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:

$\frac{D}{D\tau }=u^{\mu }{\mathrm{\nabla }}_{\mu }$,

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, ${\mathrm{\nabla }}_{\mu }$ – 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 $A^{\nu }$:

$u^{\mu }{\partial }_{\mu }{A}^{\nu }=\frac{cdt}{d\tau }\frac{\partial A^{\nu }}{c\partial t}+\frac{dx}{d\tau }\frac{\partial A^{\nu }}{\partial x}+\frac{dy}{d\tau }\frac{\partial A^{\nu }}{\partial y}+\frac{dz}{d\tau }\frac{\partial A^{\nu }}{\partial z}=$

$=\frac{dt}{d\tau }(\frac{\partial A^{\nu }}{\partial t}+\frac{dx}{dt}\frac{\partial A^{\nu }}{\partial x}+\frac{dy}{dt}\frac{\partial A^{\nu }}{\partial y}+\frac{dz}{dt}\frac{\partial A^{\nu }}{\partial z})=\frac{dt}{d\tau }\frac{d{A}^{\nu }}{dt}=\frac{d{A}^{\nu }}{d\tau }$.

Above was used material derivative in operator equation for an arbitrary function $F$:

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

where $𝐕$ is the velocity of local volume of matter, $\mathrm{\nabla }$ – 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 $x^{\mu }=(ct,x,y,z)=(ct,𝐫)$ because 4-position is not a 4-vector in curved space-time, but its differential (displacement) $d{x}^{\mu }=(cdt,dx,dy,dz)=(cdt,d𝐫)$ 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: $u^{\nu }{\mathrm{\nabla }}_{\nu }{x}^{\mu }=u^{\mu }$.

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:

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

where $J^{\nu }={\rho }_0{u}^{\nu }$ is 4-vector momentum density of matter, ${\rho }_0$ – density of matter in its rest system, ${\mathrm{\Gamma }}_{\mu \lambda }^{\nu }$ – Christoffel symbol.

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

$f_{\alpha }={\mathrm{\nabla }}_{\beta }{B_{\alpha }}^{\beta }=-u_{\alpha k}{J}^k={\rho }_0\frac{D{U}_{\alpha }}{D\tau }-J^k{\mathrm{\nabla }}_{\alpha }{U}_k={\rho }_0\frac{d{U}_{\alpha }}{d\tau }-J^k{\partial }_{\alpha }{U}_k,$

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 $𝐔$ potentials:

$U_{\alpha }=(\frac{\vartheta }{c},-𝐔).$

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:

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

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

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

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:

$\frac{D}{Dp}=\frac{d{x}^{\mu }}{dp}{\mathrm{\nabla }}_{\mu }$.

See also

• Four-gradient
• Four-force

External links

• Operator of proper-time-derivative in Russian