# Four-force

Four-force (4-force) is a four-vector, considered as a relativistic generalization of the classical 3-vector of force to the four-dimensional spacetime. As in classical mechanics, the 4-force 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 4-force acting on the particle in the field. The equality of 4-forces produced by these methods, gives the equation of motion of the particle in the given force field.

In special relativity 4-force is the derivative of 4-momentum  with respect to the proper time  of the particle: [1]

For a particle with constant invariant mass m > 0,  ,  where   is 4-velocity. This allows connecting 4-force with 4-acceleration  similarly to Newton's second law:

,

Given   is the classic 3-vector of the particle velocity;     is the Lorentz factor;

is the 3-vector of force, [2]

is the 3-vector of relativistic momentum,    is the 3-acceleration,

,

is relativistic energy.

In general relativity, the 4-force is determined by the covariant derivative of 4-momentum with respect to the proper time: [3]

,

where  are the Christoffel symbols.

## Contents

• 1 Examples
• 2 The density of 4-force
• 3 Four-force in CTG
•    3.1 Components of 4-force density
• 5 References

## Examples

4-force acting in the electromagnetic field on the particle with electric charge , is expressed as follows:

,

where  is the electromagnetic tensor

,  is 4-velocity.

## The density of 4-force

To describe liquid and extended media, in which we must find forces in different points in space, instead of 4-vector of force 4-vector of force density is used, acting locally on a small volume unit of the medium:

where   is the mass 4-current,  is the mass density in the rest reference frame relative to the matter.

In the special theory of relativity, the relations hold:

,

,

Where   is 3-vector of force density,  is 3-vector of mass current,  is the density of relativistic energy.

If we integrate (2) over the invariant volume of the matter unit, measured in the co-moving reference frame, we obtain the expression for 4-force (1):

## Four-force in CTG

If the particle is in the gravitational field, then according to the covariant theory of gravitation (CTG) gravitational 4-force equals:

,

where  is the gravitational tensor, which is expressed through the gravitational field strength and the gravitational torsion field,   is 4-momentum with lower (covariant) index, and particle mass  includes contributions from the mass-energy 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 4-force , which depends on the metric tensor through , also becomes a function of the metric. At the same time, the definition of 4-force with covariant index does not require knowledge of the metric:

In the covariant theory of gravitation 4-vector of force density is described with the help of acceleration field : [4]

where  is the acceleration stress-energy tensor  is acceleration tensor  is the 4-acceleration.

In the above expression the operator of proper-time-derivative   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 4-vector of electromagnetic current density (4-current),   is the density of electric charge of the matter unit in its rest reference frame,   is the pressure field tensor  is the gravitational stress-energy tensor is the electromagnetic stress-energy tensor,  is the pressure stress-energy tensor.

In some cases, instead of the mass 4-current the quantity    is used, where   is the density of the moving matter in an arbitrary reference frame. The quantity  is not a 4-vector, 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 4-force for these frames of reference:

In general relativity, it is believed that the stress-energy 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 stress-energy 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 4-momentum containing the energy and momentum is derived by using of Hamiltonian and not from the stress-energy tensors.

### Components of 4-force density

The expression (4) for 4-force 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, four-dimensional quantity     consists of the time component equal to the speed of light , and the spatial component in the form of particle 3-velocity vector .

Similarly, we write the charge 4-current through the charge density of moving matter  :

In addition, we express the tensors through their components, that is, the corresponding 3-vectors of the field strengths. Then the time component of the 4-force 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 4-force is the 3-vector , i.e. 4-force is as

wherein the 3-force density is:

where   is the gravitational torsion field  is the magnetic field,   is the solenoidal vector of pressure field.

Expression for the covariant 4-force can be written in terms of the components of the acceleration tensor and covariant 4-acceleration. Similarly to (3) we have:

where   is the time component of 4-acceleration,   is the 4-potential of the acceleration field,   is the acceleration field strength,   is the acceleration solenoidal vector.

Hence, the 4-acceleration 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 4-acceleration we obtain the standard expression:

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 4-acceleration inside the bodies. In particular, within the bodies there is a 4-acceleration 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 4-force density, 4-acceleration, vectors    and    are functions not only of the radius, but the distance from the axis of rotation to the point of observation.