Operator of proper-time-derivative
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 substance, – 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 substance, – 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 substance, – density of substance in its rest system, – Christoffel symbol.
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:
- Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009-2011, 858 pages, Tabl. 21, Pic. 41, Ref. 293. ISBN 978-5-9901951-1-0. (in Russian).
- Fedosin S.G. The General Theory of Relativity, Metric Theory of Relativity and Covariant Theory of Gravitation: Axiomatization and Critical Analysis. vixra.org, 26 Mar 2011.
- Fock, V. A. (1964). "The Theory of Space, Time and Gravitation". Macmillan.
- Carroll, Sean M. (2004), Spacetime and Geometry, Addison Wesley, ISBN 0-8053-8732-3, http://spacetimeandgeometry.net