# Operator of proper-time-derivative

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]

${\displaystyle ~{\frac {D}{D\tau }}=u^{\mu }\nabla _{\mu }}$,

where ${\displaystyle ~D}$ – the symbol of differential in curved spacetime, ${\displaystyle ~\tau }$ proper time, which is measured by a clock moving with test particle, ${\displaystyle ~u^{\mu }}$4-velocity of test particle or local volume of matter, ${\displaystyle ~\nabla _{\mu }}$ 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):

${\displaystyle ~{\frac {d}{d\tau }}=u^{\mu }\partial _{\mu }}$.

To prove this expression it can be applied to an arbitrary 4-vector ${\displaystyle ~A^{\nu }}$:

${\displaystyle ~u^{\mu }\partial _{\mu }A^{\nu }={\frac {c{}dt}{d\tau }}{\frac {\partial A^{\nu }}{c{}\partial t}}+{\frac {dx}{d\tau }}{\frac {\partial A^{\nu }}{\partial x}}+{\frac {dy}{d\tau }}{\frac {\partial A^{\nu }}{\partial y}}+{\frac {dz}{d\tau }}{\frac {\partial A^{\nu }}{\partial z}}=}$
${\displaystyle ~={\frac {dt}{d\tau }}\left({\frac {\partial A^{\nu }}{\partial t}}+{\frac {dx}{dt}}{\frac {\partial A^{\nu }}{\partial x}}+{\frac {dy}{dt}}{\frac {\partial A^{\nu }}{\partial y}}+{\frac {dz}{dt}}{\frac {\partial A^{\nu }}{\partial z}}\right)={\frac {dt}{d\tau }}{\frac {dA^{\nu }}{dt}}={\frac {dA^{\nu }}{d\tau }}}$.

Above was used material derivative in operator equation for an arbitrary function ${\displaystyle ~F}$:

${\displaystyle ~{\frac {dF}{dt}}={\frac {\partial F}{\partial t}}+\mathbf {V} \cdot \nabla F}$,

where ${\displaystyle ~\mathbf {V} }$ is the velocity of local volume of matter, ${\displaystyle ~\nabla }$nabla operator.

In turn, the material derivative follows from the representation of differential function ${\displaystyle ~F}$ of spatial coordinates and time:

${\displaystyle ~dF(t,x,y,z)={\frac {\partial F}{\partial t}}dt+{\frac {\partial F}{\partial x}}dx+{\frac {\partial F}{\partial y}}dy+{\frac {\partial F}{\partial z}}dz}$.

## Applications

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 ${\displaystyle ~x^{\mu }=(ct,x,y,z)=(ct,\mathbf {r} )}$  because 4-position is not a 4-vector in curved space-time, but its differential (displacement) ${\displaystyle ~dx^{\mu }=(c{}dt,dx,dy,dz)=(cdt,d\mathbf {r} )}$  is. Effect of the left side of operator of proper-time-derivative on the 4-position specifies the 4-velocity: ${\displaystyle ~{\frac {Dx^{\mu }}{D\tau }}=u^{\mu }}$ , but the right side of the operator does not so: ${\displaystyle ~u^{\nu }\nabla _{\nu }x^{\mu }\not =u^{\mu }}$ .

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]

${\displaystyle ~f^{\nu }={\frac {DJ^{\nu }}{D\tau }}=u^{\mu }\nabla _{\mu }J^{\nu }={\frac {dJ^{\nu }}{d\tau }}+\Gamma _{\mu \lambda }^{\nu }u^{\mu }J^{\lambda }}$ ,

where ${\displaystyle ~J^{\nu }=\rho _{0}u^{\nu }}$  is 4-vector momentum density of matter, ${\displaystyle ~\rho _{0}}$  – density of matter in its rest system, ${\displaystyle ~\Gamma _{\mu \lambda }^{\nu }}$  Christoffel symbol.

However in the common case the 4-force is determined with the help of 4-potential of acceleration field: [3]

${\displaystyle ~f_{\alpha }=\nabla _{\beta }{B_{\alpha }}^{\beta }=-u_{\alpha k}J^{k}=\rho _{0}{\frac {DU_{\alpha }}{D\tau }}-J^{k}\nabla _{\alpha }U_{k}=\rho _{0}{\frac {dU_{\alpha }}{d\tau }}-J^{k}\partial _{\alpha }U_{k},}$

where ${\displaystyle ~{B_{\alpha }}^{\beta }}$  is the acceleration stress-energy tensor with the mixed indices, ${\displaystyle ~u_{\alpha k}}$  is the acceleration tensor, and the 4-potential of acceleration field is expressed in terms of the scalar ${\displaystyle ~\vartheta }$  and vector ${\displaystyle ~\mathbf {U} }$  potentials:

${\displaystyle ~U_{\alpha }=\left({\frac {\vartheta }{c}},-\mathbf {U} \right).}$

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]

${\displaystyle ~a^{\nu }={\frac {Du^{\nu }}{D\tau }}=u^{\mu }\nabla _{\mu }u^{\nu }={\frac {du^{\nu }}{d\tau }}+\Gamma _{\mu \lambda }^{\nu }u^{\mu }u^{\lambda }=0}$ .

Since interval ${\displaystyle ~ds=cd\tau }$ , then equation of motion of the body along a geodesic in general relativity can be rewritten in equivalent form:

${\displaystyle ~{\frac {d}{ds}}\left({\frac {dx^{\nu }}{ds}}\right)+\Gamma _{\mu \lambda }^{\nu }{\frac {dx^{\mu }}{ds}}{\frac {dx^{\lambda }}{ds}}=0.}$

If, instead of the proper time to use a parameter ${\displaystyle ~p}$ , and equation of a curve set by the expression ${\displaystyle ~x^{\mu }(p)}$ , then there is the operator of derivative on the parameter along the curve:[5]

${\displaystyle ~{\frac {D}{Dp}}={\frac {dx^{\mu }}{dp}}\nabla _{\mu }}$ .

## References

1. 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).
2. 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.
3. 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.
4. Fock, V. A. (1964). "The Theory of Space, Time and Gravitation". Macmillan.
5. Carroll, Sean M. (2004), Spacetime and Geometry, Addison Wesley, ISBN 0-8053-8732-3