# Dissipation field

Dissipation field is a two-component vector force field, which describes in a covariant way the friction force and energy dissipation emerging in systems with a number of closely interacting particles. The dissipation field is a general field component, which is represented in the Lagrangian and Hamiltonian of an arbitrary physical system including the term with the energy of particles in the dissipation field and the term with the field energy. [1] [2] The dissipation field is included in the equation of motion by means of the dissipation field tensor and in the equation for the metric – by means the dissipation stress-energy tensor.

By energy dissipation is meant conversion of the energy of directed motion of particles into the energy of random motion of these particles and the particles of the surrounding medium, as well as conversion into the energy of intramolecular and atomic motion, while the energy of motion of fast moving fluxes of particles decreases due to the friction with slower fluxes. The typical examples of dissipation of mechanical energy are damping of motion of a jet in a liquid and heating of falling meteorites during their motion in the Earth's atmosphere.

The dissipation field is generally considered as a macroscopic field with its energy and momentum, describing the averaged interaction of particles in an arbitrary small volume of a system. The cause of the dissipation field emerging at the micro level is different interactions leading to the effects of friction and deceleration of individual particles or their fluxes. At the atomic level, the electromagnetic forces and strong gravitation are prevailing, by which the particles interact with each other and exchange their energy. The friction forces in a system of particles appear as a collective effect and are proportional not only to the velocity, but also to its derivatives with respect to time and coordinates. Since the friction force is described by the dissipation field tensor and the corresponding stress-energy tensor, the dissipation field in each small volume obtains its energy density and energy flux density. In dissipative processes, some change occurs in the internal energy of the system, mainly due to the change in the quantity of heat or in the energy of phase transitions, which can be considered as a change in the dissipation field energy. The internal energy also changes when the pressure field energy and the energy of the particles’ acceleration field change, as well as due to the change in the energy of the electromagnetic, gravitational and other fields. In turn, the dissipation field energy flux makes its contribution into the flux of internal energy and the flux of the relativistic energy of the system.

## Classical mechanics

The main processes that lead to energy dissipation are the viscous friction of the layers of liquid or gas against each other, the friction during the motion of solid bodies due to interaction with the surrounding medium, thermal conductivity and diffusion in gases and liquids. All of these processes belong to transport phenomena: friction occurs during transfer of momentum, heat conductivity occurs during the internal energy transfer, and diffusion is associated with the transfer of mass (charge, electric and magnetic moment, etc.). Friction is the main source of energy dissipation. Thermal conductivity and diffusion also make some contribution, since in real processes all kinds of transfer are intertwined with each other. In order to describe the friction forces in Euler–Lagrange equation the friction forces ${\displaystyle ~F_{j}}$ are introduced: [3]

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial L}{\partial q_{j}}}=F_{j},}$

where ${\displaystyle ~L}$ is the Lagrangian, ${\displaystyle ~q_{j}}$ are the generalized coordinates. In addition, the dissipation function ${\displaystyle ~D}$ is introduced into consideration, so that the following relation holds:

${\displaystyle ~F_{j}=-{\frac {\partial D}{\partial {\dot {q}}_{j}}}.}$

The Rayleigh dissipation function is given by the expression:

${\displaystyle D={\frac {1}{2}}\sum _{j=1}^{N}\sum _{k=1}^{N}C_{jk}{\dot {q}}_{j}{\dot {q}}_{k}.}$

If the tensor ${\displaystyle ~C_{jk}}$ does not depend on the velocities, the corresponding friction force is equal to:

${\displaystyle ~F_{j}=-\sum _{k=1}^{N}C_{jk}{\dot {q}}_{k}.}$

The dissipation function has the dimension of power and the tensor ${\displaystyle ~C_{jk}}$ must have the dimension kg/s. If the tensor is symmetrical, the friction force appears to be directed oppositely to the velocity of the particle’s motion relative to the surrounding medium.

