Previously we discussed the postulates of the special theory of relativity where we covered how the effect of light constancy produces so-called time dilation. In this blog, we will try to further our knowledge by using some quantitative processes to understand time dilation, length contraction, and even more. This will include some mathematical expressions to prove the theories in special theory.
Be aware that this is part 2 of The Special Theory, and the reader must have some basic knowledge of what we're talking
Firstly, let's try to understand the velocity of a reference body from a familiar example.
Let's say there is a car with a small machine gun loaded at the top and someone named Alice, for say, is driving the vehicle with a speed of 30m/s. The machine gun shoots the bullet at a velocity of 70m/s. Now it is time for us to add an observer to observe this scenario and we will break this reference form into 2 frames:
1. Alice's (Rocket) frame of reference :
For Alice, inside the car, she sees herself at rest. Remember when we discussed the first postulates of the special theory of relativity i.e. "the relativity of simultaneity". Alice, from her measure of the clock, sees herself at rest and observes the background in motion...In short, she sees the background (Bob) moving in her opposite direction at 30m/s and herself at rest. So when she observes the machine gun, she finds the velocity of the bullet to be only 70m/s.
2. Bob's frame (Lab) frame of reference :
Calculating the velocity of the ball from Bob's frame of reference is fairly easy. He sees himself at rest and from his perspective, he sees Alice moving at a velocity of 30 and when he sees the ball that is moving at a velocity of 70, he observes the velocity of the ball added up. In short,
velocity of the car = 30m/s
velocity of the ball = 30+70 = 100m/s
This is called the "Galilean-Newton transformation'.
So as we can see the relative velocities of the observers are not the same which is the first postulate of the Special Theory of Relativity.
Why does this happen?
We think that this happens, right? But why? Why does the velocity add up in Bob's frame... Why can't he see the speed of the ball only to be 70m/s?
>> Well this is possible because of the vector addition property.
In physics, velocities are vectors, which means they have both magnitude (speed) and direction. When you have two velocities in different directions, their combination or resultant velocity is determined through vector addition.
In the scenario we described, Alice's car is moving to the right with a velocity of 30 meters per second, and she shoots the ball with a velocity of 70 meters per second relative to her car. To determine the resultant velocity of the ball as observed by Bob, we need to add the vector representing the car's velocity to the vector representing the ball's velocity relative to the car. In this case, we have a car velocity vector of +30 meters per second (to the right) and a ball velocity vector of +70 meters per second (relative to the car). When we add these vectors, we get:
+30 meters per second (car velocity) + 70 meters per second (ball velocity) = +100 meters per second
The resultant velocity vector is +100 meters per second, which means the ball appears to have a velocity of 100 meters per second to the right from Bob's perspective. This vector addition is a fundamental concept in physics and allows us to determine the combined effect of different velocities acting in different directions. So, Bob observes the velocity of the ball to be 100 meters per second because he adds the car's velocity vector to the ball's velocity vector relative to the car, resulting in the vector sum of 100 meters per second.
But why does Alice see the velocity of the ball not changing at all?
When Alice is inside the car, her frame of reference is moving along with the car. Therefore, she sees the ball's velocity relative to her car as 70 meters per second in the direction she shot it. In her frame of reference, the ball is moving forward with a velocity of 70 meters per second.
While Alice observes Bob moving in the opposite direction, her observation of Bob's motion does not directly impact the velocity of the ball. The ball's velocity, relative to Alice's moving frame of reference (her car), remains constant at 70 meters per second, regardless of the motion of the background or her observation of Bob.As I already discussed in my previous blog, this operation of relative velocity applies to every element in our space-time except for light. This is the second principle of The Special Theory of Relativity i.e. "the velocity of light is always constant in any frame of references." In that case, no matter how fast Alice is traveling if she shoots a laser at velocity 'c', Bob also sees the velocity of that laser, precisely, 'c'.
Now that is the first significant factor that contributed to the birth of special relativity and many parts of modern physics.
Time dilation:
By finding that the speed of light is constant, there became many hindrances in the so-called laws of classical physics. Newton's idea of time becoming absolute was starting to be a suspect according to Einstein. In fact, Einstein was so amazed by the contribution of light constancy and its effect on time, that he called his best friend, Michele Beso, and said "Time is Suspect!"
