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GR Final Study Notes  special relativity  basic setup of SR  same physics in all reference  c is constant  Lorenz Transformations  Fun with 4 vectors  u = (gamma, gamma v)  E_obs = - p . u  u is normalized by u.u = -1  for a photon (or massless particle) p.p=0  Timelike, spacelike, null line elements  Principle of Equivalence  elevators  a freely falling elevator on the earth will get all the same results as the other guy  light must bend  Spacetime geometery  ds^2 = g_ab dx^a dx^b  gravitational redshift  in a static universe/system (t is ignorable) there is a killing vector xi = (1,0,0,0)   xi . u is conversed along the trajectory of the particle/light ray  E_obs / E_e = gtt(e)/gtt(obs)  Orbits of photons  effective potential formulation  conserved energy associated with killing vector (0,0,0,1) concerning angular momentum  r^2 d(phi)/d(tau) = const  Bending of light / gravitational lenses  delta theta = (4 G M)/(b c^2) where b is the impact parameter  Cosmology  scale factor a(t)  how to calculate redshits