The principle of
energies summation of an arbitrary system sets the order of
inclusion of various types of energy, associated with the system, into energy
functions that describe the state of the system. Energy summation is most
frequently used in theoretical physics, where the principle of least action is used, the
total energy of systems is calculated and the law of energy conservation is
taken into account. The principle of energies summation on the one hand is a methodological
principle, but on the other hand – is the result of complexity of systems,
consisting of matter in different states, and of the fields available in these
systems. The complexity increases due to the motion of matter and fields during
transitions of matter from one phase state to another, and during
transformation of energies of fields and matter into each other. Energy functions have different meaning
depending on their purpose. To estimate the change in the total energy of the
system we need to take into account that some components increase energy, and
others reduce it, which leads to different signs before the energy components.
If the energy functions are used to find the equations of motion, the signs
before the energy components are chosen according to the condition of
conforming to the equations of motion of matter and fields. As a result, for each energy function its own order of
energies summation is used.
Contents
o
1.6.1
EinsteinHilbert equations o 1.6.2 Equations of CTG

To calculate the energy functions in thermodynamics such
physical quantities are used as pressure , volume , absolute temperature , heat capacity , mass , amount of substance . These quantities can be well measured,
in contrast to entropy , chemical potential , amount of heat , which
are characteristic of the substance. Internal energy and its increment for
multiphase substance in a quasistatic process are given by:
,
,
where is
the increment of the amount of heat, is the work done by the system, is the number of substance phases, is the work done on the system.
Besides the internal energy in thermodynamics there are
other energy functions associated with it, such as Helmholtz free energy:
.
Accordingly, the increment of the Helmholtz free energy
is:
.
Enthalpy and its increment are as follows:
,
.
Gibbs free energy and its increment:
,
.
Grand thermodynamic potential and its increment:
,
.
Bound energy and its increment:
,
.
Two thermodynamic potentials and their increments are
also possible:
,
,
,
.
The order of addition of energy components is of such
kind that we obtain the corresponding thermodynamic potential, which has its
own meaning. Thus, the internal energy reflects the law of energy conservation,
and the change in the Helmholtz free energy in an isothermal process is
determined only by the difference in the work done by the system on the
environment and by the environment on the system.
Many relations of thermodynamics hold well not only for
gas, but also for fluids and substance in the solid state.
One of the ways to find the equations of motion of
systems and the laws of their existence is variation of the action functional,
that is, variation by different variables of the time integral of Lagrangian,
in order to determine the extreme and most probable states. Lagrangian consists of several energy components, which
in mechanics are either part of kinetic energy or of potential energy . In order to find the Lagrangian in
mechanics the difference between the kinetic and potential energies is written
as follows:
.
It is generally assumed that the Lagrangian depends only
on time, coordinates and velocities, but does not depend on the higher time
derivatives.
Since each mechanical system itself is a source of field,
in general case a term is added in the right side of the equation which is
associated with the energy of this field. In special relativity, the Lagrangian
of a particle with mass and charge in
an electromagnetic field has the following form: ^{[1]}
,
where is
the speed of light, is the spacetime interval, is the electromagnetic 4potential with lower (covariant) index, is
4displacement vector of the particle, is
the electric constant, is
the electromagnetic field tensor, is the 4volume, is
the velocity of the particle, and are scalar and vector potentials of the electromagnetic field,
respectively, and are electric field strength and magnetic induction, respectively.
In this case, the Lagrangian includes three components
with dimension of energy, which are associated with the relativistic energy of
the particle, with the energy of the particle in the electromagnetic field, and
with the electromagnetic field energy. The expressions for the energy
components and the signs before them are chosen so that by varying the action
functional we would obtain the equations of the particle’s motion in the field
and Maxwell's Equations for field strengths.
Similarly, the Lagrangian is written for a single
particle in the gravitational field in Lorentzinvariant
theory of gravitation: ^{[2]}
,
where is
the speed of gravitation, close to the speed of light, is
the gravitational fourpotential with
lower (covariant) index, is the gravitational constant, is the gravitational tensor, and are scalar and vector potentials of the gravitational field, respectively, and
are the gravitational field strength and the gravitational
torsion field, respectively, and the mass not only takes into account the sum of nucleons masses of matter, but
also the contribution of massenergy of fields interacting with the matter and
changing particle mass.
After varying the action functional we obtain the
equations of motion of the particle in the gravitational field and Maxwelllike gravitational equations for
the gravitational field
strength and the torsion
field. To use the Lagrangian in any frames of reference, it should be written
in the covariant form. In curved spacetime the interval can be expressed using
the metric tensor :
and instead of the component of 4volume during the integration over the 4volume we
should use the product , where is
the determinant of the metric tensor.
In classical mechanics, the Hamiltonian of the system of
particles can be defined with the Lagrangian: ,
where is
the generalized momentum of the ith particle, and is
its generalized velocity.
For conservative systems in which the energy is
conserved, the Hamiltonian as the function of generalized coordinates and
momenta is equal to the total energy of
the system and has the following form:
.
In this case, we see that the distinction between the
Lagrangian and Hamiltonian is in the different signs before the potential
energy of the system.
