Einstein's Special Theory of Relativity - I

Physics has evolved in such a way that we are now at the core of understanding our universe. String theories, dark matter, and quantum gravity are our next steps in understanding our universe. These upcoming theories play a vital role in the unified theory of everything, and their presence & mathematics is definitely a necessary element to understand the cosmos or the time itself before the big bang. However, no matter how important the theory is in reality, we still have much to achieve. Sure we think the upcoming hypothesis would change our way of thinking about reality, but what about the other theories? Modern Physics was not developed in a day, it had constant changes in its principle. Light being at a constant speed, and electron behaving as a wave; these notions must not be forgotten as they helped create more and more groundbreaking laws. 

The year 1905 marked a turning point in the history of physics when Albert Einstein published his Special Theory of Relativity, a groundbreaking work that would forever change our understanding of time and space. With this theory, Einstein challenged the fundamental assumptions of physics that had been held for centuries and opened up a new realm of questions and possibilities that continue to fascinate and intrigue scientists to this day.

"At its core, the Special Theory of Relativity proposed that the laws of physics are the same for all observers in uniform motion relative to one another, regardless of their individual speeds or directions. This idea led to some bizarre and mind-bending conclusions, such as the famous "twin paradox" that shows how time can appear to move at different rates for different observers."

Einstein's theory also introduced the concept of the speed of light as an absolute limit on how fast anything can travel through space. This led to the idea of time dilation, where time appears to move slower for objects that are moving faster, as well as length contraction, where objects appear to shrink in the direction of their motion.

These ideas have had profound implications not only for physics, but also for philosophy, mathematics, and even popular culture. The Special Theory of Relativity has inspired countless works of art, literature, and music, and continues to be a rich source of inspiration and fascination for people around the world.

So join me on this journey as we delve into the deep and fascinating world of Einstein's Special Theory of Relativity, and explore the mysteries and wonders that lie at the heart of modern physics.

Also, this is just part 1 of the theory of special relativity and here we will be discussing the main postulates and some important basic foundations to grasp the upcoming mathematics...

Space-time diagram

If we try to understand the motion of a body in free space then we need to focus on two things- position and time. In fact, to understand special relativity one must realize the effects of motion with time and for that, mathematicians have invented a simple method: a space-time diagram.

A space diagram is basically a graph with 4 dimensions ( length, breadth, and height {position} + time). But why do we need a 4th dimension if we can understand the motion of a body in 3rd dimension? See that's where we are wrong. For instance, if we try to know the behavior of a body let's say 'A' then we need to be sure of where it is located (position) and we also need to know about its time, when is it? 

If we are only certain about the position but not time then we cannot figure out the body's behavior, we also need to know the time too. This is the key principle to understanding the special theory and space-time diagram. So, if we plot a 2d graph X-axis being the distance(position) and Y-axis being the time, we get a space diagram. A side note that here we are neglecting the implications of The Heisenberg uncertainty principle.

Let's say we are plotting a space diagram of a rocket that is flying in space.

This is the space-time diagram of that rocket. Here we can see that at time t=0, the rocket is at a distance of 0 (origin), and at time t=1, the rocket is at a distance of 1 km, and so on. This graph is linear, the y-axis corresponds regularly with the x-axis. 

Finding velocity from the Space-time graph

If we have distance and time then we can also find the velocity of that rocket. We are very familiar with the velocity time equation:

$$ \textit{velocity} = \frac{\textit{distance}}{\textit{time}} $$

Hence we can calculate the rocket's velocity separately by dividing the distance by time.

At time (t)=1, the distance(d)=1 so the velocity equals to 1/1 = 1. At t=2, d=2 and hence v=2/2 =1km/s.

Thus it is very easy to calculate the velocity of an object when we have a distance-time graph. Now we're going into more detail to understand the graph and its slope.

A steeper slope means a higher velocity

A slope of a line is the measure of its steepness. The higher the hill, the higher the steepness. Let's look at a couple of graphs to see which figure has a higher slope. 

fig A

fig B

            

As we can see from the graphs, Fig A is more steeper than Fig B, this means that the slope of the line from Fig A is greater than the slope of the line from Fig B. 

Mathematically,

$$ \text{slope} = \frac{\text{rise}}{\text{run}} $$

 Hence, y= vertical axis(rise), and x=horizontal axis(run)

$$ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} $$

TL;DR higher steep means higher slope and vice-versa.

