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First Contact with Relativity

By: Reuben D. Budiardja







Introduction

In 1905, Albert Einstein wrote a paper with the original title " Zur Elektrodynamik bewegter Körper," which has been translated in English with the tittle " On the Electrodynamics of Moving Bodies." The paper and the content in it now are known as The Theory of Relativity.

In this paper, Einstein showed us how to view our world in a really different way from Newtonian point of view. He re-defined many things that we thought we have fully understood. In my opinion, the Theory of Relativity is one of Einstein's greatest works, and it has always been my favorite subject.

The goal of this essay is to explain Einstein's paper in a not-so-technical way, so that it can be understood more easily by people who have only a little physics background. Of course, we have to admit that such a way will have a consequent of loosing some intriguing thoughts that lead to the result of this theory. However, the writer would do any possible attempt to reduce such loosing for the sake of producing a non-so-technical and understandable essay.

This essay also reflects my study, as a writer, about the Theory of Relativity. Thus, first of all I have to admit, since I have not reached a full understanding in studying Einstein' original paper yet, this essay is not yet complete. I will only try to explain some significant concepts from this theory that lie the foundation of this theory. Hopefully, in a later time, I will be able to complete this essay as well.
 

Some Backgrounds

Michelson-Morley experiment in an attempt of detecting the ether gave no result at all. No ether wind was observed. Thus, the theory that light is propagated through ether cannot hold true. The big question at that time was: if light is truly a wave, but there is no medium in which it can propagate, then how can it reach us from the empty space?

Meanwhile, in 1860's, a well-constructed equation about electricity and magnetism came from one of the most brilliant theorists, James Clark Maxwell. Maxwell's equation describes how electric field and magnetic field give rise to each other. From his equation, Maxwell deduced that Electro-magnetic waves would travel at the speed of 186.300 mph, which is exactly the same speed that had been observed by Michelson before, while he measured the speed of light. Thus, a mere conclusion that could be drawn was that light is an Electro-magnetic wave. Then, the question would be: what is the speed of light measured relative to?

The answer comes from Einstein's Theory of Relativity. Thus, we are going to explore some basic concepts that of this theory first.
 

The Two Postulates of Relativity

Based on Newton's Law of Motion, we know that the laws of physics are the same in a uniformly moving room as they are in a room at rest. For example, if we are in a room that moving with constant velocity, such as in a ship, we cannot tell that it is moving unless we see it moving relative to other, such as the sea. In other words, the same force will produce the same acceleration in that room, and an object would not move unless force act on it. Such a room like this, where Newton's Law of motion hold, is called an inertial frame.

After Maxwell published his electrodynamics laws, Einstein pointed out that the postulate above should include Maxwell's electrodynamics. Thus, Einstein first postulate for relativity is:

Since Maxwell's equations predicted the speed of electromagnetic waves, which is light, to be a constant, the first postulate has further consequence. Thus, Einstein also proposed his second postulate: We will see that these two postulates are indeed very consistent in the theory of relativity.
 

Concept of Simultaneity

In his original paper, Einstein introduced the definition of simultaneity at the first place. This is simply because the concepts of simultaneity is a crucial thing to have, before someone goes to the next step of trying to understand the principle of relativity.

In every day life, we simply define that two events are simultaneous if they happen at the same moment, in our perception. For example, we would say, "a train arrives here at 7 o'clock" when we perceived that the time showed by our watch is at 7 o'clock when the train arrives. This concept of simultaneity seems satisfactory when we are only concerned about the time at the place the event happen, or in other words, we have the watch at the same place the train arrives. However, if our watch is not at the same place where the event happens, we will see that this kind of concept is no longer satisfactory.

It might appear possible to overcome all the difficulties attending the definition of "time" by substituting "the position of the small hand of my watch" for "time." And in fact such a definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or – what comes to the same thing – to evaluate the times of events occurring at places remote from the watch. (Einstein) For instance, let us assume that I can observe an event happening 300.000 km away. By my perception, I note that the event happened at 6.00 PM. Then, can I say that 6.00 PM in my watch is simultaneous to the events? Not really. We have to realize that in order for me to be able to observe the event, the information from the event, in this case the light, has to travel 300.000 km away from the place the event happened into my eyes, and this takes time!

A more extreme example happens when we observe a star that has thousand light-years distance. If I observe a star at midnight, is that star still at that position at this midnight? Obviously not. What we observe at that time is the event from the star a thousand years ago, in other word, we are looking at the past. Thus, we cannot say that the event, the moment the star happens to be in that position, is simultaneous with our watch. Thus, we have to re-define the definition of the simultaneity.

Now, let us use a watch that resembles the first watch as our event. To define simultaneity, we have to synchronize our watches, let us say A and B, that separated in a distance. How could we synchronize it since we have to take into account the time needed by light for travel from A to B? Einstein presented a way to do this.

"The time required by light for travel from A to B equals the time it requires to travel from B to A. Let a ray of light start from at the "A time" (ta) from A towards B, let it at the "B time " (tb) be reflected at B in the direction of A, and arrive again at A at "A time" ta'. From the first sentence, that the time required for travel from A to B equals the time it requires to travel from B to A, we can conclude that the two clocks are synchronized if:

tb – ta = ta' – tb

We see that this is the only way we can synchronize two clocks. Implicitly, this also defines what we call simultaneity. To define more that two clock, we can easily use logical conclusion:


Relativity of Time

Suppose we have 2 sets of clocks, which are synchronize each other. One set of clocks, let say A and B are staying in the ground, relatively rest to the ground, while the other set, a & b , are in a train, moving with a constant speed v relative to the ground.

