The proof that Hubble’s law is true for an observer anywhere in the universe.

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Chapter 16, Problem 8P

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The proof that Hubble’s law is true for an observer anywhere in the universe.

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Hubble's law can be stated in vector form as v = HR. Outside the local group of galaxies, all objects are moving away from us with velocities proportional to their positions relative to us. In this form, it sounds as if our location in the Universe is specially privileged. Prove that Hubble's law is equally true for an observer elsewhere in the Uni- verse. Proceed as follows. Assume we are at the origin of coordinates, one galaxy cluster is at location R, and has velocity v, = HR relative to us, and another galaxy cluster has position vector R, and velocity v, = HR, Suppose the speeds are nonrelativistic. Consider the frame of reference of an observer in the first of these galaxy clusters. (a) Show that our velocity relative to her, together with the position vector of our galaxy cluster from hers, satisfies Hubble's law. (b) Show that the position and velocity of cluster 2 rel- ative to cluster 1 satisfy Hubble's law.

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Consider the following line element, ds² = -dt² + a² (t) (dx² + dy²) + b²(t) dz², where a(t) and b(t) are distinct functions. State whether or not this line element obeys the Cosmological Principle, if applied to describe the universe on large scales. Justify your answer.