you could be off by several miles!

Now, my question is, what is the basis for this phenomenon? Why should the clocks have different times? Why should it be related to light and its frequency? This is saying that time slows down when it is exposed to intense gravity. Why is this the case?

Similarly consider the twins paradox where one twin flies off into space at the speed of light and comes back and hasn’t aged much more than the twin who is on earth. It’s not a paradox if you have relative time, but what is the physiological basis (do the cells actually undergo cell division slower?) for this phenomenon?

We must accept that time isn’t independent of space and it forms a 4-dimensional construct called space-time. General relativity shows that gravity is really not a force, but rather a consequence of the fact that space-time is curved due to the distribution of mass and energy. Bodies like the earth don’t follow a curved orbit due to a force called gravity, but they follow the nearest thing to a straight path in curved space, called a geodesic. On the surface of the earth, a geodesic is a circle (the equator, for example). In general relativity, bodies always follow straight lines in four-dimensional space-time, but they appear to us to move along curved paths in three-dimensional space. (Hawking says this is like watching an airplane over an hilly ground. Although it follows a straight line in 3D space, its shadow follows a curved path on the ground.)

I imagine this as a huge rubber sheet with bodies like the sun, the earth, etc. are laying on the sheet causing a depression (a “gravity well”). An object travelling through this sheet might just go past the end of a well, causing it follow a curved path as thought it suffered a gravitional pull. An object might also just go inside the well and follow an elliptical path around the walls of the well. An object may finally, due to friction, decay and eventually fall in the greater object at the bottom of the well.

A postulate of general relativity, which follows from logical deductions, is that if two events are close together, there is an interval between them which can be calculated by some function of their coordinates. We know, according to the mathematics of the theory, that if we choose a region of space-time where the gravitation is the same throughout the region, that we can obtain very nearly a Euclidean space. We have a second postulate which states that a body travels in a geodesic in space-time unless non-gravitional forces act on it. The third postulate says that light travels on a geodesic such that the interval between any parts of it is zero.

In the general theory, It is only neighbouring events that have a definite interval and this is INDEPENDENT of the route pursued. The interval between distant events depends on the route pursued, and it can be calculated by dividing up the route into small enough parts (each part has constant gravitional effects and thus we can calculate the interval between the two neighbouring events), and adding up the intervals for all parts. If the interval is spacelike, a body cannot travel from one event to the other. Therefore, the interval has to be timelike. The interval between neighbouring events when it is timelike is the time between them for observers who travel from one event to the other. And so the TOTAL interval between two events will be judged by people who travel from one of the other by what their clocks show to to be the time they have taken on the journey. The slower they travel, the longer they will think they have been on the journey. This is not platitude—if you travel from DC to NY and leave at 6a and arrive at 10a, the more slowly you travel, the longer you will take according to your watch. So if you travel at the speed of light (third postulate), going all across the solar system before reaching your destination, your watch would say that you had taken no time at all! If you had gone by any circuitous route, which enabled you arrive in time by travelling fast, the longer your route the less time you will take. The diminision of time is continual as you approach the speed of light.

A body when left to itself travels so that the time, measured by its clocks, is the longest. If it had travelled by any other route from one event to another, the time would be shorter. This is saying that bodies left to themselves make their journeys as slowly as they can. Russell refers to this as a law of cosmic laziness. Mathematically, they travel in geodesics, in which the total interval between any two events on the journey is GREATER than by any other alternative route. (The fact that it is greater and not less is because the sort of interval we are considering is more analogous to time than to distance.) So, if someone flies off into space and comes back to earth after a while, the time between departure and return would be less by their clocks than by the clocks on earth, since the earth, during its journey around the sun, chooses the route so that any bit of it is measured longer than any other alternative route.

“Space and time are now dynamic quantities: when a body moves, or a force acts, it affects the curvature of space and time—and in turn the curvature of space-time affects the way in which bodies move and forces act. Space and time not only affect, but are also affected by everything that that happens in the universe.” Hofstadter would call this a Strange Loop.


