9.2.E - Relativity


In 1905, Albert Einstein published (among other things) a paper called "On the Electrodynamics of Moving Bodies" in the journal Annalen der Physik. The paper presented the theory of special relativity, based on two postulates. The first said that the laws of physics apply (are the same) in all inertial frames of reference (those that are moving with constant velocity or stationary). The second postulate stated that the velocity of light in a vacuum was always measured to be 3 x 108 ms-1 (a constant) and is not relative to the motion of the observer or emitter of the light.

While the first postulate was did not really tell scientists anything they didn't already know, the second one shook the scientific community to its foundations because of the consequences for our understanding of time and space.


In this unit you will learn about

  • The Michelson-Morley experiment
  • The difference between inertial and non-intertial frames of reference
  • The fundamnetal postulates of the theory of special relativity and the consequences for space and time


Michelson-Morley Experiment

By the late 19th century physicists had pretty much accepted that light was a wave. Waves, like those of water and ligh,t needed a medium to travel through so it was assumed that electromangetic waves, such as light, needed to the same. This is how the idea of the ether came into being. It was a medium that filled all of space and allowed electromagnetic waves to travel trhough space. We could see light from stars so there must be a medium such as the ether for the light to travel through.

Based on the wave theory of the time, it was postulated that the aether posses the following properties:

  • It must be transparent for us to see through.
  • It must be able to flow easily or it could not creep into every tiny gap where light can travel.
  • It must offer no resistance to the motion of very large masses, or the planets would soon lose their energy and spiral into the sun.
  • At the same time, the aether must be very stiff, for the speed of light is large. Remember that if you make a spring stiffer or increase its tension, waves will travel more quickly along it.

In the spirit of experimental science, all hypotheses need experimental evidence to work out if they are true so Michelson and Morley, full of enthusiasm after measuring the speed of light accurately in 1879, set about to show that the ether existed by measuring the speed of light relative to it. They did this in much the same way that a swimmer's velocity can be measured relative to the stationary bank of a river or relative to the current of the river.

The Experiment


The dotted lines show the ether wind into which the Earth
and also light) was travelling at various angles depending
on the rotation angle of the Michelson Interferometer.

The approach of Michelson and Morley was to measure the relative speed at which the Earth passes through the aether. They reasoned that, if the ether is real, the Earth would at all times be moving through it like a plane through the air, producing a detectable ether wind. Depending on the angle of the light beam to th ether wind, the velocity of light relative to the ether should change.

Each year, the Earth travels a tremendous distance in its orbit around the sun, at a speed of around
30 km s-1 or more than 100,000 km per hour. Doing the same experiment at different times in the Earth's orbit around the Sun would allow them to also assess their experimental results for reliability.

The swimmer's time for each leg will depend on the direction
of their velocity relative to the river current.

The effect of the aether wind on light waves projected into it would be like the effect of a swimmer swimming into, with or against a river current as shown. The swimmer swims three different legs each of the same distance. The first leg is with the current from A to B. The seond leg is against the current from B to A and the final leg is across the current from A to D. Because of the current, to get from A to D the swimmer would have to sim initially towards C but would end up at D.

If we were to time how long it took the swimmer to complete each leg, remembering that the swimmer is moving at a constant speed and the distances are the same, we would find that A to B takes the shortest time, followed by A to D and then B to A taking the longest.

Michelson and Morley essentially did the same thing with light in place of the swimmer and an assumption that the ether was representing the river current. They expected that light would take slightly different times to travel at different angles into the ether, just like the swimmer, and that was what they were trying to measure. 'Slightly' is the key, in that, over a distance of the order of a few meters, the difference in time for the two round trips would be only on the order of a millionth of a millionth of a second. Michelson, though, already having spent a great deal of time and thought on how to measure the speed of light, had developed several techniques for measuring differences of this magnitude.

The Michelson Interfermometer

A simplified diagram of the Morley Interferometer used
by Michelson and Morley.

