Problem 5: Killing vector fields and conservation laws 9μv = 0, then show that this i) If the metric components do not d

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Problem 5: Killing vector fields and conservation laws 9μv = 0, then show that this i) If the metric components do not depend on a specific coordinate ,le. implies that P* is constant along geodesics with tangent vector PC (9pts). ii) A vector field that satisfied Killing's equation, V(KB) = 0 is called as a Killing vector field K. Show that Ka Pa is constant along geodesics with tangent vector P = d/dx (8pts). iii) For a given conserved stress-energy tensor TB and a Killing vector K, the current is written by component as ja = K&TB, Show that this current is conserved (.e., divergence-free) (8pts). Hint: Use the fact that geodesic equation is a autoparallel curve VpP = 0