The Special Theory of Relativity - Origin and Principles

1.Introduction and Galilean Relativity

Upon hearing the name - Albert Einstein, the first thought which inevitably comes in your mind is the famous equation - E = mc^2. But the equation is a direct consequence of one of the most profound theories in Physics - Einstein's Theory of Relativity. Relativity has two variants - Special Theory of Relativity and General Theory of Relativity, the former was developed for a special case of non-accelerating bodies while the latter generalizes Special Relativity to accelerating bodies. This article is going to be a two part introduction to Special Relativity. In the first part, I shall discuss it on purely theoretical grounds and the second part shall deal with the mathematical formalism of Special Relativity.

Before proceeding, it is essential to understand what Special Relativity (or just Relativity) is all about. It is not about clocks slowing down, time travel, or objects contracting. These phenomenons are merely consequences of Special Relativity. The core principle behind Relativity is to enable one to view a single event from different perspectives. The initial Theory of Relativity was discovered way before Einstein by Galileo in the 17th century. His theory (also called as Galilean Relativity) described the velocity of any object as observed by an observer in a different state of motion. The effects of Galilean Relativity are observed in everyday life. As you travel in a bus with a constant velocity and watch the trees, buildings and the roads outside run in the opposite direction, it is not just an illusion. They are actually moving in the opposite direction. There exists no ways to distinguish your viewpoint from those on the ground, that is the original Theory of Relativity. Imagine yourself travelling in a car going at 60 km/h, as you are travelling on a smooth and straight road with no bumps and potholes (in an ideal world), a car approaches to your right from behind. Initially the other car is travelling at 80 km/h but now that it approached you, the driver reduces the speed to 60 km/h and travels parallel to you for some reason. The question now is what will be the state of motion of this car with respect to you? It is easy to see that this car appears at rest for you. Now, to avoid the awkwardness you step on the accelerator and increase your speed to 90 km/h. As you look in the rear view mirror, the car appears to move away from you. The first inference that arises is that - Motion is relative. This fact is central to both Galilean and Special Relativity. For you and the person in other car both travelling at 60 km/h, both of you appeared to be at rest relative to each other. But for someone standing on the footpath, neither of you is at rest. Hence, when one speaks of motion, he/she always refers it relative to something. What appears stationary to you might be in motion for someone else. The second inference from the above situation is that the speed (or velocity) of any body is relative. Since, velocity is distance divided by time, distance too is relative. The relativity of velocity is observed from the fact that the other car appeared stationary to you when both of you had the same velocity but then it appeared to move away from you as you obtained a greater velocity. Let us now consider a slightly different scenario, you are now moving on a two-way road with some velocity. The vehicles approaching you from the opposite direction seem to move towards you with a greater velocity. More precisely, it would appear to move towards you with a velocity which is equal to the sum of both of your individual velocities. This is known as the Galilean Law of Addition of Velocities. The law included some very simple equations called as "Galilean Transformations" which gave the relations between velocities and positions of two bodies moving relative to each other.

In Relativity, and generally in all of mechanics the term "bodies" is often found to be replaced by "frames of reference". A reference frame is roughly another way of saying "observer". An "event" is then anything that happens inside these reference frames. As the word itself suggests, you view or measure any "event" with reference to that body. The frames of reference are occupied with means to measure the coordinates of such events which are usually the space and time coordinates. A ruler and a clock for example is an example of reference frame. Special Relativity involves "Inertial Frames" i.e. frames of reference moving with a constant velocity with respect to another frame. I shall now throw in a few more examples, to give a taste of how Galilean Relativity and Transformations work, which will be much more transparent with the mathematical formalism in the next part. A car is the prime example of an everyday frame of reference. Imagine yourself travelling in such a car, at a particular velocity, again on a perfectly smooth and straight road. But now, you have your friend sitting on the seat beside you. In the reference frame of the car, you and your friend are totally at rest(you are not moving inside the car). Consider again the reference frame of a third person standing on the footpath. In his frame, the car alongwith you and your friend will be in motion with some velocity. Let us complicate the situation a bit more. Your friend in the co-driver's seat decides to pick up a ball and gently throw it towards the windshield with a particular velocity of say 0.1 m/s. At this point you wonder if it was a good idea to take your friend for a ride. But you wonder something more, for you the velocity of that ball is 0.1 m/s. In other words, the ball's velocity in the car's reference frame is 0.1 m/s. What about the guy standing on footpath watching this merry sport? In his reference frame, the car, you, your friend and the ball are already travelling with the velocity of the car. When your friend throws the ball, the footpath guy should observe it to move at a speed which is the sum of car's velocity and 0.1 m/s. This is just the Galilean Law of Addition of Velocity. But, the intuition is self-explanatory without any law.