The actual Michelson-Morley experiment used more mirrors than is shown, the light being reflected back and forth several times before recombination. It was performed in the basement of a stone building close to sea level. There they set up an extended version of what has come to be known as a Michelson Interferometer. A half-silvered mirror was used to split a beam of monochromatic light into two beams travelling at right angles to one other. After leaving the splitter, the beams were each reflected back and forth between mirrors several times (to give a long path length) then recombined, producing a pattern of constructive and destructive interference. Any slight change in the amount of time the beams spent in transit would then be observed as a change in the pattern of interference.

The device was placed on a rotating bed, so that it could be rotated through the entire range of possible angles to the ether wind.

The Result

A typical interference pattern for light
showing bright and dark bands. If different
beams of light took different times to travel
the same distance, the intensity and
distances between bands on a pattern such
as this would change.

Ironically, after all this thought and preparation, the experiment became what might be called the most famous failed experiment in history. Instead of providing insight to the properties of the ether, it produced none of the effects to be expected if the Earth's motion produced an aether wind. The apparatus behaved as if there were no wind at all; as if the Earth had no motion with reference to a medium.

Something was wrong with the sensible idea that the speed of light measured by a moving receiver should be its usual speed in a vacuum, c, plus or minus the contribution from the motion of the source or receiver. So, not only did the results of this experiment suggest that there was no aether wind, they also suggested that there was something puzzling about the concept of the velocity of light relative to it's frame of reference. It seemed that regardless of the motion of the source of the light, it still travelled at the same speed. This is the same as saying if you threw a ball from a moving car in the direction of the car's motion, the ball would arrive at a point 10 m in front of a car in the same time as if it was thrown from a stationary position.

In the theory of modern geocentrism, this is taken as proof that the Earth is stationary in an ether field. This result was rather astounding and not explainable by the then current theory of wave propagation. Several explanations were attempted, among them, that the experiment had a hidden flaw (apparently Michelson's initial belief), or that the Earth's gravitational field somehow dragged the aether around with it so that there would be no ether wind.

Ernst Mach was among the first physicists to suggest that the experiment actually amounted to a disproof of the aether theory. Developments in theoretical physics had already begun to provide an alternate theory, the Fitzgerald-Lorentz contraction, which explained the null result of the experiment. The development of what became Einstein's special theory of relativity had the Fitzgerald-Lorentz contraction derived from the invariance postulate, and was also consistent with the results of the experiment.

 

Frames of Reference


A three-dimensional
coordinate system consisting
of the x, y and z dimensions.
The object is at P and its
position is measured using
x, y and z coordinates.

A coordinate system used to describe the position, velocity and acceleration of an object is called a frame of reference. If you are in a room you will use the walls, floor, and ceiling of that room as a frame of reference to judge the motion of objects in the room. These walls and the floor and the ceiling form planes that lie parallel to an x, y, z coordinate system which one might imagine in the room. It is especially easy to see this x, y, z coordinate system located with its origin at one corner where the floor and two walls meet. For example, if a ball rolled across the room, you would know that it was moving because you would see its position change relative to the floor or walls. One would, very most likely, believe that the walls of the room were standing still, and since the ball was changing position against this backdrop of still walls, one would think that the ball was in motion. In that way we would use the frame of reference of the room to determine the motion of the ball.

In a like manner, if one was in an car, and a ball was rolling around on the floor, you would use the frame of reference of the interior of the car to judge the motion of the ball. The interior of the car is, after all, quite a bit like a small room. Although its walls, ceiling, and floor are not as flat or rectilinear as that of a normal room, one could still imagine an x, y, z coordinate axes attached to it somewhere, most likely the floor, and one could use this x, y, z coordinate axes to locate the ball.

Now, let us think of the car simply as a room that might be standing still or that may be in motion. When it is in motion, it might be traveling at a steady speed down a straight path; it might be speeding up or slowing down; it might be going around a corner. Therefore, a frame of reference can be in motion and that motion can be of several different types. Anyone who has ever taken a ride in a car has been in a moving frame of reference and has experienced the several different types of motion through which this frame can move.

Actually, frames of reference are classified into two types depending upon how they are moving. Those two types are called inertial and non-inertial frames of reference.

