# 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 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.

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]

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

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:[4]

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