Invariant energy of an arbitrary physical system is a
positive quantity, which consists of all types of energies of the system, and
is equal to the relativistic energy, measured by the observer who is fixed
relative to the center of momentum of the system. The invariant energy usually
includes the rest energy of the matter; the potential energy of the proper
electromagnetic and gravitational fields associated with the system; the
internal energy of the system’s particles; the energy of the system in external
fields; the energy of emission interacting with the system. The invariant
energy of a particle equals to its rest energy and due to the principle of
mass–energy equivalence is associated with the invariant mass of the particle by the equation:
,
where is the speed of light.
The order of calculating the invariant energy through
various types of energy of the system is determined by the principle of energies summation.
Contents
o
1.3
The massive body o 1.3.1 General
relativity o 1.3.2 Covariant
theory of gravitation

In the special relativity, the invariant energy of the
particle can be calculated either through its relativistic energy and
momentum , or through the relativistic energy and
the velocity :
.
The relation holds for the photon, so that the invariant energy of the photon is zero.
In fourdimensional formalism in Minkowski space the
energy can be
calculated through the 4momentum
of the
particle:
,
where is the
metric tensor of the Minkowski space, is
4velocity,
is the Lorentz factor.
As a result, 4momentum can be represented
using the invariant energy: ^{[1]}
,
where is the 3vector of relativistic momentum.
In the curved spacetime with the metric tensor the invariant energy of the particle is found as follows:
.
If we take into account the definition of 4velocity: , where is 4displacement vector, is the differential of the proper time; and the definition of the spacetime
interval: , then again we obtain the
equality: .
In elementary particle physics the interaction of several
particles, their coalescence and decay with formation of new particles are
often considered. Conservation of the sum of 4momenta of free particles before
and after the reaction leads to the conservation laws of energy and momentum of
the system of particles under consideration. The invariant energy of the system of particles is calculated as their total relativistic energy
in the reference frame in which the center of momentum of the particle system is stationary. In this case can differ from the sum of invariant energies of the particles of the
system, since the contribution into is made not only by the rest energies of the particles, but also by the
kinetic energies of the particles and their potential energy. ^{[2]} If we observe the particles before or after the
interaction at large distances from each other, when their mutual potential
energy can be neglected, the invariant energy of the system is defined as:
,
where is the sum of relativistic energies of the system’s particles, is the vector sum of the particles’ momenta.
In determining the invariant energy of a
massive body in general
relativity (GR) there is a problem with the
contribution of the gravitational field energy, ^{[3]} since
the stressenergy tensor of gravitational field is not clearly defined,
and stressenergymomentum pseudotensor is
used instead. In case of asymptotically flat spacetime at infinity for the
estimation of the invariant energy the ADM
formalism for the massenergy of the body can be applied. ^{[4]} For
the stationary spacetime metric the Komar
mass and energy are determined. ^{[5]} There
are other approaches to determination of the massenergy, such as Bondi
energy, ^{[6]} and Hawking energy.
In the weakfield approximation the
invariant energy of a stationary body in GR is estimated as follows: ^{[7]}
where the mass
and charge of
body are obtained by integrating the corresponding density by volume, is
the energy of motion of particles inside the body, is
the gravitational constant, is
the radius of the body, is
the electric constant, is
the pressure energy.
For the masses, the relation is:
where the inertial mass of the system is
equal to the gravitational mass , the
mass
denotes the total mass of the particles that compose the body.
In covariant theory of gravitation (CTG) in
the calculation of the invariant energy the energy partition into 2 main parts
is used – for the components of the energy fields themselves and for components
associated with the energy of the particles in these fields. Calculation shows
that the sum of the components of the energy of acceleration field, pressure field, gravitational and
electromagnetic fields, for the spherical shape of the body is zero. ^{[8]}
As a result there is only a sum of the energies of the
particles in the four fields:
where is the Lorentz factor of particles, and is the scalar potential of pressure field at
the surface of system.
The ratio of the masses is as follows:
In this case the inertial mass system should be equal to 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.
A more accurate expression for the invariant energy is
presented in the following article: ^{[9]}
For the case of a relativistic uniform system,
the invariant energy can be expressed as: ^{[10]} ^{[11]}
This leads to a change in the ratio for the masses:
Here the gauge mass is related to the cosmological constant and
represents the massenergy of the matter’s particles in the fourpotentials of
the system’s fields; the inertial mass ;
the auxiliary mass is equal to the product of the particles’
mass density by the volume of the system; the mass is the sum of the invariant masses (rest
masses) of the system’s particles, which is equal in value to the gravitational
mass .
In Lorentzinvariant theory of gravitation (LITG),
in which CTG is transformed in the weakfield approximation and at a constant velocity
of motion, for the invariant energy the following formula holds:
,
where is
the relativistic energy of a moving body taking into account the contribution
of the gravitational and electromagnetic field energy, is
the total momentum of the system.
These formulas remain valid at the atomic level, with the difference
that the usual gravity replaced by strong gravitation. In the
covariant theory of gravitation based on the principle of least action is shown
that the gravitational mass of
the system increases due to the contribution of massenergy of the
gravitational field, and decreases due to the contribution of
the electromagnetic massenergy. This is the consequence of the fact that in
LITG and in CTG the gravitational stressenergy tensor is
accurately determined, which is one of the sources for the determining the
metric, energy and the equations of motion of matter and field. The acceleration stressenergy tensor, dissipation stressenergy tensor and
pressure stressenergy
tensor are also identified in
covariant form.
Vector fields such as the gravitational
and electromagnetic fields, the acceleration field,
the pressure field,
the dissipation field,
the fields of strong and weak interactions are components of general field.
This leads to the fact that the invariant energy of the system of particles and
fields can be calculated as the volumetric
integral in the centerofmomentum
frame: ^{[12]}
where and denote the time components of the
4potential of general field and the mass 4current ,
respectively, is the tensor of the general field.