Inertial Frames of Reference

An inertial frame of reference has a constant velocity. That is, it is moving at a constant speed in a straight line, or it is standing still. Understand that when something is standing still, it has a constant velocity. Its velocity is constantly zero meters per second. To say that the velocity of a frame of reference is constant is the same as saying that the frame is not accelerating. So, we could define an inertial frame of reference to be a coordinate system which is not accelerating. Such a constant velocity frame of reference is called an inertial frame because the law of inertia applies in it. That is, an object whose position is judged from this frame will tend to resist changes in its velocity; it obeys the law of inertia. An object within this frame will not spontaneously change its velocity. An object within this frame will only change its velocity if an actual non-zero net force is applied to it.

There are several ways to describe an inertial frame. An inertial frame of reference is

  • one with a constant velocity
  • non-accelerating
  • one in which the law of inertia holds
  • one in which Newton's laws of motion consistent apply
  • one where there are no fictitious forces

Consider the act of juggling, a definite demonstration of Newton's laws of motion. It is just as easy (or just as difficult) to juggle balls in a room that is standing still as it is to juggle in a bus that is travelling smoothly down a straight road at constant speed. In fact, the juggler on the bus could not determine that the bus was moving based on any clues gathered from the motion of the balls. They would move through the air within the moving bus exactly as if they were being tossed about within the still room - as long as the bus travelled smoothly down a straight road at constant speed; that is, as long as the bus moved with a constant velocity. The physics of typical mechanics is always the same when it is done within a constant velocity frame of reference. Without visual aids, such as viewing the scenery going by, and without sound clues, such as the noise of the engine and drive train, the juggler physicist on the constant velocity bus could perform no experiment to determine if the bus was moving or was parked.

Such frames of reference as our constant velocity bus are called inertial frames of reference. The bus would cease to be an inertial frame of reference while it changed its velocity. That would happen if it slowed down, or if it sped up, or if it turned around a corner. Each of these changes in velocity would constitute an acceleration. And, while the bus was accelerating, the act of juggling could get quite difficult. For example, if the bus driver slammed on the brakes while some of the balls were in flight, those balls would seem to fly forward from the juggler's perspective. From the viewpoint of the juggler it would seem as if some unknown force had pushed the balls away from her, making them fly up toward the front of the bus. The juggler, too, would feel a push toward the front of the bus. But remember, in this situation the bus is no longer an inertial frame of reference. Its velocity is changing; it is now an accelerating frame of reference. So, the law of inertia and Newton's laws of motion no longer hold. This accelerating frame is called a non-inertial frame of reference.

Non-Inertial Frames of Reference

A non-inertial frame of reference does not have a constant velocity. It is accelerating. There are several ways to imagine this motion. The frame could be:

  • Traveling in a straight line, but be speeding up or slowing down
  • Traveling along a curved path at a steady speed
  • Traveling along a curved path and also speeding up or slowing down.

Such an accelerating frame of reference is called a non-inertial frame because the law of inertia does not hold in it. That is, an object whose position is judged from this frame will seem to spontaneously change its velocity with no apparent non-zero net force acting upon it. This completely violates the law of inertia and Newton's laws of motion, since these laws claim that the only way an object can change its velocity is if an actual non-zero net force is applied to the object. Objects just do not start to move about here and there all on their own.

If you are in an automobile when the brakes are abruptly applied you will feel pushed toward the front of the car. You may actually have to extend you arms to prevent yourself from going forward toward the dashboard. However, there is really no force pushing you forward. The car, since it is slowing down is an accelerating, or non-inertial frame of reference and the law of inertia no longer holds if we use this non-inertial frame to judge your motion. If all of this is viewed relative to the ground, it becomes clear that no force is pushing you forward when the brakes are applied. The ground is stationary and, therefore, is an inertial frame. Relative to the ground, when the brakes are applied, you continue with your forward motion, just like you should according to Newton's first law of motion. The situation is this. The car is stopping, you are not; so, you head out toward the dashboard. From your point of view in the car it seems like you have spontaneously been pushed forward. Actually, there is no force acting on you. The imagined force toward the front of the car is a fictitious force.