Invariant energy of
a body is defined as the relativistic energy, measured by an observer who is
fixed relative to the body’s center of mass. The standard approach involves
summation of all the types of energy of the body:
,
where is
the rest energy of the individual matter particles, is the pressure (compression) energy of the matter understood as the
potential energy of interatomic interactions, is thermal energy, which being summed with yields the internal energy, is
the total gravitational energy of the body, including the energy of the proper
field in the body matter and beyond it and the gravitational energy in the
field from external sources, is
the total electromagnetic energy of the body, is
the energy of emission interacting with the body matter.
In general relativity this leads to the fact that a heated body
should increase its mass, and the mass of a gravitationally bound body should
be less than the total mass of the particles of matter that forms this body.
There is an alternative point of view that the energy
components are included in the equation for the invariant energy with negative
signs: ^{[3]} ^{[4]} ^{[5] [6] }^{ }^{[}^{7}^{]}
.
As a result, heated bodies should have less mass than
cold, and the mass of a star must be greater than the mass of scattered matter
from of which it was made up during the gravitational collapse.
The third
approach involves rethinking the nature and order of summation energies
in covariant theory of gravitation (CTG). Method of calculating the
invariant energy depends essentially on how to account for the scalar curvature
and the cosmological constant in the Hamiltonian. In particular, the cosmological
constant can be calibrated in such a way as to exclude the scalar curvature,
and thus find a unique expression for the Hamiltonian. ^{[8]} Another innovation is that instead
of the standard stressenergy tensor of matter, taking into account the inner
pressure, in consideration introduces two new vector fields – the acceleration field and pressure field, with the corresponding
stressenergy tensors. If we add the electromagnetic and gravitational fields,
then obtained four fields symmetrically involved in the Lagrangian and the
Hamiltonian. The calculation of the invariant energy in the spherical body
shows that the components of the energy of all four fields cancel each other.
Therefore the contribution to the invariant energy of system makes only
potential energies of the particles which are under the influence of
fields. ^{[9]} These energies are also partially
reduced, and for the invariant energy can be written:
The
relation for the mass is as follows:
where
mass and charge are calculated by integrating the
corresponding density by volume of the body with the radius .
In CTG
system mass equals the total mass of
particles , the mass equals the gravitational
mass , and excess over is due to the fact that particles
move inside the body and are under pressure in the gravitational and
electromagnetic fields.
In contrast to the invariant energy, the relativistic
energy generally includes additional energy components associated with the
motion of the system as a whole. As a result, in the formulas for the energy
dependence on velocity can be determined, such as on the velocity of the center of mass of the system. If in
Minkowski space the invariant energy is
known, then the relativistic energy in an arbitrary inertial reference frame is
found using the Lorentz transformation by the following formula:
.
EinsteinHilbert equations of general
relativity (GR) are aimed to find metric in curved spacetime and are written in
tensor form:
,
where is
the Ricci curvature tensor, is
the scalar curvature, is
the cosmological constant, and is
a stressenergy tensor with dimension of volumetric energy density, is
Newton's gravitational constant.
In GR and the tensor usually includes the stressenergy tensor of matter and the stressenergy tensor of electromagnetic field :
.
Absence of stressenergy tensor of gravitational field as
the source affecting the metric in GR is due to the fact that the gravitational
field is identified with the geometrical field in the form of the metric field,
and this field does not generate itself (absence of selfaction of the metric
field).
Equations of CTG
In
the covariant theory of gravitation (CTG) equations for the metric are
as follows: ^{[8]}
where the
coefficient is found from the equations of
motion of particles and waves in any given form of the metric, and the
tensor is the sum of four tensors:
where
is the stressenergy
tensor of gravitational field, is the acceleration stressenergy tensor, and
is the pressure stressenergy tensor.
This
means that in CTG the gravitational field is a physical field and along with
the electromagnetic field, the acceleration field and pressure field it is the
source forming the spacetime metric.
For the
case of continuously distributed matter we obtain the equality for the
cosmological constant:
where and are the mass and electromagnetic
4currents, respectively, and – 4potentials of acceleration
field and pressure field.
The
covariant derivative of the left side of equations for metric due to
calibration of the cosmological constant and the scalar curvature is zero. This
allows us to write the equation of matter motion as equality to zero of the
covariant derivative of the sum of the tensors in the right side, taken with
the contravariant indices:
.
In the concept of general
field it is assumed that all
vector fields associated with the matter are the components of this field.
4potential of the general field is
the sum of 4potentials of particular fields. ^{[10]} As a result, the sum of terms in the
Lagrangian responsible for the energy of the matter in various fields, up to a
sign is simply the product of .
With regard to energy themselves particular fields, then these energies are
included in the Lagrangian by means of the general field tensor ,
obtained as a 4curl of the 4potential of the general field. For the
Lagrangian we obtain the relation:
where and are
the constants to be determined; is
the invariant 3volume expressed in terms of the product of
space coordinates’ differentials and the square root of
the determinant of
the metric tensor, taken with a negative sign.
The relativistic energy of the system
is:
where and denote
the time components of the 4vectors and
.
The feature of the expression for the
energy is that in it the general field energy in the tensor product includes
not only energies of particular fields, but also crossterms in the form of a
sum of products of particular fields strengths in various combinations. We can
say that the energy of the particles in particular fields is included in the
energy of the system linearly, and the energy fields themselves – approximately
quadratically.