**Acceleration field** is a two-component vector field, describing
in a covariant way the four-acceleration
of individual particles and the four-force
that occurs in systems with multiple closely interacting particles. The acceleration
field is a component of the general field,
which is represented in the Lagrangian and Hamiltonian of an arbitrary physical
system by the term with the energy of particles’ motion and the term with the
field energy. ^{[1]
[2]} The acceleration field is included
in most equations of vector field.
Moreover, the acceleration field enters into the
equation of motion through the acceleration
tensor and into the equation for the metric through the acceleration stress-energy tensor.

The acceleration field was presented by Sergey Fedosin
within the framework of the metric theory
of relativity and
covariant theory of gravitation, and the
equations of this field were obtained as a consequence of the principle of
least action. ^{[3]} ^{[4]}

- 1 Mathematical description

- 1.1 Action, Lagrangian and energy
- 1.2 Equations
- 1.3 The stress-energy tensor

- 2 Specific solutions for the acceleration field
functions

- 2.1 Ideally solid particle
- 2.2 Rotation of a particle
- 2.3 The system of particles

- 3 Other approaches
- 4 The use in the general theory of relativity
- 5 See also
- 6 References
- 7 External links

The 4-potential of the
acceleration field is expressed in terms of the scalar and vector potentials:

The antisymmetric acceleration tensor is calculated with the
help of the 4-curl of the 4-potential:

The acceleration tensor
components are the components of the field strength and the components of the solenoidal vector :

We obtain the following:

In the general case the scalar
and vector potentials are found by solving the wave equations for the
acceleration field potentials.

In the covariant theory of gravitation the
4-potential of the acceleration field is part of the
4-potential of the general field , which is the sum
of the 4-potentials of particular fields, such as the electromagnetic and
gravitational fields, acceleration field, pressure
field, dissipation field, strong
interaction field, weak interaction field and other vector fields, acting on the
matter and its particles. All of these fields are somehow represented in the
matter, so that the 4-potential cannot consist of only one 4-potential . The energy density of interaction of the general
field and the matter is given by the product of the 4-potential of the general
field and the mass 4-current: . We obtain the
general field tensor from the 4-potential of the general field, using the
4-curl:

The tensor invariant in the form is up to a
constant factor proportional to the energy density of the general field. As a
result, the action function, which contains the scalar curvature and the cosmological constant , is given by the
expression: ^{[1]}

where is the Lagrange function or Lagrangian; is the time differential of the coordinate
reference system; and are the constants to be determined; is the speed of light as a measure of the
propagation speed of the electromagnetic and gravitational interactions; is the invariant 4-volume expressed in terms
of the differential of the time coordinate , the product of differentials of the space coordinates
and the square root of the determinant
of the metric
tensor, taken with a negative sign.

The variation of the action
function gives the general field equations, the four-dimensional equation of
motion and the equation for the metric. Since the acceleration field is the
general field component, then from the general field equations the
corresponding equations of the acceleration field are derived.

Given the gauge condition of the
cosmological constant in the form

is met, the system energy does
not depend on the term with the scalar curvature and is uniquely determined: ^{[4]}

where and denote the time components of the
4-vectors and
.

The system’s 4-momentum is given
by the formula:

where and denote the system’s momentum and the
velocity of the system’s center of momentum.

The four-dimensional equations of the acceleration field
are similar in their form to Maxwell equations and are as follows:

where is the
mass 4-current, is the mass density in the co-moving
reference frame, is the 4-velocity of the matter unit, is a constant, which is determined in each
problem, and it is supposed that there is an equilibrium between all fields in
the observed physical system.

The gauge condition of the
4-potential of the acceleration field:

If the second equation with the
field source is written with the covariant index in the following form:

then after substituting here the
expression for the acceleration tensor through the
4-potential of the
acceleration field we obtain the wave
equation for calculating the potentials of the acceleration field:

where is the Ricci tensor.

The continuity equation in curved
space-time is:

In Minkowski space
of the special theory of relativity, the Ricci tensor is set to zero, the form of the
acceleration field equations is simplified and they can be expressed in terms
of the field strength and the solenoidal vector :

where is the Lorentz factor, is the mass current density, is the velocity of the matter unit.

The wave equation is also simplified and can be written separately for
the scalar and vector potentials:

The equation of motion of the
matter unit in the general field is given by the formula:

.

Since , and the general field tensor is expressed in
terms of the tensors of particular fields, then the equation of motion can be
represented with the help of these tensors:

Here is the electromagnetic tensor, is the charge 4-current, is the gravitational
tensor, is the pressure
field tensor, is the dissipation
field tensor, is the strong interaction field tensor, is the weak interaction field tensor.

