Question

Problem 4: SpaceTime diagrams (25 points) Consider the two points labeled 1 and 2 in this spacetime diagram. Notice that in the Lorentz frame represented by this diagram, both events are in the future: ti > 0 and t2 > 0. (The diagonal lines with 45° slope represent the light cone.) ct (a) Prove that, regardless of the value of ß-u/c, when you boost to a different Lorentz frame, the event 1 remains in the future, that is, 0 always. Also prove that event 2 can be either in the future, t2 >0, or in the past, t2 < 0, depending on the value of (b) In a different Lorentz frame, the axes of the primed frame are tilted relative to the axes of the unprimed frame, as shown Prove that for the usual boost of β = v/c in the +x direction. the primed axes are tilted in the directions shown and both are tilted by the same size angle θ, as shown. What is the ct relation between β-v/c and the tilt-angle θ? Draw appropriate diagrams which illustrate clearly why event 2, in the primed frame, can be either in the past or the future, depending on the value of β

Answer 1

Now, equn (2) will be, i e t' = -I beta gamma x + gamma i e t et' = gamma(et - beta x) so the transformation equn satisfy Lorentz transformation. And here, x^alpha + (i e t)^alpha = invariant. Hence the geometry is called circular geometry. Here the rotation is not the actual rotation. i.e, a geometrical representation. Because if actual rotation is allowed then the frame is non - inertial, STR does not contain any non - inertial frame. Hyperbolic geometry: - Let us consider a transformation, (x' y') = (cos h theta sin h theta sin h theta cos h theta)(x y) x' = cos h theta middot x + sin h theta y equation (1) y' = sin h theta x + cos h theta y equation (2) x^2 - y^2 = x^2 - y^2 = invariant. = hyperbolic geometry. \r\n transformation equn, x' = cos h theta middot x + sin h theta middot et equation (3) et' phi' = sin h theta middot x + cos