This shows that the dissipation function represents some form of a scalar potential, which depends on the products of projections of relative velocities, and the dissipation field is considered as the corresponding scalar field. But unlike the standard scalar potentials of fundamental fields, the friction force is not in the form of a gradient of dissipative functions but in the form of a derivative with respect to the motion velocity. This approach cannot be considered as a fully covariant description of friction processes, it can only serve as a first approximation, because it does not take into consideration the friction in accelerated motion.

The Newton's law of fluid friction describes the internal friction force in the liquid layer, moving relative to other parallel layers:

${\displaystyle ~dF=-\mu {\frac {dv}{dr}}dS,}$

here ${\displaystyle ~\mu }$ is the coefficient of internal friction or the coefficient of dynamic viscosity, ${\displaystyle ~{\frac {dv}{dr}}}$ is the gradient of the velocity of the layers’ motion in the direction perpendicular to the layer surface, ${\displaystyle ~dS}$ is the surface area of the layer.

A more accurate description of motion in a viscous medium is given by the Navier–Stokes equations:

${\displaystyle \rho \left({\frac {\partial v_{i}}{\partial t}}+v_{k}{\frac {\partial v_{i}}{\partial x_{k}}}\right)=-{\frac {\partial p}{\partial x_{i}}}+{\frac {\partial }{\partial x_{k}}}\left\{\mu \left({\frac {\partial v_{i}}{\partial x_{k}}}+{\frac {\partial v_{k}}{\partial x_{i}}}-{\frac {2}{3}}\delta _{i,\;k}{\frac {\partial v_{l}}{\partial x_{l}}}\right)\right\}+{\frac {\partial }{\partial x_{k}}}\left(\xi {\frac {\partial v_{l}}{\partial x_{l}}}\delta _{i,\;k}\right)+\rho a_{mi}=}$
${\displaystyle ~=\nabla _{k}P^{ik}+\rho a_{mi},}$
${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho {\vec {v}})=0,}$

where ${\displaystyle ~\rho }$ is the mass density of the matter, ${\displaystyle ~\xi }$ is the "second viscosity", or the volume viscosity, ${\displaystyle ~\delta _{i,\;k}}$ is the Kronecker delta, ${\displaystyle ~P^{ik}}$ is the Cauchy stress tensor, ${\displaystyle ~a_{mi}}$ is the acceleration due to the mass forces (including the gravitational and electromagnetic forces and the force of inertia), while the second equation is the continuity equation.

## Relativistic hydrodynamics

Instead of the tensor ${\displaystyle ~P^{ik}}$ in the relativistic case the four-dimensional viscous stress tensor is used to describe the equation of motion of the viscous and heat-conducting media: [4]

${\displaystyle ~\tau _{ik}=-\mu \left({\frac {\partial u_{i}}{\partial x^{k}}}+{\frac {\partial u_{k}}{\partial x^{i}}}-{\frac {1}{c^{2}}}u_{k}u^{n}{\frac {\partial u_{i}}{\partial x^{n}}}-{\frac {1}{c^{2}}}u_{i}u^{n}{\frac {\partial u_{k}}{\partial x^{n}}}\right)-\left(\xi -{\frac {2}{3}}\mu \right){\frac {\partial u_{n}}{\partial x^{n}}}\left(g_{ik}-{\frac {1}{c^{2}}}u_{i}u_{k}\right),}$

where ${\displaystyle ~u^{i}}$ is the four-velocity with a contravariant index, ${\displaystyle ~u_{k}}$ is the four-velocity with a covariant index, ${\displaystyle ~\xi }$ is the coefficient of the second (volume) viscosity, ${\displaystyle ~g_{ik}}$ is the metric tensor, ${\displaystyle ~c}$ is the speed of light.

