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 the 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 the 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):
This formula and determination
of the fourforce density through the mass fourcurrent when taking into account the fields acting in the system require
correction, since they do not contain an additional contribution from the
fourmomenta of the fields themselves. ^{[4]}
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 : ^{[}^{5]}^{ }^{[6]} ^{[7]}^{}
where
is the acceleration
stressenergy tensor with mixed indices, is acceleration
tensor, and
the 4potential of the acceleration field is expressed in terms of the scalar
potential and the vector potential :
In the expression (3) 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 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:
However, in addition to the
momentum of particles, moving matter also has the momentum of the field
associated with the matter, which requires a more general definition of
fourmomentum and fourforce.^{ [8]}
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, ^{[9]} 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 the
variation of the Lagrangian of the system 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 relation (4) 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 acceleration field
strength, is the acceleration solenoidal
vector.
Using the expression for the
4potential of the accelerations field in terms of the scalar potential and the
vector potentials and the definition of material derivative, from (3) and (4)
for the scalar and vector components of the equation of motion, we obtain the
following:
Here
are the
components of the vector potential of the
acceleration field, are the
components of the velocity of the element of
matter or particle.
Equations of the matter’s motion (5)
and (6) are obtained in a covariant form and are valid in the curved spacetime.
On the lefthand side of these equations there are either potentials or the
strength and the solenoidal vector of the acceleration field. The righthand
side of the equations of motion is expressed in terms of the strengths and the
solenoidal vectors of the gravitational and electromagnetic fields, as well as
the pressure field inside the matter. Before solving these equations of motion,
first it is convenient to find the potentials of all the fields with the help
of the corresponding wave equations. Next, taking the fourcurl of the fields’
fourpotentials we can determine the strengths and the solenoidal vectors of
all the fields. After substituting them in (5) and (6), it becomes possible to
find the relation between the field coefficients, express the acceleration
field coefficient, and thus completely determine this field in the matter.
Relationship with the fouracceleration
The peculiarity of equations of motion
(5) and (6) is that they do not have a direct relationship with the fouracceleration
of the matter particle under consideration. However, in some cases it is
possible to determine the acceleration and velocity of motion, as well as the
dependence of the distance traveled on time. The simplest example is the
rectilinear motion of a uniform solid particle in uniform external fields. In
this case, the fourpotential of the acceleration field fully coincides with
the fourvelocity of the particle, so that the scalar potential , the vector
potential , where is the Lorentz
factor of the particle. Substituting the equality in (3) gives the
following:
where
is defined as the
fouracceleration.
Then the equation for the
fouracceleration of the particle follows from (3) and (4):
After multiplying by the particle’s
mass, this equation will correspond to equation (1) for the fourforce.
In the considered case of motion of a
solid particle, the fouracceleration with a covariant index can be expressed
in terms of the strength and the solenoidal vector of the acceleration field:
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. The
typical examples are the relativistic uniform system
and 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.
In the general case for extended bodies
the fouracceleration at each point of the body becomes a certain function of
the coordinates and time. As a characteristic of the physical system’s motion
we can choose the fouracceleration of the center of momentum, for the
evaluation of which it is necessary to integrate the force density over the
volume of the entire matter and divide the total force by the inertial mass of
the system. Another method involves evaluation of the fouracceleration through
the strength and the solenoidal vector of the acceleration field at the center
of momentum in the approximation of the special theory of relativity, as was
shown above.
Source: http://sergf.ru/ffen.htm