References

  • A Brief History of Time by Stephen Hawking
  • Einstein’s Law of Gravition by Bertrand Russell
  • The Two Masses by Isaac Asimov

Pseudointellectual ramblings || Ram Samudrala || me@ram.org


Lorentz-Fitzgerald Contraction

七月 14, 2012

In 1889, the Irish physicist George Francis Fitzgerald proposed
a radical explanation of why the Michelson-Morley
experiment failed to detect the luminiferous ether. His
explanation came at a time when he, like most scientists,
firmly believed in the ether. Movement through the ether,
Fitzgerald said, shortened the arm of Michelson’s interferometer
just enough to cancel the decrease in the speed
of light caused by the ether wind. This length contraction
took place along the line of motion and was almost impossible
to detect because any meter stick used to measure it
would contract, too.
Two years after Fitzgerald published his proposal, Hendrik
Lorentz, a prominent Dutch physicist who was also a
staunch believer in the ether, developed the idea further.
The shortening of objects in motion relative to an observer
became known as the Lorentz-Fitzgerald contraction.
Lorentz also came up with a general method for transforming
the space and time coordinates of events from one
inertial frame of reference to another. The equations he
derived to do this are called Lorentz transformations, and
they proved useful to Einstein as he developed the special
theory.
Lorentz’s formulas for calculating time dilation and
length contraction are identical to those Einstein developed
for special relativity. Why, then, are Lorentz and Fitzgerald
not considered to be the authors of the theory of special
relativity? The answer lies in the two men’s wrong interpretation
of the Michelson-Morley experiment. According
to Lorentz and Fitzgerald, the ether existed and the speed
of light was constant relative to it. Einstein’s bold leap forward
was to ignore the ether and accept what Maxwell’s
equations were telling him: The speed of light is the same
for every observer. It is this key conclusion that led Einstein
to relativity—and kept Lorentz and Fitzgerald from discovering
it themselves.

Nevertheless, Einstein knew he owed much to the two
men’s groundbreaking ideas and was quick to recognize
them. In an after-dinner speech he delivered in California,
Einstein credited “the ideas of Lorentz and Fitzgerald, out
of which the Special Theory of Relativity developed.”


Lorentz took the distortions that he discovered in fast-moving material objects to be laws of nature

七月 12, 2012

Time dilation. There is, however, another distortion that material objects undergo as a function of their absolute motion. That is a slowing down of clocks (and physical processes generally) at the same rate as the length contractions, or the so-called “time dilation,” which took somewhat longer for Lorentz to discover.

The Galilean transformation for time in Newtonian physics is simply t = t’ , because Newtonian physics assumes that time is the same everywhere. But by using transformation equations to describe the distortions in material objects, Lorentz found that he had to introduce a special equation for transforming time: t’ = t – vx/c2  (Goldberg, p. 94). The new factor in the transformation equation, vx/c2, implied that time on the moving frame varies with location in that frame. Lorentz called it “local time,” but he did not attribute any physical significance to it. “Local time” is not compatible with the belief in absolute space and time, and Lorentz described it as “no more than an auxiliary mathematical quantity” (Torretti, p. 45, 85), insisting that his transformation equations were merely “an aid to calculation” (Goldberg, p. 96).

The slowing down of physical processes is called “time dilation.” Lorentz discovered this distortion by tinkering with various ways of calculating the coordinates used on inertial reference frames in relative motion. Thus, it is natural to describe time dilation as the slowing down of clocks on the moving reference frame. It was included in the final version of Lorentz’s explanation, now called the “Lorentz transformation equations.” (Lorentz 1904) Those equations contained not only the length contraction and transformation for “local time”, but also the implication that clocks on moving frames are slowed down at the same rate as lengths are contracted (that is, ). The final Lorentz equation for time transformation included both the variation in local time and time dilation: .

Though Lorentz took the distortions that he discovered in fast-moving material objects to be laws of nature, he did not think that they were basic. He thought they were effects of motion on the interactions between electrons and the ether which could be explained by his electronic theory of matter, and he saw explaining this effect as the the main challenge to Newtonian physics. The transformation equations themselves never seemed puzzling to Lorentz, because he never took them to more than just a mathematical aid to calculation.


