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: $~{\frac {Dx^{\mu }}{D\tau }}=u^{\mu }$, 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:

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

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

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

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

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

If, instead of the proper time to use a parameter $~p$, 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:

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

See also

• Four-gradient
• Four-force

External links

• Operator of proper-time-derivative in Russian