Time Dilation and Special Relativity

Eric Biehn

Math 89S

Duke University

Introduction

Over the course of scientific history, there have been many monumental discoveries which

changed the way we think about our planet, solar system, and universe as a whole.  On such

discovery is the theory of special relativity.  This theory states that the universe does not run on a

universal clock; time can be experienced differently for different people.  Mankind was not

aware of this until recently, because on Earth the effects of special relativity are essentially

negligible.  However, after embracing this discovery, we have exposed ourselves to a new world

of physics that corrects the flaws in the original Newtonian methods to keep up with our 21st

century world.

Special Relativity

Special relativity essentially involves different observers being in different inertial frames of

reference.  These two inertial frames travel at constant velocities relative to each other.  An

observer measures these two inertial frames by using a coordinate system [1].  

There are two postulates involved with the theory of relativity.  The first is that all laws of

physics are the same for any inertial reference frame.  The second is that the speed of light is

constant and has the same value for all observers.  This means that our current conception of

space and time would be affected [1].  In order to account for the fact that “c” (the constant value

for light) is always constant for all observers, we begin to explore the details of special relativity.

Figure 1: The following image is an artistic depiction of what traveling at the speed of light

could potentially look like.

According to special relativity, nothing can travel faster than the speed of light.  Thus, when

velocities are large enough, Newtonian mechanics do not apply [3].  For example, if an object

traveled at a speed v with respect to a stationary observer, and another object travels at a speed of

v’ with respect to the first object, the speed u of the second object seen by an observer would be

determined by:

u= v+v '

1+ vv 'c2

Lorentz Factor and Lorentz Contraction

One component of special relativity is the Lorentz factor.  This can measure relativistic speeds.

Its equation is given by:

The Lorentz factor shows how special relativity is impacting our typical notions of physics.  A

Lorentz factor of 1 means that special relativity does not affect the object at all.  The closer to the

speed of light an object travels, the more noticeable the effects of special relativity are [4].  For

example, an Airbus A380 at a cruising speed of 250 m/s has a factor of 1.00000000000035.  As a

result, the effects of special relativity are virtually non-existent.  However, if a rocket ship were

to be traveling at 296800000 m/s (.99 times the speed of light), there would be a factor of 7.1;

thus the effects of special relativity would be extremely apparent.  This would mean that

measurements of distance and time for observers moving at this speed will differ by 6.1%.  

Figure 2: This graph shows how the Lorentz Factor begins to greatly inflate beginning at speeds

around 90% the speed of light.

Another aspect of special relativity involves the Lorentz Contraction.  This deals with space

being changed due to the effects of special relativity.  The Lorentz contraction shows that the

greater the value of gamma for the Lorentz factor, the smaller the length that is measured.  As a

result, this concept deals with the proper length, the Lorentz factor, and the observed length.

This brings up the term “length contraction” because in cases of special relativity, the length

contracts to a smaller measurement than the actual length really is [4].  The equation is given as

follows:

This Lorentz contraction deals with the different values of length that multiple observers would

measure.  When an observer moves in the same reference frame as the rod, the rod’s proper

length will be measured.  However, a stationary observer in a different frame of reference would

measure a shorter length of the rod.  

Time Dilation

When analyzing special relativity, another factor to consider is time dilation.  Essentially, time

dilation can be described by the phrase “moving clocks run slow as measured by a stationary

observer” [1].  Time dilation is given by the following equation:

Time dilation explains the phenomenon on how the passage of time depends on the person

observing it.  For the most part, this idea is confusing to people on Earth because we are all used

to time traveling at the same constant rate for the entire population.  However, with the effects of

special relativity, this is not always the case.  A stationary observer would claim that a moving

clock is running slow.  In other words, time is distorted for different observers to account for the

constant speed of light [1].

An example of time dilation is referred to as the “twin paradox”.  This concept deals with the

effects of time dilation for a pair of twins, with one twin going on a journey through space

traveling at a speed close to the speed of light, while the other remains on Earth.  Before the

journey, the twins are obviously the same age.  However, when the twin who traveled returns

from his trip, he is different in age than the twin who remained on Earth [7].  It turns out that the

twin who stays on earth is older than the twin who traveled through space.  This is due to the

effects of special relativity.  The twin who travels through space has two inertial frames: one on

the way there and one on the way back.  In order to make the change in inertial frames, the twin

has to de-accelerate and re-accelerate [7].  The difference in age is caused by the switch in

inertial frames as the traveling twin heads back to earth.

Figure 3: This diagram shows the paths traveled between the two twins.

This paradox is a fantastic example regarding time being experienced differently because the

universe does not run on a universal clock.

Experiments and Theories

There have been far too many experiments done for special relativity to list them all here;

however, there are several notable ones that give an idea of what tests for special relativity were

like.

