**Four-acceleration** (4-acceleration)
is a four-vector, considered as relativistic generalization of the classical
three-dimensional acceleration vector for the four-dimensional spacetime. Any
physical system, whether a point material particle or a connected set of
particles, has its own four-acceleration. Inside a body with continuous matter
distribution, as a rule, there are gradients of velocities of motion of typical
particles of this matter. As a result, the four-acceleration of typical
particles inside bodies, averaged over the volume with the sizes of such
particles, is a certain function of coordinates and time, leading to internal
stresses in the matter.

**Содержание**

- 1 Definition
- 2 Single particle
- 2.1 Application
- 2.2 Four-momentum and four-force
- 2.3 Expression of the four-acceleration in terms of the acceleration
field
- 3 The system of closely interacting particles
- 4 See also
- 5 References
- 6 External links

**Definition**

In the
general case, the four-acceleration of a particle is defined as the derivative
of the four-velocity with respect to the particle’s proper
time :

In the above
expression, the __operator of proper-time-derivative__ is used, which generalizes the material
derivative to the curved spacetime, ^{[1]} and the quantities represent the
Christoffel symbols.

The metric
tensor is needed to determine the four-acceleration
with a covariant index:

**Single particle**

The motion of solid-state point
physical particles, as well as of physical systems consisting of a set of
particles, moving as a single whole, is described with the help of the
four-velocity. In this case, the velocity of motion of a particle or the
velocity of the momentum center of a system is directly included into the space
component of the four-velocity.

Sometimes it
is convenient to pass on from the particle’s proper time to the coordinate time of the reference frame, in which the particle
is moving. Let’s take into account that the four-velocity can be written in the
following form:

where is the instantaneous four-position of the
particle, is the speed of light, is the three-vector of the particle’s
position, is the time component of the
four-velocity, is the
three-velocity of the particle with the corresponding components.

For the
four-acceleration components the following is then obtained:

In the
inertial reference frame, instantaneously co-moving with the moving particle,
the velocity , the Lorentz factor , the Christoffel symbols become equal to
zero, ,
and denoting in the given reference frame the proper three-acceleration
as , we obtain for the four-acceleration the
following: . Since
in this reference frame the four-velocities with contravariant and covariant
indices coincide, , then
the scalar product is . Since
the scalar product of any four-vectors is an invariant, then from the equality
of the scalar product to zero it follows that in any other reference frames the
four-velocity and four-acceleration of a particle are always perpendicular to
each other.

If we assume
that the four-velocity is directed along the world line of the particle, then
the four-acceleration at each point will be perpendicular to this line and
directed in the same way as the curvature vector of the world line.

**Application**

In case when a certain force field acts on the
particle, the acceleration of the particle will depend on both components of
this field, that is, on the field strength and the corresponding solenoidal
vector. Thus, the electric field strength, magnetic field, charge and velocity
of the particle determine the value of the Lorentz force, which accelerates the
particle in the electromagnetic field. The same situation takes place in the
covariant theory of gravitation, where there are the __gravitational
field strength__ and the __gravitational
torsion field__.

For the fields the superposition principle holds,
according to which the scalar potential of the field at some point is the
arithmetic sum of the scalar potentials of all the available field sources, and
the vector potential at this point is the vector sum of the vector potentials
of the field sources. With the help of the known field potentials it is easy to
determine the field strength and the corresponding solenoidal field vector, and
hence the corresponding acceleration of the particle. For the expression of the
force in a more general, tensor form, the concept of the __four-force__ is used, which is
proportional to the four-acceleration.

In the general relativity, the gravitational force is
reduced to the curvature of spacetime and is found through the metric tensor.
As a result, in the absence of other forces, the particle in the gravitational
field moves along the geodesic line, while the four-acceleration of the particle and the four-force are equal
to zero. Hence we obtain the geodesic equation as the equation of motion of a
particle in a given metric:

Under similar conditions in the __covariant
theory of gravitation__, the gravitational four-acceleration of a particle is
found either through the __gravitational
tensor__ or through the __gravitational
stress-energy tensor__ . In
this case the four-acceleration is not equal to zero, as long as there is a
non-zero gravitational force. The equation of motion of a solid particle is the equality
between the four-acceleration of the particle and the four-acceleration from
the gravitational field:

In the presence of other fields, acting on the
particle, the above equations of motion change. For example, in the presence of
the charge of the particle with the mass , the equation of motion in
the general relativity will be as follows: ^{[2]}

where is the electromagnetic tensor, is the electromagnetic stress-energy tensor, is the average mass density of the particle
in its proper reference frame.

