The Schwarzschild line element in the usual coordinates x = (t, r, 0, ø) is ds² = - (1 – 2GM) dt² + 1 - dr² 2GM r r²(do² + sin² Odo²). a) b) c) d) State whether this spacetime is asymptotically flat, and briefly justify your answer. The non-vanishing Christoffel symbols are = 1 - GM r(r - 2GM)' Ꮁ = sin cos 0, - GM It tr Ir tt = r(r-2GM)' GM(r-2GM) 73 IT IT Γ' 00 == -(r - 2GM), "(r-2GM) sin² 0, го re 1 Γ = Γ = 00 tan 0 Write down the t-component of the geodesic equations. (The other components are not requested.) Show that E = 12GM) t r is constant along a geodesic. Supposing that the geodesic is a timelike one, identify the natural parameter of the geodesic with respect to which we take the derivative in t, and explain how that parameter is defined. State what is the physical meaning of the constancy of E. The super-hero Hulk kicks a super-tough ball upwards so hard that the ball escapes the Earth's gravitational field. Explain why the ball's trajectory must obey ε > 1 in order for the ball to never fall back down, and determine the final velocity of the ball as a function of ε. Ignore air resistance.

The Schwarzschild line element in the usual coordinates x = (t, r, 0, ø) is ds² = - (1 – 2GM) dt² + 1 - dr² 2GM r r²(do² + sin² Odo²). a) b) c) d) State whether this spacetime is asymptotically flat, and briefly justify your answer. The non-vanishing Christoffel symbols are = 1 - GM r(r - 2GM)' Ꮁ = sin cos 0, - GM It tr Ir tt = r(r-2GM)' GM(r-2GM) 73 IT IT Γ' 00 == -(r - 2GM), "(r-2GM) sin² 0, го re 1 Γ = Γ = 00 tan 0 Write down the t-component of the geodesic equations. (The other components are not requested.) Show that E = 12GM) t r is constant along a geodesic. Supposing that the geodesic is a timelike one, identify the natural parameter of the geodesic with respect to which we take the derivative in t, and explain how that parameter is defined. State what is the physical meaning of the constancy of E. The super-hero Hulk kicks a super-tough ball upwards so hard that the ball escapes the Earth's gravitational field. Explain why the ball's trajectory must obey ε > 1 in order for the ball to never fall back down, and determine the final velocity of the ball as a function of ε. Ignore air resistance.