The density of the four-force, arising from viscosity, is calculated using the covariant derivative of the tensor ${\displaystyle ~\tau _{ik}}$ and is present in the right side of the Navier-Stokes equations. By its meaning the tensor ${\displaystyle ~\tau _{ik}}$ is the stress-energy tensor, but it cannot be derived from the principle of least action.

## The dissipation field as a vector field

The dissipation field as a two-component vector field was presented by Sergey Fedosin within the framework of the metric theory of relativity and the covariant theory of gravitation. The equations of this field were developed as a consequence of the principle of least action, [5] and a special procedure was used. [6]

### Mathematical description

The four-potential of the dissipation field is expressed in terms of the scalar ${\displaystyle ~\varepsilon }$ and vector ${\displaystyle ~\mathbf {\Theta } }$ potentials:

${\displaystyle ~\lambda _{\mu }=\left({\frac {\varepsilon }{c}},-\mathbf {\Theta } \right).}$

The antisymmetric dissipation field tensor is calculated with the four-curl of the four-potential:

${\displaystyle h_{\mu \nu }=\nabla _{\mu }\lambda _{\nu }-\nabla _{\nu }\lambda _{\mu }={\frac {\partial \lambda _{\nu }}{\partial x^{\mu }}}-{\frac {\partial \lambda _{\mu }}{\partial x^{\nu }}}.}$

The dissipation field tensor components are the vector components of the dissipation field strength ${\displaystyle ~\mathbf {X} }$ and the solenoidal vector ${\displaystyle ~\mathbf {Y} }$:

${\displaystyle ~h_{\mu \nu }={\begin{vmatrix}0&{\frac {X_{x}}{c}}&{\frac {X_{y}}{c}}&{\frac {X_{z}}{c}}\\-{\frac {X_{x}}{c}}&0&-Y_{z}&Y_{y}\\-{\frac {X_{y}}{c}}&Y_{z}&0&-Y_{x}\\-{\frac {X_{z}}{c}}&-Y_{y}&Y_{x}&0\end{vmatrix}}.}$

This yields the following:

${\displaystyle ~\mathbf {X} =-\nabla \varepsilon -{\frac {\partial \mathbf {\Theta } }{\partial t}},\qquad \qquad \mathbf {Y} =\nabla \times \mathbf {\Theta } .\qquad \qquad (1)}$

### Action, Lagrangian and energy

In the covariant theory of gravitation, the four-potential ${\displaystyle ~\lambda _{\mu }}$ of the dissipation field is part of the four-potential of the general field ${\displaystyle ~s_{\mu }}$, which is the sum of the four-potentials of particular fields, such as electromagnetic and gravitational fields, acceleration field, pressure field, dissipation field, strong interaction field, weak interaction field and other vector fields, acting on the matter and its particles. The energy density of interaction of the general field with the matter is given by the product of the four-potential of the general field and the mass four-current: ${\displaystyle ~s_{\mu }J^{\mu }}$. From the four-potential of the general field we obtain the general field tensor by applying the four-curl:

${\displaystyle ~s_{\mu \nu }=\nabla _{\mu }s_{\nu }-\nabla _{\nu }s_{\mu }.}$

The tensor invariant in the form of ${\displaystyle ~s_{\mu \nu }s^{\mu \nu }}$ is up to a constant factor proportional to the energy density of the general field. As a result, the action function that contains the scalar curvature ${\displaystyle ~R}$ and the cosmological constant ${\displaystyle ~\Lambda }$ is given by the expression: [1]

