Relative Viewpoint

Most people intuitively view the universe as a time-varying three-dimensional (TV3D) world. This is a perfectly valid view of reality as long as velocities remain small. But when velocities approach the speed of light, one person’s view of reality differs considerably from another’s.

The only way to reconcile the difference is to admit that many of the quantities previously thought to be absolute or universal in character are actually relative quantities -- quantities whose values depend upon the reference frame of the observer who measures them. Things like space, time, mass, and energy, the very foundation of the structure of our universe, are different for different observers.

But it is possible for an observer who perceives these quantities in one coordinate frame to determine how another observer perceives the same quantity in another frame. By switching from one world-view to another, it is possible to see how various things measured by one observer would appear to some other observer.

In order to distinguish between different observers, we arbitrarily call one observer the “rest” observer and the other the “moving” observer. And to distinguish between the “rest” frame of an observer and the “rest” frame of an object, we often call the observer’s reference frame the “laboratory” or “lab” frame and the object’s reference frame the “center-of-mass”, “CM”, or “proper” frame. Then, regardless of whether an object is at rest or in motion, its proper length, proper time, and proper mass will be well-defined quantities, namely, the length, time, and mass it would have if measured in a frame moving along with it.

Since you naturally picture yourself to be at rest, you should always place yourself in the rest frame and view the other frame as moving. However, you must be willing to switch back and forth between frames if you ever want to perceive what the universe looks like to the other person. Therefore, you must learn to imagine yourself to be in the other person’s shoes and to visualize what the world would look like through the other person’s eyes. Just remember, when you switch your viewpoint to the second frame, the second frame becomes the “rest frame” and the first frame becomes the “moving frame.” Never attempt to view the universe from two different frames at once. This is what causes much of the confusion, most of the paradoxes, and all of the logical inconsistencies attributed to the theory of relativity.

You can develop the ability to switch from one viewpoint to another qualitatively simply by learning to describe both world-views and how they are related to one another. But if you want quantitative results you must learn to use the Lorentz transformation equations to transform the coordinates of one observer into those of the other observer. If you do this, you will find that a certain mathematical factor keeps appearing in your results. This factor, named gamma (g), approaches unity (1) when velocities are small, but approaches infinity (¥) when velocities approach the speed of light. For a velocity of about nine-tenths the speed of light (), gamma is equal to two (g = 2).

Having said all of this, we can finally list some of the aspects of reality that are no longer absolute but relative in character.

Space is Relative
It should be obvious from the Star Trek paradox that space is different for different observers. Since Captain Kirk remains at the center of the expanding shell of light in his space and the Klingon remains at the center of the expanding shell of light in his space, the two observers must perceive and measure space differently from one another.

One of the consequences of the relativity of space is the phenomenon of length contraction. Specifically, the length of a moving object contracts (becomes shorter in the direction of motion) by a factor of gamma. This means that a spaceship traveling at 0.9 c is only half as long as one at rest (or as measured by the someone inside the spaceship). A meter stick traveling at 0.9 c is only 50 cm long.

Notice that everything in the moving spaceship is contracted by the same factor. Indeed, a rest observer would say that the reason the moving observers measure the correct proper length of the spaceship is because the contracted observers are measuring a contracted spaceship with contracted meter sticks. Since everything in the moving frame is contracted by the same factor, it is impossible for those inside that frame to measure any contraction. As far their measurements are concerned, they might as well be at rest. Indeed, they cannot distinguish their actual case from that of being at rest. This is exactly what the principle of relativity demands.

By the way, notice that contraction occurs only in the direction of motion, not perpendicular to the motion. Therefore, only the length of the spaceship changes, not its width or its height.

Time is Relative
Because space is relative, time must also be relative. Otherwise, the speed of light could not be a constant relative to all inertial observers.

One of the consequences of the relativity of time is the phenomenon of time dilation. Specifically, a moving clock runs slow by a factor of gamma. Therefore, the people in a spaceship traveling at 0.9 c age at half the rate of those at rest. If a person leaves the earth in a spaceship traveling at 0.9 c, he will arrive at a star 9 light years away in 10 years (earth time). But he will age only 5 years (proper time) while traveling to the star because he ages at half the rate of the earth.

It is interesting to see how the relativity of space and time work together to render a consistent result. If we look at this situation from the spaceship’s point of view, we see that the spaceship frame is at rest while the earth-star frame is moving at 0.9 c. Because the earth-star frame is moving, distances in that frame will be contracted by a factor of two. Therefore, the star is only 4.5 light years away from the spaceship when the trip begins. Traveling at 0.9 c, the star requires only 5 years to travel the 4.5 light years to the spaceship. Consequently, everyone agrees that the spaceship ages only 5 years during the trip, but different observers give different reasons for that result. The earth observers say time dilation is the reason, while the spaceship passengers say length contraction is the reason.

By the way, notice that when the spaceship (viewed at rest) considers the time dilation effect upon the earth (viewed in motion), it finds that the earth ages only half as fast as the spaceship. Therefore, from the spaceship point of view the earth ages only 2.5 years during the 5-year trip. Why is this result not the same as the 10-year interval calculated in the earth frame?

The answer lies in the relativity of simultaneity. The concept of simultaneity is different in the earth frame from what it is in the spaceship frame. But when simultaneity is taken into consideration, everything predicted in the spaceship frame is the same as that predicted in the earth frame. In other words, both frames are equally valid views of physical reality as they must be according to the principle of relativity. Although they depict reality differently, they always predict the same experimentally measured result.

