CHAPTER 10 : Special Theory of Relativity Demystified: Concepts, Equations, and Meaning

Have you wondered what exactly is special theory of relativity and how its applicable today from almost 120 years in real time applications? In this blog, we will traverse from the basics of general theory of relativity not just theoretically but also mathematically and experimentally. Einstein did not adjusted the equations such that it fits the experimental observation but rather opposite happened, the equation naturally was compatible and matchable with experimental calculations.

Firstly, we can start with theoretical and experimental observations. There was a lot of contradiction between Newton's way of significance of gravity and Einstein's way of explaining gravity. Newton just mentioned the accelerating effect of gravity on objects, meaning he did not considered the speed of light while deriving the equation of gravity, but Einstein considered in his special theory of relativity.

The first question that is raised while studying this theory is: what does this special theory of relativity states? It states that when an object rotates or moves faster in space its mass increases, length decreases and time slows down and spacetime separation between any two events is same. We will traverse through all of these statements in this blog with proofs.

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I) Spacetime separation

Firstly, we will first study what is this spacetime separation. Take an example of any two events happening at different places at different times in space. For e.g. I am writing here this blog at one place and one time which is one event and you the reader is reading this blog at different place at different time which is other event. So, here everything is different the coordinates as well as time.

Is there anything common in these two events?
Is there anything that is constant in these two events?

This question is the most important that was raised by Einstein. The constant thing here is speed of light. The speed of light travelled during both these two events are same.

According to Pythagoras theorem:

$$s^2=x^2+y^2+z^2$$

This equation is similar as below image attached.

In the above figure, Earth revolves around the Sun in radius R. So here R is the Radius of the Earth which remains same throughout the revolution. Here the Radius R is similar to s. The only difference is R is the physical quantity but s is not. Similar to s, the X,Y,Z  coordinates keeps changing but R remains invariant. The equation for above figure can be written as follows according to Pythagoras theorem:
                                                           $${\mathrm{R²\; =\; x}}^2+y^2+z^2$$

So, here this is the spatial distance in 3D space which is different for both events. But here if we insert here one more quantity the $s$ will be same for all and it will no longer be a physical quantity as written below:

$$s^2 = c^2 t^2 - x^2 - y^2 - z^2$$

Here, we took negative to differentiate time from spatial coordinate, as we can move up and down, forward and backward in space but could not move backward in time, it also moves forward.

Here $s$ will be same for both events even though $x, y, z$ coordinates are different. Here $s$ is the spacetime separation value. In simple words, it is a calculated value that tells how much time and location difference is there between two events. Here $ct$ is the distance covered by light during the time interval $t$.

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II) Time dilation

The second theory states about time dilation. When an object is moving at a very fast speed, time seems slower from outside. For e.g. if you keep an atomic clock inside the airplane during flight and measured time from earth's surface, there will be a small dilation in time. Airplane's atomic clock timing will lag by few seconds from Earth's clock.

Why did we took here atomic clock only? The detailed explanation is given in my 8th chapter about the atomic clock.

How does this happen? Let's understand this phenomena with the help of the equations as below:

Consider the situation where two clocks with events as:
Event 1: $(x_1, t_1)$
Event 2: $(x_2, t_2)$

$$\Delta x = x_2 - x_1 \quad , \quad \Delta t = t_2 - t_1$$

Spacetime interval:

$$s^2 = c^2(\Delta t)^2 - (\Delta x)^2$$

…… (i)

This $s$ here must be true for all observers. For light, $s=\mathrm{0\; and\; for\; objects, }$$s \neq 0$.

Consider the situation here where clock is at rest and two ticks happen at same place. So,

$$\Delta x = 0$$

Resulting in:

$$s^2 = c^2(\Delta t_0)^2$$

Here $\Delta t_0$ is the time difference when clock is at rest which means proper time.

Now observer sees clock moving with velocity $v$. So, the distance moved is:

$$\Delta x = v \Delta t$$

…… (ii)

Substitute Equation (ii) in (i):

$$c^2(\Delta t)^2 - v^2(\Delta t)^2 = c^2(\Delta t_0)^2$$

By rearranging:

$$\Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}}$$

The above equation is the final equation of time dilation, because of which moving clock runs slower.

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III) Length contraction

The third statement given by this theory is the length contraction when that object is moving faster. When an object travels faster, the length of the object's edge decreases or cut off slightly. This was proved using an experiment as follows:

There is a proper length of the rod which is calculated when the rod is at rest and the two ends of the rod is measured at the same time therefore:

$$\Delta t = 0$$

So, then $s$ will be the proper length of the rod as follows:

$$s^2 = -(\Delta x_0)^2$$

…… (iii)

When the rod is moving, at one moment, both ends of the rod will be at time $t_1$ and the other moment both the ends of the rod will be at time $t_2$. So the equation will be as follows:

$$s^2 = c^2(\Delta t)^2 - (\Delta x)^2$$

…… (iv)

By equating 3rd and 4th equation:

$$(\Delta x_0)^2 = (\Delta x)^2 - c^2(\Delta t)^2$$

…… (v)

Now, $\Delta x$ can be written as follows:

$$\Delta x = v \Delta t$$

…… (vi)

After replacing equation (vi) in (v) and rearranging we get:

$$\Delta x = \Delta x_0 \sqrt{1 - \frac{v^2}{c^2}}$$

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You must have observed in these equations gravity was not considered, Einstein then made general theory of relativity in which he described the solar system as warp like upside down bowl. The mathematical, experimental and theoretical explanation of the general theory of relativity will be given in the later chapter. Also, there is one most famous equation from special theory of relativity, which will be explained in the next chapter..

Have you wondered how gravity really works according to general theory of relativity and how it differs from Newton's description of gravity? Feel free to share your response through comments section and my social media links attached.