Calculate the components of a four-vector A and the scalar product of the four-vectors A^ and B^

Calculate the components of a four-vector Aα and the scalar product of the four-vectors A^α and B^β where

(a) A^α = (A^t , A^x , A^y , A^z ) and B^β = (B^t , B^x , B^y , B^z ) in Minkowski space defined by the line element ds2 = −dt^2 + dx^2 + dy^2 + dz^2 .

(b) A^α = (A^t , A^r , A^θ , A^φ ) and B^β = (B^t , B^r , B^θ , B^φ ) in Minkowski space defined by the line element ds2 = −dt^2 + dr^2 + r^2 dθ^2 + r^2 sin^2( θ )dφ^2 . (2)

(c) A^α = (A^t , A^r , A^θ , A^φ ) and B^β = (B^t , B^r , B^θ , B^φ ) in Schwarzschild geometry defined by the line element ds2 = − (1 – (2M)/r) dt^2 + (1 – (2M)/r)^(−1) dr^2 + r^2 (dθ^2 + sin^2( θ )dφ^2 )