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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]

 ~\frac{ D } {D \tau }= u^\mu \nabla_\mu,

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

 ~\frac{ d } {d \tau }= u^\mu \partial_\mu.

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

 ~ 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 }

 ~=\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  ~ F :

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

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

In turn, the material derivative follows from the representation of differential function  ~ F  of spatial coordinates and time:

 ~ 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.




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   ~ 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)   ~ 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:  ~ \frac{ D x^\mu } {D \tau }= u^\mu  , but the right side of the operator does not so:  ~ u^\nu \nabla_\nu x^\mu \not = u^\mu .

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]

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

where   ~ J^\nu = \rho_0 u^\nu  is 4-vector momentum density of matter,  ~ \rho_0  – density of matter in its rest system,  ~ \Gamma^\nu _{\mu \lambda}  – Christoffel symbol.

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

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

~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]

 ~a^\nu = \frac{Du^\nu } {D \tau }= u^\mu \nabla_\mu u^\nu =\frac{ du^\nu } {d \tau }+ \Gamma^\nu_{\mu \lambda} u^\mu u^\lambda=0.

Since interval  ~ds = c d\tau , then equation of motion of the body along a geodesic in general relativity can be rewritten in equivalent form:

 ~ \frac{ d } {d s }\left(\frac{ dx^\nu } {d s } \right)    + \Gamma^\nu_{\mu \lambda } \frac{ dx^\mu } {d s } \frac{ dx^\lambda } {d s }  = 0.

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

 ~\frac{ D } {D p }= \frac {d x^\mu }{dp} \nabla_\mu .


See also


  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. Федосин С.Г. Уравнения движения в теории релятивистских векторных полей. Препринт. Январь, 2018.
  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


External links


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