Suppose you seek the derivative of a quantity along a curve, such as the rate of change of a scalar by proper time along a worldline: or perhaps the rate of change of pressure by proper distance along a given spatial direction: . These derivatives are conveniently expressed as a contraction between a 1-form (the gradient of the scalar) and a tangent vector to the curve. For the first example,
where is some coordinate system, is the 1-form with components , and is the 4-velocity. is the contraction of the vector and 1-form, yielding a scalar. Schutz 2009 §3.3 gives this derivation.
A spacelike path can be parametrised by proper distance. Then , where is the unit tangent vector. An example of a paper which uses this is Gibbons 1972 , for the change in stress along a rigid cable, see the line after his Equation 4.
For a null path there is no natural parameter, at least not without additional context. But for any chosen parameter , we have as before, where is the tangent vector. Of course this applies to the other cases as well. Note all these calculations occur within a single tangent space.