${\displaystyle ~S=\int {Ldt}=\int (kR-2k\Lambda -{\frac {1}{c}}s_{\mu }J^{\mu }-{\frac {c}{16\pi \varpi }}s_{\mu \nu }s^{\mu \nu }){\sqrt {-g}}d\Sigma ,}$

where ${\displaystyle ~L}$ is the Lagrange function or Lagrangian, ${\displaystyle ~dt}$ is the time differential of the coordinate reference frame, ${\displaystyle ~k}$ and ${\displaystyle ~\varpi }$ are the constants to be determined, ${\displaystyle ~c}$ is the speed of light, as a measure of the propagation velocity of electromagnetic and gravitational interactions, ${\displaystyle ~{\sqrt {-g}}d\Sigma ={\sqrt {-g}}cdtdx^{1}dx^{2}dx^{3}}$ is the invariant four-volume, expressed in terms of the differential of the time coordinate ${\displaystyle ~dx^{0}=cdt}$, the product ${\displaystyle ~dx^{1}dx^{2}dx^{3}}$ of differentials of the space coordinates and the square root ${\displaystyle ~{\sqrt {-g}}}$ of the determinant ${\displaystyle ~g}$ of the metric tensor, taken with a negative sign.

Variation of the action function gives the general field equations, the four-dimensional equation of motion and the equation for determining the metric. Since the dissipation field is a component of the general field, then the corresponding dissipation field equations can be derived from the general field equations.

Given the gauge conditions of the cosmological constant are met in the following form:

${\displaystyle ~ck\Lambda =-s_{\mu }J^{\mu },}$

the system’s energy does not depend on the term with the scalar curvature and it becomes uniquely determined: [7]

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

where ${\displaystyle ~s_{0}}$ and ${\displaystyle ~J^{0}}$ denote the time components of the four-vectors ${\displaystyle ~s_{\mu }}$ and ${\displaystyle ~J^{\mu }}$.

The four-momentum of the system is given by the formula:

${\displaystyle ~p^{\mu }=\left({\frac {E}{c}}{,}\mathbf {p} \right)=\left({\frac {E}{c}}{,}{\frac {E}{c^{2}}}\mathbf {v} \right),}$

where ${\displaystyle ~\mathbf {p} }$ and ${\displaystyle ~\mathbf {v} }$ denote the system’s momentum and the velocity of the system’s center of mass.

### Equations

The four-dimensional equations of the dissipation field are similar by their form to Maxwell equations and have the following form:

${\displaystyle \nabla _{\sigma }h_{\mu \nu }+\nabla _{\mu }h_{\nu \sigma }+\nabla _{\nu }h_{\sigma \mu }={\frac {\partial h_{\mu \nu }}{\partial x^{\sigma }}}+{\frac {\partial h_{\nu \sigma }}{\partial x^{\mu }}}+{\frac {\partial h_{\sigma \mu }}{\partial x^{\nu }}}=0.}$
${\displaystyle ~\nabla _{\nu }h^{\mu \nu }=-{\frac {4\pi \tau }{c^{2}}}J^{\mu },}$

where ${\displaystyle ~J^{\mu }=\rho _{0}u^{\mu }}$ is the mass four-current, ${\displaystyle ~\rho _{0}}$ is the mass density in the co-moving reference frame, ${\displaystyle ~u^{\mu }}$ is the four-velocity of the matter unit, ${\displaystyle ~\tau }$ is a constant determined in each problem, and it is assumed that there is a balance between all the fields in the physical system under consideration.

The gauge condition for the four-potential of the dissipation field:

${\displaystyle ~\nabla ^{\mu }\lambda _{\mu }=0.}$

In Minkowski space of the special theory of relativity, the form of the dissipation field equations is simplified and they can be expressed in terms of the field strength ${\displaystyle ~\mathbf {X} }$ and the solenoidal vector ${\displaystyle ~\mathbf {Y} }$:

${\displaystyle ~\nabla \cdot \mathbf {X} =4\pi \tau \gamma \rho _{0},\qquad \qquad \nabla \times \mathbf {Y} ={\frac {1}{c^{2}}}\left(4\pi \tau \mathbf {J} +{\frac {\partial \mathbf {X} }{\partial t}}\right),}$
${\displaystyle ~\nabla \times \mathbf {X} =-{\frac {\partial \mathbf {Y} }{\partial t}},\qquad \nabla \cdot \mathbf {Y} =0.}$

where ${\displaystyle ~\gamma ={\frac {1}{\sqrt {1-{v^{2} \over c^{2}}}}}}$ is the Lorentz factor, ${\displaystyle ~\mathbf {J} =\gamma \rho _{0}\mathbf {v} }$ is the mass current density , ${\displaystyle ~\mathbf {v} }$ is the matter unit velocity.