There are several ways to describe a non-inertial frame. A non-inertial frame of reference is a frame of reference with a changing velocity. The velocity of a frame will change if the frame speeds up, or slows down, or travels in a curved path. A non-inertial frame of reference is

  • An accelerating frame of refer6nce
  • One in which the law of inertia does not apply
  • A frame of reference in which Newton's laws of motion do not apply
  • One in which fictitious forces arise.


TED-Ed - How fast are you moving right now?
An introduction to frames of reference and relative and absolute motion.

 

Special Relativity

Einstein saw no need for the ether and gone with the stationary ether was the notion of an absolute frame of reference against which all motion in the universe could be measured. All motion is relative, not to any stationary hitching post in the universe, but to arbitrary or defined frames of reference.

A rocket ship cannot measure its speed with respect to empty space, but only with respect to other objects in spcae. If, for example, rocket ship A drifts past rocket ship B in empty space, spaceman A and spacewoman B will each observe the relative motion and from this observation, each will be unable to determine who is moving and who is at rest, if either. This is a familiar experience to a passenger on a train who looks out his window and sees the train on the next track moving by his window. He is aware only of the relative motion between his train and the other train and cannot tell which train is moving. He may be at rest relative to the ground and the other train may be moving, or he may be moving relative to the ground and the other train may be at rest, or they both may be moving relative to the ground. The important point here is that if you were in a train with no windows, there would be no way to determine whether the train was moving with uniform velocity or was at rest.

This is a consequence of the first of Einstein's postulates of the special theory of relativity

Postulate 1
The laws of physics are the same in all intertial frames of reference.

On a jet airplane going 700 kilometers per hour, for example, coffee pours as it does when the plane is at rest. If we swing a pendulum, it swings as it would if the plane were at rest on the runway. There is no physical experiment we can perform, even with light, to determine our state of uniform motion. The laws of physics within the uniformly moving cabin are the same as those in a stationary laboratory. This postulate is sometimes called the Principle of Relativity and said another way means that one cannot detect motion in an inertial frame of reference without reference to another frame of reference. Any number of experiments can be devised to detect accelerated motion, but none can be devised, according to Einstein, to detect a state of uniform motion. Therefore, absolute motion has no meaning.

It would be very peculiar if the laws of mechanics varied for observers moving at different speeds. It would mean, for example, that a snooker player on a smooth-moving ocean liner would have to adjust her style of play to the speed of the ship, or even to the seasons as the Earth varies in its orbital speed about the sun. It is our common experience that no such adjustment is necessary and, according to Einstein, this same insensitivity to motion extends to electromagnetism. No experiment, mechanical or electrical or optical has ever revealed absolute motion. That is what the first postulate of relativity means.

One of the questions that Einstein is supposed to have asked of his schoolteacher was "What would a light beam look like if you traveled along beside it?" According to classical physics, the beam would be at rest to such an observer. The more Einstein thought about this, the more convinced he became that one could not move with a light beam. He finally came to the conclusion that no matter how close a person comes to the speed of light, he would still measure the light approaching or leaving him at 300,000,000 ms-1. This was the second postulate of his special theory of relativity.

Postulate 2
The speed of light in free space has the same measured value regardless of the motion of the source or the motion of the observer; that is, the speed of light is invariant.

 

The speed of light is invariant and
not relative!

To illustrate this statement, consider a rocket ship departing from the space station shown. A flash of light traveling at 300,000,000 ms-1, or c, is emitted from the station. Regardless of the velocity of the rocket, an observer in the rocket sees the flash of light pass her at the same speed c. If a flash is sent to the station from the moving rocket, observers on the station will measure the speed of the flash to be c. The speed of light is measured to be the same regardless of the speed of the source or receiver. All observers who measure the speed of light will find it has the same value c.

The more you think about this, the more you think it doesn't make sense. We will see though that the explanation has to do with the relationship between space and time and how these quantities can vary.

If apply Newtonian mechanics to particular problems the result is quite different to that of applying special relativity as shown below.

Another example of the constancy of the speed of electromagnetic radiation
received by two spacecraft with different velocities.

