Fourforce
Fourforce (4force) is a fourvector, considered as a relativistic generalization of the classical 3vector of force to the fourdimensional spacetime. As in classical mechanics, the 4force can be defined in two ways. The first one measures the change in the energy and momentum of a particle per unit of proper time. The second method introduces force characteristics – strengths of field, and with their help in certain energy and momentum of the particle is calculated 4force acting on the particle in the field. The equality of 4forces produced by these methods, gives the equation of motion of the particle in the given force field.
In special relativity 4force is the derivative of 4momentum with respect to the proper time of the particle: ^{[1]}
For a particle with constant invariant mass m > 0, , where is 4velocity. This allows connecting 4force with fouracceleration similarly to Newton's second law:
 ,
Given is the classic 3vector of the particle velocity; is the Lorentz factor;
is the 3vector of force, ^{[2]}
is the 3vector of relativistic momentum, is the 3acceleration,
 ,
is relativistic energy.
In general relativity, the 4force is determined by the covariant derivative of 4momentum with respect to the proper time: ^{[3]}
 ,
where are the Christoffel symbols.
Contents
Examples[edit]
4force acting in the electromagnetic field on the particle with electric charge , is expressed as follows:
 ,
where is the electromagnetic tensor

, is 4velocity.
The density of 4force[edit]
To describe liquid and extended media, in which we must find forces in different points in space, instead of 4vector of force 4vector of force density is used, acting locally on a small volume unit of the medium:
where is the mass 4current, is the mass density in the rest reference frame relative to the matter.
In the special theory of relativity, the relations hold:
 ,
 ,
where is 3vector of force density, is 3vector of mass current, is the density of relativistic energy.
If we integrate (2) over the invariant volume of the matter unit, measured in the comoving reference frame, we obtain the expression for 4force (1):
Fourforce in CTG[edit]
If the particle is in the gravitational field, then according to the covariant theory of gravitation (CTG) gravitational 4force equals:
 ,
where is the gravitational tensor, which is expressed through the gravitational field strength and the gravitational torsion field, is 4momentum with lower (covariant) index, and particle mass includes contributions from the massenergy of fields associated with the matter of the particle.
In CTG gravitational tensor with covariant indices is determined directly, and for transition to the tensor with contravariant indices in the usual way the metric tensor is used which is in general a function of time and coordinates:
Therefore the 4force , which depends on the metric tensor through , also becomes a function of the metric. At the same time, the definition of 4force with covariant index does not require knowledge of the metric:
In the covariant theory of gravitation 4vector of force density is described with the help of acceleration field: ^{[4]}
where is the acceleration stressenergy tensor, is acceleration tensor, is the 4acceleration.
In the above expression the operator of propertimederivative is used, which generalizes the material derivative (substantial derivative) to the curved spacetime. ^{[2]}
If there are only gravitational and electromagnetic forces and pressure force, then the following expression is valid:
where is the metric tensor, is the 4vector of electromagnetic current density (4current), is the density of electric charge of the matter unit in its rest reference frame, is the pressure field tensor, is the gravitational stressenergy tensor, is the electromagnetic stressenergy tensor, is the pressure stressenergy tensor.
In some cases, instead of the mass 4current the quantity is used, where is the density of the moving matter in an arbitrary reference frame. The quantity is not a 4vector, since the mass density is not an invariant quantity in coordinate transformations. After integrating over the moving volume of the matter unit due to the relations and we obtain:
For inertial reference systems in the last expression we can bring beyond the integral sign. This gives 4force for these frames of reference:
In general relativity, it is believed that the stressenergy tensor of matter is determined by the expression , and for it , that is the quantity consists of four timelike components of this tensor. The integral of these components over the moving volume gives respectively the energy (up to the constant, equal to ) and the momentum of the matter unit. However, such a solution is valid only in approximation of inertial motion, as shown above. In addition, according to the findings in the article, ^{[5]} the integration of timelike components of the stressenergy tensor for energy and momentum of a system in general is not true and leads to paradoxes such as the problem of 4/3 for the gravitational and electromagnetic fields.
Instead of it, in the covariant theory of gravitation 4momentum containing the energy and momentum is derived by using of Hamiltonian and not from the stressenergy tensors.
Components of 4force density[edit]
The expression (4) for 4force density can be divided into two parts, one of which will describe the bulk density of energy capacity, and the other describe total force density of available fields. We assume that speed of gravity is equal to the speed of light. In order do not depend on the metric tensor, we can write (4) with the lower, covariant index:
In this relation we make a transformation:
where denotes interval, is the differential of coordinate time, is the mass density of moving matter, fourdimensional quantity consists of the time component equal to the speed of light , and the spatial component in the form of particle 3velocity vector .
Similarly, we write the charge 4current through the charge density of moving matter :
In addition, we express the tensors through their components, that is, the corresponding 3vectors of the field strengths. Then the time component of the 4force density with covariant index is:
where is the gravitational field strength, is the electromagnetic field strength, is the pressure field strength.
The spatial component of covariant 4force is the 3vector , i.e. 4force is as
wherein the 3force density is:
where is the gravitational torsion field, is the magnetic field, is the solenoidal vector of pressure field.
Expression for the covariant 4force can be written in terms of the components of the acceleration tensor and covariant 4acceleration. Similarly to (3) we have:
where is the time component of 4acceleration, is the 4potential of the acceleration field, is the acceleration field strength, is the acceleration solenoidal vector.
Hence, the 4acceleration with covariant index can be expressed through its scalar and vector components:
In special relativity and substituting the vectors and for a particle, for the covariant 4acceleration we obtain the standard expression:
If the mass of the particle is constant, then for the force acting on the particle, we can write:
where is the relativistic energy, is the 3vector of relativistic momentum of the particle.
For a body with a continuous distribution of matter vectors and are substantially different from the corresponding instantaneous vectors of specific particles in the vicinity of the observation point. These vectors represent the averaged value of 4acceleration inside the bodies. In particular, within the bodies there is a 4acceleration generated by the various forces in matter. A typical example are the space bodies, where the major forces are the force of gravity and the internal pressure generally oppositely directed. Upon rotation of the bodies the 4force density, 4acceleration, vectors and are functions not only of the radius, but the distance from the axis of rotation to the point of observation.
See also[edit]
References[edit]
 ↑ Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press. ISBN 0198539525.
 ↑ ^{2.0} ^{2.1} Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 9785990195110. (in Russian).
 ↑ Landau L.D., Lifshitz E.M. (1975). The Classical Theory of Fields. Vol. 2 (4th ed.). ButterworthHeinemann. ISBN 9780750627689.
 ↑ Fedosin S.G. About the cosmological constant, acceleration field, pressure field and energy. Jordan Journal of Physics. Vol. 9 (No. 1), pp. 130 (2016).
 ↑ Fedosin S.G. The generalized Poynting theorem for the general field and solution of the 4/3 problem. Preprint, February 2016.