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 mass 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 mass 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 vacuum permittivity, is
the pressure energy.
For the
masses, the relation is:
where the
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]}
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: ^{[10]}
where and
denote
the time components of the 4potential of
general field and the mass 4current , respectively, is the
tensor of the general field.