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2. Under a Lorentz transformation, the electric and magnetic ﬁelds transform into each other. Give a simple physical explanation of a situation when a static electric ﬁeld between two charges gives rise to magnetic ﬁeld when viewed by a moving observer. 7. From the condition expressing the invariance of the metric, derive the explicit form of the Lorentz transformation for a boost v = +v xˆ . 8. In the spacetime diagram, display the timelike, spacelike, and lightlike regions. Also, draw in the worldline for some inertial observer.
For an Euclidean space, we can have the Cartesian coordinates with a set of orthonormal bases ei · ej = δij . Namely, the metric is simply given by the identity matrix, [g] = 1. 5 Consider the scalar product of two vectors i = j ei · ej V i U j = i, j Making it more explicit U j ej V i ei · V·U= gij V i U j . 36) i, j g11 g21 1 2 V · U = V ,V ,... .. g12 g22 .. 1 ... U U 2 . . . 37) The metric is needed to relate the scalar product to the vector components. For the case V = U, the above equation is an expression for the (squared) length of the vector.
When observed in another frame O , this interval also has a vanishing value s 2 = 0, because the velocity of light remains the same in the new frame O . 29) s 2 = F s2 , where F is the proportional factor, and it can in principle depend on the coordinates and the relative velocity of these two frames: F = F(x, t, v). e. 2 The new kinematics of space and time point in space and in time) implies that there cannot be any dependence of x and t. That space is isotropic means that the proportional factor cannot depend on the direction of their relative velocity v.