The acceleration
stress-energy tensor is calculated with the help of the acceleration
tensor:

.

We find as part of the tensor the 3-vector of the energy-momentum flux , which is similar
in its meaning to the Poynting vector and the Heaviside vector. The vector can be represented through the vector product
of the field strength and the solenoidal vector :

here the index is

The covariant derivative of the
stress-energy tensor of the acceleration field with mixed indices specifies the 4-force density:

where denotes the proper time differential in the
curved spacetime.

The stress-energy tensor of the
acceleration field is part of the stress-energy tensor of the general field :

where is the electromagnetic stress–energy
tensor, is the gravitational
stress-energy tensor, is the pressure
stress-energy tensor, is the dissipation
stress-energy tensor, is the strong interaction stress-energy
tensor, is the weak interaction stress-energy tensor.

Through the tensor the stress-energy tensor of the acceleration
field enters into the equation for the metric:

where is the Ricci tensor, is the gravitational
constant, is a certain constant, and the gauge
condition of the cosmological constant is used.

The four-potential of any vector field, the global vector potential of
which is equal to zero in the proper reference frame K', that is, in the center-of-momentum
frame, in case of rectilinear motion in the laboratory reference frame K, can
be presented as follows: ^{[3]} ^{[5]}

where is for the electromagnetic field and for the remaining fields; and are the invariant mass density and the charge
density in the comoving reference frame, respectively; is the invariant energy density of the
interaction, calculated as product of the four-potential of the field and the
corresponding four-current; is the covariant four-velocity that
determines the motion of the center of momentum of the physical system in K.

In the special relativity (SR), in the center-of-momentum frame K' the energy density is ,
where is the Lorentz factor, and for the
acceleration field, while the physical system is moving in K, the
four-potential of the acceleration field will equal .

In case when the physical system is stationary in K, we will have , and consequently, the scalar potential will
be .
If in the physical system, on the average, there are directed fluxes of matter
or rotation of matter, the vector potential
of the acceleration field is no longer equal
to zero.

If the
four-potential of acceleration field in K' is
known, then in the laboratory reference frame K the four-potential is determined
using the matrix connecting the coordinates and
time of both frames: ^{[6]}

In the
special case of the system’s motion at the constant velocity represents the Lorentz
transformation matrix.

In the approximation, when a particle
is regarded as an ideally solid object, the matter inside the particle is
motionless. It means that the Lorentz factor
of this matter in the center-of-momentum
frame K'
is equal to unity, so that the four-potential of the acceleration field
becomes equal to the four-velocity of motion of the center of momentum:

In the SR, the expression for
4-velocity is simplified and we can write:

The acceleration tensor components
according to (1) will equal:

Since in the solid-state motion equation for the
four-acceleration with a covariant index the relation holds

then in SR we obtain the following:

and the equations for the Lorentz
factor and for the 3-acceleration :

Multiplying equation (6) by the
velocity , substituting the quantity from equation (5) to (6), taking into account
relation we find the well-known expression for the
derivative of the Lorentz factor using the scalar product of the velocity and
acceleration in SR:

We can prove the validity of
equation (6) by substituting in its right-hand side the expression
for the strength and solenoidal vector:

Indeed, the use of the material derivative gives the following:

In addition

Substituting these relations in
(7), taking into account the expression we
obtain the identity:

If the components of the particle
velocity are the functions of time and they do not directly depend on the space
coordinates, then the solenoidal vector vanishes in such a motion.

In the SR is the
relativistic energy, is the
3-vector of relativistic momentum. If the mass
of a particle
is constant, then multiplying (7) by the mass, we arrive to following equation
for the force:

**Rotation of a particle**

For a small ideally solid particle, we can neglect the motion of the
matter inside the particle and can assume that the four-potential of the
acceleration field is equal to the four-velocity of the particle’s center of
momentum. Let us assume that the particle rotates about the axis
OZ of the coordinate system at the distance
from the axis at the constant
angular velocity counterclockwise, as viewed from
the side, in which the OZ axis is directed. Then we can assume that the linear
velocity of the particle depends only on the coordinates and , and for the velocity’s projections on the axes of the coordinate
system we can write: ,
while the square of the velocity equals
. For the Lorentz factor in the SR we obtain the following:

With this in mind, the potentials and field strengths of the
acceleration field can be written as follows:

If we substitute , , and in (6), we can determine the
acceleration components of the particle and the acceleration amplitude:

The acceleration is directed towards the center of rotation and
represents centripetal acceleration. Using now the classic vector description,
we have also for the time and coordinates of reference frame at the center of
rotation:

where and are two coordinates of the
cylindrical coordinate system, is the vector from the center of
rotation to the particle, is the axial vector of the
differential of the rotation angle directed along the axis OZ.

