**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: ^{[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:

- 1 Applications
- 2 See also
- 3 References
- 4 External links

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 the 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]}

where is the acceleration
stress-energy tensor with the mixed indices, is the acceleration
tensor, and the 4-potential of acceleration field
is expressed in terms of the scalar
and vector potentials:

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:^{ [}^{4}^{]}

.

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]}

.

- Four-gradient
- Four-force

- 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. International Journal of Theoretical and Applied Physics, Vol.
4, No. 1, pp. 9-26
(2014). http://dx.doi.org/10.5281/zenodo.890781.
- 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.
- Carroll, Sean M. (2004),
*Spacetime and Geometry*, Addison Wesley, ISBN 0-8053-8732-3

Source:
http://sergf.ru/oden.htm