### Want to know how Special Relativity works in all its glory ? Read on then.

**Preface:**

In this many part series, I aim to present a proper representation of the Special Theory of Relativity, that was proposed by Einstein in 1905 (yes the title does bear a resemblance to Sean Carroll’s article on General Relativity). Here, I plan to start from the ground up, first explaining the foundations of Special Relativity (SR), and then gradually introducing the math as and when required. I am writing this for people with a background in physics (one year atleast, with a little bit of calculus) and will thus include equations and derivations at that level. I hope to provide all the mathematical rigor that comes with a deeper understanding of SR, mainly the concept of four-vectors, which will aid a transition to the General Relativity. Hope you understand this beautiful and magical theory. Best of luck.

**What exactly do we mean by relativity ?**

Relativity, in physics, means exactly the same thing as it does in English (shocking!). Every person, animal, or actually anything that is capable of “looking” at the events taking place around him sees them from his own unique perspective.

A simple example is that of when you are moving in a car. In your perspective, you are at rest, and the whole world around you is in motion. A car going at the same speed as you will appear to be at rest. A person standing on the sidewalk appears to (magically) move past you without even moving a foot. The surrounding cars moving in the same direction as you appear to be moving slower/faster to you depending on whether you are overtaking them or they you.The cars that are approaching you from the opposite direction, however, appear to be very fast(even above the speed limit!!).

However, for the person standing on the sidewalk, the picture can be entirely different. He will, likewise, consider himself at rest, and the rest of the world moving around him. He will see you go past him at speed. He will see cars going in the opposite direction move with the same speed as you are (approximately). And everything not moving with respect to him he will consider to be at rest.

This everyday experience was captured brilliantly by Galileo in his theory of relativity as early as the year 1632. To see the equations that Galileo put forth, lets see a few very important definitions which will be needed throughout our discussion.

*Event : *It is a physical situation, or occurrence, at a given point in space and at a particular instant in time. For example, a traffic light changing from red to green, or Chelsea defeating Liverpool.

*Observer: *An observer is simply an entity that is capable of recording information regarding the event that we are observing (such as its position, velocity, duration for which it occurred, etc.). Note that an observer is not a person, or an entity as such, but is instead spread over the entire frame of reference(see next definition) attached to that observer. For example, from the above examples, the person standing on the pavement is an observer, as is the person traveling in the car. An observer is thus an entity that can assign co-ordinate positions to each event it observes, along with the corresponding time at which the event took place.

*Frame of Reference : *It is the co-ordinate system fixed to each observer. The observer can use this set of co-ordinates to express the position of an event he observes and also indicate the instant of time at which the event occurred. For example, in the above example, we can associate a co-ordinate system to the person on the pavement, as well as to the person in the car which is fixed with respect to them individually, using which they measure the positions of the cars they observe.

On to Galileo then !

**Galilean Relativity:**

Galileo put forth his theory by quoting an example of a ship moving at a constant velocity. We will also do the same, however, in a bit of an idealized situation. Consider two observers, one who is at rest (with respect to us) denoted by , and the other, denoted by , is moving away with a constant velocity along the X-axis.

Let the clocks of both the frames start ticking when the origin of both S and S’ coincide. Then, according to Galileo, the measurements made in the x-direction by the frame S will exceed those made in the frame S’ by and amount , which is the distance S’ has moved in the x-direction. That is,

And in the absence of any indication to the contrary, our normal experience also dictates

To convert velocity components measured in the S frame to their corresponding values in the S’ frame, we just take the derivatives wrt time:

where is the velocity with which any object in S is moving

and is the velocity of the same object measured in S’.

And similarly for the other co-ordinates and .