In the matter inside the body, several fields can act
simultaneously on the particle, for example, the gravitational and
electromagnetic fields, the __pressure field__, and the __dissipation
field__. In the covariant theory of gravitation, the gravitational field is
considered as a vector field, just like an electromagnetic field. If we assume
that other fields in macroscopic bodies are described by the vector fields and
are the __general field__ components, then the equation
of motion of the solid particle in the specified four fields
has the form: ^{[3]}

Here is the __pressure
field tensor__, is the __dissipation
field tensor__, is the __pressure
stress-energy tensor__, and is the __dissipation
stress-energy tensor__.

**Four-momentum
and four-force**

The four-momentum of a particle is determined as the
product of the particle’s mass by the four-velocity:

where is the relativistic energy, is the three-vector of the relativistic
momentum of the particle.

In order to calculate the four-force we need to apply
the operator of proper-time-derivative to the four-momentum:

If the mass is constant, it can be taken out beyond
the sign of the differential:

In this case, the four-force is proportional to the
mass and four-acceleration. If we determine that is the power as the rate of change of the
particle’s energy, and is the three-force, written also in Cartesian
coordinates and acting on the particle, then the components of the four-force
will be expressed in terms of the power, the three-force components and the
four-acceleration components as follows:

**Expression of
the four-acceleration in terms of the acceleration field**

The __acceleration field__ is characterized by its own
four-potential, the __acceleration tensor__ and the __acceleration
stress-energy tensor __ . The acceleration tensor components are the
components of two three-vectors – the field strength and the solenoidal vector of the acceleration field. For a solid
particle, the four-acceleration with a covariant index can be expressed in terms of the
following quantities: ^{[4]}

The equality for the four-acceleration with the
right-hand side (3), containing the stress-energy tensor of the acceleration
field, follows from the principle of least action. The equality for the
four-acceleration with the left-hand side (3) is proved as follows. For a point
solid particle, the four-potential of the acceleration field is the
four-velocity of the particle with the covariant index , and the acceleration tensor will equal:

Multiplying this equality by gives the required relation:

In this case, we took into account the equality and its covariant derivative:

With the help of the metric tensor we can pass on to
the four-acceleration with a contravariant index:

After substituting (4) into the right-hand side of (2)
we can see that there is a relationship between the power and the three-force on the one hand, and the scalar product and the sum
on the other hand.

In a flat spacetime, the Christoffel symbols become
equal to zero, , and
the metric tensor has only the diagonal components that are equal in the
absolute value to 1. In this case we obtain the following:

Thus, the power of work done by an arbitrary force and
the force itself are expressed in terms of the velocity of motion , the strength and the solenoidal vector of the acceleration field.

The expressions for the vectors and follow from the definition of the
four-potential and the acceleration tensor:

In the flat spacetime, the scalar potential of the
particle’s acceleration field is , and the vector potential of the acceleration
field is . Finding the vectors and with the help of these potentials, and substituting
them into the expressions for the power and force, we obtain the following:

The vectors and are expressed in terms of partial
derivatives, characteristic of the four-vector algebra, and similarly we obtain
the expressions for the power and force. In addition, there are two other
expressions for the force, ^{[2]} ^{[1]} with the use of the
three-acceleration of the particle:

**The system of
closely interacting particles**

The four-acceleration concept for a sufficiently large
system of particles differs significantly from the four-acceleration of a
single point particle. Multiparticle interactions in the system result
in a new quality, when not the motion of a particular physical particle becomes
important, but the motion of certain typical particles, which characterize the
system under consideration on the average. As a result of averaging of the
kinetic energies and momenta of individual particles, such macroscopic
parameters emerge as temperature and pressure. A peculiarity of a typical
particle is that its root mean square speed of motion becomes the
function of location in the system. In simple physical systems consisting of
particles of one phase or having sufficiently uniform composition, the
pressure, temperature and velocity of typical particles at the center of the
system usually reach the maximum value.

