1 Two dimensional Minkowski space-time We study the two dimensional space-time R

Question

1 Two dimensional Minkowski space-time We study the two dimensional space-time R2 with "length" u of a vector u = 1 given by /u/2 = t2-2. Here t represents time , z represents spacial x coordinate and we are working with units where the speed of light is 1 (e.g. take unit of time-second and unit of distance-1 light by sents spacial ®coordinate and …[t] second) Let G=the transformations of R2 that preserves the above notion of" length"-These are called Lorentz transformations.Coordinate systems of any two inertial observers must be related by Lorentz transformations. 1. (a) Lets take a new basis (A, B) of R2 such that A = t+x and B = t-ie18- 1,1 rite the distance formula in terms of A,B gth (b) Show that the transformation in (4, B) should satisty the formula B where r >0 t'l (c) Derive the following transformation formula in (t,z) coordinates = Y, Wand w r+ is called the"velocity". r r.

Explanation / Answer

Ans(1.a):

Given that distance formula for u=[x,t]^T is |u|^2=t^2-x^2
Now compare [A,B]^T with [x,t]^T, we get x=A and t=B
so distance ( length) formula becomes
|u|^2=B^2-A^2...(i)
Given A=t+x and B=t-x
plug it into (i)
|u|^2=(t-x)^2-(t+x)^2
|u|^2=t^2+x^2-2tx-t^2-x^2-2tx
|u|^2=-4tx

Hence required distance formula is |u|^2=-4tx or you can also use |u|^2=(t-x)^2-(t+x)^2

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