If we also use the gauge condition in the form of ${\displaystyle ~\partial ^{\mu }\lambda _{\mu }={\frac {1}{c^{2}}}{\frac {\partial \varepsilon }{\partial t}}+\nabla \cdot \mathbf {\Theta } =0}$ and relation (1), we can obtain the wave equations for the dissipation field potentials from the field equations:

${\displaystyle ~{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varepsilon }{\partial t^{2}}}-\Delta \varepsilon =4\pi \tau \gamma \rho _{0},}$
${\displaystyle ~{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {\Theta } }{\partial t^{2}}}-\Delta \mathbf {\Theta } ={\frac {4\pi \tau }{c^{2}}}\mathbf {J} .}$

In curved space the equation of motion of the matter unit in the general field is given by the formula:

${\displaystyle ~s_{\mu \nu }J^{\nu }=0}$.

Since ${\displaystyle ~J^{\nu }=\rho _{0}u^{\nu }}$, and the general field tensor is expressed in terms of tensors of particular fields, then the equation of motion can be represented using these tensors: [8]

${\displaystyle ~-u_{\mu \nu }J^{\nu }=F_{\mu \nu }j^{\nu }+\Phi _{\mu \nu }J^{\nu }+f_{\mu \nu }J^{\nu }+h_{\mu \nu }J^{\nu }+\gamma _{\mu \nu }J^{\nu }+w_{\mu \nu }J^{\nu }.\qquad (2)}$

Here ${\displaystyle ~u_{\mu \nu }}$ is the acceleration tensor, ${\displaystyle ~F_{\mu \nu }}$ is the electromagnetic tensor, ${\displaystyle ~j^{\nu }}$ is the charge four-current, ${\displaystyle ~\Phi _{\mu \nu }}$ is the gravitational tensor, ${\displaystyle ~f_{\mu \nu }}$ is the pressure field tensor, ${\displaystyle ~\gamma _{\mu \nu }}$ is the strong interaction field tensor, ${\displaystyle ~w_{\mu \nu }}$ is the weak interaction field tensor.

### Stress–energy tensor

The dissipation stress-energy tensor is calculated with the help of the dissipation field tensor:

${\displaystyle ~Q^{ik}={\frac {c^{2}}{4\pi \tau }}\left(-g^{im}h_{nm}h^{nk}+{\frac {1}{4}}g^{ik}h_{mr}h^{mr}\right).}$

The tensor ${\displaystyle ~Q^{ik}}$ includes the three-vector of energy-momentum flux ${\displaystyle ~\mathbf {Z} }$, which is similar in its meaning to the Poynting vector and the Heaviside vector. The vector ${\displaystyle ~\mathbf {Z} }$ can be represented through the vector product of the field strength ${\displaystyle ~\mathbf {X} }$ and the solenoidal vector ${\displaystyle ~\mathbf {Y} }$:

${\displaystyle ~\mathbf {Z} =cQ^{0i}={\frac {c^{2}}{4\pi \tau }}[\mathbf {X} \times \mathbf {Y} ],}$

here the index is ${\displaystyle ~i=1,2,3.}$

The covariant derivative of the stress-energy tensor of the dissipation field determines the density of the dissipation four-force:

${\displaystyle ~f^{\alpha }=-\nabla _{\beta }Q^{\alpha \beta }={h^{\alpha }}_{k}J^{k}.\qquad \qquad (3)}$

The stress-energy tensor of the dissipation field is part of the stress-energy tensor of the general field ${\displaystyle ~T^{ik}}$, but in the general case the tensor ${\displaystyle ~T^{ik}}$ also contains the cross-terms with the products of strengths and solenoidal vectors of particular fields:

