Problem B.4: Special Relativity - Part II Space and time are interconnected according to special relativity. Because of that, coordinates have four components (three position coordinates x, y, z, one time coordinate t) and can be ex- pressed as a vector with four rows as such: ct The spaceship from problem A.4 (Special Relativity - Part I) travels away from the Earth into the deep space outside of our Milky Way. The Milky Way has a very circular shape and can be ex- pressed as all vectors of the following form (for all 0 < p < 2m): ct sin y COs o (a) How does the shape of the Milky Way look like for the astronauts in the fast-moving space- ship? To answer this question, apply the Lorentz transformation matrix (see A.4) on the circular shape to get the vectors (ct', a', y', z') of the shape from the perspective of the moving spaceship. (b) Draw the shape of the Milky Way for a spaceship with a velocity of 20%, 50%, and 90% of the speed of light in the figure below (Note: The ring shape for a resting spaceship is already drawn.): 1.00 0.75- 0.50 - 0.25 0.00 -0.25- -0.50 -0.75 -1.00 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 z'

Problem B.4: Special Relativity - Part II Space and time are interconnected according to special relativity. Because of that, coordinates have four components (three position coordinates x, y, z, one time coordinate t) and can be ex- pressed as a vector with four rows as such: ct The spaceship from problem A.4 (Special Relativity - Part I) travels away from the Earth into the deep space outside of our Milky Way. The Milky Way has a very circular shape and can be ex- pressed as all vectors of the following form (for all 0 < p < 2m): ct sin y COs o (a) How does the shape of the Milky Way look like for the astronauts in the fast-moving space- ship? To answer this question, apply the Lorentz transformation matrix (see A.4) on the circular shape to get the vectors (ct', a', y', z') of the shape from the perspective of the moving spaceship. (b) Draw the shape of the Milky Way for a spaceship with a velocity of 20%, 50%, and 90% of the speed of light in the figure below (Note: The ring shape for a resting spaceship is already drawn.): 1.00 0.75- 0.50 - 0.25 0.00 -0.25- -0.50 -0.75 -1.00 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 z'