The second postulate says that a person travelling toward or away from a light source will measure the same speed of light as that for a person at rest. Consider the two spacecraft receiving microwave communication from a satellite. One is at rest relative to the satellite while the other is moving at a velocity, v, which is half the speed of light, c/2. Common sense tells us that the spaceship that is stationary relative to the satellite will receive the message travelling at c. We might also expect that the moving spaceship would receive the signal at c + c/2 = 1.5c. This is not what the second postulate predicts. According to the theory of special relativity, the signal will arrive at both spaceships with a velocity of c.

Standard Metre

The constancy of the speed of light has allowed us to define the standard metre in a more useful way. Until the invention of atomic clocks, the standard metre was a metal rod kept in Paris. Australia had a copy in Sydney. With the modern definition of a metre, you can more readily set up a standard metre in all parts of the world. The speed of light in a vacuum is now defined to be exactly 299 792 458 ms-1, which is the best measured value to date. The metre is then defined as the distance light travels in a vacuum during a time of exactly 1/299 792 458 s. This new standard will never need to be revised. Improvements in measuring time will simply improve the measurement of the metre.

 

The Relativity of Simultaneity


A thought experiment demonstrating the relativity of simultaneity.

An interesting consequence of Einstein's second postulate occurs with the concept of simultaneity. We say that two events are simultaneous if they occur at the same time. In the style of one of Einstein's famous thought experiments, consider a light source in the exact center of the compartment of a rocket ship (or a train if you like). When the light source is switched on, light spreads out in all directions at speed c. Because the light source is equidistant from the front and back ends of the compartment, an observer inside the compartment finds that light reaches the front end at the same instant it reaches the back end. This occurs whether the ship is at rest or moving at constant velocity. The events of hitting the back end and hitting the front end occur simultaneously. But time in one frame of reference may be different from time in another frame.

To an outside observer who views the same two events in another frame of reference, say from a planet not moving with the ship, these same two events are not simultaneous. As light travels out from the source, this observer sees the ship move forward, so the back of the compartment moves toward the beam while the front moves away from it. The beam going to the back of the compartment therefore has a shorter distance to travel than the beam going forward. Since the speed of light is the same in both directions, this observer sees the event of light hitting the back of the compartment before seeing the event of light hitting the front of the compartment.

A little thought will show that an observer in another rocket ship that passes the ship in the opposite direction would report that the light reaches the front of the compartment first.

Two events that are simultaneous in one frame of reference need not be simultaneous in another frame of reference.

Both observers must see light travelling at the same speed, 3,000,000 ms-1, and this is what causes the non-simultaneity for the observer outside the space ship. This non-simultaneity of events in one frame that are simultaneous is another is a purely relativistic result and a consequence of light always having the same speed for all observers.

 

Time Dilation

A simple light clock for measuring equal
intervals of time.

Time is not an absolute but is relative, depending on the motion of an observer relative to what is being observed. Imagine, for example, that we are somehow able to observe a flash of light bouncing back and forth between a pair of parallel mirrors. If the distance between the mirrors is fixed, then the arrangement constitutes a sort of 'light clock' because the back and forth trips of the light flash take equal intervals of time.

Suppose our light clock is inside a transparent high-speed spaceship as shown in the diagram below (a). If we were on the ship and watching the light clock we would see the flash of light reflecting straight up and down between the two mirrors just as it would if the spaceship were at rest. Our observations will show no relativistic effects because there is no relative motion between us and our light clock because we and the light clock are in the same reference frame in spacetime. Relativistic effects such as time dilation can only be observed from a different frame of reference.

If, instead, we stand at some relative rest position and observe the spaceship whizzing by us at an appreciable speed - say, half the speed of light - things are quite different. We no longer share the same reference frame, for in this case there is relative motion between the observer and the observed. We will not see the path of the light in simple up-and-down motion as before. Because the light flash keeps up with the horizontally moving light clock, we will see the flash following a diagonal path (diagram below).

