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:


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

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:


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

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 $$ ~ x^\mu=(ct,x,y,z)=(ct, \mathbf{r} )$$ 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 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:


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

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:


 * $$ ~ f_\alpha = \nabla_\beta {B_\alpha}^\beta = - u_{\alpha k} J^k = \rho_0 \frac {DU_\alpha }{D \tau}- J^k \nabla_\alpha U_k = \rho_0 \frac {dU_\alpha }{d \tau} - J^k \partial_\alpha U_k ,$$

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:


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

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:


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