**Acceleration field** is a two-component vector field, describing
in a covariant way the four-acceleration
of individual particles and the four-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:

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.

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 four-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 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 :

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.

In 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:

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 four-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 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: ^{[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 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 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