# 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 substance, $~ \nabla_\mu$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 substance, $~ \nabla$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 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 substance, $~ \rho_0$ – density of substance in its rest system, $~ \Gamma^\nu _{\mu \lambda}$Christoffel symbol.

In general relativity freely falling body in a gravitational field moves along a geodesic, and 4-acceleration of body in this case is equal to zero:[3]

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

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:[4]

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