${\displaystyle ~T^{ik}=k_{1}W^{ik}+k_{2}U^{ik}+k_{3}B^{ik}+k_{4}P^{ik}+k_{5}Q^{ik}+k_{6}L^{ik}+k_{7}A^{ik}+cross\quad terms,}$

where ${\displaystyle ~k_{1}{,}k_{2}{,}k_{3}{,}k_{4}{,}k_{5}{,}k_{6}{,}k_{7}}$ are some coefficients, ${\displaystyle ~W^{ik}}$ is the electromagnetic stress-energy tensor, ${\displaystyle ~U^{ik}}$ is the gravitational stress-energy tensor, ${\displaystyle ~B^{ik}}$ is the acceleration stress-energy tensor, ${\displaystyle ~P^{ik}}$ is the pressure stress-energy tensor, ${\displaystyle ~L^{ik}}$ is the strong interaction stress-energy tensor, ${\displaystyle ~A^{ik}}$ is the weak interaction stress-energy tensor.

By means of the tensor ${\displaystyle ~T^{ik}}$, the stress-energy tensor of the dissipation field becomes part of the equation for the metric:

${\displaystyle ~R^{ik}-{\frac {1}{4}}g^{ik}R={\frac {8\pi G\beta }{c^{4}}}T^{ik},}$

where ${\displaystyle ~R^{ik}}$ is the Ricci tensor, ${\displaystyle ~G}$ is the gravitational constant, ${\displaystyle ~\beta }$ is a certain constant, and the gauge condition for the cosmological constant is used.

### Application

In the case when a certain vector potential of a particle is equal to zero in the rest frame of the particle, the four-potential of this vector field in an arbitrary frame of reference can be represented as follows: [6]

${\displaystyle ~L_{\mu }={\frac {k_{f}\varepsilon _{p}}{\rho _{0}c^{2}}}u_{\mu },}$

where ${\displaystyle ~k_{f}={\frac {\rho _{0}}{\rho _{0q}}}}$ for electromagnetic field and ${\displaystyle ~k_{f}=1}$ for other fields, ${\displaystyle ~\rho _{0}}$ and ${\displaystyle ~\rho _{0q}}$ are the mass density and accordingly charge density in comoving reference frame, ${\displaystyle ~\varepsilon _{p}}$ is the energy density of the particle in the given field, ${\displaystyle ~u_{\mu }}$ is the covariant four-velocity.

For the dissipation field ${\displaystyle ~\varepsilon _{p}=\alpha \rho _{0}}$ and ${\displaystyle ~k_{f}=1}$, and according to the definition, for the four-potential of the dissipation field of one particle we have the following:

${\displaystyle ~\lambda _{\mu }=\left({\frac {\varepsilon }{c}},-\mathbf {\Theta } \right)={\frac {\alpha }{c^{2}}}u_{\mu },}$

where ${\displaystyle ~\alpha }$ is the dissipation function. For an arbitrary particle, the components of the four-potential in the framework of the special relativity (STR) take the form: ${\displaystyle ~\varepsilon =\gamma \alpha ,}$ ${\displaystyle ~\mathbf {\Theta } ={\frac {\gamma \alpha }{c^{2}}}\mathbf {v} ,}$

and hence, the vector potential is directed along the particle’s velocity. If the vector potential components are the functions of time and do not directly depend on the space coordinates, then for such motion according to (1) the solenoidal vector ${\displaystyle ~\mathbf {Y} }$ vanishes.

