Q3: (i) In the weak field limit, where g+h, show that hoo=2p/c, and that the Einstein equations reduce to the Newton's equation (Poisson equation). (ii) What are the assumptions made by Schwarzschild to solve Einstein field equations? (iii) Use the Schwarzschild line element, compare the Newtonian and Einstein's general relativity poten- tials in the limit when r→ 0 and r→ ∞. (iv) Show that frequency of light under gravity experiences a red shift, when emitted at a given point and received at a different point, separated by a distance h. How is the frequency shift is related to h? Q4: (i) If the Lie derivative of the vector A along the direction is defined by CA =A-AP, find the following Lie derivatives: (a) T (b) LeKad (c) L₂C% (ii) Show that Ceska = 0. (iii) If T=0, then (7) = 0, if is a Killing vector. What is the physical meaning of this equation? (iv) Show that the Minkowski metric, ds² = c2d2-dr², under the transformation, et sinhr and x=cosh 7, y = y, z, where p = a/c², and a is the proper acceleration of the observer, becomes dp² ds²=- dr² + dat (cosh² - sinh² = 1)