In special relativity, it is customary to introduce the dimensionless parameters 1

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1 Math 3181 Dr. Franz Rothe January 21, SPR\4080_spr16h1.tex Name: Homework has to be turned in this handout. For extra space, use the back pages, or put blank pages between. The homework can be done in groups up to three students due Wednesday Jan 20 or Thursday Jan 21, 3 p.m. 1 Homework Special relativity Let (ct, x, y, z) be the coordinates for any event, measured in the inertial system S. Let (ct, x, y, z ) be the coordinates for the same event, measured in the inertial system S. Assume that the origins of systems S and S are equal, and that the system S moves with velocity v in +x direction relative to system S. We want to determine the linear transformation 90 (1.1) t = At + Bx x = Dt + Ex y = y and z = z In special relativity, it is customary to introduce the dimensionless parameters β := v c and γ := 1 1 β 2 Problem 1.1. From the relative velocity of the two systems, and from the postulate of constancy of the velocity of light c, one gets the three assumptions x = vt if and only if x = 0; x = ct if and only if x = ct ; x = ct if and only if x = ct. Use these three assumptions to determine the constants B, D and E in terms of A and the relativity parameters β and γ. 1

2 Definition 1 (Isochronic Lorentz-, proper Lorentz transformation). An isochronic Lorentz transformation maps the cone of light rays which point into the future into itself. A proper Lorentz transformation maps the cone of light rays which point into the future into itself, and has the determinant +1. Problem 1.2. In many texts and lectures, it is customary to begin with the stronger postulate of invariance of the Minkowski metric: (1.2) c 2 t 2 x 2 y 2 z 2 = c 2 t 2 x 2 y 2 z 2 instead of the weaker postulate of constancy of the velocity of light. Use the invariance of the Minkowski metric and the fact that x = vt x = 0, to determine the constants A, B, D and E in the transformation (1.1), in terms of the relativity parameters β and γ. There are four solutions, with different signs of these constants. Write down all four solutions. What is the meaning of these solutions? Which one of these four solutions is a proper Lorentz transformation. 2

3 Here is another way to determine the constant A left open in problem 1.1. We begin with the assumptions (i) The proper Lorentz transformations are a group. (ii) Among the Lorentz transformations are the boosts as well as the usual 3-dimension rotations. (iii) The rotations without boost leave the time invariant. (iv) Any isochronic Lorentz transformation that is diagonal leaves the time invariant. Thus both the boost and the rotation L = S = A βa 0 0 βa A about the z-axis by 180 are Lorentz transformations. Problem 1.3. Calculate the matrix products SLS and LSLS. Give a convincing argument that A 2 (1 β 2 ) = 1 holds in the physically meaningful case. Determine the sign of A occurring for a proper Lorentz transformation. Once more, write down the matrix for a boost in +x direction. 3

4 For a graphic representation of the usual boost (1.3) ct = γct βγx x = βγct + γx one uses the same scale for the lengths ct and x, and a right angle between the ct-axis and the x-axis. Problem 1.4. Convince yourself that the same angle occurs between the ct- and ct -axis as between the x- and x -axis; the angle between the x - and the ct -axis is acute. Draw the future light ray x = ct, t 0. Draw the hyperbola of all points which have the invariant space-like squared distance 1 from the origin. 4

5 To illustrate the Lorentz contraction, we imagine a rigid tube of (comoving) length d with mirrors on both ends. Now the tube is moving with velocity v relative to the S-system. Thus the tube is at rest in the comoving S system. I calculate at first with coordinates from the S-system. The mirrors at the end of the tube move along the lines x = vt and x = vt + L Let a light ray OA be sent from the mirror at the right end to the mirror at the left end, and reflected into a light ray AB from the mirror at the left end to the mirror at the right end. The equations of these light rays are x = ct and x = ct + a. Problem 1.5. Determine constant a. Determine the S-system coordinates of the reflection events A and B. Problem 1.6. Use the Lorentz transformation (1.3) to determine the S -system coordinates of the reflection events A and B. Problem 1.7. Convince yourself that (1.4) L = 1 β 2 d < d which is the famous Lorentz-FitzGerald contraction. There are no forces of any kind involved. 5

6 To illustrate the time dilation, we imagine a rigid tube of (comoving) length d with mirrors on both ends. Again the tube is moving with velocity v relative to the S-system, but turned by 90. Thus the tube is at rest in the comoving S system, and lying on the y -axis. I calculate at first with coordinates (ct, x, y ) from the comoving S -system. The mirrors at the end of the tube move along the lines x = 0, y = 0 and x = 0, y = d Let a light ray OA be sent from the mirror at the right end to the mirror at the left end, and reflected into a light ray AB from the mirror at the left end to the mirror at the right end. The light goes on to be reflected forth and back, and each reflection give a tick of this mirror-clock. In the comoving system, the equations of these light rays are OA : x = 0, y = ct AB : x = 0, y = 2d ct The emission and reflection events have S -coordinates O = (0, 0, 0), A = (cd, 0, d), B = (2cd, 0, 0). Thus cd is the proper time interval between the ticks of the mirror clock. Problem 1.8. Determine the S-system coordinates (ct, x, y) of the reflection events A and B. Determine the equations of the light rays OA and AB. Problem 1.9. Convince yourself that times of the reflection events A and B in the S-system and γcd and 2γcd. Since γ > 1, we get longer time intervals between the ticks of the clock. The moving clock is slowing down by the relativity parameter γ. 6