In practice, one need not be concerned with the tangent spaces. The vector space structure of Minkowski space allows for the canonical identification of vectors in tangent spaces at points (events) with vectors (points, events) in Minkowski space itself. See e.g. Lee (2003, Proposition 3.8.) or Lee (2012, Proposition 3.13.) These identifications are routinely done in mathematics. They can be expressed formally in Cartesian coordinates as ( x 0 , x 1 , x 2 , x 3 ) ↔ ↔ x 0 e 0 | p + x 1 e 1 | p + x 2 e 2 | p + x 3 e 3 | p ↔ ↔ x 0 e 0 | q + x 1 e 1 | q + x 2 e 2 | q + x 3 e 3 | q with basis vectors in the tangent spaces defined by e μ μ | p = ∂ ∂ ∂ ∂ x μ μ | p or e 0 | p = ( 1 0 0 0 ) , etc .