As we can see, in case of such a motion with acceleration the vector
product is not equal to zero, just as the
three-vector of the energy-momentum flux of
the acceleration field inside the particle.

Due to interaction of a number of particles with each
other by means of various fields, including interaction at a distance without
direct contact, the acceleration field in the matter changes and is different
from the acceleration field of individual particles at the observation point. As a result, the density of the
4-force in the system of
particles is given by the strength and the solenoidal vector, which represent
the typical average characteristics of the matter motion. For example, in a
gravitationally bound system there is a radial gradient of the vector and if the system is moving or rotating,
there is a vector From (4) there follows the general expression
for the density
of the 4-force with covariant index:

where denotes a four-dimensional space-time
interval. For a stationary case, when the potentials of the acceleration field
are independent of time, under the assumption that wave equation (2) for the scalar potential
in the SR is transformed
into the equation:

The solution of this equation for
a fixed sphere with the particles randomly moving in it has the form: ^{[7]}

where is the Lorentz factor for the velocities of the particles in the center of the
sphere, and due to the smallness of the argument the sine is expanded to the
second order terms. From the formula it follows that the average velocities of
the particles are maximal in the center and decrease when approaching the
surface of the sphere.

In such a system, the scalar
potential becomes the function of the radius, and the
vector potential and the solenoidal vector are equal to zero. The acceleration field
strength is found with the help of (1). Then we can calculate
all the functions of the acceleration field, including the energy of particles
in this field and the energy of the acceleration field itself. ^{[8]} For cosmic bodies the main contribution to the
four-acceleration in the matter makes the gravitational force and the pressure
field.

At the same time the relativistic
rest energy of the system is automatically derived, taking into account the
motion of particles inside the sphere. For the system of particles with the
acceleration field, pressure field, gravitational and electromagnetic fields
the given approach allowed solving the 4/3 problem and showed where and in what
form the energy of the system is contained. The relation for the acceleration
field constant in this problem was found:

where is the electric
constant, and are the total charge and mass of the system.

The solution of the wave equation
for the acceleration field within the system results in temperature
distribution according to the formula: ^{[7]}

where is the temperature in the center, is the mass of the particle, for which the
mass of the proton is taken (for systems which are based on hydrogen or
nucleons in atomic nuclei), is the mass of the system within the current
radius , is the Boltzmann constant.

This dependence is well satisfied
for a variety of space objects, including gas clouds and Bok globules, the
Earth, the Sun and neutron stars.

In articles ^{[9]} ^{[10]}
the ratio of the field’s coefficients for the fields was specified as follows:

where is the pressure
field constant.

If we introduce the parameter as the number
of nucleons per ionized gas particle, then the acceleration field constant is
expressed as follows:

For the temperature inside the
cosmic bodies in the gravitational equilibrium model we find the dependence on
the current radius:

where is the mass of
one gas particle, which is taken as the unified atomic mass unit, and the
coefficients and are included
into the dependence of the mass density on the radius in the relation

Under the
assumption that the system’s typical particles have the mass , and that it is typical
particles that define the temperature and pressure, for the acceleration field
constant we obtain the following: ^{[11]}

The Lorentz
factor of the particles in the center of the system is also determined: ^{[12]}

The wave equation (3) for the
vector potential of the acceleration field was used to represent the
relativistic equation of the fluid’s motion in the form of the Navier–Stokes equations in hydrodynamics
and to describe the motion of the viscous compressible and charged
fluid. ^{[1}^{3}^{]}

Taking into
account the acceleration field and pressure field, within the framework of the __relativistic
uniform system__, it is possible to refine the virial theorem, which in the relativistic form is written as follows: ^{[14]}

where the
value exceeds the kinetic energy of the
particles by a factor equal to the Lorentz
factor of the particles at the center
of the system. Under normal conditions we can assume that , then we can see that in the
virial theorem the kinetic energy is related to the potential energy not by the
coefficient 0.5, but rather by the coefficient close to 0.6. The difference
from the classical case arises due to considering the pressure field and the
acceleration field of particles inside the system, while the derivative of the
virial scalar function is not equal to zero and should be considered
as the material
derivative.

