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):
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]}^{ }^{[5]} ^{[6]}^{}
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:
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, ^{[7]} 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