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The line element of space-time outside a static and spherically symmetric mass distribution of mass M is c²d7² (1-a)c²dt². dr.² 1-a where a = rs /r and rs = 2GM/c² is the Schwarzschild radius. a) Using the Euler-Lagrange equations, show that (1 – 3)t and r² are constants of motion for a particle that moves in the equatorial plane (0 = π/2) around this object. What does the dot symbolize in the notation shown? = b) Assuming that (1 - 3)i = k and r²6 particle, the quantity c) 7² + h² 7-2 = h, where k and h are constants, show that for a massive 2GM - r²(d0² + sin² 0do²), r 2GMh² 73c2 (1) is a constant of motion. Demonstrate that using n = h²/(r²c²) and R = r/r, the equation (1) can be written as +$=k² - 1, c² 17 1 where = is known as the effective potential. Explain what the quantity n physically R² R R³ represents and why is so named.