2. Electrodynamics in the 19th century

Let us take a quick detour to the developments taking place in the field of Electromagnetism in the 19th and early 20th century. The notorious Maxwell Equations introduced by James Clerk Maxwell in 1865, demonstrated the wave nature of light and of electromagnetic radiations. But they also showed something more fundamental, something that would become so important that it radically altered our perception of space and time. The equations mathematically proved that the speed of electromagnetic radiations in vacuum is always constant - which is roughly 3 lakh km per second. Physicists at the time were very curious to decipher the wave mechanics of electromagnetic radiations. It was known at the time that a material medium is obligatory for wave propagation. Sound waves travel through the air medium, water waves travel in water, transverse waves on a string use the string itself as a medium and so forth. What about light then? They deduced that for light to travel as a wave, a hypothetical medium must permeate through all of free space. This hypothetical medium was named as - "luminiferous aether" or simply "ether". Physicists then embarked on the task of experimentally finding this ether and its properties. This is where "Galilean Relativity" steps in the game. If electromagnetic radiations travel through the ether, then they must have a velocity relative to the "ether". Imagine the car scenario that was mentioned earlier, when your friend in car throws a ball at the windshield and a person from the footpath observes the event. For him the total velocity of the ball should be equal to the car's velocity plus the ball's velocity. Now imagine the same scenario but this time, replace the ball by light waves, the car by "ether" and the guy on footpath by Michelson and Morley - two American physicists at Case Western Reserve University.(You get the idea, skip to the last two lines of this para to cut the details). It was reasoned at the time that if the ether did exist then the motion of Earth through space and through the ether should create an "ether wind". Light moving through this ether will either travel upstream through the wind or downstream. This wind should have an effect on the velocity of light. The effect would be maximum for light travelling in the direction of this wind and minimum for light travelling perpendicular to this wind. Michelson and Morley proposed an experiment called the "Michelson-Morley Interferometer" to detect inconsistencies in the speed of light in two different directions. It came as a big surprise when the results were totally different from the predictions. The velocity of light in the experiment remained constant regardless of its direction of propagation. This and many other experiments disproved the existence of "ether". It was then realized that the electromagnetic waves propagate in the electromagnetic field by self-sustained, oscillating electric and magnetic fields. But the experiment also implied something far more bizarre. It implied that the velocity of light should always remain the same in any reference frame. Whether you are at rest or in motion, the speed of light will never change.

3. Postulates of Special Relativity and the Relativity of Time

Revisiting our car example, but this time we are going to bring in the postulate of the constancy of light speed. In the example as your friend throws a ball at the windshield, the observer on the footpath observes the ball's speed as a sum of the car's speed and the speed with which your friend threw the ball (which we supposed to be 0.1 m/s). In your frame of reference however, the speed of ball was just 0.1 m/s. Now, you hand over a torch to your friend and he shines it on the windshield. In the car’s reference frame, the light travels from the torch to the windshield at its usual velocity of 3 lakh km per second. If we employ the Law of Addition of Velocities then, the speed of light should be greater than 3 lakh km per second in the reference frame of the guy on footpath. Unfortunately, Galilean Relativity runs into a problem here. There is contradiction between the theoretical predictions of Galilean Relativity and the experimental observations of the Michelson Morley Experiment and many other experiments which proved that light always travels at a constant velocity. This inconsistency demanded either of the two propositions to be changed. It is obvious that the experimental evidence always prevails in such situations. Hence, a modification of Galilean Relativity was necessary to prevent the constancy of light speed to be violated. Furthermore, the modification should be such that it should preserve the Principle of Consistency. It means that the new theory should account for the successful predictions of the old theory which it seeks to modify. Rather, the old theory must exist as a limiting case of the new theory. Such limiting cases are prevalent in Einstein’s Relativity as well as in Quantum Mechanics, where in the proper circumstances they can be approximated to Galilean Relativity and Classical Mechanics, respectively.

Finally, Einstein came along in 1905 and published his paper – “On the Electrodynamics of Moving Bodies”[1] which presented the ideas of Special Relativity. His idea was based on two main postulates – The first postulate was that the speed of light is constant in all frames of reference. The second postulate stated that the laws of physics are “invariant” i.e. they don’t change for different frames of reference. It was already known that the first postulate violates Galilean Relativity. Einstein’s genius was to realize the fact it necessarily implies that time is “relative”. Before 1905, Time was thought to be an absolute entity. Sir Isaac Newton along with Galileo believed that every observer in the universe should have the same perception of time regardless of their position or state of motion. The transformation equations of Galilean Relativity stated that position of an object is different for different reference frames i.e. “space” is relative. But, the equations showed that “time” is the same in all reference frames. There is a universal clock for our entire universe and every observer should agree with the time given by this clock. Einstein disproved this notion with his famous “thought experiments”[2] i.e. experiments conducted in his own school of thought. No fancy laboratories or machines, just his remarkable brain and the power of imagination. Firstly, if light always travels at a constant speed then there is a fundamental limit on how fast information from an event reaches us. Since, the easiest way we receive information is from light itself entering our eyes, implies that the information from any event should take a finite amount of time before reaching us. The light emitted from the surface of Sun takes nearly 8 minutes and 20 seconds to reach Earth. Next time, when you look at the Sun understand that you are looking at an 8 minutes older version of it. There is always a lag in the information you receive. If the Sun were to explode at this instant, we would observe it 8 minutes later. But for someone located closer to the star, the light should reach them before us. So for them it had already exploded, but for us it is yet to explode. All of this Sun blabber means only one thing, the distinction between Past, Present and Future doesn’t exist. An event which occurred in the past for you is yet to occur for someone else.

