The exact definition of verticality is that the inner product of two vectors is zero. In the case of special relativity, the definition of vector inner product here is correct with the chapter on vector space in higher algebra mentioned above.
Mathematically, if two vectors a (x 1, x2, x3, x4, ... xn) b (y 1, y2, y3, y4, ... yn) are represented, then the inner product can be calculated as ab' = x1y1. If the value is 0, it is orthogonal (that is, vertical).
However, in general relativity, the space to be considered is not a general space, but a Riemannian space, so we can't refer to the vector space of higher algebra. Its vertical definition is also that the inner product is 0, but the expression of the inner product has changed, because the curvature of space needs to be considered and it needs to be corrected in the measurement with space (measurement determines the nature of space).
At this time, the mathematical expression is GAB', where g is a tensor, which is a second-order tensor in general relativity and contains 16 elements. It is a symmetric 4*4 matrix. Its elements are determined by the curvature of space points, and the determination of curvature depends on mathematical analysis. So some people upstairs say that quoting mathematical analysis is funny, which is obviously ignorance.
There are more complicated formulas, so I won't list them. If you don't understand, you can talk to me.