Invariant energy

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 $$~E_0 $$ of a particle equals to its rest energy and due to the principle of mass–energy equivalence is associated with the invariant mass $$~ M $$ of the particle by the equation:
 * $$~E_0= M c^2$$,

where $$~c$$ 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.

One particle
In the special relativity, the invariant energy of the particle can be calculated either through its relativistic energy $$~E$$ and momentum $$~ \mathbf {p}$$, or through the relativistic energy and the velocity $$~v$$:
 * $$~E_0= \sqrt {E^2-p^2 c^2}=E \sqrt {1-\frac {v^2}{c^2}}$$.

The relation $$~ E=p c$$ holds for the photon, so that the invariant energy of the photon is zero.

In four-dimensional formalism in Minkowski space the energy $$~E_0$$ can be calculated through the 4-momentum $$~ p^{\mu}= M u^{\mu} $$ of the particle:
 * $$~E_0= c \sqrt { \eta_{\mu \nu} p^{\mu} p^{\nu}}= M c \sqrt {u_{\mu} u^{\mu}}= M c^2$$,

where $$~\eta_{\mu \nu} $$ is the metric tensor of the Minkowski space, $$ ~ u^\mu = \left(\gamma c, \gamma {\mathbf {v}}\right)$$ is 4-velocity, $$ ~ \gamma = \frac{1}{\sqrt{1-(\frac{v}{c})^2}}$$ is the Lorentz factor.

As a result, 4-momentum can be represented using the invariant energy:
 * $$ ~ p^\mu =\frac { E_0 u^\mu }{c^2} = \left(\frac {\gamma E_0}{c},\gamma M{\mathbf {v}}\right) = \left(\frac {\gamma E_0}{c}, \mathbf {p} \right) $$,

where $$ ~\mathbf {p} $$ is the 3-vector of relativistic momentum.

In the curved spacetime with the metric tensor $$~ g_{\mu \nu}$$ the invariant energy of the particle is found as follows:
 * $$~E_0= c \sqrt { g_{\mu \nu} p^{\mu} p^{\nu}}= Mc \sqrt { g_{\mu \nu} u^{\mu} u^{\nu}}$$.

If we take into account the definition of 4-velocity: $$~ u^{\mu} = \frac {dx^{\mu} }{d\tau} $$, where $$~ dx^{\mu} $$ is 4-displacement vector, $$~ d\tau $$ is the differential of the proper time; and the definition of the spacetime interval: $$~ ds = \sqrt { g_{\mu \nu} dx^{\mu} dx^{\nu} }= c d\tau $$, then again we obtain the equality: $$~E_0= Mc^2$$.

The system of particles
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 4-momenta 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 $$~E_{0c}$$ 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 $$~E_{0c}$$ can differ from the sum of invariant energies of the particles of the system, since the contribution into $$~E_{0c}$$ is made not only by the rest energies of the particles, but also by the kinetic energies of the particles and their potential energy. 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:
 * $$~E_{0c}= \sqrt {E_c^2 - p_c^2 c^2}$$,

where $$~E_c = \sum_i E_i $$ is the sum of relativistic energies of the system’s particles, $$~\mathbf {p}_c =\sum_i \mathbf {p}_i $$ is the vector sum of the particles’ momenta.

General relativity
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, since the stress-energy tensor of gravitational field is not clearly defined, and stress-energy-momentum pseudotensor is used instead. In case of asymptotically flat spacetime at infinity for the estimation of the invariant energy the ADM formalism for the mass-energy of the body can be applied. For the stationary spacetime metric the Komar mass and energy are determined. There are other approaches to determination of the mass-energy, such as Bondi energy, and Hawking energy.