Michelson-Morley Experiment

The Michelson-Morley Experiment was carried out in 1887 by Albert Michelson and Edward

Morley.  The purpose of this experiment was to detect relative motion of matter through the

“aether” in which they believed light traveled through.  In order to do so, they compared the

speed of light detected in perpendicular directions.  Upon conducting the experiment, the

expected difference in speed between the light in different directions of movement through their

respective “aether” was found to be negligent.  Thus, the speeds detected for each beam of light

were equal [6].  This puzzled the scientists, because they did not understand how they could both

get equivalent speed measurements for the different beams of light in this particular scenario.

The experiment itself ended up being evidence against the aether theory, and it began a method

of research that eventually lead to special relativity.  

Lorentz Aether Theory

The Lorentz Ether Theory was developed by Hendrik Lorentz with help from Henri Poincaré

between 1892 and 1906.  The theory introduced the idea of a strict separation between matter (in

the form of electrons) and “aether”.   This model proposes that aether is a completely motionless

surrounding that is not set in motion by matter [4].  Lorentz also proposed the theories behind

length contraction and the Lorentz transformation, which are important components about what

we know regarding special relativity today.  

Einstein’s Thought Experiment

Einstein stated that a helpful factor that led him to his theory of special relativity was a thought

experiment he conducted at the age of 16.  The thought experiment involved Einstein

envisioning himself moving next to a light beam at the speed of light.  Einstein figured that if he

was able to catch up to this beam of light, he would be able to observe the light frozen in space.

However, he knew that light could not be frozen in space or it would no longer be light.  This led

Einstein to realize that light cannot be slowed down and it must always be moving at a constant

speed.  Thus, when approaching speeds of light, another factor would have to change [5].  This

thought experiment eventually led Einstein to piece together his special theory of relativity.

Theoretical Application of Special Relativity

The effects of special relativity on earth are so small that we do not notice or feel the effects of it

in our everyday life.  Thus, the concepts are difficult to grasp for many people because the idea is

quite unfamiliar to us.  However, if the speed of light were 10 m/s, the effects of special

relativity would be experienced every day, and we would become very aware of its effects.  If

the speed of light were 10 m/s, it would take extremely large amounts of energy to travel at

speeds close to the speed of light.  However, for the sake of the example, we will assume that we

can travel at the same speeds we do on Earth today (as long as they do not exceed the speed of

light).

For starters, time dilation would play a major part in our everyday lives [2].  If a car were to be

driving down a road at 8 m/s, for every minute the person in the inertial frame of the car

experiences, an observer at rest would experience it to be 100 seconds rather than 60 seconds,

shown by the equation:

t '= 60 s

√1−(8 m

s)

2

(10 ms)

2

t '=100 s

Thus, when people and vehicles are in motion, they would experience time differently than those

at rest.  

In addition, the world would visually look much different than it does today as a result of the

Doppler Shift caused by traveling at speeds near the speed of light.  This effect is caused by

relativistic motion of a source and an observer.  The closer one travels to the speed of light, the

faster wavelengths of light appear [2].  As a result, the world would appear much different than

we see it now.  For example, if somebody was driving in a car and they saw a yellow sign ahead

of them, the sign would appear to be purple.  This is because the wavelengths of the color purple

are closer together.  However, as they pass the yellow sign and look back at it, it will appear to

be red, because the wavelengths of the color red are further apart [2]. This means that the world

would look entirely different when someone was in motion than it would when they are standing

still.  It would be a beautiful sight; however, we would likely have to re-think the traffic light

system for roads.

Figure 4: This image shows the spectrums of color resulting from the Doppler shift when

traveling at relative speeds.

Furthermore, everything seen by humans on Earth would be delayed.  Because light can only

travel 10 meters in one second, gazing at an object moving 50 meters away would essentially be

looking at it 5 seconds in the past.  Thus, this hypothetical world would have many visual delays

[2].

Conclusion

The discovery of special relativity forever changed our knowledge of the workings of the

universe. Through concepts such as length contraction and time dilation, the theory of special

relativity leads to occurrences that we are not accustomed to dealing with on Earth. In the future,

our technology of space travel may become advanced enough allowing concepts such as the twin

paradox to actually happen to humans. Until then, we can still study these concepts and marvel

at the wonders surrounding the nature of our universe.

Sources

O'connor, JJ. "Special Relativity."   Special Relativity . Mathematics Physics Index, n.d. Web. 8

Dec. 2016. [1]

“Speed of Light 100 mph” 10 June 2010. The Physicist.  Web. 8 Dec. 2016. [2]

Weisstein, Eric. "Special Relativity." Special Relativity. N.p., n.d. Web. 8 Dec. 2016. [3]

"Lorentz Ether Theory." Wikipedia. Wikimedia Foundation, n.d. Web. 8 Dec. 2016. [4]

Norton, John. "Chasing a Beam of Light: Einstein's Most Famous Thought Experiment." Einstein's Most Famous Thought Experiment. University of Pittsburgh, n.d. Web. 8 Dec. 2016. [5]

"Michelson-Morley Experiment."   Michelson-Morley Experiment . Science World, n.d. Web. 8 Dec. 2016.[6]

Possel, Markus. "The Case of the Travelling Twins." The Case of the Travelling Twins. N.p., n.d. Web. 8 Dec. 2016. [7]