Now, we may think why do we need slope in general. Well, it seems that finding a slope gives you tons of information about the line or curve. If we measure the slope of the curve we find the tangent to that curve, calculating derivatives, finding the speed, and so more. 

Velocity from the space-time graph

Mathematically, velocity = distance/time

Now you may have got a hint from this equation, the slope of distance and time graph gives out velocity. In other words, if we find the rise/run from the space-time graph we'll get the velocity. 

Earlier, we also talked about the relation between the steepness and the slope. Similarly, if we plot two space-time graphs with different slopes then we can compare its velocity without even calculating any operations.

From these two graphs, we can clearly see that the slope of the first graph is more. Thus we can say that the graph A has a higher velocity than B. If you want then you can also check the velocity by the earlier process.

∴ Velocity of graph A > Velocity of graph B. 

We don't have to calculate the velocity to compare it with others, instead, we can compare it in a graph. 

We are now familiar with space-time diagrams and the determination of velocities from its slope. Let's delve into the integral aspects of the special theory of relativity which are basically two postulates. These two postulates are essential parts to comprehend the special theory and their formation is what made the theory a groundbreaking philosophy.

Postulates of The Special Theory of Relativity 

Postulate no. 1
 / Principle of Relativity 

- The laws of physics are the same for all observers in uniform motion relative to one another.

This postulate is also known as the principle of relativity. It states that the laws of physics are the same for all observers who are moving at a constant velocity relative to each other. This means that there is no preferred reference frame and that there is no way to determine who is "at rest" or "in motion" without reference to some external object or frame of reference.

Imagine two trains traveling alongside each other in the same direction at the same speed. Inside each train, there are two people - Alice and Bob. Alice is on the first train, while Bob is on the second train. Both trains are moving at a constant velocity relative to the ground, so they are in a state of uniform motion.

Now, from Alice's perspective, Bob's train is moving, while her own train is stationary. From Bob's perspective, it is Alice's train that is moving, and his own train is stationary. Both Alice and Bob can use the laws of physics to describe what they observe inside their own train, and they will find that these laws work in the same way, regardless of the motion of the other train.

This scenario illustrates the first postulate of the Special Theory of Relativity - the laws of physics are the same for all observers in uniform motion relative to one another. Alice and Bob are both in uniform motion relative to each other, and they can use the laws of physics to describe what they observe. There is no way to determine which train is "at rest" or "in motion" without reference to some external object or frame of reference.

Another example:

Imagine two spaceships in space, Alice is in spaceship 1 which is at rest and Bob is in spaceship 2 which is traveling along the right side from Alice. Now when Alice observes Bob, she thinks that Bob is in motion (moving to the right) and she, herself, is at rest. But when we go to Bob's observation when Bob is observing he sees that he is at rest and Alice is moving towards his left. 

Which one of these scenarios is correct? Well both of them are correct in their own sense. When there is no other outside factor to compare the motion of the body, we are moot! Thus, Alice sees Bob in motion and Bob sees Alice in motion.

This is the first postulate also called "The Principle of Relativity" which tells us that the motion is relative...

Postulate no. 2 
/The Electromagnetic Theory

Before postulate no. 2 we need to address the elephant in the room, the main backbone of the special theory of relativity... 

Long before Einstein's theory of the photoelectric effect, light was considered an electromagnetic wave from Maxwell's equation. However there was still a lot of debate to prove light was a particle and Albert Einstein, from his photoelectric effect, proved this notion. Hence this began the first of the several hints to produce Quantum Physics. 

Both of the theories were correct in their own way; Maxwell's equation exactly predicted the behavior of light by identifying it as an electromagnetic wave and Einstein's equation helped to understand X-rays and the solar effect. Despite being a competition, there was something that lacked Maxwell's theory of light being a wave: medium...

If the light is a wave then what is its medium? We knew that classical mechanics required a medium to propagate waves; sound waves needed air medium to travel, water waves needed water to travel. But how can light travel in a vacuum without any medium, in other words, what was an electromagnetic wave's medium?

This was a huge blow to classical mechanics. Maxwell's equations predicted light's behavior precisely and there were several other experiments like the double slit experiment that accurately proved light to be a wave. So we were 100% sure that light must be a wave and so began the search for the grand suspicious medium: The Ether.