An analysis
of the integral theorem of generalized virial makes it possible to find, on the
basis of field theory, a formula for the root-mean-square speed of typical
particles of a system without using the notion of temperature: ^{[15]}

The integral
__field
energy theorem__ for acceleration field in a curved
space-time is as follows:^{[6]}

In the relativistic
uniform system, the scalar potential of the acceleration field is
related to the scalar potential of the pressure field: ^{[16]}

The
relativistic expression for pressure is as follows:

where is the mass density of moving
matter, is the speed of light, is the Lorentz factor. In the limit of low velocities, this relationship
turns into the standard formula of the kinetic theory of gases.

Studying the Lorentz covariance of the 4-force, Friedman
and Scarr found incomplete covariance of the
expression for the 4-force in the form ^{[1}^{7}^{]}

This led them to conclude that
the four-acceleration in SR must be expressed with the help of a certain
antisymmetric tensor :

Based on the analysis of various
types of motion, they estimated the required values of the acceleration tensor
components, thereby giving indirect definition to this tensor. From comparison with (4) it follows that the tensor up to a sign and a constant multiplier
coincides with the acceleration tensor in case when rectilinear motion of a solid
body without rotation is considered. Then indeed the four-potential of the
acceleration field coincides with the four-velocity, . As a
result, the quantity on the right-hand side of (4) vanishes, since
the following relations hold true: , . With this in mind, in (4) we can raise the
index and
cancel the mass density, which gives the following:

Mashhoon and Muench considered transformation
of inertial reference frames, co-moving with the accelerated reference frame,
and obtained the relation: ^{[1}^{8}^{]}

The tensor has the same properties as the acceleration
tensor

**The use in the general theory of relativity**

The action function in the general relativity (GR) can be represented as
the sum of the four terms, which are responsible, respectively, for the
spacetime metric, the matter in the form of substance, the electromagnetic
field and the pressure field:

Additional terms can be included in the action function, if other fields
must be taken into account. The first, second and third terms of the action
have the standard form: ^{[19]}

where is the electromagnetic four-potential.

The term , which is responsible for the contribution of pressure into the action
function, is different in the works of different authors, depending on how the
pressure is related to the elastic energy and whether the pressure field is
considered to be a scalar field or a vector field. It should be noted that in
the GR, the gravitational field is included in the action function not
directly, but indirectly, by means of the metric tensor. In this case, as a
rule, the pressure field is considered to be a scalar field.

In contrast, in the __covariant theory of gravitation__ (CTG), the term associated with the acceleration
field is used instead of the term , and the action function can be written as follows: ^{[4]}

Here

where is the four-potential of the __pressure
field__, is the coefficient of the
pressure field, is the __pressure field tensor__,
.

In the case of rectilinear motion
of a rigid body without rotation, the following relations will hold: , , and in the term the relation
is obtained. In this particular case it is
clear that the term differs from the term by an additional term associated with the
energy of the acceleration field.

This is due to the fact that in the covariant theory of gravitation the
acceleration field is considered to be a vector field, whereas as in the
general relativity the acceleration field is actually used
as a scalar field that does not depend on the particles’ velocities. In both
theories, the acceleration field allows us to determine the contribution of the
rest energy of the particles into the total energy of the system of particles
and fields. However, the use of the acceleration field as a scalar field in the
general relativity does not agree in its form with the vector nature of the
electromagnetic field. Indeed, in the limiting case, when only the particles’
accelerations and electromagnetic forces are taken into account, the
acceleration must be two-component, as is the case for the acceleration due to
the action of the two-component Lorentz force. But this is possible only in the
case, when the acceleration field is a vector field. The situation can be
improved if, in addition to the gravitational field function, we ascribe to the
metric field in the general relativity the
function of the vector component of the acceleration field, but this makes the
equations of the theory even more complex and complicated.

It should be noted that in the general case of arbitrary
motion of the matter the relation is no longer satisfied and CTG does not coincide
any more with GR in the method of describing the rest energy of a physical
system. This means that in GR the motion of the matter is considered in a
simplified way, as rectilinear motion of a solid body, whereas in CTG the use
of the four-potential o f the acceleration
field allows us to take into account the internal motion of the matter in each
selected volume element. For example, when a particle moves round a circle, the
four-potential of the particle’s matter will depend on the
location of this matter with respect to the circle line, since the velocity of
the particle’s matter depends on the radius of rotation.

