Question

Write down the Lorentz transformation equations and derive the expression for length

contraction and time dilation.(3 marks)​

Answer 1

Answer:

One has to understand first that Lorentz transformation is usually in the denominator. So that square root of thing even though it is below, is written in the numerator using a symbol called "gamma" because gamma increases as velocity increases. (I mean you have something in the denominator that keeps going small as you increase velocity in any manner, so it'd be more appropriate to keep it as something in the numerator). Anyway, gamma is in the top because derivations of it make it so. In a more simple manner, it can be derived that the Lorentz factor sits there because of the postulates of Special relativity and Pythagoras theorem. Further reading for that derivation is here. Dilation, Length Contraction and Simultaneity (from Einstein Light) From this you see that the times dilatation is more for an object moving with respect to an observer.

Now to find out why the gamma is in the opposite side of the equation when one relates to distances, this again can be proved using the fact that the speed of light is same for all observes. So we have that " c " for person stationary (i.e an observer) is same to that " c " for a person moving with velocity " v " with respect to the observer.

So we have that c=c irrespective of any velocity. Also we know that the moving object experiences time dilatation, so the equation turns out to be

d/t=d′/t′ where the letters with a hyphen up there signifying it ain't all equal all there. Now from the link there you'll know that t'= gamma times t, where t is the time the observer feels. So the equation becomes d/t=d′/γ∗t

But before cancelling those t, let's remember that there's a problem here, it doesn't in exact definition say that they've seen it for the same time, it shows that while one person sees it for say 5 seconds, the other sees it for more or less than 5 seconds, so instead of cancelling it for the "t", we'll cancel it such that both observer and moving object sees the light move for 5 seconds, in their own frame, because this is what will help us understand how the length has contracted for one object with respect to another. So we get, d/γ=d′ (because we cancelled another L from the left hand side so as to ensure both of the observers, one moving and the other stationary see it for the same time, but yet keep the equation proper). So here we see that length contracts.

Explanation:

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