# Einstein's Theory of Special Relativity Made Relatively Simple

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**Einstein's Theory of Special Relativity Made Relatively**

**Simple!**

*by*

*Christopher P Benton, PhD*

**Young Einstein**

Albert Einstein was born in 1879 and died in 1955. He didn't start talking until he was three, and

at age nine he still didn't talk very well. Everyone thought he was retarded. However, he got

smarter.

**The Ether**

In 1820, Thomas Young performed an experiment that indicated that light is composed of waves.

However, common sense told the physicists of that day that every wave needs some sort of

medium to wave through. For example, ocean waves wave through water, and sound waves

wave through air (as well as water and other materials). Thus, physicists believed that light

waves also needed some medium to wave through. They called this medium the

**ether**.

In 1831 two American scientists, Michelson and Morley, set up an experiment to detect the ether.

The idea behind their experiment was that as the earth moves through space it would at times be

moving with the ether and at other times against the ether. Suppose the earth were moving

against the ether. Then the situation would be analogous to moving upstream in a boat. In that

case, anything you threw downstream would move away from you faster than something that

you threw upstream. Thus, physicists reasoned that as the earth moved through the ether, the

speed at which light moved in one direction would be different from the speed in another

direction. The experiment of Michelson and Morley was designed to detect this difference in

speed, and thus, confirm the existence of the ether.

However, when performed, the Michelson-Morley experiment detected no variation in the speed

of light. As a result, scientists gradually discarded the idea of the ether (since it couldn't be

detected), and they began to accept the idea that the speed of light is the same in all directions.

Thus, Einstein began his theory of special relativity with two assumptions:

1. The Principle of Relativity: One cannot tell by any experiment whether one is at rest or

moving uniformly (that is, moving in a straight line with constant velocity). In other words,

**there is no such thing as absolute rest**. All motion or rest is only in relation to other

observed objects (i.e. I can consider myself not to be moving with respect to the earth while

at the same time I am moving very rapidly with respect to the sun).

2. The Constancy of the Velocity of Light: The speed of light in a vacuum has the same value

*c*with respect to any observer either at rest or moving uniformly (

**= 186,282.397 miles per**

*c*second).

What Einstein then discovered was that in order for all observers to measure the same velocity

for a beam of light, time would have to "flow" differently for different observers, and space

would sometimes have to contract.

**Experiment 1:**Mrs. Einstein is standing in a field and Mr. Einstein is riding on a railroad car

that is moving with velocity

*v*. Mr. Einstein shines a flashlight in the direction in which he is

moving.

Question: What happens?

Answer: Because of the principle of the constancy of the velocity of light, each observer will

measure the light beam from the flashlight as traveling at the same speed. This may be contrary

to what you expected as you might have thought that the observer in the field would have seen

the beam moving at (the speed of light) + (the speed of the train). Nevertheless, this is not what

is observed in practice. What actually occurs in the real world is that no one ever measures light

moving faster or slower than c ≈ 186,000 miles per second (in a vacuum).

**Experiment 2:**Mrs. Einstein is standing in a field. Next to her is a light clock. That is, two

mirrors that are reflecting a beam of light back and forth, and the journey from one mirror to the

other and back again counts as one tick of the clock. Also, Mrs. Einstein is wearing a watch that

is synchronized with her light clock.

Standing on a railroad car is Mr. Einstein. He also has a light clock, and his clock is

synchronized with Mrs. Einstein's and his own wristwatch. The railroad car is not moving.

Question: What happens?

Answer: Nothing unusual happens. Mr. Einstein's watch and clock stay perfectly synchronized

with Mrs. Einstein's.

**Experiment 3:**We now have the same set up except that the railroad car is now moving to the

left with a velocity

*v*.

Question: What happens?

Answer: From Mr. Einstein's perspective, the beam of light keeps going up and down between

the mirrors, but from Mrs. Einstein's perspective, the light now has to travel a diagonal path from

one mirror to the other. Since Mrs. Einstein still measures the speed of light as c, she is now

going to observe Mr. Einstein's light clock as ticking slower than hers since the light now has a

longer distance to travel. However, since Mr. Einstein still experiences his watch as being

synchronized with his clock, Mrs. Einstein will see his watch slow down along with his clock!

