Operator of proper-time-derivative is a differential operator and the relativistic generalization of material derivative (substantial derivative) in four-dimensional spacetime.
In coordinate notation, this operator is written as follows: 
where – the symbol of differential in curved spacetime, – proper time, which is measured by a clock moving with test particle, – 4-velocity of test particle or local volume of matter, – covariant derivative.
In flat Minkowski spacetime operator of proper-time-derivative is simplified, since the covariant derivative transforms into 4-gradient (the operator of differentiation with partial derivatives with respect to coordinates):
To prove this expression it can be applied to an arbitrary 4-vector :
Above was used material derivative in operator equation for an arbitrary function :
where is the velocity of local volume of matter, – nabla operator.
In turn, the material derivative follows from the representation of differential function of spatial coordinates and time:
Operator of proper-time-derivative is applied to different four-dimensional objects – to scalar functions, 4-vectors and 4-tensors. One exception is 4-position (4-radius), which in four-Cartesian coordinates has the form because 4-position is not a 4-vector in curved space-time, but its differential (displacement) is. Effect of the left side of operator of proper-time-derivative on the 4-position specifies the 4-velocity: , but the right side of the operator does not so: .
where is 4-vector momentum density of matter, – density of matter in its rest system, – Christoffel symbol.
In general relativity freely falling body in a gravitational field moves along a geodesic, and four-acceleration of body in this case is equal to zero: 
Since interval , then equation of motion of the body along a geodesic in general relativity can be rewritten in equivalent form:
If, instead of the proper time to use a parameter , and equation of a curve set by the expression , then there is the operator of derivative on the parameter along the curve: