It is a recurring theme in special relativity to take the expression A ⋅ ⋅ B = A 0 B 0 − − A 1 B 1 − − A 2 B 2 − − A 3 B 3 = C in one reference frame, where C is the value of the inner product in this frame, and: A ′ ⋅ ⋅ B ′ = A ′ 0 B ′ 0 − − A ′ 1 B ′ 1 − − A ′ 2 B ′ 2 − − A ′ 3 B ′ 3 = C ′ in another frame, in which C ′ is the value of the inner product in this frame. Then since the inner product is an invariant, these must be equal: A ⋅ ⋅ B = A ′ ⋅ ⋅ B ′ that is: C = A 0 B 0 − − A 1 B 1 − − A 2 B 2 − − A 3 B 3 = A ′ 0 B ′ 0 − − A ′ 1 B ′ 1 − − A ′ 2 B ′ 2 − − A ′ 3 B ′ 3