Consider Minkowski spacetime in Cartesian coordinates x = (t, x, y, z), such that ds² = dt² + dx² + dy² + dz². (a) Consider the vector with components V" = (1, -1, 0, 0). Determine Vμ and V. V. (b) Consider now the coordinate system x = (u, v, y, z) such that u =t-x, v=t+x. Write down the line element, the metric, the Christoffel symbols and the Riemann curvature tensor in the new coordinates. [See the Appendix of this document.] (c) Determine V', that is, write the object in question A1.a in the coordinate system x'". Verify explicitly that V. V is invariant under the coordinate transformation.

Consider Minkowski spacetime in Cartesian coordinates x = (t, x, y, z), such that ds² = dt² + dx² + dy² + dz². (a) Consider the vector with components V" = (1, -1, 0, 0). Determine Vμ and V. V. (b) Consider now the coordinate system x = (u, v, y, z) such that u =t-x, v=t+x. Write down the line element, the metric, the Christoffel symbols and the Riemann curvature tensor in the new coordinates. [See the Appendix of this document.] (c) Determine V', that is, write the object in question A1.a in the coordinate system x'". Verify explicitly that V. V is invariant under the coordinate transformation.