4. Relativity of Simultaneity

The consequences of the first postulate are quite clear to understand for bodies at rest with respect to the light emitting source. What about a body in motion? Let us consider the original thought experiment which Einstein imagined. You are inside a railway carriage moving at a considerable velocity. There is a raging thunderstorm outside. A friend of yours is standing on a platform at the side of railway track. You decide to stand at the door of railway carriage and watch outside. Just as you are watching, two lightning bolts strike on the ground in front of you at two different places simultaneously. The two locations are at equal distances from you and your friend. Quite conveniently, it happens so that you just cross your friend on the railway platform as the lightning strike occurs as shown in the diagram below. At this point neither you nor your friend knows anything about Special Relativity. You are still living in the era of Galilean Relativity. However, it is just known to both of you, that light always travels at a constant speed. Owing to this fact, the light coming from both lightning strikes will take a certain amount of time to reach you and your friend at the platform. But, there is a crucial difference between the state of motion of you and your friend. You are in motion with respect to the platform, whereas your friend is at rest. Since, you are in motion along with the railway carriage, you are moving towards the light coming from one lightning bolt and away from the light coming from the other bolt. Consequently, the light from one bolt will reach you before the light coming from the other or, “you will reach the light from one bolt before the light from the other bolt can catch up with you”. (I rephrased the sentence just to emphasize on the fact that the speed of light coming from both bolts is the same). So for you, the lightning strikes weren’t simultaneous. But for your friend on the platform, who is completely at rest with respect to the platform, the lightning strikes were indeed simultaneous. [3]

The results of this thought experiment are definitely pesky and hard to wrap your mind around. This is so because the Newtonian perception of time is deeply ingrained in our brains. However, this simple thought experiment marked the end of the era of absolute time. Not only time was rendered relative, the thought experiment also proved that the word “simultaneous” holds no meaning in Special Relativity. At this point of time one is definitely tempted to ponder upon the validity of such results. In fact, one might feel these results to be paradoxical. Such thoughts are often justified; even prominent physicists at the time of Einstein struggled with Special Relativity. Going back to the thought experiment, one thing to note is that there is no conflict of thoughts between you and your friend on the platform. You agree with the fact that the lighting strikes are non-simultaneous for you but simultaneous for your friend. Why? – Because in your reference frame, you are at rest and your friend is moving in the opposite direction to you (as shown in the diagram below). In your reference frame, the lightning strikes were non-simultaneous because they “were” actually non-simultaneous. For you, the light from the two bolts wasn’t generated at the same time. Hence, the light from the bolt towards which you were moving was generated first later followed by the light from second bolt. How do you know that the lightning bolts appeared simultaneous to your friends on the platform? – As said before, in your reference frame – you are at rest inside the railway carriage and everything outside along with your friend on the platform is in motion. So for you as the first lightning bolt strikes, your friend is actually moving away from it. As the second lightning bolt strikes in your reference frame, your friend is moving towards the light coming from this bolt. Hence, the event which occurred a little late for you, actually occurs a little early for your friend. In your reference frame, his backward motion accounts for the delay in the two lightning strikes. This viewpoint is very crucial and is often misunderstood when learning Relativity for the first time. Notice, how there is a symmetry in the interpretations from different reference frames owing to the fact that motion is always relative. In your friend’s reference frame, the lightning struck simultaneously. In his reference frame, he was at rest and you along with the railway carriage were in motion. For him, the reason those two bolts didn’t appear simultaneous to you is because you were moving towards one bolt and away from the other. Thus, even if there is no conflict in the simultaneity of those two events between you and your friend, there exists a conflict in the interpretations behind them. Everything ultimately boils down to the relativity of motion and constancy of light speed.

5. Conclusions

After learning all these thought experiments one is tempted to ask the following question (like I did) – “Is there an absolute underlying reality?” Since, this entire humble jumble took place just by allowing observers into the scene. Is there a reality independent of observers? We then run into a plethora of philosophical and metaphysical arguments. Such observer and consciousness oriented arguments are found in Quantum Physics too. Personally, I believe that there is no such reality. In this thought experiment we talk of observers, but one can very well forego such talks and just consider frames of reference. As we will see later, the entire Universe is filled with reference frames. These frames stretch, curve, wobble and do all sorts of dance in the presence of energy and mass. Such effects are now observed experimentally and that is how we know Relativity must be right and that the loss of simultaneity is actually the reality and that motion, time and length are purely relative. I shall now conclude this article. In the next part, all the open ends will be tied with the mathematical structure of Special Relativity, with the introduction of its consequences like time dilation, length contraction, etc.

6. References and Sources

[1] http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf

[2] https://fs.blog/2017/06/thought-experiment/

[3] https://www.youtube.com/watch?v=-jJ5PPcLUw8