In the weak-field approximation the invariant energy of a stationary body in GR is estimated as follows:
 * $$~E_{0}= Mc^2=m_b c^2 + E_k - \frac {6G m^2_b}{5a}+ \frac {3 q^2_b}{20 \pi \varepsilon_0 a}+E_p,$$

where the mass $$ ~ m_b $$ and charge $$ ~ q_b $$ of body are obtained by integrating the corresponding density by volume, $$ ~ E_k $$ is the energy of motion of particles inside the body, $$ ~ G $$ is the gravitational constant, $$ ~ a $$ is the radius of the body, $$ ~ \varepsilon_0 $$ is the electric constant, $$ ~ E_p $$ is the pressure energy.

For the masses, the relation is:
 * $$ ~ M = m_g <m_b <m', $$

where the inertial mass of the system $$ ~ M $$ is equal to the gravitational mass $$ ~ m_g $$, the mass $$ ~ m '$$ denotes the total mass of the particles that compose the body.

Covariant theory of gravitation
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. As a result there is only a sum of the energies of the particles in the four fields:
 * $$~E_{0}= Mc^2 \approx m_b c^2 \gamma_s - \frac {3G m^2_b}{10a}+ \frac {3 q^2_b}{40 \pi \varepsilon_0 a} + m_b \wp_s ,$$

where $$~ \gamma_s $$ is the Lorentz factor of particles, and $$~ \wp_s $$ is the scalar potential of pressure field at the surface of system.

The ratio of the masses is as follows: $$~m' = M < m_ b = m_g. $$

In this case the inertial mass system $$ ~ M $$ should be equal to the total mass of particles $$ ~ m'$$, the mass $$ ~ m_b $$ equals the gravitational mass $$ ~ m_g $$ and excess $$ ~ m_b $$ over $$ ~ M $$ 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:
 * $$~E_{0}= Mc^2 \approx m_b c^2 \gamma_s - \frac {G m^2_b}{2a}+ \frac {q^2_b}{8 \pi \varepsilon_0 a} + m_b \wp_s .$$

For the case of a relativistic uniform system, the invariant energy can be expressed as:


 * $$~E_{0}= Mc^2 \approx m_b c^2 - \frac {1}{10\gamma_c } \left( 7- \frac {27}{2 \sqrt {14}} \right) \left( \frac {G m^2_b}{a}- \frac {q^2_b}{4 \pi \varepsilon_0 a} \right) . $$

This leads to a change in the ratio for the masses:
 * $$~m' < M < m < m_b = m_g .$$

Here the gauge mass $$~m' $$ is related to the cosmological constant and represents the mass-energy of the matter’s particles in the four-potentials of the system’s fields; the inertial mass $$~M $$; the auxiliary mass $$~m $$ is equal to the product of the particles’ mass density by the volume of the system; the mass $$~m_b $$ is the sum of the invariant masses (rest masses) of the system’s particles, which is equal in value to the gravitational mass $$~m_g $$.

In Lorentz-invariant theory of gravitation (LITG), in which CTG is transformed in the weak-field approximation and at a constant velocity of motion, for the invariant energy the following formula holds:
 * $$~ E_{0}= \sqrt {E^2-p^2 c^2}$$,

where $$~E $$ is the relativistic energy of a moving body taking into account the contribution of the gravitational and electromagnetic field energy, $$~p $$ 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 $$ ~ m_g $$ of the system increases due to the contribution of mass-energy of the gravitational field, and decreases due to the contribution of the electromagnetic mass-energy. This is the consequence of the fact that in LITG and in CTG the gravitational stress-energy 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 stress-energy tensor, dissipation stress-energy tensor and pressure stress-energy 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 center-of-momentum frame:


 * $$~E = \int {( s_0 J^0 + \frac {c^2 }{16 \pi \varpi } s_{ \mu\nu} s^{ \mu\nu} ) \sqrt {-g} dx^1 dx^2 dx^3}, $$

where $$~ s_0 $$ and $$~ J^0$$ denote the time components of the 4-potential  $$~ s_{\mu } $$ of general field and the mass 4-current $$~ J^{\mu } $$, respectively, $$~ s_{ \mu\nu} $$ is the tensor of the general field.