Real-World Relativity: The GPS Navigation System

二月 12, 2010

People often ask me “What good is Relativity?” It is a commonplace to think of Relativity as an abstract and highly arcane mathematical theory that has no consequences for everyday life. This is in fact far from the truth. Consider for a moment that when you are riding in a commercial airliner, the pilot and crew are navigating to your destination with the aid of the Global Positioning System (GPS). Further, many luxury cars now come with built-in navigation systems that include GPS receivers with digital maps, and you can purchase hand-held GPS navigation units that will give you your position on the Earth (latitude, longitude, and altitude) to an accuracy of 5 to 10 meters that weigh only a few ounces and cost around $100. GPS was developed by the United States Department of Defense to provide a satellite-based navigation system for the U.S. military. It was later put under joint DoD and Department of Transportation control to provide for both military and civilian navigation uses. The current GPS configuration consists of a network of 24 satellites in high orbits around the Earth. Each satellite in the GPS constellation orbits at an altitude of about 20,000 km from the ground, and has an orbital speed of about 14,000 km/hour (the orbital period is roughly 12 hours – contrary to popular belief, GPS satellites are not in geosynchronous or geostationary orbits). The satellite orbits are distributed so that at least 4 satellites are always visible from any point on the Earth at any given instant (with up to 12 visible at one time). Each satellite carries with it an atomic clock that “ticks” with an accuracy of 1 nanosecond (1 billionth of a second). A GPS receiver in an airplane determines its current position and heading by comparing the time signals it receives from a number of the GPS satellites (usually 6 to 12) and triangulating on the known positions of each satellite. The precision is phenomenal: even a simple hand-held GPS receiver can determine your absolute position on the surface of the Earth to within 5 to 10 meters in only a few seconds (with differential techiques that compare two nearby receivers, precisions of order centimeters or millimeters in relative position are often obtained in under an hour or so). A GPS receiver in a car can give accurate readings of position, speed, and heading in real-time! To achieve this level of precision, the clock ticks from the GPS satellites must be known to an accuracy of 20-30 nanoseconds. However, because the satellites are constantly moving relative to observers on the Earth, effects predicted by the Special and General theories of Relativity must be taken into account to achieve the desired 20-30 nanosecond accuracy. Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture). Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion. Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth’s mass is less than it is at the Earth’s surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day. The combination of these two relativitic effects means that the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38 microseconds per day (45-7=38)! This sounds small, but the high-precision required of the GPS system requires nanosecond accuracy, and 38 microseconds is 38,000 nanoseconds. If these effects were not properly taken into account, a navigational fix based on the GPS constellation would be false after only 2 minutes, and errors in global positions would continue to accumulate at a rate of about 10 kilometers each day! The whole system would be utterly worthless for navigation in a very short time. This kind of accumulated error is akin to measuring my location while standing on my front porch in Columbus, Ohio one day, and then making the same measurement a week later and having my GPS receiver tell me that my porch and I are currently about 5000 meters in the air somewhere over Detroit. The engineers who designed the GPS system included these relativistic effects when they designed and deployed the system. For example, to counteract the General Relativistic effect once on orbit, they slowed down the ticking frequency of the atomic clocks before they were launched so that once they were in their proper orbit stations their clocks would appear to tick at the correct rate as compared to the reference atomic clocks at the GPS ground stations. Further, each GPS receiver has built into it a microcomputer that (among other things) performs the necessary relativistic calculations when determining the user’s location. Relativity is not just some abstract mathematical theory: understanding it is absolutely essential for our global navigation system to work properly!


Rethinking relativity: Is time out of joint?

十月 27, 2009

Bean found her evidence lurking in existing data collected by the Cosmic Evolution Survey, a multi-telescope imaging project that includes the longest survey yet by the Hubble Space Telescope. COSMOS, which detected more than 2 million galaxies over a small patch of sky, takes advantage of gravity’s ability to bend light. Massive objects like galaxy clusters bend the light of more distant objects so that it is directed towards or away from Earth. This effect, called gravitational lensing, is at its most dramatic when it creates kaleidoscopic effects like luminous rings or the appearance of multiple copies of a galaxy.


