The reason for writing this article comes from a quote of Einstein. Einstein , the person who is regarded as one of the smartest people to have ever lived , had said ” If you cannot explain it simply , you don’t understand it well enough”. So here I am , trying to see if I do understand , while helping the general crowd to realise the impact this guy had on physics.
However , before we start , I am going to assume you know a few things.. the first being high school mathematics , and the second being that you know that light is the fastest thing in the universe and that nothing can travel faster than light.
And one more thing… The special theory of relativity applies only to an ‘Inertial reference frame’ .
What that means , is that it applies only to things that are NOT accelerating . They are either ‘stationary’ or moving with a constant velocity. Also , it does not apply to things moving in a circular path or a curve. The reason for this is that if you move in a curve , you have something called a centripetal acceleration. Which means you are accelerating ‘inwards’.
Also if you consider yourself to be in a car , lets call the car S ( Get excited , you can choose to be in a Bugatti Veyron if you want)
and there is a car in front of you , lets call that car S’ . Now , all you can tell is that S’ is moving away from you at a velocity v. But that the same time , a person in S’ could say that you are moving away from it with a velocity v.. Or maybe you both are moving away from each other with a ‘combined’ velocity v …There is nothing you can do to tell the difference.
Imagine the car you are in to be at rest for the moment and S’ is a little bit behind you in the next lane .. now your car , that is S , has a wire coming out at the side and S’ has the same. When S’ reaches you , a current is passed and the clocks on both of the cars are ‘synchronized’ or lets say the both hit 12′ o clock. And this is t= 0 . What you might think here is that both the clocks will forever be synchronized. Well , just wait and watch.
Now , after the clocks were synchronized , S’ moved some distance away from you . Now imagine a flash of light occurs and you in S and the other guy in S’ see it.
Now , you say that the light flashed at a distance of ‘x’ from you when the time was ‘t’
And S’ says that the light flashed at a distance ‘ x’ ‘ from it and the time measured ‘ t’ ‘
Now , you can say easily that x = x’ + vt’ ( for you , S’ is moving at the velocity v and t’ is the time S’ recorded when the light flash occurred )
x’ = x – vt
This seems pretty decent until now…
Let us now look to an experiment conducted my Michelson and Morley …. well , lets look at an analogy that is easier to understand.
Consider yourself to be running on the top of the train because you are a cool , badass individual.
Lets say , for the sake of understanding that you run at 20 m/s . and the train runs at 50 m/s .
The dude on the platform , while still in awe at what you’re doing will tell you that he sees you move at
50+20 = 70 m/s.
which is quite obvious , actually.
If you run in the direction opposite to that of train , the guy on the platform will say he sees you move at
50 – 20 = 30 m/s.
Now , lets say Michelson and Morley weren’t as badass as you , so instead they took a flash light ( speed ‘c’ or about 3 x 10 ^8 )and shone it in the direction of the train and asked the dude on the platform at what speed was the light appearing to travel at?
To the surprise of the whole world , he said it it was ‘c’.
They expected him to say that it would be c + 50 m/s . Which would be the obvious answer.
It turns out that in whatever way you measure the speed of light , the answer is always ‘c’ .
This was completely mind boggling and counter intuitive.. One might even say they had some erroneous clocks of a scale (or ruler .. whatever you call it ).
However , the fact that the speed of light was always measured to be ‘c’ was the conclusion of the Michelson and Morley’s experiment.
Now , lets to back a little , remember when the clocks in S and S’ were synchronized when the wires protruding from both cars touched? Lets say that when the wires touched , a signal went from your car ,S, to a point far down the road and it caused the light , which both of you saw , to appear.
When the light does , in fact appear , S’ has moved a little down the road .
Now , you , in the ‘S’ car will say that the light appeared at a distance ‘x’ , when time was ‘t’ and therefore ,
and the S’ guy will say that the light appeared at a distance x’ when the time was t’ and therefore ,
about here is when your mind should explode, because since x and x’ aren’t equal , t and t’ cannot be equal because , we know from Michelson and Morley , that both the observers , you in the car S and the other guy in car S’ , will measure the same speed for light , c.
Now you say that there must be an error made by S’ while measuring either distance and/or time . So you say that
x’ = ( x-vt)z
where z is the error in measurement.
But , at the same time , S’ says that
x= (x’ + vt’ )z
We use the same z because you both call each other wrong , but you cannot say that the other is more wrong or less wrong than you are , obviously.
Simple mathematics will let you find the error ‘z’ .
All you have to do is multiply the equations , resulting in
xx’ = z^2 (x’ +vt’ )(x – vt)
xx’ = z^2 ( x’x + vt’x – x’vt – v^2 t’t )
now , wherever you see t or t’, substitute
t = x/c
t’ = x’/c
and then you get
xx’ = z^2 ( xx’ + v(x’/c)x – x’ v (x/c) – v^2 (x/c)(x’/c) )
Now , you can divide all of that by xx’
1 = z^2 ( 1 + v/c – v/c – v^2/c^2 )
z= 1/ [sqrt ( 1- v^2/c^2 ) ]
and that is called the Lorentz factor.
You can now work out the values of x and x’ using the previous equations and substituting the value of z
Now , while studying classical mechanics , we ignore the lorentz factor because if v is really small , the value of z is very very near to 1. So , z is only relevant if v is really high, generally nearing the speed of light, c.
You can also , work this relation for time, you will get
t’ =[ t – (v/c^2) x ]z and t = [t’ +(v/c^2)x’ ]z
Now , to summarize , when speeds are relativistic , i.e when the ‘z’ becomes large enough (relatively) , the distance measured by you and the time measured by you will depend on the speed between you , in the car S and the guy in the car S’
In other words , the clocks moving away from you move slower , and the distances appear shorter.
We will take this a little further in part 2, We will try to understand what ‘Time dilation , length contraction , simultaneity and other mumbo jumbo means and we will also see what E = mc^2 means. So stay tuned 🙂
In the meantime , you might want to check out lectures by Professor Leonard Susskind on Stanford university’s youtube channel . They are amazing, and you will learn a lot .
– Rishikesh Jani