Here, I want to draw your attention to a few key points which emerge from Galileo’s relativity:

- Note that the time remains invariant (i.e. it does not change) between the two frames. The time measured by both observers in S and S’ is the same. Thus, the clocks throughout the reference frames are synchronized and tick off at the same rate.
- If we restrict ourselves to any one frame, there is no way to know whether our frame is in motion relative to any other frame of reference, without looking outside our frame of reference. This means, that if we are in the frame S’, which is actually a spaceship with no windows(design issues !), then no matter what experiment we perform inside our spaceship, provided there is no influence from the “outside”, it will not give a result suggesting that our reference frame(i.e. the spaceship) is moving (this is true for ANY experiment conducted from ANY branch of physics). This curious phenomena stems from the fact that the frame S’ is moving with uniform speed, and in such frames, Newton’s laws are valid, especially his first law which says that ” a body at rest or in uniform motion continues to maintain its state of motion unless acted upon by an external force”. This is the effect that helps us discern whether we are in motion or not in everyday life, which is violated whenever there is an acceleration (braking of a car, or turning on the road). This can understood, since if is constant, then is also constant.
- This also leaves Newton’s second law unaffected:

{as }

Thus, the second law will be valid according to measurements of both observers, provided we add the conditions that and themselves are invariant.

- Newton’s third law, is also unaffected, as we are assuming the forces to be invariant, as explained above.
- Thus, we can conclude the last three points by saying that the laws of physics are the same in all reference frames.

Thus, the most important fact to come out of Galilean relativity is that the notion of *absolute* velocity is wrong. There is no absolute velocity in this universe; everything is moving relative to everything else, and there is no single “absolute” rest frame from which the velocities of all these frames can be deduced.

This fact is so important, that it was correspondingly included in Einstein’s SR, and is still held valid today.

**Inertial Frames and Observers:**

From the above discussion, it becomes clear that when a frame is moving with constant velocity, no “new” phenomena are generated due to its motion. However, this is not the case when the frame is accelerating, which results in the appearance of new forces which we call pseudo-forces (for example centrifugal force).

As an illustration of the above claim, consider yourself drinking soup in a train. If the train is moving with constant velocity, then there is no noticeable difference in the bowl of soup. However, if the train is accelerating, either speeding up or turning on a bend, then we can see the soup climb up one side of the bowl.

Thus, it is reasonable and useful to single out a class of preferred observers: those who are unaccelerated. These observers are called *inertial *observers and each has a constant velocity with respect to any other one. The associated frames of these inertial observers are called as *inertial *reference frames.

A reference frame (or co-ordinate system) can be called as inertial if it meets the following three criterion:

- The distance between point (coordinates and point (coordinates ) is independent of time.
- The clocks that sit at every point ticking off the time are synchronized and all run at the same rate.
- The geometry of space at any constant time is Euclidean.{that is, there is no curvature in space.}

From here on, unless mentioned, all frames mentioned will be inertial frames.

**Why was there a need for a theory like Special Relativity ?**

So all was fine and fun till a man called James Clerk Maxwell came onto the scene.

It all started in the late 19th century. Maxwell proposed his famous four equations of electromagnetism. Hertz, then, on the basis of these equations had proved the existence of electromagnetic waves which were found to propagate at the speed of light, i.e. approximately .

Now, at that time, there was a popular theory of the luminiferous aether going around in physics, which proposed the “aether” as the medium through which the light propagates in space. To test this hypothesis, Albert A. Michelson and Edward Morley devised their famous Michelson-Morley experiment to detect the changes in speed of light due to the motion of the earth through the aether. Simply put, the light sent vertically upward from the earth and reflected back to the detector should arrive earlier than light emitted in the direction of earth’s revolution around the sun, as it is perpendicular to the direction of earth and should not have any effect on the speed of light (just like our analysis in Galilean relativity). Click on the following links for their respective wikipedia pages.

However, the results of this experiment were negative, which gave first strong indication of the absence of the hypothesized “aether”, and proved that there was no such thing as “absolute motion” or “absolute rest frame”. But along with that, it also produced a startling result that the speed of light was the same for all observers.

This formed the second foundation of Einstein’s SR: the speed of light in free space has the same value in all inertial frames of reference.

Thus, we now have the two foundations on which Einstein based his Special Relativity upon.

**The Fundamental Postulates of Special Relativity:**

*Principle of relativity (Galileo) :*No experiment can measure the absolute velocity of an observer; the results of any experiment performed by an observer do not depend on his speed relative to other observers who are not involved in the experiment. Thus, all the laws of physics have the same form for every inertial observer.*Universality of the speed of light :*The speed of light relative to any unaccelerated observer is , regardless of the motion of the light’s source relative to the observer.

*Coming up next time : The consequences of Einstein’s two postulates, which include time dilation, length contraction, and Lorentz transformations.*

Here is the link for the next part: