A hypothesis once used to explain the Hubble relation is the ``tired light hypothesis.'' The tired light hypothesis states that the universe is not expanding, but that photons simply lose energy as they move through space (by some unexplained means), with the energy loss per unit distance being given by the law
drdE=−kE,(2.42)
where k is a constant. Show that this hypothesis gives a distance--redshift relation that is linear in the limit z≪1. What must the value of k be in order to yield a Hubble constant of H0=68kms−1Mpc−1?