The Light Clock Experiment
If we're to learn about time dilation then perhaps the best example to analyze it quantitatively is by discussing a very famous example known as the 'Light clock experiment'
Let's say that Alice is on a train with a very high velocity carrying an atomic clock. The clock has a photon bouncing up and down, so when the photon bounces upward and returns backward that period is 1 unit. This is the example of an atomic clock that Alice is carrying...
Now we will discuss this scenario in 2 frames of reference:
1. Alice's frame of reference(Rocket frame)
As we already have discussed from the example before, when moving with contact velocity, Alice sees herself at rest and finds the motion of the light clock to be constant. In other words, her motion is not affected by the motion of the train.
2. Bob's frame of reference(Lab frame):
For Bob, however, the case is different. He observes that when the photon is moving from down to up with speed c, the upward part of the clock is moving to the right at a very high velocity, so he observes that the photon hits the upper part a little later than usual. Note that to observe this phenomenon, the speed of the train has to be extremely high (but less than 'c' of course). When the velocity is extremely high he observes the path of the light being altered.
It might be difficult to analyze this so I manage to put an animation of what I am describing.
 |
| For Alice's frame of reference |
Now for Alice, from her frame of reference, she is stationary and finds her atomic clock moving at the same speed. In short, she sees no differences in the motion of her light clock.
But Bob, however, as we see from our animation, observes the distance between the path of the light elongated. Because Alice is in high motion, Bob sees that when the light gets near the 'upper part' of the atomic clock, the distance is a little bit shifted. Of course, for this to work, Alice needs to have a very high speed (closer to the speed of light). Now since the speed of light is constant from the principle of light constancy, Bob observes the light clock hitting the path more slowly (because of the distance). This is the qualitative description of the time dilation but it will be very reliable if we describe this principle from quantitative purpose.
 |
| From Bob's perspective |
Here is a snapshot of Alice's light clock from her frame of reference
\( \Delta t_a = \frac{\text{distance travelled by the light path(2l)}}{\text{velocity of light(c)}} \)
Now here is a snapshot of Alice's light clock from Bob's frame of reference:
\(\Delta t_a = \frac{\text{distance travelled by the light path}(2l)}{\text{velocity of light}(c)}\)
where \(x = v \times \Delta t\)
\(t_{\text{}_B} = \frac{2D}{c}\)
Now let's try to analyze the figure more clearly by breaking it into some parts:
We know,
This is the ultimate equation for time dilation. As we can see, when something is traveling at a very high speed the observer sees their time dilated but the motion body doesn't find their time to be dilated. This creates a discrepancy in time resulting in dilation.
Let's plug in some numbers to get the exact value of time dilation
when v=0.01c
We count c=1 for the comfortability of the expression
γ= 1.00005
Similarly,
V=0.6c , t(rest frame)=0.001s then
γ=1.25 and t(moving frame)= 1/1.25 *0.01 = 1.25*10^-3 s
Another where t(moving frame)=60 min, γ =1.25
then, t(rest frame)=γ*t(moving frame)= 48
------------------------------------
This shows that the running clock runs slow from the rest(lab) frame of reference.
Now why don't we see time dilation daily? Well, the answer is obvious... we need higher velocity. If we use a lower velocity like 2000m/s (which is still pretty high), we would get an almost negligible and very small value.
Thus to experience time dilation, we have to have extremely high velocity.
This is another reason we cannot travel at the speed of light because if we replace our velocity with 'c' then 'γ' would be infinite which is not theoretically possible.
Thus, traveling at the speed of light is impossible even theoretically.
Length Contraction:
When a body is traveling at a very high speed, then not only time dilates but the length of the moving body is also contracted from the observer's frame of reference. This phenomenon of contraction of length whilst traveling at very high speed is called 'length contraction'.
Let's say that Alice is on a rest ship and Bob is moving to the left with velocity(V) on Alice's frame of reference. They both have two clocks one at the end and the other at the front of the ship.
To measure the length of the moving ship(Bob's ship) we can use time and velocity as our unit. When Bob's front ship exactly matches Alice's front it shows, that they both observe their time.
