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\textit{Oh, an empty article!} GR Final Study Notes  You can get started 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 \textbf{double clicking} this text block 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  z + 1 = a(te)/a(to)  X = int([0,1], dt/a(t)) (horizon)  Specific Notes:  Principle of equivalence  weight in elevator = m(g+a)  so if a = -g then w = 0  therefor gravity is not a real force  andbegin editing. You  can also click be seen as a curvature in spacetime  ch11 - grav lenses eq (11.1)  Rs = 2 G M / c^2  ch18 - past light cones (who sees whom)  eddington finkelstein  t = v - r  incoming light rays are at 45 degrees regardless of potion  if you’re inside  the \textbf{Insert} button below singularity (r = 2M) then outgoing rays  curve  to add new block elements. Or you can \textbf{drag the positive side on the outside  curve to the negative side if inside  Red shift  omega _ inf = omega _e (1-2M/r_e )^(1/2)  to compare two positions set e = {1,2}  and drop an image} right onto this text. Happy writing! then divide the two;  o_1 / o_2 = (1-2M/r_2)/(1-2M/r_1)  Gravitational Collapse  the schwarzchild radius intersects the collapsing star and becomes a black hole  the geodesic goes completely vertical at r=2M  Expanding universe  metric  -> dx/dt = 1/a  -> geodesics are of the form x(t) = sqrt(t*t_0)  past light cone is much bigger than if it was in flat space-time