Let’s consider a scalar field, say temperature of a rod varying with time i.e . (something like the following)
We will take this setup and put it on a really fast train moving at a constant velocity (also known as performing a ‘Lorentz boost’).
Now the temperature of the bar in this new frame of reference is given by where,
Temperature is a scalar field and therefore irrespective of which frame of reference you are on, the temperature at each point on the rod will remain the same on both the frames i.e
Therefore we can say that Temperature (a scalar field) is Lorentz invariant. Now what other quantities can we make from T that would also be Lorentz invariant ?
Well, let’s give it a try:
But just for fun let’s just square the terms and see if we can churn something out of that:
We immediately notice that:
Therefore in addition to realizing that is Lorentz invariant, we have also found another quantity that is also Lorentz invariant. This quantity is also written as .
** There is a very important reason why this quantity did not work out. This post was inspired in part by Micheal Brown’s answer on stackexchange . I request the interested reader to check that post for a detailed explanation.