Bob, when his ship aligns with Alice, sees the time in his clock at (tb₁), and Alice in her frame observes the time at (ta₁). Now when Bob's ship passes Alice(Bob's back aligns with Alice's front) he measures the time as tb₂ and Alice has her time at ta₂.
The figure of length measuring...
In fig1, photo is taken at tA1 and tB1
In fig2, photo is taken at tA2 and tB2
'Alice's' result for the length of Bob is then:
(length of Bob in Alice's frame)= velocity*(tA2-tA1) = V*ΔtA
'Bob's' result for the length of his clock is:
(length of Bob in his frame)= velocity*(tB2-tB1)= V*ΔtB
This is how we can measure the length of a body that is in motion and if the time were to be absolute we wouldn't find any differences in the length. However, since the time is different for both of the people, the length must differ too. So,
Since ΔtA and ΔtB are different, the length is different.
This is the qualitative description of length contraction and now let's focus on the quantitative part of length contraction, exactly how much length is contacted when traveling in motion.
We know,
(length of Bob in Alice frame)=vΔtA
= v* 1/γ*(ΔtB) ------ from time dilation
Lmoving = 1/γ * Lrest
This shows that when a body is moving with a very high constant velocity, the observer finds the time dilated which also results in length contraction. So for Alice, she sees Bob's ship to be smaller than what Bob observes by his own frame.
Now this raises a question, What about breadth and width?
We can say that the length is just a form of breadth and width, in fact, we can assume length to be breadth and others which completely works. However, when a body is in motion only the unit which is horizontal to the direction of motion contracts.
By that I mean, if a body is moving upward with a very high velocity then only the height of the body aligns with the direction of motion and that is where we observe the contraction. When a body is moving to the left/right the length is aligned horizontally with velocity and that is where length contraction happens.
The length contraction is the effect of the time dilation. If two clocks that are at rest are synchronized with each other then they may not be synchronized in the moving reference frame. The fact that time is relative not only affects the length contraction but it has many effects on other parts of physics like leading clocks lagging, paradoxes like the twin paradox, and the relativity of simultaneity. Let us now embark on our journey to these effects...
The Muon
Before we discuss other principles or paradoxes of the Special Theory, let's talk about the most common example that supports relativity, 'muons'.
When the cosmic rays from the stars and galaxy hit the Earth's atmosphere it creates muons. Muons are the elementary particles that travel at 0.998c and their lifetime is around 2.2 microseconds. Since muons have extremely high velocity relativity plays a huge role. When a cosmic ray hits the Earth's surface, muons are created and the muons travel to the Earth (10000m) and when their average life is finished, they disintegrate. This phenomenon is also referred to as a 'meteor shower'.
But when we calculate the distance a muon can travel with its mean life and speed, we find that it can only travel to 660m. If that's true then how can Muon hit the Earth when the distance between the surface of the Earth and atmosphere is around 10,000m? This was a huge paradox back in the early 1900s- there were several answers but none of them provided more accurate data than relativity.
We know,
(lifetime)rest = 1/γ * (lifetime)moving ---- from time dilation
where
This is the main reason we as an observer see the muon hitting the Earth's surface. But what about from the Muon's perspective? When we described the relativity of simultaneity we found that the reference frame does play a role in time and length change. Thus from Muon's perspective;
The Muons, from their frame, are at rest and see Earth traveling towards them. During this, length contraction takes place in a way that the 10km distance is contracted to 660m.
For Muon- its time is constant, but the length is contracted (length contraction)
For Earth- its length is constant, but the time is delayed (time dilation)
We still have lots to do like the Lorentz Transformation, the invariant interval, paradoxes, and many more. However, these topics require some rigorous mathematics and other topics to be introduced. Please make sure to comment if the reader is interested in more parts on The Special Theory of Relativity. I will try my best to continue this journey so that anyone can learn relativity for free. For now, this is it! Have a great day!
By the way, thanks to Nitesh Das Yadav and Mukesh Poudel for helping me write this blog.
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Wow an awesome read!!. waiting for more !
Appreciate it!
Kudos to your hard work, indeed it's very helpful article man.
Thank you for your kind words! These feedback really do help me to write more!!!!