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 electric
constant, 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.