What the Global Positioning System Tells Us about Relativity

三月 5, 2009

Tom Van Flandern, Univ. of Maryland & Meta Research From the book ‘Open Questions in Relativistic Physics’ (pp. 81-90), edited by Franco Selleri, published by Apeiron, Montreal (1998) 1. What is the GPS? The Global Positioning System (GPS) consists of a network of 24 satellites in roughly 12-hour orbits, each carrying atomic clocks on board. The orbital radius of the satellites is about four Earth-radii (26,600 km). The orbits are nearly circular, with a typical eccentricity of less than 1%. Orbital inclination to the Earth’s equator is typically 55 degrees. The satellites have orbital speeds of about 3.9 km/s in a frame centered on the Earth and not rotating with respect to the distant stars. Nominally, the satellites occupy one of six equally spaced orbital planes. Four of them occupy each plane, spread at roughly 90-degree intervals around the Earth in that plane. The precise orbital periods of the satellites are close to 11 hours and 58 minutes so that the ground tracks of the satellites repeat day after day, because the Earth makes one rotation with respect to the stars about every 23 hours and 56 minutes. (Four extra minutes are required for a point on the Earth to return to a position directly under the Sun because the Sun advances about one degree per day with respect to the stars.) The on-board atomic clocks are good to about 1 nanosecond (ns) in epoch, and about 1 ns/day in rate. Since the speed of light is about one foot per nanosecond, the system is capable of amazing accuracy in locating anything on Earth or in the near-Earth environment. For example, if the satellite clocks are fully synchronized with ground atomic clocks, and we know the time when a signal is sent from a satellite, then the time delay for that signal to reach a ground receiver immediately reveals the distance (to a potential accuracy of about one foot) between satellite and ground receiver. By using four satellites to triangulate and determine clock corrections, the position of a receiver at an unknown location can be determined with comparable precision. 2. What relativistic effects on GPS atomic clocks might be seen? General Relativity (GR) predicts that clocks in a stronger gravitational field will tick at a slower rate. Special Relativity (SR) predicts that moving clocks will appear to tick slower than non-moving ones. Remarkably, these two effects cancel each other for clocks located at sea level anywhere on Earth. So if a hypothetical clock at Earth’s north or south pole is used as a reference, a clock at Earth’s equator would tick slower because of its relative speed due to Earth’s spin, but faster because of its greater distance from Earth’s center of mass due to the flattening of the Earth. Because Earth’s spin rate determines its shape, these two effects are not independent, and it is therefore not entirely coincidental that the effects exactly cancel. The cancellation is not general, however. Clocks at any altitude above sea level do tick faster than clocks at sea level; and clocks on rocket sleds do tick slower than stationary clocks. For GPS satellites, GR predicts that the atomic clocks at GPS orbital altitudes will tick faster by about 45,900 ns/day because they are in a weaker gravitational field than atomic clocks on Earth’s surface. Special Relativity (SR) predicts that atomic clocks moving at GPS orbital speeds will tick slower by about 7,200 ns/day than stationary ground clocks. Rather than have clocks with such large rate differences, the satellite clocks are reset in rate before launch to compensate for these predicted effects. In practice, simply changing the international definition of the number of atomic transitions that constitute a one-second interval accomplishes this goal. Therefore, we observe the clocks running at their offset rates before launch. Then we observe the clocks running after launch and compare their rates with the predictions of relativity, both GR and SR combined. If the predictions are right, we should see the clocks run again at nearly the same rates as ground clocks, despite using an offset definition for the length of one second. We note that this post-launch rate comparison is independent of frame or observer considerations. Since the ground tracks repeat day after day, the distance from satellite to ground remains essentially unchanged. Yet, any rate difference between satellite and ground clocks continues to build a larger and larger time reading difference as the days go by. Therefore, no confusion can arise due to the satellite clock being located some distance away from the ground clock when we compare their time readings. One only needs to wait long enough and the time difference due to a rate discrepancy will eventually exceed any imaginable error source or ambiguity in such comparisons. 3. Does the GPS confirm the clock rate changes predicted by GR and SR? The highest precision GPS receiver data is collected continuously in two frequencies at 1.5-second intervals from all GPS satellites at five Air Force monitor stations distributed around the Earth. An in-depth discussion of the data and its analysis is beyond the scope of this paper. [1] This data shows that the on-board atomic clock rates do indeed agree with ground clock rates to the predicted extent, which varies slightly from nominal because the orbit actually achieved is not always precisely as planned. The accuracy of this comparison is limited mainly because atomic clocks change frequencies by small, semi-random amounts (of order 1 ns/day) at unpredictable times for reasons that are not fully understood. As a consequence, the long-term accuracy of these clocks is poorer than their short-term accuracy. Therefore, we can assert with confidence that the predictions of relativity are confirmed to high accuracy over time periods of many days. In ground solutions with the data, new corrections for epoch offset and rate for each clock are determined anew typically once each day. These corrections differ by a few ns and a few ns/day, respectively, from similar corrections for other days in the same week. At much later times, unpredictable errors in the clocks build up with time squared, so comparisons with predictions become increasingly uncertain unless these empirical corrections are used. But within each day, the clock corrections remain stable to within about 1 ns in epoch and 1 ns/day in rate. The initial clock rate errors just after launch would give the best indication of the absolute accuracy of the predictions of relativity because they would be least affected by accumulated random errors in clock rates over time. Unfortunately, these have not yet been studied. But if the errors were significantly greater than the rate variance among the 24 GPS satellites, which is less than 200 ns/day under normal circumstances, it would have been noticed even without a study. So we can state that the clock rate effect predicted by GR is confirmed to within no worse than ±200 / 45,900 or about 0.7%, and that predicted by SR is confirmed to within ±200 / 7,200 or about 3%. This is a very conservative estimate. In an actual study, most of that maximum 200 ns/day variance would almost certainly be accounted for by differences between planned and achieved orbits, and the predictions of relativity would be confirmed with much better precision. 12-hour variations (the orbital period) in clock rates due to small changes in the orbital altitude and speed of the satellites, caused by the small eccentricity of their orbits, are also detected. These are observed to be of the expected size for each GPS satellite’s own orbit. For example, for an orbital eccentricity of 0.01, the amplitude of this 12-hour term is 23 ns. Contributions from both altitude and speed changes, while not separable, are clearly both present because the observed amplitude equals the sum of the two predicted amplitudes.


