**Acceleration field** is a two-component vector field, describing
in a covariant way the 4-acceleration of individual particles and the
4-acceleration 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]} 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. ^{[2]} ^{[3]}

- 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 One particle
- 2.2 The system
of particles

- 3 Other approaches
- 4 See also
- 5 References
- 6 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 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: ^{[3]}

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
mass.

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:

In Minkowski space of the special
theory of relativity 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.

Using as well the gauge condition
in the form of and relation (1), we can obtain from
the field equations the wave equations for the acceleration field 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 and the 4-acceleration :

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 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 , but in the general case the tensor contains also the cross-terms with the
products of strengths and solenoidal vectors of particular fields:

where are
certain coefficients, 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 for a single particle can be
represented as: ^{[2]}

where for
electromagnetic field and for
other fields, and are the mass density and accordingly charge density
in comoving reference frame, is the field energy density of the
particle, is the covariant four-velocity.

For the acceleration field , , and according
to the definition, for the four-potential of the acceleration field of one
particle we have the following:

i. e. for the single particle the field 4-potential is the 4-velocity
with a covariant index. In the special relativity (SR) we can write:

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

Since in the equation of motion
for the 4-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 :

Substituting the quantity from equation (5) to (6),
multiplying equation (6) by the velocity 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 it 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.

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. The acceleration field 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 4-acceleration 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 is
transformed into the equation:

The solution of this equation for
a fixed sphere with the particles randomly moving in it has the form: ^{[4]}

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. ^{[5]} For
cosmic bodies the main contribution to the 4-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 vacuum permittivity, 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: ^{[4]}

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 ^{[6]} ^{[7]} 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

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. ^{[8]}

Studying the Lorentz covariance of
the 4-force, Friedman and Scarr found incomplete
covariance of the expression for the 4-force in the form ^{[9]}

This led them to conclude that the
4-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 is
up to a sign and a constant factor equal to the acceleration tensor with mixed indices.

Mashhoon and Muench
considered transformation of inertial reference frames, co-moving with the
accelerated reference frame, and obtained the relation: ^{[10]}

The tensor has the same properties as the acceleration
tensor

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

^{1.0}^{1.1}Fedosin S.G. The Concept of the General Force Vector Field. OALib Journal, Vol. 3, P. 1-15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459.- 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, 771 – 779 (2014). http://dx.doi.org/10.12988/astp.2014.47101.
^{3.0}^{3.1}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).^{4.0}^{4.1}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, P. 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, P.
1-16, (2015).
- 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, P.
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. Preprint, February 2016.
- 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, P. 13-24 (2015). http://dx.doi.org/10.5541/ijot.5000034003
.
- 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.

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