Notice that the flash travels a longer distance as it moves between the mirrors in our position of space-time than it does in the reference frame of an observer riding with the ship. Since the speed of light is the same in all reference frames (Einstein's second postulate), the flash must travel for a longer time between the mirrors in our frame than in the reference frame of an observer on board. This follows from the definition of speed, simply stated, as a ratio of distance to time. The longer diagonal distance must be divided by a correspondingly longer time interval to yield an unvarying value for the speed of light. Thus, from our relative position of rest, we measure a longer time interval between ticks when a clock is in motion than when it is at rest. We have considered a light clock in our example, but the same is true for any kind of clock. Moving clocks appear to run slow. This stretching out of time is called time dilation, which has nothing to do with the mechanics of clocks, but instead arises from the nature of time itself. Time dilation applies to all properly functioning timepieces.

This limmerick illustrates the nature of time dilation quite nicely.

There was a young lady named Bright
Who travelled much faster than light.
She departed one day
In a relative way
And returned on the previous night.

The equation for time dilation for different frames of reference in space-time is shown below.

Equation - Time Dilation

tv

t0

v

c

the observed (dilated) time from one reference frame

the rest (normal) time from the other reference frame

the velocity of the object

the velocity of light (3 x 108 ms-1)

s

s

ms-1

ms-1

Use this equation to calculate the dilated time for a reference frame which is moving relative to another reference frame.

 


Relativistic effects are only really observed
at speeds in excess of half of the speed of
light.

To consider some numerical values, assume that v is 50% the speed of light. Then we substitute 0.5c for v in the time-dilation equation and after some arithmetic find that tv = 1.15t0. This means that if we viewed a clock on a spaceship traveling at half the speed of light, we would see the second hand take 1.15 minutes to make a revolution, whereas if the spaceship were at rest, we would see it take 1 minute. If the spaceship passes us at 87% the speed of light, tv = 2.0t0. We would measure time events on the spaceship taking twice the usual intervals, for the hands of a clock on the ship would tum only half as fast as those on our own clock. Events on the ship would seem to take place in slow motion. At 99.5 the speed of light, tv = 10t0. We would see the second hand of the spaceship's clock take 10 minutes to sweep through a revolution requiring 1 minute on our clock.

To put these figures another way, at 0.995 c, the moving clock would appear to run at a tenth of our rate; it would tick only 6 seconds while our clock ticks 60 seconds. At 0.87c, the moving clock ticks at half rate and shows 30 seconds to our 60 seconds; at 0.50c, the moving clock ticks 1/1.15 as fast and ticks 52 seconds to our 60 seconds. We see that moving clocks run slow.

Nothing is unusual about a moving clock itself; it is simply ticking to the rhythm of a different time. The faster a clock moves, the slower it appears to run as viewed by an observer not moving with the clock. If it were possible to make a clock fly by us at the speed of light, the clock would not appear to be running at all. We would measure the interval between ticks to be infinite. The clock would be ageless! If we could move with such an imaginary clock, however, the clock would not show any slowing down of time. To us the clock would be operating normally. This is because there would be no motion of the clock relative to us. We and the clock would share the same frame in spacetime.

If a person whizzing past us checked a clock in our reference frame, he would find our clock to be running as slowly as we find his to be. We each see each other's clock running slow. There is really no contradiction here, for it is physically impossible for two observers in relative motion to refer to one and the same realm of spacetime. The measurements made in one realm of spacetime need not agree with the measurements made in another realm of spacetime. The measurement that all observers always agree on, however, is the speed of light.

Time dilation has been confirmed in the laboratory innumerable times with particle accelerators. The lifetimes of fast-moving radioactive particles increase as their speed goes up, and the amount of increase is just what Einstein's equation predicts. Time dilation has been confirmed also for not-so-fast motion. In 1971 four cesium-beam atomic clocks were twice flown on regularly scheduled commercial jet flights around the world, once eastward and once westward, to test Einstein's theory of relativity with macroscopic clocks. The clocks indicated different times after their round trips. Relative to the atomic time scale of the U.S. Naval Observatory, the observed time differences, in billionths of a second, were in accord with relativistic prediction. Now, with atomic clocks orbiting the earth as part of the global positioning system, adjustments for the effects of time dilation are essential in order to use signals from the clocks to pinpoint locations on earth.