Due to the interaction of a set of particles with each other by means of various fields, including interaction at a distance without direct contact, the dissipation field in the matter changes and is different from the dissipation field of a single particle at the observation point. The dissipation field in the system of particles is specified by the field strength and solenoid vector, which represent the typical averaged characteristics of the matter’s motion. As a rule, in a gravitationally-bound system the radial gradients of different field strengths appear, including the dissipation field strength ${\displaystyle ~\mathbf {X} ,}$ and if some part of the particles is moving synchronously or rotating, then the vector ${\displaystyle ~\mathbf {Y} }$ appears. From (2) and (3) we derive a general expression for the four-force density with a covariant index, which arises from the dissipation field:

${\displaystyle ~(f_{\mu })_{d}=h_{\mu \nu }J^{\nu }=\rho _{0}{\frac {cdt}{ds}}\left({\frac {1}{c}}\mathbf {X} \cdot \mathbf {v} {,}\qquad -\mathbf {X} -[\mathbf {v} \times \mathbf {Y} ]\right),}$

where ${\displaystyle ~ds}$ denotes the four-dimensional space-time interval.

The relativistic equation of motion of viscous compressible matter, taking into account the four-potential of the dissipation field, the dissipation field tensor and the stress-energy tensor of the dissipation field, within the limits of low curvature of spacetime can be represented as follows: [5]

${\displaystyle ~{\frac {d}{dt}}[\gamma \mathbf {v} (1+{\frac {p_{0}}{\rho _{0}c^{2}}}+{\frac {\alpha }{c^{2}}})]=\mathbf {a_{m}} -{\frac {1}{\gamma }}\nabla ({\frac {p_{0}}{\rho _{0}}})-{\frac {\varsigma }{4\pi \eta \gamma ^{2}c^{2}\rho _{0}^{2}}}({\frac {\partial ^{2}(\gamma \mathbf {v} )}{\partial t^{2}}}-c^{2}\Delta (\gamma \mathbf {v} ))+{\frac {\omega }{\gamma \rho _{0}}}\nabla (\nabla \cdot \mathbf {v} ),}$

where ${\displaystyle ~\mathbf {a_{m}} =\mathbf {\Gamma } +\mathbf {v} \times \mathbf {\Omega } +{\frac {\rho _{0q}}{\rho _{0}}}(\mathbf {E} +\mathbf {v} \times \mathbf {B} )}$ is the acceleration of gravitational and electromagnetic mass forces, ${\displaystyle ~\mathbf {\Gamma } }$ is the gravitational field strength, ${\displaystyle ~\mathbf {\Omega } }$ is the gravitational torsion field, ${\displaystyle ~\mathbf {E} }$ is the electric field strength, ${\displaystyle ~\mathbf {B} }$ is the magnetic field.

At low velocities, for the Lorentz factor we can assume ${\displaystyle ~\gamma =1}$. Under ordinary conditions, we can also neglect the contribution of the pressure ${\displaystyle ~p_{0}}$ and the dissipation function ${\displaystyle ~\alpha }$ on the left side of the equation. Determining the dynamic viscosity by the expression ${\displaystyle ~\mu ={\frac {\varsigma }{4\pi \eta \gamma ^{2}\rho _{0}}}}$, where ${\displaystyle ~\eta }$ is the acceleration field coefficient, and denoting ${\displaystyle ~\omega =\xi +{\frac {\mu }{3}}}$, we obtain the following:

${\displaystyle ~{\frac {d\mathbf {v} }{dt}}={\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {v} =\mathbf {a_{m}} -\nabla ({\frac {p_{0}}{\rho _{0}}})+{\frac {\mu }{\rho _{0}}}\Delta \mathbf {v} -{\frac {\mu }{\rho _{0}c^{2}}}{\frac {\partial ^{2}\mathbf {v} }{\partial t^{2}}}+{\frac {1}{\rho _{0}}}(\xi +{\frac {\mu }{3}})\nabla (\nabla \cdot \mathbf {v} ).}$

The main difference of this equation from the Navier–Stokes equation is a small additional term, which contains the second time derivative of the flux velocity ${\displaystyle ~\mathbf {v} }$ and the square of the speed of light in the denominator.