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
vacuum permittivity, 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.