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 4acceleration 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.
4force acting in the
electromagnetic field on the particle with electric charge , is expressed as
follows:
,
where is the electromagnetic tensor

, is 4velocity.
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):
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.
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.
Source: http://sergf.ru/ffen.htm