In order to describe the motion of a typical particle
the same equations can be used, as for a physical particle, with the difference
that these equations should be averaged. This requires averaging of the field
strengths and solenoidal vectors of all the acting fields over the volume of a
typical particle at its location. In fact, averaging is performed by using the
corresponding field equations, while the approximation of continuous medium is
often used.

In many cases, the dependences of the
fields as well as of the mass and charge densities on the coordinates and time
are known in the system, but distribution of the particles’ four-velocity is
unknown. Then,
in the concept of __general field__ and covariant theory of gravitation,
for calculating the basic parameters of the system, the expressions presented
in the article equation of vector field
may be required:

1) The equation for calculating the metric:

where is the Ricci tensor, is the scalar curvature, is the __gravitational
constant__, is a certain constant, and the gauge
condition of the cosmological constant is used.

2) The acceleration field equations for calculating
the vectors and as the components of the acceleration tensor :

where is the mass four-current, is the acceleration field constant.

Sometimes it is easier to use first the wave equation
to calculate the four-potential of the acceleration field

and then to apply the four-curl to in order to determine .

3) The equation of motion (1), if typical
particles of the system are considered as solid particles.

4) The gauge condition of the four-potential of the
acceleration field and the continuity equation for the mass four-current,
respectively:

The connection between the metric and the four-velocity is also
contained in the invariant:

As it was said, all physical quantities in these
equations should refer to typical particles and therefore should be averaged.
This also applies to the Ricci tensor, the scalar curvature, and the
cosmological constant inside the body. ^{[5]}

In items 1) and 2), the vectors and , necessary to calculate the four-acceleration
in relation (3), are expressed in terms of the metric and the acceleration
field coefficient . With the help of item 3) the
coefficient is expressed in terms of the coefficients of other
fields. In item 4) additional restrictions are imposed on the acceptable type
of functions, including the particles’ velocities and their potentials. Due to
the complexity and interdependence of the equations, they often have to be
solved simultaneously.

The situation becomes more complicated in the case
when the system’s typical particles are not solid and have nonzero vector field
potentials. For example, the particles can have the spin and magnetic moment,
leading to the vector potential of the acceleration field and to the vector
potential of the electromagnetic field. In this case,
the four-potential of the acceleration field
is no longer equal to the particle’s
four-velocity . Instead of the four-acceleration, the
covariant derivative of the stress-energy tensor of the accelerated field with
mixed indices defines the density of the __four-force__: ^{[6]}

From the foregoing we can see that if the
four-velocity in the matter is not initially specified, it should be found from
the field equations and then be used to calculate the four-acceleration by the
formula:

From comparison of (5) and (3) it follows that the four-acceleration
in the matter inside a typical particle can be estimated in the first
approximation as follows:

In this expression, the four-acceleration depends on
the four-velocity of the matter’s motion inside the particle,
on the four-potential of the acceleration field as a function of
the coordinates inside the particle and on the coordinates of the particle
itself inside the system. In case of using the general relativity instead of
the covariant theory of gravitation, the described above order of calculating
the four-acceleration in general remains the same. An exception is that in the
general relativity, the gravitational force and its contribution into the
four-acceleration are included in the metric, which changes both the equation
for the metric and the equation of motion.

**See also**

- Four-vector
- 4-velocity
__Four-force__- 4-momentum
__Acceleration field__

**References**

^{1.0}^{1.1}Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0. (in Russian).^{2.0}^{2.1}Ландау Л. Д., Лифшиц Е. М. Теория поля. – Издание 7-е, исправленное. – М.: Наука, 1988. – 512 с. – (Теоретическая физика, том II).- 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. About the
cosmological constant, acceleration field, pressure field and energy.
Jordan Journal of Physics. Vol. 9, No. 1, pp. 1-30 (2016). http://dx.doi.org/10.5281/zenodo.889304.
- 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. 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.

**External links**

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