Many scientists started searching for so-called this ether, some of them tried hundreds of times to figure out the perfect equation that would demand the medium of light. Many experiments were conducted to evidence the ether but none of them were successful. Even Einstein was confused about the notion of ether and how light acts both as a wave and a corpuscle (particle). 

This all came to an end when an experiment known as the "Michelson Morley Experiment" found that there is no such thing as ether.

Now I apologize for not explaining further about The Michelson Morley Experiment. It will be extremely lengthy if we address this experiment. However, there are tons of videos online where you can easily understand the objectives and results of the Michelson-Morley experiment. If you want to learn more about this experiment and other hypotheses to prove Ether's existence like Stellar aberration and so on, make sure to comment on this channel. 

Thus from Michelson-Morley Experiment, it was found that there is no such medium called Ether. Light doesn't need any medium to travel in space. 

The foundation of light not having a medium made Einstein realize the constancy of light's speed. He said that the speed of light is constant no matter what relative speed an observer has. He believed that no matter how fast an observer is traveling, they will always observe the speed of light as 'c'.

To understand this principle of light constancy, we need to follow a diagram:

Let's say that Alice is in a car that is moving with velocity v to the right. On top of her car, there is a small tennis ball machine that shoots a ball with velocity '2v' to the right. When Alice is driving the car, she shoots the ball accordingly. Now imagine there's an observer there, Bob, who observed this scenario. If we try to describe their frames of reference :

From Bob's perspective:

For the observer, when Bob sees Alice driving a car to the right, he sees the velocity of the car 'v'. And when the tennis ball is shooted at 2v, Bob sees the velocity added up. Hence for Bob the velocity of the tennis ball = Vel. of the ball (v)+Vel. of the car (2v). 

Bob sees the velocity of the ball at 3v.

From Alice's perspective: 

Now Alice, however, doesn't observe the same as Bob. She sees herself as rest and instead sees Bob moving towards his left. Remember what we discussed from the first postulate, the frame of reference is relative for all observers. So when Alice from her perspective sees herself at rest, she finds the moving ball only at 2v.

This creates a discrepancy, Alice measures the velocity of the ball as 2v but Bob measures the velocity of the ball as 3v. They both are correct according to their frame of reference. 

This phenomenon of the addition of relative velocity applies to everybody in this universe except light.

Einstein predicted that no matter how much speed the object is relative to the observer, the speed of light is always constant. This means that if we replaced the tennis ball machine with a laser light and let Alice and Bob measure its velocity, they both would find the speed of light 'c'. This is the second postulate called "The principle of light constancy", the speed of light is always constant even if it is relative to the observer. Now why is it that the velocities add up to almost every object but not light, in other words, why is the speed of light constant on any frame of reference. Unfortunately, there is not enough evidence of why light's speed is constant but physicists have several theories to explain this strange behavior.   

Now these are the only topics I will be covering in this part. Here, we discussed the two main postulates of the special theory of relativity, and in the upcoming parts we will be going into more detail: the maths behind the special theory of relativity. In the next article, we will address the mathematics behind time dilation, length contraction, and so on and so forth.

To sum up,
The special theory of relativity, formulated by Albert Einstein, is based on two postulates. The first postulate states that the laws of physics are the same in all inertial reference frames, meaning that the fundamental principles of nature apply universally, regardless of an observer's motion. The second postulate is that the speed of light in a vacuum is constant for all observers, regardless of their relative motion.

By combining these postulates, Einstein derived remarkable phenomena such as time dilation and length contraction. Time dilation refers to the fact that time can appear to pass slower for an object moving relative to another object at rest. Length contraction, on the other hand, describes how objects moving at high speeds can appear shorter along their direction of motion when observed from a stationary frame.

Building upon the foundation of special relativity, Einstein developed the theory of general relativity. General relativity extends the principles of special relativity to include gravity. It proposes that gravity is not a force in the traditional sense but rather a curvature of spacetime caused by mass and energy. In this theory, massive objects like planets and stars deform the fabric of spacetime, causing objects to follow curved paths.

General relativity beautifully explains phenomena like the bending of light around massive objects, the slowing of time in strong gravitational fields, and the existence of black holes. It revolutionized our understanding of gravity and laid the groundwork for modern cosmology, where the structure and evolution of the universe are studied in the context of curved spacetime.

Stay tuned for Part-2