Now using the definition of the simultaneity above, we see that A and B are still synchronized to each other for the observer on the ground. However, according to the observer on the ground, the clocks a & b behave with the following manner:

and 

Thus, according to the observer on the ground, the clocks on the train are no longer synchronized. However, if there is an observer on the train, he would find that his clocks are in fact still synchronized, because he is relatively at rest in his inertial frame of reference (the train). This concept then introduces what we call as individual time.
 

Further Implication: Time Dilation

As have been explained above, we can merely draw a conclusion that the concept of simultaneity in not invariant when we move from one inertial frame of reference to another. In other words, simultaneity, two events happen at the same time, in one inertial frame is not necessarily simultaneity when it is seen form another inertial frame. Hence, there should be factor involve to distinct the simultaneity in difference inertial frame. In this section, we will see that and learn what is recognize as time dilation

Again, let suppose there are two different inertial frames. The first inertial frame is Amir's, which is in relative rest to the ground. The second one is Abud's, which is moving with constant speed in a car relative to the ground. Let say, before Abud's went with his car, he has synchronized his clock with Amir's (her) in one same inertial frame.

Suppose their clock is build from two mirrors facing each other, with a blip of light bouncing back and forth. One of the mirrors is equipped by a photocell, so that it can catch a bit of the light, and count it. Let say, if the photocell catch the blip of the light once, then one blip second (as our unit) has passed. Let's us forget about that complexity to build such a light clock, and assume that some mechanisms will maintain the intensity of the blip, and assume that the clocks work perfectly.

Now, when Abud is moving with her car, Amir is standing at the edge of the road, in relative rest, as an observer to Abud's inertial frame. What will she see? Will she see that their clocks still synchrony? No. Instead, Amir will see that Abud's clock blip of light is moving zigzag, because of his motion with his car.

Now, let us take a look the distance traveled by the blip of light in Abud's clock so that the photocell clicks. If the clock is at rest, the distance traveled will be 2ct or w; with t is the time needed to travel between 2 mirrors. Now, according to Amir's observation against Abud's clock, the distance will be as follows.

From the diagram, we see a right triangle form by the blip of light path in one zig. From Phytagoras theorem, we know that

Then, half of distance traveled by the blip of light of Abud's clock to get one blip second, according to Amir, is:

thus:

Then, if Amir observe her identical clocks with Abud's, she will see that Abud's clock slow down compare to hers. To get the half blip second of Abud's clock according to Amir's, we solve the equation above for t.

To get one blip second, we multiply the equation by two:

2w/c is one blip second for Amir's clock, which is relatively at rest. Hence, we clearly see from the equation, that Abud's clock is slowed down by the factor according to Amir from her inertial frame.

However, this is not the whole story. Abud's inertial frame of reference is also valid as he looks at Amir's clock. He will see that Amir's clock is the one that slows down by the factor compared to his.

Here Einstein put the basic concepts of time that revolt the Newtonian view that time is invariant. Time indeed is different for observer in different inertial frame by the factor , with v is relative to the observer. Relativity theory states that time is not invariant, instead, it is just another variant dimension, such as length and width.

A further implication of this is that two clocks in one inertial frame that are synchronized, will no longer synchronized as one clock moves to the other. The clock that moves will lag behind the other by a factor of , "t being the time occupied in the distance traveled (Einstein 49)". This result does not only work for a straight line or polygonal line motion, but also "valid for a continuously curved line". Hence, we can conclude that a "balance clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under identical conditions" (Einstein 50).
 

Relativity of Length

Suppose Abud's clock is equipped by a device that will mark the road every one blip second, and he is moving with a constant speed v m / blip second. Now, the question is: how far is the distance between marks on the road?

According to Abud's the, the distance will be v meters apart. However, Amir will have a different measurement. Since Abud's clock is running slow compare to her, Amir will notice that Abud's device will give a mark at the interval blip second. Since Abud and Amir agree about the speed of the car, which is v m / blip second, Amir will see that the distance between marks are meters apart, a longer distance.

Who is right? Amir is right, because the mark is in her inertial frame, so that she can really measure it with a tape measurement. The conclusion to this situation is that, as a result of his motion, Abud will see that everything is shortened in his direction of motion by a factor . This is the relativity of length.

This result indeed also applies to Amir. Since we can say that Amir is relatively moving toward Abud, she will also see that Abud and his car is shortened in the direction of motion by factor . Thus, for an observer in the ground that relatively at rest to the ground, he will see that an object is shortened in the direction of motion of that object by the factor .

"From reasons of symmetry it is now evident that the length of a given rod moving perpendicularly to its axis, measure in the stationary system, must depend only on the velocity and not on the direction and the sense of the motion." (Einstein, 47) Thus, a length of a moving object according an observer at rest will be:

with L is the length of the object in relative rest.
 

Conclusion

As have been said in the introduction, this essay is not yet complete. Another variant that will change according to an observer in difference inertial frame is the mass of an object, which is also depend on the velocity of the object.

Einstein's Theory of Relativity has many further implications in the area regarding the dimension and mass of a moving object. Some of its is a different way of adding velocities. We see from the two equations about relativity of time and length above that if the velocity of a moving object is c, which is the speed of light, the equations become meaningless. Thus, we can conclude that c is an impossible speed to be reached.
 

References

Einstein, Albert., et all. The Principle of Relativity. New York: Dover Publication, Inc., 1952.
Gamow, George. Mr.Tompkins in Paperpack. Press Syndicate University of Cambridge: 1968.
Gardner, Martin. Relativity Simply Explained. New York: Dover Publication Inc., 1996
Jaffe, Bernard. Michelson and The Speed of Light. New York: Doubleday & Company, Inc. 1960.

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