a) Derive xα=gαβxβ. [Hint: Use (2-5) and (2-6), or alternatively, use the matrix form of the contravariant metric tensor along with column vectors in terms of Cartesian coordinates.] Note that this relation and (2-5) hold in general for any 4D vector, not just the position vector.
b) Express ∂μ∂μ in terms of i) contravariant and covariant 4D components, and ii) in terms of time t and Cartesian coordinates Xi. The operation ∂μ∂μ=∂μ∂μ is called the d'Alembertian operator, and is the 4D Minkowski coordinates analogue of the 3D Laplacian operator ∂i∂i=∂i∂i of Cartesian coordinates.
c) Then find ∂μ∂μ(xαxα), where physical length of the interval of xα is xαxα, i) by expressing all terms in t and Xi, and ii) solely using 4D component notation. (For the last part, note, from a), that ∂xα/∂xβ=gαβ and from (2-5), ∂xα/∂xβ=gαβ.)