Conclusion: If someone moves in a straight line with velocity

*v*with respect to you, then you will

observe time passing more slowly for them.

With a little algebra we can compute exactly how much time will slow down. Let,

*L*= length or height of the light clocks

*v*= Mr. Einstein's velocity

*t*= time between ticks on Mrs. Einstein's clock

*t*′ = time between ticks on Mr. Einstein's clock as Mrs. Einstein observes it

*c*= speed of light.

We will make frequent use of the formula distance = rate

**x**time. We now compute the distance

that the light travels in two ways.

*ct*′

*ct*′

2

*L*

2

*L*

*vt*′

*vt*′

2

2

On the one hand, the distance traveled is

*ct*′ (distance = rate

**x**time). But on the other hand,

using the Pythagorean Theorem, we have that the distance is

2

2

2

2

2

2

2

2

2

⎛ ′ ⎞

′

+

′

+

′

2

*vt*

2

*v t*

4

*L*

*v t*

2 4

*L*

*v t*

2

2

*L*+

= 2

*L*+

= 2

=

= 4

*L*+ 2 ′

⎜

⎟

2

*v t*

⎝ 2 ⎠

4

4

4

Hence,

2

2

2

*ct*′ =

4

*L*+

*v t*′

which implies that 2 2

2

2

2

*c t*′ = 4

*L*+

*v t*′ ,

which implies 2 2

2

2

2

*c t*′ −

*v t*′ = 4

*L*,

which implies 2

*t*′ ( 2

2

*c*−

*v*)

2

= 4

*L*,

2

2

4

*L*

4

*L*

which implies 2

*t*′ =

=

,

2

2

2

*c*−

*v*

⎛

⎞

2

*v*

*c*⎜1−

⎟

2

*c*

⎝

⎠

2

2

4

*L*

4

*L*

2

*L*

(2

*L*)

*c*

which implies

*t*′ =

=

=

=

.

2

2

2

⎛

⎞

2

2

*v*

2

*v*

*v*

*v*

*c*⎜1−

⎟

*c*

1 −

*c*1 −

1 −

2

2

2

*c*

⎝

⎠

*c*

*c*

2

*c*

Since

*t*= (2

*L*)

*c*(i.e. time = distance/rate), this gives us

*t*

*t*′ =

2

*v*

1 − 2

*c*

Therefore, if

*t*is the time between ticks on Mrs. Einstein's watch, then she will observe a longer

interval of

*t*

*t*′ =

2

*v*

1 − 2

*c*

between ticks on Mr. Einstein's watch! Notice that this difference is not very much unless one is

traveling at an extremely fast velocity.

**Experiment 4:**Super Einstein is flying through space with his twin brother, Murray, who is

186,000 miles behind him. Every time he wants to make an acceleration of 10 mph, he uses a

flashlight to signal his brother so that they will accelerate together and stay the same distance

apart.

From Einstein's perspective, he and his brother are 186,000 miles apart, and by letting 1 second

pass on his clock before accelerating, he allows the light to reach his brother right at the moment

of acceleration. As a result, from Einstein's point of view he and his brother accelerate together

and remain a constant 186,000 miles apart. This is what Einstein sees, but if we are standing still

with respect to Einstein and his brother, then what will we see?

Answer: From our perspective, two things happen. First, we say that the beam of light has less

than 186,000 miles to travel since his brother is traveling toward it. Second, we are going to see

Einstein's clock as running slow. Thus, for two reasons we are going to see Einstein's brother get

the signal to accelerate before a full second has passed on Einstein's clock and he begins his own

acceleration. Hence, the distance between Einstein and his brother gets shorter. However, if we

stop our analysis at this point, then we are going to wind up with a contradiction, because if

Einstein and his brother keep accelerating, and if we keep seeing his brother accelerate first, then

eventually distance between them will become so small that Murray will run into Einstein.

However, from Einstein's perspective, he and his brother stay a constant 186,000 miles apart!