This all seems very strange to us only because it is not our common experience to deal with measurements made at relativistic speeds or atomic-clock-type measurements at ordinary speeds. The theory of relativity does not make common sense. But common sense, according to Einstein, is that layer of prejudices laid down in the mind prior to the age of 18. If we spent our youth zapping through the universe in high-speed spaceships, we would probably be quite comfortable with the results of relativity.

Twin Paradox

A twin travelling at a relativistic speeds will appear younger to his
twin from another reference frame if they return to the smae point
in space.

Einstein himself suggested one of the strangest effects of relativity. Suppose that two twins were chosen for an an experiment. one was to fly into space to another planet and the other was to stay on Earth. Because of time dilation, it would appear that the twin on board the spaceship would be much younger than the earthbound twin on return.

This problem is often considered a paradox because the principle of relativity demands that no inertial frame of reference be preferred over others. In other words, relativity's effects should be reversible just by looking at them from a different viewpoint. To the earthbound twin, it would appear that the clocks on the spaceship were running slow. To the twin in the spaceship, they would see earthbound clocks running slow. The apparent paradox is this: if both points of view are valid then each twin will see the other as older than themselves. How can this be possible?

The answer is that this particular problem is not reversible because while the earthbound twin is always in one frame of reference, the twin in the spaceship exists in two frames of reference during the trip separated by an accelerating reference frame when it turns around. The spaceship has experienced two different realms of time. Hence the frame of reference of one twin is not equivalent to the frame of reference of the other.

All things considered, when the travelling twin returned, they would be older. The twins can meet again at the same place in space but only at the expense of time.

 

Length Contraction

As objects move through space-time, space as well as time undergoes changes in measurement. The lengths of objects appear to be contracted when they move by us at relativistic speeds. This length contraction is really a space contraction, quite different from the contraction of matter suggested by Fitzgerald and worked out by Lorentz. Nevertheless, because Einstein's formula is the same as Lorentz's, we call the effect the Lorentz contraction. It is expressed mathematically as shown below.

Equation - Length Contraction

lv

l0

v

c

the observed (contracted) length from one reference frame

the rest (normal) length from the other reference frame

the velocity of the object

the velocity of light (3 x 108 ms-1)

m

m

ms-1

ms-1

Use this equation to calculate the contracted length for a reference frame which is moving relative to another reference frame.

 


Length contraction only occurs in the direction of motion.
The ball is only contracted horizontally and not vertically.

Suppose that an object is at rest and v = 0. Upon substitution we find lv = l0, as we would expect. At 87% the speed of light, an object would appear to be contracted to half its original length. At 99.5% the speed of light, it would seem to contract to one-tenth its original length.

If the object moved at c, its length would be zero. This is one of the reasons we say that the speed of light is the upper limit for the speed of any moving object.

Another nerdy limmerick about length contraction goes like this:

There was a young fencer named Fisk,
Whose thrust was exceedingly brisk.
So fast was his action
The Lorentz contraction
Reduced his sword to a disk.

As the diagram indicates, contraction takes place only in the direction of motion. If an object is moving horizontally, no contraction takes place vertically.


Matter does not contract at relativistic speeds. It is the contraction of
space that occurs.

Do objects really contract at relativistic speeds? Well, if we attempt to check this out (in principle) and travel alongside the moving object with a meter stick, we notice nothing at all unusual about the length of the object. An observer at rest watching this experiment would report that the reason we don't measure the contraction, which is obvious to him, is that our meter stick is shortened as well. But because we are moving with the object, it appears to us that there is no contraction. This is because our relative velocity with respect to the object being measured is zero. The object doesn't contract.

A measure of the object from another reference frame contracts. We are simply measuring the distortion of space when we measure such a contraction, just as we measure the distortion of time itself when we find clocks running slow. The distortions are of space and time between different space-times, not of objects and events within individual realms of space-time.

