In coordinate notation, this operator is written as follows: [1]
,
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: .
In covariant theory of gravitation operator of proper-time-derivative is used to determine the density of 4-force acting on a solid point particle in curved spacetime:[2]
,
where is 4-vector momentum density of matter,
– density of matter in its rest system,
– Christoffel symbol.
However in the common case the 4-force is determined with the help of 4-potential of acceleration field: [3]
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:[5]
↑ Fedosin S.G. Equations of Motion in the Theory of Relativistic Vector Fields. International Letters of Chemistry, Physics and Astronomy, Vol. 83, pp. 12-30 (2019). https://doi.org/10.18052/www.scipress.com/ILCPA.83.12.
↑ Fock, V. A. (1964). "The Theory of Space, Time and Gravitation". Macmillan.