I am puzzled by Roberto's solution of this exercise. He draws the axes of the primed system as though they were rotated (in the Euclidean sense) with respect to those of the unprimed system. It seems to me that if the unprimed axes are drawn vertical and horizontal, as he shows them, and the t' axis is rotated by theta toward the x axis, then the x' axis should be rotated by theta TOWARD the t axis, not away.

With this configuration and tan(theta) = v, the desired result can be obtained as follows. Let l' be laid out as OA along the x' axis. Drop a perpendicular (constant x) to the x axis and call the line segment along the x axis OB. Follow the (negative) t' direction from A to the X axis ( i.e., constant x') and call its intersection with the latter C. Then "l" is the length of the segment 0C.

Distances constructed with the Minkowski metric are invariant, so we can determine the length of l' according to the unprimed coordinates. Denoting by X the length of OB, the t coordinate of A is vX and from the metric (l')**2 = X**2*(1 - v**2) or X=l'/SQRT(1-v**2). By similar triangles, the length of BC is Xv**2. The desired distance, l, is OC = OB - BC, which is seen to be l'*SQRT(1-v**2), as claimed.

I apologize for my awkward notation due to my inability to use LATEX.

16 Mar 2011, 17:14

Roberto

Joined: 03 Jun 2010, 15:18 Posts: 136

Re: Exercise [18.12] comment

In my understanding, my solution and Smith's are both correct, just follow different approaches. I attached a short note to clarify the difference between the two approaches.