4. Lorentz contraction exercise [from Brandenberger] In retrospect, this problem should be called: What do you...

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4. Lorentz contraction exercise [from Brandenberger] In retrospect, this problem should be called: "What do you see when you look out the windows of a fast-moving spaceship?" (a) Suppose frame S moves with velocity e relative to frame S. A projectile in frame S' is fired with velocity at an angle with respect to the forward direction of motion (). What is this angle measured in S? What if the projectile is a photon? (b) An observer A at rest relative to the fixed distant stars sees an isotropic distri- bution of stars in a galaxy which occupies some region of her sky. The number of stars seen within an element of solid angle di? is Pan- dn In where N is the total number of stars that A can see. Another observer B moves uniformly along the axis relative to 4 with velocity v. Letting and be respectively the polar (with respect to :) and azimuthal angle in the inertial frame of B, what is the distribution function P() such that P'(,)d? is the number of stars seen by B in the solid angle di?' =sin'de'de'. (e) Check that when integrating the distribution function over the sphere in the coordinates of B you obtain N! Discuss the behavior of the distribution P in the limiting cases when the velocity v goes to 0 or to 1. 4. Lorentz contraction exercise [from Brandenberger] In retrospect, this problem should be called: "What do you see when you look out the windows of a fast-moving spaceship?" (a) Suppose frame S moves with velocity e relative to frame S. A projectile in frame S' is fired with velocity at an angle with respect to the forward direction of motion (). What is this angle measured in S? What if the projectile is a photon? (b) An observer A at rest relative to the fixed distant stars sees an isotropic distri- bution of stars in a galaxy which occupies some region of her sky. The number of stars seen within an element of solid angle di? is Pan- dn In where N is the total number of stars that A can see. Another observer B moves uniformly along the axis relative to 4 with velocity v. Letting and be respectively the polar (with respect to :) and azimuthal angle in the inertial frame of B, what is the distribution function P() such that P'(,)d? is the number of stars seen by B in the solid angle di?' =sin'de'de'. (e) Check that when integrating the distribution function over the sphere in the coordinates of B you obtain N! Discuss the behavior of the distribution P in the limiting cases when the velocity v goes to 0 or to 1.