**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
- 3 Application
- 4 Four-momentum and four-force
- 5 Expression of the four-acceleration in terms of the acceleration
field
- 6 The system of closely interacting particles
- 7 See also
- 8 References
- 9 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
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 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 is the equality between the four-acceleration of the particle and the
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
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. 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, 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 is
proved as follows. For a point 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 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 velocity 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 solving the corresponding field equations,
while the approximation of continuous medium is often used.

In the
concept of the __general field__ and the covariant theory of gravitation, the
following expressions may be required to calculate the four-acceleration in the
matter:

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).

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 4-potential of the field of accelerations
is also contained in the invariant of the acceleration field:

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.

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).
- Fedosin S.G. The calibration of energy and metric in the covariant
theory of gravitation. Preprint, April 2017.

**External
links**

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