How can we resolve this seeming contradiction? Only by making a very bizarre assumption. In

order to keep Murray from running into his brother the shortening of the distance between

Einstein and his brother must be compensated for by a contraction of length in a direction

parallel to the direction in which Einstein and his brother are moving! In other words, from our

point of view, Einstein and his brother are getting shorter so that some distance always remains

between them in spite of their accelerations.

In the next experiment we will derive a formula that will show us by just how much lengths

contract.

**Experiment 5:**Next to Einstein, a light clock lies on its side on a railroad car as the car moves to

the left with velocity

*v*.

Question: How is the length of the clock affected?

Answer: We know that the time between ticks on the moving clock, as we see it, will be

*t*

*t*′ =

.

2

*v*

1 − 2

*c*

Let,

*t*= time it takes the light to go from the first mirror to the second (right to left), as we perceive it.

1

*t*= time that it takes the light to go from the second mirror back to the first, as we perceive it.

2

*L*′ = length of the light clock, as we perceive it.

We can analyze the situation as follows:

*vt*1

*L*′ +

*vt*

*v*1

*vt*2

*L*′ −

*vt*

*v*2

From the above picture we see that the distance the light traveled in time

*t*was

*L*′ +

*vt*.

1

1

However, this distance is also equal to

*ct*(rate

**x**time). Also, the distance the light traveled on

1

the return trip was

*L*′ −

*vt*=

*ct*. Solving for

*t*, we have

2

2

1

*L*′ +

*vt*=

*ct*

1

1

which implies that

*L*′ =

*ct*−

*vt*,

1

1

which implies

*L*′ = (

*c*−

*v*)

*t*,

1

*L*′

which implies

*t*=

.

1

*c*−

*v*

Similarly,

*L*′ −

*vt*=

*ct*

2

2

which implies that

*L*′ =

*ct*+

*vt*,

2

2

which implies

*L*′ = (

*c*+

*v*)

*t*,

2

*L*′

which implies

*t*=

.

2

*c*+

*v*

Thus,

*t*′ =

*t*+

*t*

1

2

*L*′

*L*′

=

+

*c*−

*v*

*c*+

*v*

*L c*

′ +

*L v*

′ +

*L c*

′ −

*L v*

′

=

2

2

*c*−

*v*

2

*L c*

′

=

,

2

2

*c*−

*v*

*t*

2

*L c*

′

which implies that

*t*′ =

=

.

2

2

2

*c*−

*v*

*v*

1 − 2

*c*

Since

*t*= (2

*L*)

*c*(

*see experiment 3*),

*t*

(2

*L*)

*c*

2

*L*

2

*L c*

′

*t*′ =

=

=

=

,

2

2

2

2

2

*c*−

*v*

*v*

*v*

*v*

1 −

1 −

*c*1 −

2

2

2

*c*

*c*

*c*

2

*L*

2

*L c*

′

2

*L c*

′

which implies that

=

=

,

2

2

2

2

*c*−

*v*

*v*

⎛

⎞

2

*v*

*c*1 −

*c*⎜1−

⎟

2

2

*c*

*c*

⎝

⎠

2

*L*

2

*L*′

which implies

=

,

2

2

*v*

⎛

*v*⎞

*c*1 −

*c*⎜1−

⎟

2

2

*c*

*c*

⎝

⎠

*L*

*L*′

which implies

=

,

2

2

*v*

⎛

*v*⎞

1 −

⎜1−

⎟

2

2

*c*

*c*

⎝

⎠

2

⎛

*v*⎞

*L*⎜1−

⎟

2

*c*

⎝

⎠

which implies

*L*′ =

,

2

*v*

1 − 2

*c*

2

*v*

which implies

*L*′ =

*L*1 −

.

2

*c*

2

*v*

Conclusion: Since 1 −

is less than 1, this shows that we will measure the length of Einstein’s

2

*c*

light clock as being less than ours. Thus, when an object is moving in a straight line with a fixed

velocity v, we will see its length, as measured in the direction in which it is moving, shorten.

# Document Outline

- Conclusions.pdf
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