SECTION B Answer ONLY TWO questions in Section B [Expect to use one single-sided A4 page for each Section-B sub question.] Question B1 Consider the line element where w is a constant. ds²=-dt²+e2wt dx², a) Determine the components of the metric and of the inverse metric. [2 marks] b) Determine the Christoffel symbols. [See the Appendix of this document.] [10 marks] c) Write down the geodesic equations. [5 marks] d) Show that e2wt it is a constant of geodesic motion. [4 marks] e) Solve the geodesic equations for null geodesics. [4 marks] The Schwarzschild line element in the usual coordinates x" = (t, r, 0, ) is a) ds² = - 1- 2GM dr.2 dt² + 2GM +r² (d02+ sin² Odo²). 1 r State whether this spacetime is asymptotically flat, and briefly justify your answer. b) The non-vanishing Christoffel symbols are [4 marks] Ittr = GM r(r - 2GM)' Ir tt = GM(r-2GM) 1-3 GM Irr = − r(r - 2GM)' Tee (r-2GM), 1 = (r-2GM) sin² 0, re= го sin cos 0, r 1 1 ΓΦ TO = Гроф r tan 0 Write down the t-component of the geodesic equations. (The other components are not requested.) c) Show that E = = (1-2GM) t [5 marks] 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 ε. [8 marks] d) 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. [8 marks]

Check Your understanding Using the three displacement vectors A , B , and F in Figure 2.13, choose a convenient scale, and use a ruler and a protractor to find vector G given by the vector equation G=A+2BF . The three displacement vectors A , B , and C in Figure are specified by their magnitudes A=10.0,B=7.0 and C=8.0 , respectively, and by their respective direction angles with the horizontal direction =35,=110 , and =30 . The physical units of the magnitudes are centimeters. Choose a convenient scale and use a ruler and a protractor to find the following vector sums: (a) R=A+B , (b) D=AB , and (c) S=A3B+C .

Consider the equation s=s0+v0t+a0t2/2+j0t3/6+s0t4/24+ct5/120 , were s is a length and t is a time. What are the dimensions and SI units of (a) s0 , (b) v0 , (c) a0 , (d) j0 , (e) s0, and (f) c ?

A small plane flies 40.0 km ma direction 60 north of east and then flies 30.0 km in a direction 15 north of east. Use the analytical method to find the total distance the plane covers from the starting point, and the geographic direction of its displacement vector. What is its displacement vector?

Check Your Understanding For the vectors given in Figure 2.13, find the scalar products AB and FC . The three displacement vectors A , B , and C in Figure 2.13 are specified by their magnitudes A=10.0,B=7.0 and C=8.0 , respectively, and by their respective direction angles with the horizontal direction =35,=110 , and =30 . The physical units of the magnitudes are centimeters. Choose a convenient scale and use a ruler and a protractor to find the following vector sums: (a) R=A+B , (b) D=AB , and (c) S=A3B+C .

For the three-dimensional vectors in the following figure, find (a) GH , (b) GH , and (c) GH .

Check Your understanding Three displacement vectors A , B , and F in (Figure 2.13) are specified by their magnitudes A=10.0,B=7.0 and F=20.00 , respectively, and by their respective direction angles with the horizontal direction =35,=110 , and =110 . The physical units of the magnitudes are centimeters. Use the analytical method to find vector G=A+2BF . Verify that G=28.15cm and that G=68.65 . The three displacement vectors A , B , and C in Figure are specified by their magnitudes A=10.0,B=7.0 and C=8.0 , respectively, and by their respective direction angles with the horizontal direction =35,=110 , and =30 . The physical units of the magnitudes are centimeters. Choose a convenient scale and use a ruler and a protractor to find the following vector sums: (a) R=A+B , (b) D=AB , and (c) S=A3B+C .

A surveyor measures the distance across a river that flows straight north by the following method. Starting directly across from a tree on the opposite bank, the surveyor walks 100 m along the river to establish a baseline. She then sights across to the tree and reads that the angle from the baseline to the tree is 35 . How wide is the river?

Assuming the +x-axis is horizontal to the right for the vectors in the previous figure, find the following vector products: AC , (b) AF , (c) DC , (d) A(F+2C) , (e) iB , (f) jB , (g)

A skater glides along a circular path of radius 5.00 m in clockwise direction. ‘When he coasts around one- half of the circle, starting from the west point, find (a) the magnitude of his displacement vector and (b) how far he actually skated. (c) What is the magnitude of his displacement vector when he skates all the way around the circle and comes back to the west point?