 

Mass Dilation

You will remember that all of the laws of physics apply in all intertial frames of reference. This indicates that momentum must be conserved in all collisions, however, time dilation would seem to indicate that momentum is not conserved at relativistic speeds (recall that p = mv and v = s/t).

Einstein believed very strongly that momentum must be conserved in all inertial frames of reference. In order to solve this dilemma he suggested that the mass of an object must dilate or increase at relativistic speeds by a factor that compensates for the time dilation on the speed measurement.

If a constant net force was applied to a 1 kg mass, its speed would increase and continue to increase so long as the net force was applied. Since no object can go faster than the speed of light, the objects speed could not increase indefinitely. It is the increase in mass of the object at relativistic speeds that causes the acceleration of the object to decrease. The faster it goes, the heavier it gets and from Newton's Second Law (F=ma), the larger the mass the smaller the acceleration. The increase in mass is caused by the conversion of some of the work (energy) into mass (E = mc2) rather than more kinetic energy.

This is what prevents the original 1 kg mass from exceeding the speed of light.

The relationship is given in the following equation.

Equation - Mass Dilation

mv

m0

v

c

the observed (dilated) mass from one reference frame

the rest (normal) mass from the other reference frame

the velocity of the object

the velocity of light (3 x 108 ms-1)

kg

kg

ms-1

ms-1

Use this equation to calculate the dilated mass for a reference frame which is moving relative to another reference frame.

 

Equivalence of Mass and Energy

Einstein not only linked space and time, he also linked mass and energy. A piece of matter, even if at rest and even if not interacting with anything else, has “energy of being.” This is called its rest energy. Einstein concluded that it takes energy to make mass and that energy is released if mass disappears. Mass is, in effect, a kind of potential energy. Mass stores energy, just as a boulder rolled to the top of a hill stores energy. If the mass of something decreases, as it can do in nuclear reactions, energy is released, just as the boulder rolling to the bottom of the hill releases energy.

The amount of rest energy, E, is related to the mass, m, by the most celebrated equation of the twentieth century

This equation gives the total energy content of a piece of stationary matter of mass, m. Recall that tiny decreases of nuclear mass in both nuclear fission and nuclear fusion produced enormous releases of energy, all in accord with the equation above. To the general public, it is synonymous with nuclear energy. If we were to weigh a fully-fueled nuclear power plant, then weigh it again a week later, we'd find it weighs slightly less. Part of the fuel's mass, about 1 part in a thousand, has been converted to energy. Now, interestingly enough, weigh a coal-burning power plant and all the coal and oxygen it consumes in a week, and then weigh it again with all the carbon dioxide and other combustion products that come out during the week, we'll also find it all weighs slightly less. Again, mass has been converted to energy. About 1 part in a billion has been converted. If both plants produce the same amount of energy, the mass change will be the same for both whether energy is released by nuclear or chemical mass conversion makes no difference. The chief difference lies in the amount of energy released in each individual reaction and the amount of mass involved. Fissioning of a single uranium nucleus releases 10 million times as much energy as the combustion of carbon to produce a single carbon dioxide molecule. Hence a few truckloads of uranium fuel will power a fission plant while a coal-buming plant consumes many hundred-car trainloads of coal.

The equation E = mc2 is not restricted to chemical and nuclear reactions. A change in energy of any object at rest is accompanied by a change in its mass. The filament of a light bulb energized with electricity has more mass than when it is tumed off. A hot cup of tea has more mass than the same cup of tea when cold. A wound-up spring clock has more mass than the same clock when unwound. But these examples involve incredibly small changes in mass—too small to be measured. Even the much larger changes of mass in radioactive change were not measured until after Einstein predicted the mass-energy equivalence. Now, however, mass-to-energy and energy-to-mass conversions are measured routinely.

In ordinary units of measurement, the speed of light, c, is a large quantity and its square is even larger. Hence, a small amount of mass stores a large amount of energy. The quantity c2 is a conversion factor. It converts the measurement of mass to the measurement of equivalent energy. One kilogram of matter has an energy equal to 90 quadrillion joules. Even a speck of matter with a mass of only 1 milligram has a rest energy of 90 billion joules.