You might have heard about Special relativity and how it revolutionized the way we perceive time. Special relativity showed us that time is indeed relative and it actually depends on the speed at which you are moving but how is this relation true? Why is time dependent on an observer’s relative speed? What has it got to do with the speed of light? This article answers all of these questions and tries to provide a mathematical model of time dilation.

## What is Relative speed?

Before we start talking about time, it is important for us to understand an important concept: relative speed. It's a well-known concept that speed is relative. What I mean by this is that if you imagine a car moving at 100 km/h then you might not be technically correct. You will have to say that the car is moving at the speed of 100 km/h relative to something. Imagine if the car is moving at 50 km/h relative to the ground and you start running beside the car at the speed of 10 km/h then will you also perceive the car’s speed as 50 km/h? Will there be a difference between what you perceive the car’s speed when you were stationary than when you were moving beside it? Yes, there will be. When you are stationary, you will perceive the car’s speed as 50 km/h but when you are running at the speed of 10 km/h beside it, then you will perceive its speed as only 40 km/h. This is also the reason why you feel as if you are stationary inside a flying aeroplane. You are moving with the aeroplane so relative to you, the aeroplane’s speed is 0. These concepts illustrate that speed is relative, which means speed depends on the frame of reference of each observer. So it's always vital when you're talking about speed to recognize that you can only ever frame the idea of speed for ordinary objects, that we encounter, as the object has this and that speed relative to this or that object. You need to specify the reference in order that the speed that you are specifying has any meaning at all.

So Einstein showed us that just like speed, time itself is relative. He answered questions which no one was asking. He showed us that for different observers in different frames of reference, time indeed runs differently. He showed us that time is dependant on the relative speed of the observers. But how is that? How can we say that time is relative? Well, we can look at a thought experiment: light clocks.

## A thought experiment involving light clocks

What is a light clock? A light clock is a contraption in which we have two mirrors that are facing one another and a ball of light bounces in between them and every time the ball goes up and down you can think of that as tick-tock and we can say, every time the light bounces back, a second passes.

Conceptually a light clock is no different from any other kind of clock which means any conclusion that we reach about the nature of time that makes use of this light clock applies to any Clock. The reason why we are using a light clock here instead of a traditional clock is that it's easier to observe this effect in a light clock. Now let's introduce a similar light clock but this time, this clock is in motion. Imagine in your mind that I have one of these light clocks and I'm going to walk with it. From your perspective think about the trajectory that the light will travel. The light will start here. hit the top of the mirror here and then it hits the bottom mirror here so from your perspective the light will have undergone a diagonal up and a diagonal down trajectory as I'm walking with the light.

Now the since the light beam is travelling diagonally, it has to travel more in order to return to its initial position, what does this tell us? It tells us that the stationary clock and the moving clock won't agree with each other on the time elapsed. It is going to take longer for the moving clock as the light beam will have to travel up and down diagonally. In other words, the period between the first second of the moving clock will take longer than the first second of the stationary clock.

Does that mean one of them is wrong? Absolutely not. Both of those observers are right because time is relative. This is a completely certified fact. We have observed time dilation experimentally. We have conducted experiments using ultra-precise clocks, atomic clocks, and all those experiments show that time dilation indeed exists. The clock paradox effect also has been substantiated by experiments comparing the elapsed time of an atomic clock on Earth with that of an atomic clock flown in an aeroplane. We have experimental data to prove that time dilation is a real phenomenon and not just some numbo jumbo.

## A mathematical model of time dilation

So a natural question arises, how can we calculate this time difference? Well, Einstein gave us an equation to actually work out the dilated time.

T = To/ ( 1- v^2/c^2) ^½

Where: T = Time observed in another frame of reference To = Time observed in observer’s own frame of reference V= relative speed at which the observer is moving C = speed of light

Now if you closely look at the equation then you will realize that as the speed at which the observer is moving gets bigger and bigger, the value of T increases. This shows that as the relative speed increases, the dilated time increases. So to put this in context, imagine that you decide to go around the world at the speed of 2.5 * 10^8 and you spend 10 hours travelling at this speed. Your friend on the other hand decides to stay at home and binge-watch the new season of peaky blinders. So what will be the time difference between you and your friend when you return back? Essentially, how long will your friend think that you were out travelling when you finally come back? So if he sees you leaving on your adventure on a bright Sunday morning then how many hours would have passed between that Sunday morning and the day you return back? So we can simply calculate this by inputting the values in our equation: T = 10 / ( 1- 2.5 *10^8/3 * 10^8) ^½ T= 24.5 hours So the time your friend measures will be 24.5 hours whereas the time you measure will only be 10 hours and both of you are correct! And as the speed at which you are travelling gets higher, this time difference increases even more. And this time difference reaches infinite if you are travelling at the speed of light which means that time will essentially stop if you are moving at the speed of light! If you want to further explore what happens at different relative speeds, you can try this amazing website.

Before Einstein, we used to think time is absolute. We used to believe that time runs at the same pace for everyone and that it is an intrinsic property of our universe. But Einstein showed us that time is actually dependent on an observer’s frame of reference. He showed us that time is relative. He transformed the way we look at the time and made us really think that sometimes what we think is intuitively correct is actually not right. Today, time dilation is used in satellites, Military, Aerospace and so many other industries. It surrounds us; it is a part of our lives but we just don't experience its effect because the speed at which we generally travel is pretty low. But an important thing to note is that speed is not the only factor that affects time. General relativity, a theory of gravitation developed by Albert Einstein between 1907 and 1915, also predicts that gravity affects the passage of time and experimental data has confirmed this assertion too. There will be more on how general relativity affects time soon on this blog.

Further interesting resources you can try: Loss of simultaneity (train thought experiment)