This example serves to illustrate an important aspect of the theory of relativity. When properly analyzed, all paradoxes and apparent inconsistencies disappear. In the final analysis, relativity is a perfectly self-consistent theory that projects an unambiguous picture of reality.

Velocity is Relative
Since velocity is a relative concept in classical physics, it is also a relative quantity in special relativity. (The only exception to this statement, of course, is the speed of light, a universal constant.) But the relativistic equation for the addition of velocities is different from the classical equation. There is an additional factor in the expression that prevents the sum of two velocities from ever exceeding the speed of light.

In classical physics, if you take an object at rest and boost its speed to half the speed of light, then repeat the process again and again, each time boosting the speed by the same amount relative to the new reference frame, you will get the velocity sequence 0, 0.5 c, 1.0 c, 1.5 c, 2.0 c, etc., which quickly exceeds the speed of light. But in relativity you get the sequence, 0, 0.5 c, 0.8 c, 0.93 c, 0.98 c, etc., which gets closer and closer to the speed of light but never quite reaches it. According to the relativistic addition of velocities equation, the speed of light is an absolute speed limit. Nothing but massless particles (like light) can ever attain that speed. (And massless particles can exist at no other speed except the speed of light.) Indeed, from the boosted object’s point of view, no matter how many times it is boosted, it will still measure itself to be at rest, just as far from attaining the speed of light as it was in the beginning.

Mass is Relative
Everything we have discussed so far has been a consequence of the relativity of space and time. When mass is added to the mixture a whole new set of additional relativistic quantities emerge. Even mass itself becomes a relative quantity whose value depends upon the frame of reference of the observer.

For example, relativistic mass can be defined as the ratio of momentum to velocity (where momentum is that relativistic quantity that obeys the law of conservation of momentum) and proper mass as the limit of that ratio as the velocity approaches zero. Then both classical and relativistic physics picture mass as a quantitative measure of inertia. Both assert that mass determines the tendency of an object to keep on moving (its momentum) for a given velocity. Furthermore, relativistic mass reduces to classical mass in the limit of small velocities (as required by the correspondence principle).

But unlike classical mass, relativistic mass changes with velocity. Specifically, a moving mass increases by a factor of gamma. So an object traveling at 60% the speed of light will have a mass 25% greater than normal because gamma will be equal to 1.25.

Also unlike classical mass, relativistic mass does not obey a conservation law. In classical physics, mass can neither be created nor destroyed, but in relativistic physics both are possible. More specifically, mass can be converted into energy and energy into mass. In fact, mass can be conceptualized as being nothing more than an extremely concentrated form of localized energy able to be viewed at rest.

Momentum is Relative
Since momentum is relative in classical physics, it is also relative in relativistic physics. In fact, there is no difference between relativistic momentum and classical momentum if one uses the relativistic mass in the equations. In both cases the momentum of a particle is equal to its mass times its velocity. Of course, if one uses the proper mass in the equations, the relativistic momentum differs from the classical momentum. It is larger by a factor of gamma because relativistic mass increases by a factor of gamma.

Energy is Relative
Since energy is relative in classical physics, it is also relative in relativistic physics. However, the theory of relativity recognizes the fact that mass and energy are actually two manifestations of the same physical quantity. Therefore, under the proper conditions it is possible to convert mass into energy and energy into mass. The conversion factor from mass to energy is the speed of light squared. Hence, we get Einstein’s famous mass-energy equation E = mc², where E is the total energy associated with an object and m is the object’s relativistic mass. Since the speed of light squared is a very large number (in standard MKS units c² »10¹⁷ m²/s²), this equation shows that mass is simply an extremely concentrated form of localized energy.

If we apply Einstein’s mass-energy equation to an object at rest, then m becomes the rest mass and E becomes the total energy of the object when it is at rest. This would include every form of energy except kinetic energy, namely, potential energy, chemical energy, nuclear binding energy, and any other form of energy that might be associated with the object at rest.

Reversing this relationship leads to the conclusion that the rest mass of a system of static particles includes every form of energy present in the system. Therefore, a compressed spring weighs more than an uncompressed spring because it has the additional mass equivalent of its potential energy. And a helium nucleus has less mass than the four nucleons that comprise it (two protons and two neutrons) because of the binding energy holding everything together.

Since kinetic energy is the energy due to motion, the relativistic kinetic energy of an object is simply the difference between its total energy and its rest energy. Therefore, the mass equivalent of the kinetic energy is simply the difference between the relative mass of an object and its rest mass. And finally, the total relativistic energy of a system of particles is equal to the energy equivalent of the rest masses of its constituent particles plus the energy of interaction between the particles plus the kinetic energy of the particles. Therefore, if a system of particles is observed in its center-of-mass frame, we conclude that the proper mass of that system is equal to the mass equivalent of the total relativistic energy associated with that system. And if a system of particles is observed in any other frame of reference, the relative mass of the system is equal to the mass equivalent of the total relativistic energy in that frame of reference.

Lorentz Transform – How to transform from one reference frame to another.
Space – How space differs from one observer to another.
Time – How time differs from one observer to another.
Velocity – How velocities differ from one observer to another.
Mass – How mass differs from one observer to another.
Momentum – How momentum differs from one observer to another.
Energy – How energy differs from one observer to another.

ModPhy1/Unit1/Special Relativity/