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]


where – the symbol of differential in curved spacetime, proper time, which is measured by a clock moving with test particle, 4-velocity of test particle or local volume of matter, 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):


To prove this expression it can be applied to an arbitrary 4-vector :


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


where is the velocity of local volume of matter, nabla operator.

In turn, the material derivative follows from the representation of differential function of spatial coordinates and time:


Applications edit

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   because 4-position is not a 4-vector in curved space-time, but its differential (displacement)   is. Effect of the left side of operator of proper-time-derivative on the 4-position specifies the 4-velocity:  , but the right side of the operator does not so:  .

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:[2]


where   is 4-vector momentum density of matter,   – density of matter in its rest system,   Christoffel symbol.

However in the common case the 4-force is determined with the help of 4-potential of acceleration field: [3]


where   is the acceleration stress-energy tensor with the mixed indices,   is the acceleration tensor, and the 4-potential of acceleration field is expressed in terms of the scalar   and vector   potentials:


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: [4]


Since interval  , then equation of motion of the body along a geodesic in general relativity can be rewritten in equivalent form:


If, instead of the proper time to use a parameter  , and equation of a curve set by the expression  , then there is the operator of derivative on the parameter along the curve:[5]


See also edit

References edit

  1. Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009-2011, 858 pages, Tabl. 21, Pic. 41, Ref. 293. ISBN 978-5-9901951-1-0. (in Russian).
  2. Fedosin S.G. The General Theory of Relativity, Metric Theory of Relativity and Covariant Theory of Gravitation: Axiomatization and Critical Analysis. International Journal of Theoretical and Applied Physics, Vol. 4, No. 1, pp. 9-26 (2014). http://dx.doi.org/10.5281/zenodo.890781.
  3. Fedosin S.G. Equations of Motion in the Theory of Relativistic Vector Fields. International Letters of Chemistry, Physics and Astronomy, Vol. 83, pp. 12-30 (2019). https://doi.org/10.18052/www.scipress.com/ILCPA.83.12.
  4. Fock, V. A. (1964). "The Theory of Space, Time and Gravitation". Macmillan.
  5. Carroll, Sean M. (2004), Spacetime and Geometry, Addison Wesley, ISBN 0-8053-8732-3

External links edit