- General field
- Pressure field
- Dissipation
field
- Covariant theory of gravitation
- Metric theory of
relativity
- Acceleration tensor
- Acceleration stress-energy tensor
- Four-force
- Equation of
vector field

^{1.0}^{1.1}Fedosin S.G. The Concept of the General Force Vector Field. OALib Journal, Vol. 3, pp. 1-15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459.- Fedosin
S.G. Two components of the macroscopic general field. Reports in Advances
of Physical Sciences, Vol. 1, No. 2, 1750002, 9 pages (2017). http://dx.doi.org/10.1142/S2424942417500025.
^{3.0}^{3.1}Fedosin S.G. The procedure of finding the stress-energy tensor and vector field equations of any form. Advanced Studies in Theoretical Physics, Vol. 8, No. 18, pp. 771-779 (2014). http://dx.doi.org/10.12988/astp.2014.47101.^{4.0}^{4.1}^{4.2}Fedosin S.G. About the cosmological constant, acceleration field, pressure field and energy. Jordan Journal of Physics. Vol. 9, No. 1, pp. 1-30 (2016).- Fedosin S.G. Equations of
Motion in the Theory of Relativistic Vector Fields. International Letters
of Chemistry, Physics and Astronomy, Vol. 83, pp. 12-30 (2019). https://doi.org/10.18052/www.scipress.com/ILCPA.83.12.
^{6.0}^{6.1}Fedosin S.G. The Integral Theorem of the Field Energy. Gazi University Journal of Science. Vol. 32, No. 2, pp. 686-703 (2019). http://dx.doi.org/10.5281/zenodo.3252783.^{7.0}^{7.1}Fedosin S.G. The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and Acceleration Field. American Journal of Modern Physics. Vol. 3, No. 4, pp. 152-167 (2014). http://dx.doi.org/10.11648/j.ajmp.20140304.12.- Fedosin S.G. Relativistic
Energy and Mass in the Weak Field Limit. Jordan Journal of Physics. Vol. 8, No. 1, pp. 1-16 (2015). http://dx.doi.org/10.5281/zenodo.889210.
- Fedosin S.G. Estimation
of the physical parameters of planets and stars in the gravitational
equilibrium model. Canadian Journal of Physics, Vol. 94, No. 4, pp.
370-379 (2016). http://dx.doi.org/10.1139/cjp-2015-0593.
- Fedosin S.G. The generalized Poynting theorem for the general field
and solution of the 4/3 problem. International Frontier Science Letters,
Vol. 14, pp. 19-40 (2019). https://doi.org/10.18052/www.scipress.com/IFSL.14.19.
- Fedosin S.G. The binding energy and the total energy of a
macroscopic body in the relativistic uniform model. Middle East Journal of
Science, Vol. 5, Issue 1, pp. 46-62 (2019). http://dx.doi.org/10.23884/mejs.2019.5.1.06.
- Fedosin S.G. Energy and metric gauging in the covariant theory of
gravitation. Aksaray University Journal of
Science and Engineering, Vol. 2, Issue 2, pp. 127-143 (2018). http://dx.doi.org/10.29002/asujse.433947.
- Fedosin
S.G. Four-Dimensional
Equation of Motion for Viscous Compressible and Charged Fluid with Regard
to the Acceleration Field, Pressure Field and Dissipation Field.
International Journal of Thermodynamics. Vol. 18, No. 1, pp. 13-24 (2015).
http://dx.doi.org/10.5541/ijot.5000034003.
- Fedosin S.G. The virial
theorem and the kinetic energy of particles of a macroscopic system in the
general field concept. Continuum Mechanics and Thermodynamics. Vol. 29,
Issue 2, pp. 361-371 (2017). https://dx.doi.org/10.1007/s00161-016-0536-8.
- Fedosin S.G. The integral theorem of
generalized virial in the relativistic uniform model. Continuum Mechanics and Thermodynamics, Vol.
31, Issue 3, pp. 627-638 (2019). https://dx.doi.org/10.1007/s00161-018-0715-x.
- Fedosin S.G. The potentials of the acceleration field and pressure field in
rotating relativistic uniform system. Continuum Mechanics and
Thermodynamics, Vol. 33, Issue 3, pp. 817-834 (2021). https://doi.org/10.1007/s00161-020-00960-7.
- Yaakov Friedman and Tzvi Scarr. Covariant Uniform
Acceleration. Journal of Physics: Conference Series Vol. 437 (2013)
012009 doi:10.1088/1742-6596/437/1/012009.
- Bahram Mashhoon and Uwe Muench. Length measurement in
accelerated systems. Annalen der Physik. Vol. 11, Issue 7, P. 532–547, 2002.
- Fock, V. A. (1964). "The
Theory of Space, Time and Gravitation". Macmillan.

·
Acceleration
field in Russian

Source:
http://sergf.ru/afen.htm