Special Relativity – Experimental Verification

三月 3, 2009

Special Relativity – Experimental Verification

Like any scientific theory, the theory of relativity must be confirmed by experiment. So far, relativity has passed all its experimental tests. The special theory predicts unusual behavior for objects traveling near the speed of light. So far no human has traveled near the speed of light. Physicists do, however, regularly accelerate subatomic particles with large particle accelerators like the recently canceled Superconducting Super Collider (SSC). Physicists also observe cosmic rays which are particles traveling near the speed of light coming from space. When these physicists try to predict the behavior of rapidly moving particles using classical Newtonian physics, the predictions are wrong. When they use the corrections for Lorentz contraction, time dilation, and mass increase required by special relativity, it works. For example, muons are very short lived subatomic particles with an average lifetime of about two millionths of a second. However when they are traveling near the speed of light physicists observe much longer apparent lifetimes for muons. Time dilation is occurring for the muons. As seen by the observer in the lab time moves more slowly for the muons traveling near the speed of light.

Time dilation and other relativistic effects are normally too small to measure at ordinary velocities. But what if we had sufficiently accurate clocks? In 1971 two physicists, J. C. Hafele and R. E. Keating used atomic clocks accurate to about one billionth of a second (one nanosecond) to measure the small time dilation that occurs while flying in a jet plane. They flew atomic clocks in a jet for 45 hours then compared the clock readings to a clock at rest in the laboratory. To within the accuracy of the clocks they used time dilation occurred for the clocks in the jet as predicted by relativity. Relativistic effects occur at ordinary velocities, but they are too small to measure without very precise instruments.

The formula E=mc2 predicts that matter can be converted directly to energy. Nuclear reactions that occur in the Sun, in nuclear reactors, and in nuclear weapons confirm this prediction experimentally.

Albert Einstein’s special theory of relativity fundamentally changed the way scientists characterize time and space. So far it has passed all experimental tests. It does not however mean that Newton’s law of physics is wrong. Newton’s laws are an approximation of relativity. In the approximation of small velocities, special relativity reduces to Newton’s laws.

Resources

Books

Cutnell, John D., and Kenneth W. Johnson. Physics. 3rd ed. New York: Wiley, 1995.

Einstein, Albert. Relativity. New York: Crown, 1961.

Mould, R.A. Basic Relativity. Springer Verlag, 2001.

Hawking, Stephen. Black Holes and Baby Universes and Other Essays. New York: Bantam, 1993.

Schrödinger, Edwin. Space-Time Structure, Reprint Edition. Cambridge University Press, 2002.

Paul A. Heckert
K. Lee Lerner

KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

General relativity
—The part of Einstein’s theory of relativity that deals with accelerating (noninertial) reference frames.

Lorentz contraction
—An effect that occurs in special relativity; to an outside observer the length appears shorter for an object traveling near the speed of light.

Reference frames
—A system, consisting of both a set of coordinate axes and a clock, for locating an object’s (or event’s) position in both space and time.

Space-time
—Space and time combined as one unified concept.

Special relativity
—The part of Einstein’s theory of relativity that deals only with nonaccelerating (inertial) reference frames.

Time dilation
—An effect that occurs in special relativity; to an outside observer time appears to slow down for an object traveling near the speed of light.